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Research Methods in Biomechanics
by D. Gordon E. Robertson, Graham E. Caldwell, Joseph Hamill, Gary Kamen and Saunders Whittlesey
440 Pages
Research Methods in Biomechanics, Second Edition, demonstrates the range of available research techniques and how to best apply this knowledge to ensure valid data collection. In the highly technical field of biomechanics, research methods are frequently upgraded as the speed and sophistication of software and hardware technologies increase. With this in mind, the second edition includes up-to-date research methods and presents new information detailing advanced analytical tools for investigating human movement.
Expanded into 14 chapters and reorganized into four parts, the improved second edition features more than 100 new pieces of art and illustrations and new chapters introducing the latest techniques and up-and-coming areas of research. Also included is access to biomechanics research software designed by C-Motion, Visual3D Educational Edition, which allows users to explore the full range of modeling capabilities of the professional Visual3D software in sample data files as well as display visualizations for other data sets. Additional enhancements in this edition include the following:
• Special features called From the Scientific Literature highlight the ways in which biomechanical research techniques have been used in both classic and cutting-edge studies.
• An overview, summary, and list of suggested readings in each chapter guide students and researchers through the content and on to further study.
• Sample problems appear in select chapters, and answers are provided at the end of the text.
• Appendixes contain mathematical and technical references and additional examples.
• A glossary provides a reference for terminology associated with human movement studies.
Research Methods in Biomechanics, Second Edition, assists readers in developing a comprehensive understanding of methods for quantifying human movement. Parts I and II of the text examine planar and three-dimensional kinematics and kinetics in research, issues of body segment parameters and forces, and energy, work, and power as they relate to analysis of two- and three-dimensional inverse dynamics. Two of the chapters have been extensively revised to reflect current research practices in biomechanics, in particular the widespread use of Visual3D software. Calculations from these two chapters are now located online with the supplemental software resource, making it easier for readers to grasp the progression of steps in the analysis.
In part III, readers can explore the use of musculoskeletal models in analyzing human movement. This part also discusses electromyography, computer simulation, muscle modeling, and musculoskeletal modeling; it presents new information on MRI and ultrasound use in calculating muscle parameters. Part IV offers a revised chapter on additional analytical procedures, including signal processing techniques. Also included is a new chapter on movement analysis and dynamical systems, which focuses on how to assess and measure coordination and stability in changing movement patterns and the role of movement variability in health and disease. In addition, readers will find discussion of statistical tools useful for identifying the essential characteristics of any human movement.
The second edition of Research Methods in Biomechanics explains the mathematics and data collection systems behind both simple and sophisticated biomechanics. Integrating software and text, Research Methods in Biomechanics, Second Edition, assists both beginning and experienced researchers in developing their methods for analyzing and quantifying human movement.
Part I. Kinematics
Chapter 1. Planar Kinematics
D. Gordon E. Robertson and Graham E. Caldwell
Description of Position
Degrees of Freedom
Kinematic Data Collection
Linear Kinematics
Angular Kinematics
Summary
Suggested Readings
Chapter 2. Three-Dimensional Kinematics
Joseph Hamill, W. Scott Selbie, and Thomas M. Kepple
Collection of Three-Dimensional Data
Coordinate Systems and Assumption of Rigid Segments
Transformations between Coordinate Systems
Defining the Segment LCS for the Lower Extremity
Pose Estimation: Tracking the Segment LCS
Joint Angles
Joint Angular Velocity and Angular Acceleration of Cardan Joint Angles
Summary
Suggested Readings
Part II. Kinetics
Chapter 3. Body Segment Parameters
D. Gordon E. Robertson
Methods for Measuring and Estimating Body Segment Parameters
Two-Dimensional (Planar) Computational Methods
Three-Dimensional (Spatial) Computational Methods
Summary
Suggested Readings
Chapter 4. Forces and Their Measurement
Graham E. Caldwell, D. Gordon E. Robertson, and Saunders N. Whittlesey
Force
Newton’s Laws
Free-Body Diagrams
Types of Forces
Moment of Force, or Torque
Linear Impulse and Momentum
Angular Impulse and Momentum
Measurement of Force
Summary
Suggested Readings
Chapter 5. Two-Dimensional Inverse Dynamics
Saunders N. Whittlesey and D. Gordon E. Robertson
Planar Motion Analysis
Numerical Formulation
Human Joint Kinetics
Applications
Summary
Suggested Readings
Chapter 6. Energy, Work, and Power
D. Gordon E. Robertson
Energy, Work, and the Laws of Thermodynamics
Conservation of Mechanical Energy
Ergometry: Direct Methods
Ergometry: Indirect Methods
Mechanical Efficiency
Summary
Suggested Readings
Chapter 7. Three-Dimensional Kinetics
W. Scott Selbie, Joseph Hamill, and Thomas Kepple
Segments and Link Models
3-D Inverse Dynamics Analysis
Presentation of the Net Moment Data
Joint Power
Interpretation of Net Joint Moments
Sources of Error in Three-Dimensional Calculations
Summary
Suggested Readings
Part III. Muscles, Models, and Movement
Chapter 8. Electromyographic Kinesiology
Gary Kamen
Physiological Origin of the Electromyographic Signal
Recording and Acquiring the Electromyographic Signal
Analyzing and Interpreting the Electromyographic Signal
Applications for Electromyographic Techniques
Summary
Suggested Readings
Chapter 9. Muscle Modeling
Graham E. Caldwell
The Hill Muscle Model
Muscle-Specific Hill Models
Beyond the Hill Model
Summary
Suggested Readings
Chapter 10. Computer Simulation of Human Movement
Saunders N. Whittlesey and Joseph Hamill
Overview: Modeling As a Process
Why Simulate Human Movement?
General Procedure for Simulations
Control Theory
Limitations of Computer Models
Summary
Suggested Readings
Chapter 11. Musculoskeletal Modeling
Brian R. Umberger and Graham E. Caldwell
Musculoskeletal Models
Control Models
Analysis Techniques
Summary
Suggested Readings
Part IV. Further Analytical Procedures
Chapter 12. Signal Processing
Timothy R. Derrick and D. Gordon E. Robertson
Characteristics of a Signal
Fourier Transform
Time-Dependent Fourier Transform
Sampling Theorem
Ensuring Circular Continuity
Smoothing Data
Summary
Suggested Readings
Chapter 13. Dynamical Systems Analysis of Coordination
Richard E.A. van Emmerik, Ross H. Miller, and Joseph Hamill
Movement Coordination
Foundations for Coordination Analysis
Quantifying Coordination: Relative Phase Methods
Quantifying Coordination: Vector Coding
Overview of Coordination Analysis Techniques
Summary
Suggested Readings
Chapter 14. Analysis of Biomechanical Waveform Data
Kevin J. Deluzio, Andrew J. Harrison, Norma Coffey, and Graham E. Caldwell
Biomechanical Waveform Data
Principal Component Analysis
Functional Data Analysis
Comparison of PCA and FDA
Summary
Suggested Readings
D. Gordon E. Robertson, PhD, an emeritus professor and a fellow of the Canadian Society for Biomechanics, wrote Introduction to Biomechanics for Human Motion Analysis. He taught undergraduate- and graduate-level biomechanics at the University of Ottawa and previously at the University of British Columbia, Canada. He conducts research on human locomotion and athletic activities and authors the analogue data analysis software BioProc3.
Graham E. Caldwell, PhD, an associate professor and a fellow of the Canadian Society for Biomechanics, teaches undergraduate- and graduate-level biomechanics at the University of Massachusetts at Amherst and previously held a similar faculty position at the University of Maryland. He won the Canadian Society for Biomechanics New Investigator Award and in 1998 won the Outstanding Teacher Award for the School of Public Health and Health Sciences at the University of Massachusetts at Amherst. He served as an associate editor for Medicine and Science in Sports and Exercise.
Joseph Hamill, PhD, is a professor and fellow of the Research Consortium, International Society of Biomechanics in Sports, Canadian Society for Biomechanics, American College of Sports Medicine, and National Academy of Kinesiology. He coauthored the popular undergraduate textbook Biomechanical Basis of Human Movement. He teaches undergraduate- and graduate-level biomechanics and is director of the Biomechanics Laboratory at the University of Massachusetts at Amherst. He serves on the editorial boards of several prestigious professional journals. He is adjunct professor at the University of Edinburgh in Scotland and the University of Limerick in Ireland and a distinguished research professor at Republic Polytechnic in Singapore.
Gary Kamen, PhD, is a professor and fellow of the American Alliance for Health, Physical Education, Recreation and Dance; American College of Sports Medicine; and National Academy of Kinesiology. He authored an undergraduate textbook on kinesiology, Foundations of Exercise Science, as well as a primer on electromyography, Essentials of Electromyography. He was president of the Research Consortium of AAPHERD and teaches undergraduate and graduate courses in exercise neuroscience and motor control in the department of kinesiology at the University of Massachusetts at Amherst.
Saunders (Sandy) N. Whittlesey, PhD, a graduate of the University of Massachusetts at Amherst, is a self-employed technology consultant specializing in athletic training, sporting goods, and clinical applications.
Additional Contributors
Norma Coffey, PhD, a postdoctoral researcher in statistics at the National University of Ireland at Galway, has expertise is functional data analysis and worked extensively with the Biomechanics Research Unit at the University of Limerick. Her current area of research involves applying functional data analysis techniques to time-course gene expression data.
Timothy R. Derrick, PhD, a professor in the department of kinesiology at Iowa State University, has an extensive background in signal processing and conducts research on impacts to the human body particularly from the ground during running activities.
Kevin Deluzio, PhD, is a professor in the department of mechanical and materials engineering at Queen's UUniversity in Kingston, Canada, and held a similar position at Dalhousie University. He studies human locomotion to investigate the biomechanical factors of musculoskeletal diseases such as knee osteoarthritis. He is also interested in the design and evaluation of noninvasive therapies as well as surgical treatments such as total-knee replacement.
Andrew (Drew) J. Harrison, PhD, is a senior lecturer in biomechanics in the department of physical education and sport sciences at the University of Limerick in Ireland and a fellow of the International Society for Biomechanics in Sport. He is the director of the Biomechanics Research Unit at the University of Limerick. His research focuses on biomechanics of sport performance and sport injuries.
Thomas M. Kepple, PhD, is an instructor in the department of health, nutrition, and exercise sciences at the University of Delaware. He worked for many years as a biomechanist at the National Institutes of Health on motion capture technology and gait laboratory instrumentation.
Ross H. Miller, PhD, an assistant professor in the department of kinesiology at the University of Maryland, has published papers on static optimization and forward dynamics as well as methods on nonlinear techniques of data analysis.
Scott Selbie, PhD, is an adjunct professor at Queen's University, Canada, and at the University of Massachusetts at Amherst. He is a graduate of Simon Fraser University, Canada. He is the director of research at C-Motion, developers of the Visual3D software, and president of HAS-Motion in Canada.
Brian R. Umberger, PhD, is an associate professor teaching biomechanics at the undergraduate and graduate levels in the department of kinesiology at the University of Massachusetts at Amherst. In 2010, he received the Outstanding Teacher Award for the School of Public Health and Health Sciences at the University of Massachusetts at Amherst. In his research, he uses a combination of experimental, modeling, and simulation approaches to study the biomechanics and energetics of human locomotion.
Richard E.A. van Emmerik, PhD, is a professor in the kinesiology department at the University of Massachusetts at Amherst, where he teaches motor control at the undergraduate and graduate levels. In his research, he applies principles from complex and nonlinear dynamical systems to the study of posture and locomotion.
“From how to understand and build concepts to new chapters on new techniques and research in the works, this provides a fine college-level analysis of the math and data collection systems behind biomechanics, and makes for a fine reference for any research interested in analyzing human movement.”
-- Midwest Book Review
Using EMG to diagnose and treat low back pain
The diagnosis and treatment of low back pain are important issues.
EMG and Low Back Pain
The diagnosis and treatment of low back pain are important issues. In the United States, for example, back pain accounts for two-thirds of all workers compensation costs. Peripheral muscle problems may be one issue; atrophy of the multifidus is frequently observed in patients with low back pain (Hides et al. 1994). EMG is becoming an increasingly useful tool for the diagnosis and treatment of low back pain.
The trunk musculature plays an important role in tasks such as lifting and throwing. During these kinds of tasks as well as in other activities that might threaten the postural control of the individual and perturb balance, the nervous system implements strategies to activate trunk muscles while performing voluntary movements involving remote muscle groups. However, the prevalence of back pain in the general population, and the need to identify ergonomically efficient and safe ways to use back muscles, require that we understand the nature of activation in muscles controlling the trunk. Here, too, EMG is an important tool.
A number of issues render EMG recordings from the trunk musculature difficult to obtain and interpret. From an anatomical viewpoint, the architecture of the back muscles is complex. For example, it is generally accepted that injury or disorder of the multifidus frequently results in back pain. Both multifidus and erector spinae have distinct superficial and deep portions (Bustami 1986; Macintosh et al. 1986) with different histochemical and biomechanical characteristics (Bogduk et al. 1992; Dickx et al. 2010). In some respects, it is fortunate that most of the muscle mass of the multifidus lies in the more superficial portion, so that surface EMG has a greater possibility of characterizing the fibers that produce large amounts of force. However, the multifidus superficial portion has a different role than the deeper fibers. Fibers in the superficial portion cross numerous spinal segments and are thus in a better position to produce back extension. However, the deeper fibers are relatively short, crossing perhaps one or two segments, and thus protect segments of the lumbar spine from inappropriate shear or torsion torques (Macintosh and Valencia 1986).
In addition to issues concerning the activation of the back extensor muscles, a number of important questions require resolution by EMG analysis in which considerable electrocardiographic (ECG) artifact may be present. ECG artifact is particularly problematic during relatively low force contractions, exaggerating the ratio between the signal of interest and the interfering ECG signal. Recording EMG signals from abdominal, knee extensor, and other trunk muscles during normal human movements, for example, can result in the placement of electrodes well within recording range of the electrocardiogram. The relatively large ECG signal can exaggerate the amplitude of EMG activity, and the low-frequency characteristics of the ECG wave can also alter the frequency characteristics of the recorded EMG activity.
For example, the absence of sufficient trunk activity can produce instability resulting in low back pain (van Dieën et al. 2003). However, the amplitude of EMG activity required to ensure stability is small and thus can be affected by the ECG signal (Cholewicki et al. 1997). The best solution is to place the electrodes in a location from which no or minimal ECG artifact can be recorded. However, this is frequently difficult if not impossible. Consequently, numerous algorithms have been suggested to remove the ECG artifact.
One frequently implemented technique to remove this artifact is to compute a template of the ECG signal and subtract it from the electromyogram. This has been proved to be a reasonably successful procedure and has been implemented in studies recording from the diaphragm (Bartolo et al. 1996) as well as rectus abdominis (Hof 2009). Hof (2009) described a technique in which the ECG signal is recorded simultaneously with the EMG signal of interest, and template subtraction is then implemented (figure 8.15).
Digital filtering is another useful alternative for ECG artifact removal. Drake and Callaghan (2006) used a FIR (finite impulse response) filter with a hamming window, although they concluded that the most efficient filtering result could be obtained using a somewhat simpler fourth-order, 30 Hz high-pass cutoff Butterworth filter. Template subtraction improved extraction of the ECG signal but required a large amount of time.
Alternative ECG removal techniques include the use of adaptive filters (Lu et al. 2009; Marque et al. 2005), wavelet-independent component analysis (Taelman et al. 2007), and wavelet-based adaptive filters (Zhan et al. 2010). Irrespective of the technique used for ECG signal artifact removal, it is apparent that the resultant improvement in signal interpretation can be important. Hu and colleagues (2009), for example, found that the use of independent component analysis to remove artifact resulted in an improved ability to discriminate between patients with low back pain and normal subjects during both sitting and standing tasks.
Analysis of the EMG activity in back muscles has produced some interesting results, in part reflecting the unique anatomical issues discussed here. As long ago as 1962, Morris and colleagues noted that “the three muscles of the erector spinae group considered here . . .
do not always show parallel activity, and one may be active while the other two are inactive” (p. 519). This may suggest that the mere demonstration of greater or lesser EMG amplitude may not be indicative of abnormality in a particular muscle. Potentially more important than EMG amplitude is the timing of EMG activity. Deep and superficial portions of the multifidus are differentially active during arm movements (Moseley et al. 2002). One suggestion is that the manner in which the multifidus is differentially controlled in people with back pain compared with those without may be the source of chronic back pain (MacDonald et al. 2009).
Similarly, the analysis of EMG frequency characteristics in patients with back pain compared with control subjects has suggested that there may be differences between muscles and even differences within muscles. Patients with low back pain frequently exhibit alterations in the electromyographic response to fatigue in the back extensor muscles (Biedermann et al. 1991). This is one area in which spectral analysis of the EMG signals has been helpful. As in other muscles, median EMG frequency decreases in the trunk extensors with fatigue (Demoulin et al. 2007). It is interesting to note that Kramer and colleagues (2005) found that the magnitude of median frequency decrease was greater in healthy subjects than in individuals with back pain. Support for importance of electrode placement is obtained from the observations of Sung and colleagues (2009). Normal subjects and patients with back pain underwent an isometric contraction protocol that induced fatigue of the back extensors. EMG frequency measurements made in both the thoracic and the lumbar portions of the erector spinae indicated that the thoracic portion had a significantly lower median frequency than the lumbar portion in patients with low back pain. However, median frequency was lower in the lumbar portion than in the thoracic portion in control subjects. Thus, in the analysis of trunk musculature that might be involved in the development of back pain, EMG studies using wire electrodes frequently may be required to record from different portions of the muscle.
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Using musculoskeletal optimization models to generate control signals
A popular method for generating control signals in musculoskeletal models is to use some form of optimization.
Optimization Models
A popular method for generating control signals in musculoskeletal models is to use some form of optimization. In some cases, when the optimization criterion is derived from motor control principles, this is tantamount to developing a theoretical control model. In other cases, the optimization criterion may result in a coordinated movement pattern yet have little physiological basis. In general, the optimization approach is used in an effort to determine which set of model control signals will produce a result that optimizes (minimizes or maximizes) a given criterion measure. The criterion measure is known as a cost function, objective function, or performance criterion. The cost function can be relatively simple (e.g., find the solution that yields minimal muscle force) or complex (e.g., determine a nonlinear combination of maximal muscle force at minimal metabolic cost). It can be directly related to muscle function (e.g., minimize muscular work) or to some aspect of the motion under study (e.g., maximize vertical jump height). Alternatively, the cost function may be formulated to minimize the differences between model outputs (e.g., joint angles, ground reaction forces) and corresponding experimental measures. This latter case is often referred to as a tracking problem in that the goal is to find a solution that causes the model to follow, or track, the experimental data.
In all cases, the cost function serves as a guiding constraint that determines the selection of one particular set of optimal muscular controls from among many different possible solutions. Two major optimization approaches that have been used to control musculoskeletal models are referred to in the literature as static optimization and dynamic optimization. The meaning of static in this context is that the cost function is evaluated at each time step during a movement, independent of any prior or subsequent time steps. In contrast, dynamic optimization is dynamic in the sense that the entire movement sequence must be simulated to determine the value of the cost function.
Static Optimization
Initial attempts to predict individual muscle forces used static optimization models in conjunction with inverse dynamics analysis of an actual motor performance. The inverse dynamics analysis permits the calculation of net joint moments at incremental times throughout the movement (see chapter 5). In most applications, numerical optimization is used to find the set of muscle forces that balances these joint moments while also satisfying a selected cost function (figure 11.9). Thus, with this approach the muscle forces themselves, rather than neural signals, are the controls. The time-independent nature of static optimization allows solutions to be obtained with relatively little computational cost, but there are some drawbacks. One issue in early applications was sudden, nonphysiological switching on and off of muscle forces caused by the independence of solutions for sequential time increments (i.e., the optimization model balanced the joint moments separately for each time interval using very different sets of muscles). This problem can be avoided by careful selection of the initial guess in the optimization problem, such as by using the solution from one time step as the initial guess for the next time step. A stronger approach to address this issue is to use muscle models that invoke physiological realism through force-length and force-velocity relations and time-dependent stimulation-activation dynamics (chapter 9). When more detailed muscle models are included in a static optimization model, muscle activations become the control variables, rather than muscle forces.
Early studies using static optimization were plagued also by two other kinds of nonphysiological results. The first was the prediction of model forces that were too high for actual muscles to produce. This difficulty was easily addressed by defining physiologically valid maximal force constraints for each muscle in the musculoskeletal model. The second problem was that solutions would often select only one muscle to balance the net joint moment rather than choose a more realistic muscular synergy. Depending on the exact formulation, the muscle with the largest moment arm (if minimizing muscle force) or a favorable combination of moment arm and muscle strength (if minimizing muscle stress) would be selected to fully balance the joint moment, with zero force predicted in other synergist muscles. Mathematically, synergism can be produced by using nonlinear cost functions (e.g., minimizing the sum of squared or cubed muscle forces), although the physiological rationale for specific nonlinear cost functions is not always clear. A widely used nonlinear cost function proposed by Crowninshield and Brand (1981a) involves minimizing the sum of cubed muscle stresses. It was originally argued that this particular cost function would lead to solutions that maximize muscle endurance, making it appropriate for predicting muscle forces in submaximal tasks such as walking. However, the Crowninshield and Brand criterion has since been used to solve for muscle forces in a range of activities, some of which are unlikely candidates for maximizing muscle endurance (e.g., jumping, landing).
An additional issue with static optimization models is that they are essentially a decomposition of experimental joint moments into individual muscle forces. Therefore, the predicted muscle forces will be subject to any errors in the experimental joint moments in addition to any shortcomings associated with the approximation made in creating the musculoskeletal model. To complete this section, we present a static optimization example motivated by a classic review article by Crowninshield and Brand (1981b), which demonstrates the process of distributing an empirically determined elbow joint moment across a set of muscles spanning the elbow joint.
The net joint moment to be balanced in example 11.1 was a flexor moment, and only muscles with flexor moment arms were included in the model (figure 11.10). If an elbow extensor muscle had been included, there is no cost function of the type presented here that would predict force in an antagonist muscle. Any force in an elbow extensor would itself contribute to a higher cost function value and would also require greater elbow flexor forces to balance the target 10 N·m joint moment, further increasing the cost function value. The total lack of coactivation in this example is contrary to the common observation that during heavy activation of the elbow flexors, there is some activity in the triceps muscle group. The extensor coactivation likely helps stabilize the joint during heavy exertion but will not be predicted using traditional static optimization techniques in simple one DOF models. Although the example presented here focuses on obtaining numerical results for muscle forces, the interested reader is referred to the review by Crowninshield and Brand (1981b), which presents an interesting graphical interpretation of the solution to this static optimization problem.
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Dynamic Optimization
Ongoing work with static optimization models has led to the development of dynamic optimization or optimal control models, which are applied in conjunction with forward dynamics models of human motion. In contrast to the inverse dynamics approach, which uses experimental data from an actual performance to calculate net joint moments, forward dynamics analyses simulate the motion of the body from a given set of joint moments or muscle forces (see chapter 10). Dynamic optimization models are therefore often designed to find the muscle stimulation patterns that result in an optimal motion (figure 11.9). As mentioned earlier, the optimal motion may be one that maximizes (or minimizes) a performance criterion, such as maximizing jump height; or, the optimal motion may be defined as one that best reproduces a set of experimental data. Regardless, the variables that are optimized are usually the muscle stimulation patterns that control the motion of the musculoskeletal model. These stimulation patterns are used as the inputs to muscle models that predict individual muscle forces, which are then multiplied by the appropriate moment arms to compute the active muscle moments. The active muscle moments are combined with passive moments to compute the net joint moments, which actuate the skeletal model and produce movement. Thus, dynamic optimization leads to the synthesis of body segment kinematics associated with optimal performance, which is fundamentally different from the static optimization approach.
The optimal kinematics predicted from a dynamic optimization can be compared with experimental motion data; however, the experimental data are not required to obtain the solution, as they are with static optimization. This feature of dynamic optimization permits one to address questions for which no experimental data exist, such as testing the feasibility of possible forms of locomotion in extinct species (e.g., Nagano et al. 2005). Moreover, because dynamic optimization uses forward dynamics models that simulate the whole movement performance and provide complete muscle force time histories (rather than solutions at independent time increments), many of the problems associated with static optimization models are overcome. However, these advantages come at a computational cost, because dynamic optimization often requires a more detailed musculoskeletal model and the entire movement must be simulated for each possible solution. Thus, solving a dynamic optimization problem can easily require an order of magnitude more time than is required to solve a comparable static optimization problem. Although computational costs are difficult to compare in an objective manner, it is not uncommon for static optimization solutions to take seconds or minutes to obtain, whereas dynamic optimization solutions can take hours, days, or even weeks, when run on standard computer workstations.
In many cases, users of dynamic optimization must define the goal of the movement mathematically. This definition is easiest for movements in which the performance criterion to be optimized is clear in a mechanical sense. In vertical jumping, for example, the cost function can be stated as maximization of the vertical displacement of the body center of mass during the flight phase. If the model is constructed with appropriate constraints (realistic muscle properties and joint range of motion), jump heights and body segment motions approximating those of human jumpers can be attained (Pandy and Zajac 1991; van Soest et al. 1993). However, for many human movements the performance criterion is not as clear. In walking, for instance, the goal may seem to be getting from point A to point B in a certain amount of time. Unfortunately, this does not provide enough information to solve for a set of muscle stimulation patterns that will result in realistic walking. Successful simulations of walking have been generated using a cost function that minimizes the metabolic cost of transport (Anderson and Pandy 2001; Umberger 2010); however, the cost function formulation is considerably more complicated than for vertical jumping. Minimum-energy solutions also require the inclusion of an additional model for predicting the metabolic cost of muscle actions (e.g., Umberger et al. 2003). In cases where the performance criterion is not clear, dynamic optimization can serve as an effective approach for testing various theoretical criteria to see how well each can produce the desired movement patterns.
Movements for which the underlying criterion is difficult to define, such as walking or pedaling, have often been simulated by formulating and solving a tracking problem. This approach has a similar appeal to the EMG-based techniques described earlier, in that the resulting motion should closely approximate the way in which humans actually move. However, it is unclear which of several possible experimental measures (kinematics, kinetics, EMG) are the most important for the model to track. Also, because of errors in the experimental data and differences between the musculoskeletal model and the experimental subjects, it may be impossible for the model to track the data perfectly. This approach also limits much of the predictive ability of the dynamic optimization approach, as it is only possible to consider conditions for which experimental data are available.
The tracking approach has seen frequent use in hybrid solution algorithms, which seek to retain the advantages of forward dynamics and dynamic optimization while achieving the computational efficiency of static optimization. Two examples are computed muscle control (Thelen et al. 2003) and direct collocation (Kaplan and Heegaard 2001). Computed muscle control works by solving a static optimization problem at each time step within a single forward dynamics movement simulation. By using feedback on the kinematics and muscle activations at each time step of the numerical integration, computed muscle control can generate a forward dynamics simulation that optimally tracks a set of experimental data without solving a dynamic optimization problem that would require thousands of forward dynamics simulations. In contrast, direct collocation works by converting the differential equations of motion into a set of algebraic constraint equations and treating both the control variables (muscle stimulations) and state variables (positions and velocities) as unknowns in the optimization problem. Although they differ considerably in implementation, both techniques appear to be able to produce results that are similar to dynamic optimization tracking solutions, with a computational cost closer to that of static optimization.
Although static and dynamic optimization have both become popular means for controlling musculoskeletal models, several issues concerning their use must be addressed. Do humans actually produce movements based on one given performance criterion, or does the performance objective change during the movement? If the optimized model results differ from those of a human performer, is it because the model is too simple or not constrained properly, or is the human not performing optimally? Finally, EMG data have illustrated various degrees of simultaneous antagonistic cocontraction during some movement sequences. Optimization models tend not to yield solutions predicting muscle antagonism around a joint, although some antagonism is predicted when models contain muscles that contribute to more than one joint moment. However, optimization models that seek to minimize or maximize specific cost functions will not predict antagonistic cocontraction associated with joint stability or stiffness. In the case of walking, for example, there is no single cost function that could simultaneously predict the relatively minimal muscle coactivation in healthy subjects and the substantial antagonism exhibited by patients with cerebral palsy. Despite these drawbacks, optimization models have increased the understanding of human biomechanics and will continue to do so in the future.
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The state space in the presentation of biomechanical data
Biomechanical data can be represented graphically in a variety of different ways.
The State Space
Biomechanical data can be represented graphically in a variety of different ways. A traditional and very informative type of graph is to display the changes in kinetic or kinematic parameters over time. Figure 13.4 presents an example of the changes in the left and right thigh angular displacements over a time period of 10 s while a subject is walking on a treadmill. Information about peak flexion and extension angles can be obtained from these graphs and quantified. A closer inspection of the graphs makes it quite evident that the coordination between these segments is primarily antiphase. However, especially during the stance phase there are more subtle changes in the patterns of coordination that are harder to obtain from these plots.
In the dynamical systems approach, the reconstruction of the so-called state space is essential in identifying the important features of the behavior of a system, such as its stability and ability to change and adapt to different environmental and task constraints. The state space is a representation of the relevant variables that will help identify these features. To help us understand and quantify coordination between joints or segments, it can be very useful to represent the system in a state space that is based on an angle-angle relative motion plot. Figure 13.5 presents a state space of the relative motion of the time series of the thigh and leg shown earlier in figure 13.4. This angle-angle plot can reveal regions where coordination changes take place as well as parts of the gait cycle where there is relative invariance in coordination patterns. These coordinative changes in the angle-angle plots can be further quantified by vector coding techniques that are discussed later.
Another form of state space is where the position and velocity of a joint or segment are plotted relative to each other. This state-space representation is also often referred to as the phase plane. An example of a position-velocity phase-plane plot is shown in figure 13.6, where the angular displacement of the thigh is plotted against the angular velocity. This is a higher-dimensional state space because the time derivative of position is used to identify the pattern. The phase-plane representation is a first and critical step in the quantification of coordination using continuous relative phase techniques.
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Attractors in State Space
In most forms of human movement, the dynamics in state space are limited to distinct regions, as can be seen in the phase-plane plot in figure 13.6, where the pattern in state space is limited to a fairly narrow, cyclical band. In dynamical systems, preferred regions in state space onto which the dynamics tend to settle are called attractors. Attractors can come in a variety of different forms. A seemingly simple kind of attractor is the point attractor. In this case, the dynamics in the system tend to converge onto one relatively fixed value in the state space. Figure 13.7, c and d, provides an example of a point attractor. The point attractor dynamics are shown in figure 13.7c by a relatively consistent value of discrete relative phase throughout the gait cycle. The discrete relative phase is based on the occurrence of peak flexion in the left and right thigh angular displacements, and the time series in 13.7c show that there is a consistent antiphase or 180° coordination pattern. This antiphase coordination can already be observed by comparing the individual time series in figure 13.4 but is more objectively quantified by the state-space plot in figure 13.7d, where so-called return maps (plotting the coordination at xn vs. xn+1, where n is the cycle number) identify a fixed-point attractor in state space. The relative phase dynamics are of the fixed-point type in this case, as a perturbation (sudden change in treadmill speed and return to the original speed; figure 13.7) results in a return to the original antiphase dynamics with a relative short latency. The perturbation here consisted of a brief (5 s) increase in treadmill speed from 1.2 m/s to 2.0 m/s, with a subsequent return to 1.2 m/s.
Whereas the coordination between the right and left legs can be identified in the form of a point attractor with a relatively fixed value of coordination from cycle to cycle, the dynamics of the individual limb segments show a very different pattern. These are clearly cyclical, as can be seen in the phase planes in figure 13.7, a and b. Attractors of this form are called limit-cycle attractors, and the dynamics converge onto a cyclical region in state space. These limit-cycle attractors are typically identified on the basis of a very narrow, overlapping band of the trajectories in the state space. However, the existence of the narrow band is a necessary but not sufficient condition to characterize the dynamics as a limit-cycle. An essential feature of the limit-cycle attractor is stability with respect to perturbation; to classify as a limit-cycle attractor, the system should also show resistance to perturbations. An example of this is given in figure 13.7, a and b. The regular cyclic pattern in the phase plane (solid line) represents the steady-state gait patterns while a subject was walking at a speed of 1.2 m/s. The dashed trajectories represent the perturbation phase of the trial when speed was suddenly increased to 2.0 m/s and the consequent return back to the original pattern. This return to the preperturbation dynamic is an essential feature of the limit-cycle attractor. The patterns in the phase plane can also serve as an energy plot. The convergence and divergence of trajectories in the phase plane can identify the loss and gain of energy in the system.
Higher-dimensional state spaces (three dimensions and up) can also reveal more complex types of attractors: quasiperiodic and chaotic attractors. The chaotic attractor demonstrates both stable attraction to a region in state space and variability. Figure 13.8 shows an example of the Lorenz attractor, a well-known chaotic attractor that emerges from dynamic interactions in fluid and air flow systems (Strogatz 1994). These dual features of stability and adaptability can be associated with a higher pattern complexity that is now commonly regarded as reflective of healthy and expert systems (see figure 13.3). As an example, increased heart rate variability is considered an important indicator of healthy heart function, reflecting a degree of complexity in organization in which disruptions can be compensated for more easily (Glass 2001; Lipsitz 2002).
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Using EMG to diagnose and treat low back pain
The diagnosis and treatment of low back pain are important issues.
EMG and Low Back Pain
The diagnosis and treatment of low back pain are important issues. In the United States, for example, back pain accounts for two-thirds of all workers compensation costs. Peripheral muscle problems may be one issue; atrophy of the multifidus is frequently observed in patients with low back pain (Hides et al. 1994). EMG is becoming an increasingly useful tool for the diagnosis and treatment of low back pain.
The trunk musculature plays an important role in tasks such as lifting and throwing. During these kinds of tasks as well as in other activities that might threaten the postural control of the individual and perturb balance, the nervous system implements strategies to activate trunk muscles while performing voluntary movements involving remote muscle groups. However, the prevalence of back pain in the general population, and the need to identify ergonomically efficient and safe ways to use back muscles, require that we understand the nature of activation in muscles controlling the trunk. Here, too, EMG is an important tool.
A number of issues render EMG recordings from the trunk musculature difficult to obtain and interpret. From an anatomical viewpoint, the architecture of the back muscles is complex. For example, it is generally accepted that injury or disorder of the multifidus frequently results in back pain. Both multifidus and erector spinae have distinct superficial and deep portions (Bustami 1986; Macintosh et al. 1986) with different histochemical and biomechanical characteristics (Bogduk et al. 1992; Dickx et al. 2010). In some respects, it is fortunate that most of the muscle mass of the multifidus lies in the more superficial portion, so that surface EMG has a greater possibility of characterizing the fibers that produce large amounts of force. However, the multifidus superficial portion has a different role than the deeper fibers. Fibers in the superficial portion cross numerous spinal segments and are thus in a better position to produce back extension. However, the deeper fibers are relatively short, crossing perhaps one or two segments, and thus protect segments of the lumbar spine from inappropriate shear or torsion torques (Macintosh and Valencia 1986).
In addition to issues concerning the activation of the back extensor muscles, a number of important questions require resolution by EMG analysis in which considerable electrocardiographic (ECG) artifact may be present. ECG artifact is particularly problematic during relatively low force contractions, exaggerating the ratio between the signal of interest and the interfering ECG signal. Recording EMG signals from abdominal, knee extensor, and other trunk muscles during normal human movements, for example, can result in the placement of electrodes well within recording range of the electrocardiogram. The relatively large ECG signal can exaggerate the amplitude of EMG activity, and the low-frequency characteristics of the ECG wave can also alter the frequency characteristics of the recorded EMG activity.
For example, the absence of sufficient trunk activity can produce instability resulting in low back pain (van Dieën et al. 2003). However, the amplitude of EMG activity required to ensure stability is small and thus can be affected by the ECG signal (Cholewicki et al. 1997). The best solution is to place the electrodes in a location from which no or minimal ECG artifact can be recorded. However, this is frequently difficult if not impossible. Consequently, numerous algorithms have been suggested to remove the ECG artifact.
One frequently implemented technique to remove this artifact is to compute a template of the ECG signal and subtract it from the electromyogram. This has been proved to be a reasonably successful procedure and has been implemented in studies recording from the diaphragm (Bartolo et al. 1996) as well as rectus abdominis (Hof 2009). Hof (2009) described a technique in which the ECG signal is recorded simultaneously with the EMG signal of interest, and template subtraction is then implemented (figure 8.15).
Digital filtering is another useful alternative for ECG artifact removal. Drake and Callaghan (2006) used a FIR (finite impulse response) filter with a hamming window, although they concluded that the most efficient filtering result could be obtained using a somewhat simpler fourth-order, 30 Hz high-pass cutoff Butterworth filter. Template subtraction improved extraction of the ECG signal but required a large amount of time.
Alternative ECG removal techniques include the use of adaptive filters (Lu et al. 2009; Marque et al. 2005), wavelet-independent component analysis (Taelman et al. 2007), and wavelet-based adaptive filters (Zhan et al. 2010). Irrespective of the technique used for ECG signal artifact removal, it is apparent that the resultant improvement in signal interpretation can be important. Hu and colleagues (2009), for example, found that the use of independent component analysis to remove artifact resulted in an improved ability to discriminate between patients with low back pain and normal subjects during both sitting and standing tasks.
Analysis of the EMG activity in back muscles has produced some interesting results, in part reflecting the unique anatomical issues discussed here. As long ago as 1962, Morris and colleagues noted that “the three muscles of the erector spinae group considered here . . .
do not always show parallel activity, and one may be active while the other two are inactive” (p. 519). This may suggest that the mere demonstration of greater or lesser EMG amplitude may not be indicative of abnormality in a particular muscle. Potentially more important than EMG amplitude is the timing of EMG activity. Deep and superficial portions of the multifidus are differentially active during arm movements (Moseley et al. 2002). One suggestion is that the manner in which the multifidus is differentially controlled in people with back pain compared with those without may be the source of chronic back pain (MacDonald et al. 2009).
Similarly, the analysis of EMG frequency characteristics in patients with back pain compared with control subjects has suggested that there may be differences between muscles and even differences within muscles. Patients with low back pain frequently exhibit alterations in the electromyographic response to fatigue in the back extensor muscles (Biedermann et al. 1991). This is one area in which spectral analysis of the EMG signals has been helpful. As in other muscles, median EMG frequency decreases in the trunk extensors with fatigue (Demoulin et al. 2007). It is interesting to note that Kramer and colleagues (2005) found that the magnitude of median frequency decrease was greater in healthy subjects than in individuals with back pain. Support for importance of electrode placement is obtained from the observations of Sung and colleagues (2009). Normal subjects and patients with back pain underwent an isometric contraction protocol that induced fatigue of the back extensors. EMG frequency measurements made in both the thoracic and the lumbar portions of the erector spinae indicated that the thoracic portion had a significantly lower median frequency than the lumbar portion in patients with low back pain. However, median frequency was lower in the lumbar portion than in the thoracic portion in control subjects. Thus, in the analysis of trunk musculature that might be involved in the development of back pain, EMG studies using wire electrodes frequently may be required to record from different portions of the muscle.
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Using musculoskeletal optimization models to generate control signals
A popular method for generating control signals in musculoskeletal models is to use some form of optimization.
Optimization Models
A popular method for generating control signals in musculoskeletal models is to use some form of optimization. In some cases, when the optimization criterion is derived from motor control principles, this is tantamount to developing a theoretical control model. In other cases, the optimization criterion may result in a coordinated movement pattern yet have little physiological basis. In general, the optimization approach is used in an effort to determine which set of model control signals will produce a result that optimizes (minimizes or maximizes) a given criterion measure. The criterion measure is known as a cost function, objective function, or performance criterion. The cost function can be relatively simple (e.g., find the solution that yields minimal muscle force) or complex (e.g., determine a nonlinear combination of maximal muscle force at minimal metabolic cost). It can be directly related to muscle function (e.g., minimize muscular work) or to some aspect of the motion under study (e.g., maximize vertical jump height). Alternatively, the cost function may be formulated to minimize the differences between model outputs (e.g., joint angles, ground reaction forces) and corresponding experimental measures. This latter case is often referred to as a tracking problem in that the goal is to find a solution that causes the model to follow, or track, the experimental data.
In all cases, the cost function serves as a guiding constraint that determines the selection of one particular set of optimal muscular controls from among many different possible solutions. Two major optimization approaches that have been used to control musculoskeletal models are referred to in the literature as static optimization and dynamic optimization. The meaning of static in this context is that the cost function is evaluated at each time step during a movement, independent of any prior or subsequent time steps. In contrast, dynamic optimization is dynamic in the sense that the entire movement sequence must be simulated to determine the value of the cost function.
Static Optimization
Initial attempts to predict individual muscle forces used static optimization models in conjunction with inverse dynamics analysis of an actual motor performance. The inverse dynamics analysis permits the calculation of net joint moments at incremental times throughout the movement (see chapter 5). In most applications, numerical optimization is used to find the set of muscle forces that balances these joint moments while also satisfying a selected cost function (figure 11.9). Thus, with this approach the muscle forces themselves, rather than neural signals, are the controls. The time-independent nature of static optimization allows solutions to be obtained with relatively little computational cost, but there are some drawbacks. One issue in early applications was sudden, nonphysiological switching on and off of muscle forces caused by the independence of solutions for sequential time increments (i.e., the optimization model balanced the joint moments separately for each time interval using very different sets of muscles). This problem can be avoided by careful selection of the initial guess in the optimization problem, such as by using the solution from one time step as the initial guess for the next time step. A stronger approach to address this issue is to use muscle models that invoke physiological realism through force-length and force-velocity relations and time-dependent stimulation-activation dynamics (chapter 9). When more detailed muscle models are included in a static optimization model, muscle activations become the control variables, rather than muscle forces.
Early studies using static optimization were plagued also by two other kinds of nonphysiological results. The first was the prediction of model forces that were too high for actual muscles to produce. This difficulty was easily addressed by defining physiologically valid maximal force constraints for each muscle in the musculoskeletal model. The second problem was that solutions would often select only one muscle to balance the net joint moment rather than choose a more realistic muscular synergy. Depending on the exact formulation, the muscle with the largest moment arm (if minimizing muscle force) or a favorable combination of moment arm and muscle strength (if minimizing muscle stress) would be selected to fully balance the joint moment, with zero force predicted in other synergist muscles. Mathematically, synergism can be produced by using nonlinear cost functions (e.g., minimizing the sum of squared or cubed muscle forces), although the physiological rationale for specific nonlinear cost functions is not always clear. A widely used nonlinear cost function proposed by Crowninshield and Brand (1981a) involves minimizing the sum of cubed muscle stresses. It was originally argued that this particular cost function would lead to solutions that maximize muscle endurance, making it appropriate for predicting muscle forces in submaximal tasks such as walking. However, the Crowninshield and Brand criterion has since been used to solve for muscle forces in a range of activities, some of which are unlikely candidates for maximizing muscle endurance (e.g., jumping, landing).
An additional issue with static optimization models is that they are essentially a decomposition of experimental joint moments into individual muscle forces. Therefore, the predicted muscle forces will be subject to any errors in the experimental joint moments in addition to any shortcomings associated with the approximation made in creating the musculoskeletal model. To complete this section, we present a static optimization example motivated by a classic review article by Crowninshield and Brand (1981b), which demonstrates the process of distributing an empirically determined elbow joint moment across a set of muscles spanning the elbow joint.
The net joint moment to be balanced in example 11.1 was a flexor moment, and only muscles with flexor moment arms were included in the model (figure 11.10). If an elbow extensor muscle had been included, there is no cost function of the type presented here that would predict force in an antagonist muscle. Any force in an elbow extensor would itself contribute to a higher cost function value and would also require greater elbow flexor forces to balance the target 10 N·m joint moment, further increasing the cost function value. The total lack of coactivation in this example is contrary to the common observation that during heavy activation of the elbow flexors, there is some activity in the triceps muscle group. The extensor coactivation likely helps stabilize the joint during heavy exertion but will not be predicted using traditional static optimization techniques in simple one DOF models. Although the example presented here focuses on obtaining numerical results for muscle forces, the interested reader is referred to the review by Crowninshield and Brand (1981b), which presents an interesting graphical interpretation of the solution to this static optimization problem.
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Dynamic Optimization
Ongoing work with static optimization models has led to the development of dynamic optimization or optimal control models, which are applied in conjunction with forward dynamics models of human motion. In contrast to the inverse dynamics approach, which uses experimental data from an actual performance to calculate net joint moments, forward dynamics analyses simulate the motion of the body from a given set of joint moments or muscle forces (see chapter 10). Dynamic optimization models are therefore often designed to find the muscle stimulation patterns that result in an optimal motion (figure 11.9). As mentioned earlier, the optimal motion may be one that maximizes (or minimizes) a performance criterion, such as maximizing jump height; or, the optimal motion may be defined as one that best reproduces a set of experimental data. Regardless, the variables that are optimized are usually the muscle stimulation patterns that control the motion of the musculoskeletal model. These stimulation patterns are used as the inputs to muscle models that predict individual muscle forces, which are then multiplied by the appropriate moment arms to compute the active muscle moments. The active muscle moments are combined with passive moments to compute the net joint moments, which actuate the skeletal model and produce movement. Thus, dynamic optimization leads to the synthesis of body segment kinematics associated with optimal performance, which is fundamentally different from the static optimization approach.
The optimal kinematics predicted from a dynamic optimization can be compared with experimental motion data; however, the experimental data are not required to obtain the solution, as they are with static optimization. This feature of dynamic optimization permits one to address questions for which no experimental data exist, such as testing the feasibility of possible forms of locomotion in extinct species (e.g., Nagano et al. 2005). Moreover, because dynamic optimization uses forward dynamics models that simulate the whole movement performance and provide complete muscle force time histories (rather than solutions at independent time increments), many of the problems associated with static optimization models are overcome. However, these advantages come at a computational cost, because dynamic optimization often requires a more detailed musculoskeletal model and the entire movement must be simulated for each possible solution. Thus, solving a dynamic optimization problem can easily require an order of magnitude more time than is required to solve a comparable static optimization problem. Although computational costs are difficult to compare in an objective manner, it is not uncommon for static optimization solutions to take seconds or minutes to obtain, whereas dynamic optimization solutions can take hours, days, or even weeks, when run on standard computer workstations.
In many cases, users of dynamic optimization must define the goal of the movement mathematically. This definition is easiest for movements in which the performance criterion to be optimized is clear in a mechanical sense. In vertical jumping, for example, the cost function can be stated as maximization of the vertical displacement of the body center of mass during the flight phase. If the model is constructed with appropriate constraints (realistic muscle properties and joint range of motion), jump heights and body segment motions approximating those of human jumpers can be attained (Pandy and Zajac 1991; van Soest et al. 1993). However, for many human movements the performance criterion is not as clear. In walking, for instance, the goal may seem to be getting from point A to point B in a certain amount of time. Unfortunately, this does not provide enough information to solve for a set of muscle stimulation patterns that will result in realistic walking. Successful simulations of walking have been generated using a cost function that minimizes the metabolic cost of transport (Anderson and Pandy 2001; Umberger 2010); however, the cost function formulation is considerably more complicated than for vertical jumping. Minimum-energy solutions also require the inclusion of an additional model for predicting the metabolic cost of muscle actions (e.g., Umberger et al. 2003). In cases where the performance criterion is not clear, dynamic optimization can serve as an effective approach for testing various theoretical criteria to see how well each can produce the desired movement patterns.
Movements for which the underlying criterion is difficult to define, such as walking or pedaling, have often been simulated by formulating and solving a tracking problem. This approach has a similar appeal to the EMG-based techniques described earlier, in that the resulting motion should closely approximate the way in which humans actually move. However, it is unclear which of several possible experimental measures (kinematics, kinetics, EMG) are the most important for the model to track. Also, because of errors in the experimental data and differences between the musculoskeletal model and the experimental subjects, it may be impossible for the model to track the data perfectly. This approach also limits much of the predictive ability of the dynamic optimization approach, as it is only possible to consider conditions for which experimental data are available.
The tracking approach has seen frequent use in hybrid solution algorithms, which seek to retain the advantages of forward dynamics and dynamic optimization while achieving the computational efficiency of static optimization. Two examples are computed muscle control (Thelen et al. 2003) and direct collocation (Kaplan and Heegaard 2001). Computed muscle control works by solving a static optimization problem at each time step within a single forward dynamics movement simulation. By using feedback on the kinematics and muscle activations at each time step of the numerical integration, computed muscle control can generate a forward dynamics simulation that optimally tracks a set of experimental data without solving a dynamic optimization problem that would require thousands of forward dynamics simulations. In contrast, direct collocation works by converting the differential equations of motion into a set of algebraic constraint equations and treating both the control variables (muscle stimulations) and state variables (positions and velocities) as unknowns in the optimization problem. Although they differ considerably in implementation, both techniques appear to be able to produce results that are similar to dynamic optimization tracking solutions, with a computational cost closer to that of static optimization.
Although static and dynamic optimization have both become popular means for controlling musculoskeletal models, several issues concerning their use must be addressed. Do humans actually produce movements based on one given performance criterion, or does the performance objective change during the movement? If the optimized model results differ from those of a human performer, is it because the model is too simple or not constrained properly, or is the human not performing optimally? Finally, EMG data have illustrated various degrees of simultaneous antagonistic cocontraction during some movement sequences. Optimization models tend not to yield solutions predicting muscle antagonism around a joint, although some antagonism is predicted when models contain muscles that contribute to more than one joint moment. However, optimization models that seek to minimize or maximize specific cost functions will not predict antagonistic cocontraction associated with joint stability or stiffness. In the case of walking, for example, there is no single cost function that could simultaneously predict the relatively minimal muscle coactivation in healthy subjects and the substantial antagonism exhibited by patients with cerebral palsy. Despite these drawbacks, optimization models have increased the understanding of human biomechanics and will continue to do so in the future.
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The state space in the presentation of biomechanical data
Biomechanical data can be represented graphically in a variety of different ways.
The State Space
Biomechanical data can be represented graphically in a variety of different ways. A traditional and very informative type of graph is to display the changes in kinetic or kinematic parameters over time. Figure 13.4 presents an example of the changes in the left and right thigh angular displacements over a time period of 10 s while a subject is walking on a treadmill. Information about peak flexion and extension angles can be obtained from these graphs and quantified. A closer inspection of the graphs makes it quite evident that the coordination between these segments is primarily antiphase. However, especially during the stance phase there are more subtle changes in the patterns of coordination that are harder to obtain from these plots.
In the dynamical systems approach, the reconstruction of the so-called state space is essential in identifying the important features of the behavior of a system, such as its stability and ability to change and adapt to different environmental and task constraints. The state space is a representation of the relevant variables that will help identify these features. To help us understand and quantify coordination between joints or segments, it can be very useful to represent the system in a state space that is based on an angle-angle relative motion plot. Figure 13.5 presents a state space of the relative motion of the time series of the thigh and leg shown earlier in figure 13.4. This angle-angle plot can reveal regions where coordination changes take place as well as parts of the gait cycle where there is relative invariance in coordination patterns. These coordinative changes in the angle-angle plots can be further quantified by vector coding techniques that are discussed later.
Another form of state space is where the position and velocity of a joint or segment are plotted relative to each other. This state-space representation is also often referred to as the phase plane. An example of a position-velocity phase-plane plot is shown in figure 13.6, where the angular displacement of the thigh is plotted against the angular velocity. This is a higher-dimensional state space because the time derivative of position is used to identify the pattern. The phase-plane representation is a first and critical step in the quantification of coordination using continuous relative phase techniques.
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Attractors in State Space
In most forms of human movement, the dynamics in state space are limited to distinct regions, as can be seen in the phase-plane plot in figure 13.6, where the pattern in state space is limited to a fairly narrow, cyclical band. In dynamical systems, preferred regions in state space onto which the dynamics tend to settle are called attractors. Attractors can come in a variety of different forms. A seemingly simple kind of attractor is the point attractor. In this case, the dynamics in the system tend to converge onto one relatively fixed value in the state space. Figure 13.7, c and d, provides an example of a point attractor. The point attractor dynamics are shown in figure 13.7c by a relatively consistent value of discrete relative phase throughout the gait cycle. The discrete relative phase is based on the occurrence of peak flexion in the left and right thigh angular displacements, and the time series in 13.7c show that there is a consistent antiphase or 180° coordination pattern. This antiphase coordination can already be observed by comparing the individual time series in figure 13.4 but is more objectively quantified by the state-space plot in figure 13.7d, where so-called return maps (plotting the coordination at xn vs. xn+1, where n is the cycle number) identify a fixed-point attractor in state space. The relative phase dynamics are of the fixed-point type in this case, as a perturbation (sudden change in treadmill speed and return to the original speed; figure 13.7) results in a return to the original antiphase dynamics with a relative short latency. The perturbation here consisted of a brief (5 s) increase in treadmill speed from 1.2 m/s to 2.0 m/s, with a subsequent return to 1.2 m/s.
Whereas the coordination between the right and left legs can be identified in the form of a point attractor with a relatively fixed value of coordination from cycle to cycle, the dynamics of the individual limb segments show a very different pattern. These are clearly cyclical, as can be seen in the phase planes in figure 13.7, a and b. Attractors of this form are called limit-cycle attractors, and the dynamics converge onto a cyclical region in state space. These limit-cycle attractors are typically identified on the basis of a very narrow, overlapping band of the trajectories in the state space. However, the existence of the narrow band is a necessary but not sufficient condition to characterize the dynamics as a limit-cycle. An essential feature of the limit-cycle attractor is stability with respect to perturbation; to classify as a limit-cycle attractor, the system should also show resistance to perturbations. An example of this is given in figure 13.7, a and b. The regular cyclic pattern in the phase plane (solid line) represents the steady-state gait patterns while a subject was walking at a speed of 1.2 m/s. The dashed trajectories represent the perturbation phase of the trial when speed was suddenly increased to 2.0 m/s and the consequent return back to the original pattern. This return to the preperturbation dynamic is an essential feature of the limit-cycle attractor. The patterns in the phase plane can also serve as an energy plot. The convergence and divergence of trajectories in the phase plane can identify the loss and gain of energy in the system.
Higher-dimensional state spaces (three dimensions and up) can also reveal more complex types of attractors: quasiperiodic and chaotic attractors. The chaotic attractor demonstrates both stable attraction to a region in state space and variability. Figure 13.8 shows an example of the Lorenz attractor, a well-known chaotic attractor that emerges from dynamic interactions in fluid and air flow systems (Strogatz 1994). These dual features of stability and adaptability can be associated with a higher pattern complexity that is now commonly regarded as reflective of healthy and expert systems (see figure 13.3). As an example, increased heart rate variability is considered an important indicator of healthy heart function, reflecting a degree of complexity in organization in which disruptions can be compensated for more easily (Glass 2001; Lipsitz 2002).
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Using EMG to diagnose and treat low back pain
The diagnosis and treatment of low back pain are important issues.
EMG and Low Back Pain
The diagnosis and treatment of low back pain are important issues. In the United States, for example, back pain accounts for two-thirds of all workers compensation costs. Peripheral muscle problems may be one issue; atrophy of the multifidus is frequently observed in patients with low back pain (Hides et al. 1994). EMG is becoming an increasingly useful tool for the diagnosis and treatment of low back pain.
The trunk musculature plays an important role in tasks such as lifting and throwing. During these kinds of tasks as well as in other activities that might threaten the postural control of the individual and perturb balance, the nervous system implements strategies to activate trunk muscles while performing voluntary movements involving remote muscle groups. However, the prevalence of back pain in the general population, and the need to identify ergonomically efficient and safe ways to use back muscles, require that we understand the nature of activation in muscles controlling the trunk. Here, too, EMG is an important tool.
A number of issues render EMG recordings from the trunk musculature difficult to obtain and interpret. From an anatomical viewpoint, the architecture of the back muscles is complex. For example, it is generally accepted that injury or disorder of the multifidus frequently results in back pain. Both multifidus and erector spinae have distinct superficial and deep portions (Bustami 1986; Macintosh et al. 1986) with different histochemical and biomechanical characteristics (Bogduk et al. 1992; Dickx et al. 2010). In some respects, it is fortunate that most of the muscle mass of the multifidus lies in the more superficial portion, so that surface EMG has a greater possibility of characterizing the fibers that produce large amounts of force. However, the multifidus superficial portion has a different role than the deeper fibers. Fibers in the superficial portion cross numerous spinal segments and are thus in a better position to produce back extension. However, the deeper fibers are relatively short, crossing perhaps one or two segments, and thus protect segments of the lumbar spine from inappropriate shear or torsion torques (Macintosh and Valencia 1986).
In addition to issues concerning the activation of the back extensor muscles, a number of important questions require resolution by EMG analysis in which considerable electrocardiographic (ECG) artifact may be present. ECG artifact is particularly problematic during relatively low force contractions, exaggerating the ratio between the signal of interest and the interfering ECG signal. Recording EMG signals from abdominal, knee extensor, and other trunk muscles during normal human movements, for example, can result in the placement of electrodes well within recording range of the electrocardiogram. The relatively large ECG signal can exaggerate the amplitude of EMG activity, and the low-frequency characteristics of the ECG wave can also alter the frequency characteristics of the recorded EMG activity.
For example, the absence of sufficient trunk activity can produce instability resulting in low back pain (van Dieën et al. 2003). However, the amplitude of EMG activity required to ensure stability is small and thus can be affected by the ECG signal (Cholewicki et al. 1997). The best solution is to place the electrodes in a location from which no or minimal ECG artifact can be recorded. However, this is frequently difficult if not impossible. Consequently, numerous algorithms have been suggested to remove the ECG artifact.
One frequently implemented technique to remove this artifact is to compute a template of the ECG signal and subtract it from the electromyogram. This has been proved to be a reasonably successful procedure and has been implemented in studies recording from the diaphragm (Bartolo et al. 1996) as well as rectus abdominis (Hof 2009). Hof (2009) described a technique in which the ECG signal is recorded simultaneously with the EMG signal of interest, and template subtraction is then implemented (figure 8.15).
Digital filtering is another useful alternative for ECG artifact removal. Drake and Callaghan (2006) used a FIR (finite impulse response) filter with a hamming window, although they concluded that the most efficient filtering result could be obtained using a somewhat simpler fourth-order, 30 Hz high-pass cutoff Butterworth filter. Template subtraction improved extraction of the ECG signal but required a large amount of time.
Alternative ECG removal techniques include the use of adaptive filters (Lu et al. 2009; Marque et al. 2005), wavelet-independent component analysis (Taelman et al. 2007), and wavelet-based adaptive filters (Zhan et al. 2010). Irrespective of the technique used for ECG signal artifact removal, it is apparent that the resultant improvement in signal interpretation can be important. Hu and colleagues (2009), for example, found that the use of independent component analysis to remove artifact resulted in an improved ability to discriminate between patients with low back pain and normal subjects during both sitting and standing tasks.
Analysis of the EMG activity in back muscles has produced some interesting results, in part reflecting the unique anatomical issues discussed here. As long ago as 1962, Morris and colleagues noted that “the three muscles of the erector spinae group considered here . . .
do not always show parallel activity, and one may be active while the other two are inactive” (p. 519). This may suggest that the mere demonstration of greater or lesser EMG amplitude may not be indicative of abnormality in a particular muscle. Potentially more important than EMG amplitude is the timing of EMG activity. Deep and superficial portions of the multifidus are differentially active during arm movements (Moseley et al. 2002). One suggestion is that the manner in which the multifidus is differentially controlled in people with back pain compared with those without may be the source of chronic back pain (MacDonald et al. 2009).
Similarly, the analysis of EMG frequency characteristics in patients with back pain compared with control subjects has suggested that there may be differences between muscles and even differences within muscles. Patients with low back pain frequently exhibit alterations in the electromyographic response to fatigue in the back extensor muscles (Biedermann et al. 1991). This is one area in which spectral analysis of the EMG signals has been helpful. As in other muscles, median EMG frequency decreases in the trunk extensors with fatigue (Demoulin et al. 2007). It is interesting to note that Kramer and colleagues (2005) found that the magnitude of median frequency decrease was greater in healthy subjects than in individuals with back pain. Support for importance of electrode placement is obtained from the observations of Sung and colleagues (2009). Normal subjects and patients with back pain underwent an isometric contraction protocol that induced fatigue of the back extensors. EMG frequency measurements made in both the thoracic and the lumbar portions of the erector spinae indicated that the thoracic portion had a significantly lower median frequency than the lumbar portion in patients with low back pain. However, median frequency was lower in the lumbar portion than in the thoracic portion in control subjects. Thus, in the analysis of trunk musculature that might be involved in the development of back pain, EMG studies using wire electrodes frequently may be required to record from different portions of the muscle.
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Using musculoskeletal optimization models to generate control signals
A popular method for generating control signals in musculoskeletal models is to use some form of optimization.
Optimization Models
A popular method for generating control signals in musculoskeletal models is to use some form of optimization. In some cases, when the optimization criterion is derived from motor control principles, this is tantamount to developing a theoretical control model. In other cases, the optimization criterion may result in a coordinated movement pattern yet have little physiological basis. In general, the optimization approach is used in an effort to determine which set of model control signals will produce a result that optimizes (minimizes or maximizes) a given criterion measure. The criterion measure is known as a cost function, objective function, or performance criterion. The cost function can be relatively simple (e.g., find the solution that yields minimal muscle force) or complex (e.g., determine a nonlinear combination of maximal muscle force at minimal metabolic cost). It can be directly related to muscle function (e.g., minimize muscular work) or to some aspect of the motion under study (e.g., maximize vertical jump height). Alternatively, the cost function may be formulated to minimize the differences between model outputs (e.g., joint angles, ground reaction forces) and corresponding experimental measures. This latter case is often referred to as a tracking problem in that the goal is to find a solution that causes the model to follow, or track, the experimental data.
In all cases, the cost function serves as a guiding constraint that determines the selection of one particular set of optimal muscular controls from among many different possible solutions. Two major optimization approaches that have been used to control musculoskeletal models are referred to in the literature as static optimization and dynamic optimization. The meaning of static in this context is that the cost function is evaluated at each time step during a movement, independent of any prior or subsequent time steps. In contrast, dynamic optimization is dynamic in the sense that the entire movement sequence must be simulated to determine the value of the cost function.
Static Optimization
Initial attempts to predict individual muscle forces used static optimization models in conjunction with inverse dynamics analysis of an actual motor performance. The inverse dynamics analysis permits the calculation of net joint moments at incremental times throughout the movement (see chapter 5). In most applications, numerical optimization is used to find the set of muscle forces that balances these joint moments while also satisfying a selected cost function (figure 11.9). Thus, with this approach the muscle forces themselves, rather than neural signals, are the controls. The time-independent nature of static optimization allows solutions to be obtained with relatively little computational cost, but there are some drawbacks. One issue in early applications was sudden, nonphysiological switching on and off of muscle forces caused by the independence of solutions for sequential time increments (i.e., the optimization model balanced the joint moments separately for each time interval using very different sets of muscles). This problem can be avoided by careful selection of the initial guess in the optimization problem, such as by using the solution from one time step as the initial guess for the next time step. A stronger approach to address this issue is to use muscle models that invoke physiological realism through force-length and force-velocity relations and time-dependent stimulation-activation dynamics (chapter 9). When more detailed muscle models are included in a static optimization model, muscle activations become the control variables, rather than muscle forces.
Early studies using static optimization were plagued also by two other kinds of nonphysiological results. The first was the prediction of model forces that were too high for actual muscles to produce. This difficulty was easily addressed by defining physiologically valid maximal force constraints for each muscle in the musculoskeletal model. The second problem was that solutions would often select only one muscle to balance the net joint moment rather than choose a more realistic muscular synergy. Depending on the exact formulation, the muscle with the largest moment arm (if minimizing muscle force) or a favorable combination of moment arm and muscle strength (if minimizing muscle stress) would be selected to fully balance the joint moment, with zero force predicted in other synergist muscles. Mathematically, synergism can be produced by using nonlinear cost functions (e.g., minimizing the sum of squared or cubed muscle forces), although the physiological rationale for specific nonlinear cost functions is not always clear. A widely used nonlinear cost function proposed by Crowninshield and Brand (1981a) involves minimizing the sum of cubed muscle stresses. It was originally argued that this particular cost function would lead to solutions that maximize muscle endurance, making it appropriate for predicting muscle forces in submaximal tasks such as walking. However, the Crowninshield and Brand criterion has since been used to solve for muscle forces in a range of activities, some of which are unlikely candidates for maximizing muscle endurance (e.g., jumping, landing).
An additional issue with static optimization models is that they are essentially a decomposition of experimental joint moments into individual muscle forces. Therefore, the predicted muscle forces will be subject to any errors in the experimental joint moments in addition to any shortcomings associated with the approximation made in creating the musculoskeletal model. To complete this section, we present a static optimization example motivated by a classic review article by Crowninshield and Brand (1981b), which demonstrates the process of distributing an empirically determined elbow joint moment across a set of muscles spanning the elbow joint.
The net joint moment to be balanced in example 11.1 was a flexor moment, and only muscles with flexor moment arms were included in the model (figure 11.10). If an elbow extensor muscle had been included, there is no cost function of the type presented here that would predict force in an antagonist muscle. Any force in an elbow extensor would itself contribute to a higher cost function value and would also require greater elbow flexor forces to balance the target 10 N·m joint moment, further increasing the cost function value. The total lack of coactivation in this example is contrary to the common observation that during heavy activation of the elbow flexors, there is some activity in the triceps muscle group. The extensor coactivation likely helps stabilize the joint during heavy exertion but will not be predicted using traditional static optimization techniques in simple one DOF models. Although the example presented here focuses on obtaining numerical results for muscle forces, the interested reader is referred to the review by Crowninshield and Brand (1981b), which presents an interesting graphical interpretation of the solution to this static optimization problem.
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Dynamic Optimization
Ongoing work with static optimization models has led to the development of dynamic optimization or optimal control models, which are applied in conjunction with forward dynamics models of human motion. In contrast to the inverse dynamics approach, which uses experimental data from an actual performance to calculate net joint moments, forward dynamics analyses simulate the motion of the body from a given set of joint moments or muscle forces (see chapter 10). Dynamic optimization models are therefore often designed to find the muscle stimulation patterns that result in an optimal motion (figure 11.9). As mentioned earlier, the optimal motion may be one that maximizes (or minimizes) a performance criterion, such as maximizing jump height; or, the optimal motion may be defined as one that best reproduces a set of experimental data. Regardless, the variables that are optimized are usually the muscle stimulation patterns that control the motion of the musculoskeletal model. These stimulation patterns are used as the inputs to muscle models that predict individual muscle forces, which are then multiplied by the appropriate moment arms to compute the active muscle moments. The active muscle moments are combined with passive moments to compute the net joint moments, which actuate the skeletal model and produce movement. Thus, dynamic optimization leads to the synthesis of body segment kinematics associated with optimal performance, which is fundamentally different from the static optimization approach.
The optimal kinematics predicted from a dynamic optimization can be compared with experimental motion data; however, the experimental data are not required to obtain the solution, as they are with static optimization. This feature of dynamic optimization permits one to address questions for which no experimental data exist, such as testing the feasibility of possible forms of locomotion in extinct species (e.g., Nagano et al. 2005). Moreover, because dynamic optimization uses forward dynamics models that simulate the whole movement performance and provide complete muscle force time histories (rather than solutions at independent time increments), many of the problems associated with static optimization models are overcome. However, these advantages come at a computational cost, because dynamic optimization often requires a more detailed musculoskeletal model and the entire movement must be simulated for each possible solution. Thus, solving a dynamic optimization problem can easily require an order of magnitude more time than is required to solve a comparable static optimization problem. Although computational costs are difficult to compare in an objective manner, it is not uncommon for static optimization solutions to take seconds or minutes to obtain, whereas dynamic optimization solutions can take hours, days, or even weeks, when run on standard computer workstations.
In many cases, users of dynamic optimization must define the goal of the movement mathematically. This definition is easiest for movements in which the performance criterion to be optimized is clear in a mechanical sense. In vertical jumping, for example, the cost function can be stated as maximization of the vertical displacement of the body center of mass during the flight phase. If the model is constructed with appropriate constraints (realistic muscle properties and joint range of motion), jump heights and body segment motions approximating those of human jumpers can be attained (Pandy and Zajac 1991; van Soest et al. 1993). However, for many human movements the performance criterion is not as clear. In walking, for instance, the goal may seem to be getting from point A to point B in a certain amount of time. Unfortunately, this does not provide enough information to solve for a set of muscle stimulation patterns that will result in realistic walking. Successful simulations of walking have been generated using a cost function that minimizes the metabolic cost of transport (Anderson and Pandy 2001; Umberger 2010); however, the cost function formulation is considerably more complicated than for vertical jumping. Minimum-energy solutions also require the inclusion of an additional model for predicting the metabolic cost of muscle actions (e.g., Umberger et al. 2003). In cases where the performance criterion is not clear, dynamic optimization can serve as an effective approach for testing various theoretical criteria to see how well each can produce the desired movement patterns.
Movements for which the underlying criterion is difficult to define, such as walking or pedaling, have often been simulated by formulating and solving a tracking problem. This approach has a similar appeal to the EMG-based techniques described earlier, in that the resulting motion should closely approximate the way in which humans actually move. However, it is unclear which of several possible experimental measures (kinematics, kinetics, EMG) are the most important for the model to track. Also, because of errors in the experimental data and differences between the musculoskeletal model and the experimental subjects, it may be impossible for the model to track the data perfectly. This approach also limits much of the predictive ability of the dynamic optimization approach, as it is only possible to consider conditions for which experimental data are available.
The tracking approach has seen frequent use in hybrid solution algorithms, which seek to retain the advantages of forward dynamics and dynamic optimization while achieving the computational efficiency of static optimization. Two examples are computed muscle control (Thelen et al. 2003) and direct collocation (Kaplan and Heegaard 2001). Computed muscle control works by solving a static optimization problem at each time step within a single forward dynamics movement simulation. By using feedback on the kinematics and muscle activations at each time step of the numerical integration, computed muscle control can generate a forward dynamics simulation that optimally tracks a set of experimental data without solving a dynamic optimization problem that would require thousands of forward dynamics simulations. In contrast, direct collocation works by converting the differential equations of motion into a set of algebraic constraint equations and treating both the control variables (muscle stimulations) and state variables (positions and velocities) as unknowns in the optimization problem. Although they differ considerably in implementation, both techniques appear to be able to produce results that are similar to dynamic optimization tracking solutions, with a computational cost closer to that of static optimization.
Although static and dynamic optimization have both become popular means for controlling musculoskeletal models, several issues concerning their use must be addressed. Do humans actually produce movements based on one given performance criterion, or does the performance objective change during the movement? If the optimized model results differ from those of a human performer, is it because the model is too simple or not constrained properly, or is the human not performing optimally? Finally, EMG data have illustrated various degrees of simultaneous antagonistic cocontraction during some movement sequences. Optimization models tend not to yield solutions predicting muscle antagonism around a joint, although some antagonism is predicted when models contain muscles that contribute to more than one joint moment. However, optimization models that seek to minimize or maximize specific cost functions will not predict antagonistic cocontraction associated with joint stability or stiffness. In the case of walking, for example, there is no single cost function that could simultaneously predict the relatively minimal muscle coactivation in healthy subjects and the substantial antagonism exhibited by patients with cerebral palsy. Despite these drawbacks, optimization models have increased the understanding of human biomechanics and will continue to do so in the future.
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The state space in the presentation of biomechanical data
Biomechanical data can be represented graphically in a variety of different ways.
The State Space
Biomechanical data can be represented graphically in a variety of different ways. A traditional and very informative type of graph is to display the changes in kinetic or kinematic parameters over time. Figure 13.4 presents an example of the changes in the left and right thigh angular displacements over a time period of 10 s while a subject is walking on a treadmill. Information about peak flexion and extension angles can be obtained from these graphs and quantified. A closer inspection of the graphs makes it quite evident that the coordination between these segments is primarily antiphase. However, especially during the stance phase there are more subtle changes in the patterns of coordination that are harder to obtain from these plots.
In the dynamical systems approach, the reconstruction of the so-called state space is essential in identifying the important features of the behavior of a system, such as its stability and ability to change and adapt to different environmental and task constraints. The state space is a representation of the relevant variables that will help identify these features. To help us understand and quantify coordination between joints or segments, it can be very useful to represent the system in a state space that is based on an angle-angle relative motion plot. Figure 13.5 presents a state space of the relative motion of the time series of the thigh and leg shown earlier in figure 13.4. This angle-angle plot can reveal regions where coordination changes take place as well as parts of the gait cycle where there is relative invariance in coordination patterns. These coordinative changes in the angle-angle plots can be further quantified by vector coding techniques that are discussed later.
Another form of state space is where the position and velocity of a joint or segment are plotted relative to each other. This state-space representation is also often referred to as the phase plane. An example of a position-velocity phase-plane plot is shown in figure 13.6, where the angular displacement of the thigh is plotted against the angular velocity. This is a higher-dimensional state space because the time derivative of position is used to identify the pattern. The phase-plane representation is a first and critical step in the quantification of coordination using continuous relative phase techniques.
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Attractors in State Space
In most forms of human movement, the dynamics in state space are limited to distinct regions, as can be seen in the phase-plane plot in figure 13.6, where the pattern in state space is limited to a fairly narrow, cyclical band. In dynamical systems, preferred regions in state space onto which the dynamics tend to settle are called attractors. Attractors can come in a variety of different forms. A seemingly simple kind of attractor is the point attractor. In this case, the dynamics in the system tend to converge onto one relatively fixed value in the state space. Figure 13.7, c and d, provides an example of a point attractor. The point attractor dynamics are shown in figure 13.7c by a relatively consistent value of discrete relative phase throughout the gait cycle. The discrete relative phase is based on the occurrence of peak flexion in the left and right thigh angular displacements, and the time series in 13.7c show that there is a consistent antiphase or 180° coordination pattern. This antiphase coordination can already be observed by comparing the individual time series in figure 13.4 but is more objectively quantified by the state-space plot in figure 13.7d, where so-called return maps (plotting the coordination at xn vs. xn+1, where n is the cycle number) identify a fixed-point attractor in state space. The relative phase dynamics are of the fixed-point type in this case, as a perturbation (sudden change in treadmill speed and return to the original speed; figure 13.7) results in a return to the original antiphase dynamics with a relative short latency. The perturbation here consisted of a brief (5 s) increase in treadmill speed from 1.2 m/s to 2.0 m/s, with a subsequent return to 1.2 m/s.
Whereas the coordination between the right and left legs can be identified in the form of a point attractor with a relatively fixed value of coordination from cycle to cycle, the dynamics of the individual limb segments show a very different pattern. These are clearly cyclical, as can be seen in the phase planes in figure 13.7, a and b. Attractors of this form are called limit-cycle attractors, and the dynamics converge onto a cyclical region in state space. These limit-cycle attractors are typically identified on the basis of a very narrow, overlapping band of the trajectories in the state space. However, the existence of the narrow band is a necessary but not sufficient condition to characterize the dynamics as a limit-cycle. An essential feature of the limit-cycle attractor is stability with respect to perturbation; to classify as a limit-cycle attractor, the system should also show resistance to perturbations. An example of this is given in figure 13.7, a and b. The regular cyclic pattern in the phase plane (solid line) represents the steady-state gait patterns while a subject was walking at a speed of 1.2 m/s. The dashed trajectories represent the perturbation phase of the trial when speed was suddenly increased to 2.0 m/s and the consequent return back to the original pattern. This return to the preperturbation dynamic is an essential feature of the limit-cycle attractor. The patterns in the phase plane can also serve as an energy plot. The convergence and divergence of trajectories in the phase plane can identify the loss and gain of energy in the system.
Higher-dimensional state spaces (three dimensions and up) can also reveal more complex types of attractors: quasiperiodic and chaotic attractors. The chaotic attractor demonstrates both stable attraction to a region in state space and variability. Figure 13.8 shows an example of the Lorenz attractor, a well-known chaotic attractor that emerges from dynamic interactions in fluid and air flow systems (Strogatz 1994). These dual features of stability and adaptability can be associated with a higher pattern complexity that is now commonly regarded as reflective of healthy and expert systems (see figure 13.3). As an example, increased heart rate variability is considered an important indicator of healthy heart function, reflecting a degree of complexity in organization in which disruptions can be compensated for more easily (Glass 2001; Lipsitz 2002).
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Using EMG to diagnose and treat low back pain
The diagnosis and treatment of low back pain are important issues.
EMG and Low Back Pain
The diagnosis and treatment of low back pain are important issues. In the United States, for example, back pain accounts for two-thirds of all workers compensation costs. Peripheral muscle problems may be one issue; atrophy of the multifidus is frequently observed in patients with low back pain (Hides et al. 1994). EMG is becoming an increasingly useful tool for the diagnosis and treatment of low back pain.
The trunk musculature plays an important role in tasks such as lifting and throwing. During these kinds of tasks as well as in other activities that might threaten the postural control of the individual and perturb balance, the nervous system implements strategies to activate trunk muscles while performing voluntary movements involving remote muscle groups. However, the prevalence of back pain in the general population, and the need to identify ergonomically efficient and safe ways to use back muscles, require that we understand the nature of activation in muscles controlling the trunk. Here, too, EMG is an important tool.
A number of issues render EMG recordings from the trunk musculature difficult to obtain and interpret. From an anatomical viewpoint, the architecture of the back muscles is complex. For example, it is generally accepted that injury or disorder of the multifidus frequently results in back pain. Both multifidus and erector spinae have distinct superficial and deep portions (Bustami 1986; Macintosh et al. 1986) with different histochemical and biomechanical characteristics (Bogduk et al. 1992; Dickx et al. 2010). In some respects, it is fortunate that most of the muscle mass of the multifidus lies in the more superficial portion, so that surface EMG has a greater possibility of characterizing the fibers that produce large amounts of force. However, the multifidus superficial portion has a different role than the deeper fibers. Fibers in the superficial portion cross numerous spinal segments and are thus in a better position to produce back extension. However, the deeper fibers are relatively short, crossing perhaps one or two segments, and thus protect segments of the lumbar spine from inappropriate shear or torsion torques (Macintosh and Valencia 1986).
In addition to issues concerning the activation of the back extensor muscles, a number of important questions require resolution by EMG analysis in which considerable electrocardiographic (ECG) artifact may be present. ECG artifact is particularly problematic during relatively low force contractions, exaggerating the ratio between the signal of interest and the interfering ECG signal. Recording EMG signals from abdominal, knee extensor, and other trunk muscles during normal human movements, for example, can result in the placement of electrodes well within recording range of the electrocardiogram. The relatively large ECG signal can exaggerate the amplitude of EMG activity, and the low-frequency characteristics of the ECG wave can also alter the frequency characteristics of the recorded EMG activity.
For example, the absence of sufficient trunk activity can produce instability resulting in low back pain (van Dieën et al. 2003). However, the amplitude of EMG activity required to ensure stability is small and thus can be affected by the ECG signal (Cholewicki et al. 1997). The best solution is to place the electrodes in a location from which no or minimal ECG artifact can be recorded. However, this is frequently difficult if not impossible. Consequently, numerous algorithms have been suggested to remove the ECG artifact.
One frequently implemented technique to remove this artifact is to compute a template of the ECG signal and subtract it from the electromyogram. This has been proved to be a reasonably successful procedure and has been implemented in studies recording from the diaphragm (Bartolo et al. 1996) as well as rectus abdominis (Hof 2009). Hof (2009) described a technique in which the ECG signal is recorded simultaneously with the EMG signal of interest, and template subtraction is then implemented (figure 8.15).
Digital filtering is another useful alternative for ECG artifact removal. Drake and Callaghan (2006) used a FIR (finite impulse response) filter with a hamming window, although they concluded that the most efficient filtering result could be obtained using a somewhat simpler fourth-order, 30 Hz high-pass cutoff Butterworth filter. Template subtraction improved extraction of the ECG signal but required a large amount of time.
Alternative ECG removal techniques include the use of adaptive filters (Lu et al. 2009; Marque et al. 2005), wavelet-independent component analysis (Taelman et al. 2007), and wavelet-based adaptive filters (Zhan et al. 2010). Irrespective of the technique used for ECG signal artifact removal, it is apparent that the resultant improvement in signal interpretation can be important. Hu and colleagues (2009), for example, found that the use of independent component analysis to remove artifact resulted in an improved ability to discriminate between patients with low back pain and normal subjects during both sitting and standing tasks.
Analysis of the EMG activity in back muscles has produced some interesting results, in part reflecting the unique anatomical issues discussed here. As long ago as 1962, Morris and colleagues noted that “the three muscles of the erector spinae group considered here . . .
do not always show parallel activity, and one may be active while the other two are inactive” (p. 519). This may suggest that the mere demonstration of greater or lesser EMG amplitude may not be indicative of abnormality in a particular muscle. Potentially more important than EMG amplitude is the timing of EMG activity. Deep and superficial portions of the multifidus are differentially active during arm movements (Moseley et al. 2002). One suggestion is that the manner in which the multifidus is differentially controlled in people with back pain compared with those without may be the source of chronic back pain (MacDonald et al. 2009).
Similarly, the analysis of EMG frequency characteristics in patients with back pain compared with control subjects has suggested that there may be differences between muscles and even differences within muscles. Patients with low back pain frequently exhibit alterations in the electromyographic response to fatigue in the back extensor muscles (Biedermann et al. 1991). This is one area in which spectral analysis of the EMG signals has been helpful. As in other muscles, median EMG frequency decreases in the trunk extensors with fatigue (Demoulin et al. 2007). It is interesting to note that Kramer and colleagues (2005) found that the magnitude of median frequency decrease was greater in healthy subjects than in individuals with back pain. Support for importance of electrode placement is obtained from the observations of Sung and colleagues (2009). Normal subjects and patients with back pain underwent an isometric contraction protocol that induced fatigue of the back extensors. EMG frequency measurements made in both the thoracic and the lumbar portions of the erector spinae indicated that the thoracic portion had a significantly lower median frequency than the lumbar portion in patients with low back pain. However, median frequency was lower in the lumbar portion than in the thoracic portion in control subjects. Thus, in the analysis of trunk musculature that might be involved in the development of back pain, EMG studies using wire electrodes frequently may be required to record from different portions of the muscle.
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Using musculoskeletal optimization models to generate control signals
A popular method for generating control signals in musculoskeletal models is to use some form of optimization.
Optimization Models
A popular method for generating control signals in musculoskeletal models is to use some form of optimization. In some cases, when the optimization criterion is derived from motor control principles, this is tantamount to developing a theoretical control model. In other cases, the optimization criterion may result in a coordinated movement pattern yet have little physiological basis. In general, the optimization approach is used in an effort to determine which set of model control signals will produce a result that optimizes (minimizes or maximizes) a given criterion measure. The criterion measure is known as a cost function, objective function, or performance criterion. The cost function can be relatively simple (e.g., find the solution that yields minimal muscle force) or complex (e.g., determine a nonlinear combination of maximal muscle force at minimal metabolic cost). It can be directly related to muscle function (e.g., minimize muscular work) or to some aspect of the motion under study (e.g., maximize vertical jump height). Alternatively, the cost function may be formulated to minimize the differences between model outputs (e.g., joint angles, ground reaction forces) and corresponding experimental measures. This latter case is often referred to as a tracking problem in that the goal is to find a solution that causes the model to follow, or track, the experimental data.
In all cases, the cost function serves as a guiding constraint that determines the selection of one particular set of optimal muscular controls from among many different possible solutions. Two major optimization approaches that have been used to control musculoskeletal models are referred to in the literature as static optimization and dynamic optimization. The meaning of static in this context is that the cost function is evaluated at each time step during a movement, independent of any prior or subsequent time steps. In contrast, dynamic optimization is dynamic in the sense that the entire movement sequence must be simulated to determine the value of the cost function.
Static Optimization
Initial attempts to predict individual muscle forces used static optimization models in conjunction with inverse dynamics analysis of an actual motor performance. The inverse dynamics analysis permits the calculation of net joint moments at incremental times throughout the movement (see chapter 5). In most applications, numerical optimization is used to find the set of muscle forces that balances these joint moments while also satisfying a selected cost function (figure 11.9). Thus, with this approach the muscle forces themselves, rather than neural signals, are the controls. The time-independent nature of static optimization allows solutions to be obtained with relatively little computational cost, but there are some drawbacks. One issue in early applications was sudden, nonphysiological switching on and off of muscle forces caused by the independence of solutions for sequential time increments (i.e., the optimization model balanced the joint moments separately for each time interval using very different sets of muscles). This problem can be avoided by careful selection of the initial guess in the optimization problem, such as by using the solution from one time step as the initial guess for the next time step. A stronger approach to address this issue is to use muscle models that invoke physiological realism through force-length and force-velocity relations and time-dependent stimulation-activation dynamics (chapter 9). When more detailed muscle models are included in a static optimization model, muscle activations become the control variables, rather than muscle forces.
Early studies using static optimization were plagued also by two other kinds of nonphysiological results. The first was the prediction of model forces that were too high for actual muscles to produce. This difficulty was easily addressed by defining physiologically valid maximal force constraints for each muscle in the musculoskeletal model. The second problem was that solutions would often select only one muscle to balance the net joint moment rather than choose a more realistic muscular synergy. Depending on the exact formulation, the muscle with the largest moment arm (if minimizing muscle force) or a favorable combination of moment arm and muscle strength (if minimizing muscle stress) would be selected to fully balance the joint moment, with zero force predicted in other synergist muscles. Mathematically, synergism can be produced by using nonlinear cost functions (e.g., minimizing the sum of squared or cubed muscle forces), although the physiological rationale for specific nonlinear cost functions is not always clear. A widely used nonlinear cost function proposed by Crowninshield and Brand (1981a) involves minimizing the sum of cubed muscle stresses. It was originally argued that this particular cost function would lead to solutions that maximize muscle endurance, making it appropriate for predicting muscle forces in submaximal tasks such as walking. However, the Crowninshield and Brand criterion has since been used to solve for muscle forces in a range of activities, some of which are unlikely candidates for maximizing muscle endurance (e.g., jumping, landing).
An additional issue with static optimization models is that they are essentially a decomposition of experimental joint moments into individual muscle forces. Therefore, the predicted muscle forces will be subject to any errors in the experimental joint moments in addition to any shortcomings associated with the approximation made in creating the musculoskeletal model. To complete this section, we present a static optimization example motivated by a classic review article by Crowninshield and Brand (1981b), which demonstrates the process of distributing an empirically determined elbow joint moment across a set of muscles spanning the elbow joint.
The net joint moment to be balanced in example 11.1 was a flexor moment, and only muscles with flexor moment arms were included in the model (figure 11.10). If an elbow extensor muscle had been included, there is no cost function of the type presented here that would predict force in an antagonist muscle. Any force in an elbow extensor would itself contribute to a higher cost function value and would also require greater elbow flexor forces to balance the target 10 N·m joint moment, further increasing the cost function value. The total lack of coactivation in this example is contrary to the common observation that during heavy activation of the elbow flexors, there is some activity in the triceps muscle group. The extensor coactivation likely helps stabilize the joint during heavy exertion but will not be predicted using traditional static optimization techniques in simple one DOF models. Although the example presented here focuses on obtaining numerical results for muscle forces, the interested reader is referred to the review by Crowninshield and Brand (1981b), which presents an interesting graphical interpretation of the solution to this static optimization problem.
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Dynamic Optimization
Ongoing work with static optimization models has led to the development of dynamic optimization or optimal control models, which are applied in conjunction with forward dynamics models of human motion. In contrast to the inverse dynamics approach, which uses experimental data from an actual performance to calculate net joint moments, forward dynamics analyses simulate the motion of the body from a given set of joint moments or muscle forces (see chapter 10). Dynamic optimization models are therefore often designed to find the muscle stimulation patterns that result in an optimal motion (figure 11.9). As mentioned earlier, the optimal motion may be one that maximizes (or minimizes) a performance criterion, such as maximizing jump height; or, the optimal motion may be defined as one that best reproduces a set of experimental data. Regardless, the variables that are optimized are usually the muscle stimulation patterns that control the motion of the musculoskeletal model. These stimulation patterns are used as the inputs to muscle models that predict individual muscle forces, which are then multiplied by the appropriate moment arms to compute the active muscle moments. The active muscle moments are combined with passive moments to compute the net joint moments, which actuate the skeletal model and produce movement. Thus, dynamic optimization leads to the synthesis of body segment kinematics associated with optimal performance, which is fundamentally different from the static optimization approach.
The optimal kinematics predicted from a dynamic optimization can be compared with experimental motion data; however, the experimental data are not required to obtain the solution, as they are with static optimization. This feature of dynamic optimization permits one to address questions for which no experimental data exist, such as testing the feasibility of possible forms of locomotion in extinct species (e.g., Nagano et al. 2005). Moreover, because dynamic optimization uses forward dynamics models that simulate the whole movement performance and provide complete muscle force time histories (rather than solutions at independent time increments), many of the problems associated with static optimization models are overcome. However, these advantages come at a computational cost, because dynamic optimization often requires a more detailed musculoskeletal model and the entire movement must be simulated for each possible solution. Thus, solving a dynamic optimization problem can easily require an order of magnitude more time than is required to solve a comparable static optimization problem. Although computational costs are difficult to compare in an objective manner, it is not uncommon for static optimization solutions to take seconds or minutes to obtain, whereas dynamic optimization solutions can take hours, days, or even weeks, when run on standard computer workstations.
In many cases, users of dynamic optimization must define the goal of the movement mathematically. This definition is easiest for movements in which the performance criterion to be optimized is clear in a mechanical sense. In vertical jumping, for example, the cost function can be stated as maximization of the vertical displacement of the body center of mass during the flight phase. If the model is constructed with appropriate constraints (realistic muscle properties and joint range of motion), jump heights and body segment motions approximating those of human jumpers can be attained (Pandy and Zajac 1991; van Soest et al. 1993). However, for many human movements the performance criterion is not as clear. In walking, for instance, the goal may seem to be getting from point A to point B in a certain amount of time. Unfortunately, this does not provide enough information to solve for a set of muscle stimulation patterns that will result in realistic walking. Successful simulations of walking have been generated using a cost function that minimizes the metabolic cost of transport (Anderson and Pandy 2001; Umberger 2010); however, the cost function formulation is considerably more complicated than for vertical jumping. Minimum-energy solutions also require the inclusion of an additional model for predicting the metabolic cost of muscle actions (e.g., Umberger et al. 2003). In cases where the performance criterion is not clear, dynamic optimization can serve as an effective approach for testing various theoretical criteria to see how well each can produce the desired movement patterns.
Movements for which the underlying criterion is difficult to define, such as walking or pedaling, have often been simulated by formulating and solving a tracking problem. This approach has a similar appeal to the EMG-based techniques described earlier, in that the resulting motion should closely approximate the way in which humans actually move. However, it is unclear which of several possible experimental measures (kinematics, kinetics, EMG) are the most important for the model to track. Also, because of errors in the experimental data and differences between the musculoskeletal model and the experimental subjects, it may be impossible for the model to track the data perfectly. This approach also limits much of the predictive ability of the dynamic optimization approach, as it is only possible to consider conditions for which experimental data are available.
The tracking approach has seen frequent use in hybrid solution algorithms, which seek to retain the advantages of forward dynamics and dynamic optimization while achieving the computational efficiency of static optimization. Two examples are computed muscle control (Thelen et al. 2003) and direct collocation (Kaplan and Heegaard 2001). Computed muscle control works by solving a static optimization problem at each time step within a single forward dynamics movement simulation. By using feedback on the kinematics and muscle activations at each time step of the numerical integration, computed muscle control can generate a forward dynamics simulation that optimally tracks a set of experimental data without solving a dynamic optimization problem that would require thousands of forward dynamics simulations. In contrast, direct collocation works by converting the differential equations of motion into a set of algebraic constraint equations and treating both the control variables (muscle stimulations) and state variables (positions and velocities) as unknowns in the optimization problem. Although they differ considerably in implementation, both techniques appear to be able to produce results that are similar to dynamic optimization tracking solutions, with a computational cost closer to that of static optimization.
Although static and dynamic optimization have both become popular means for controlling musculoskeletal models, several issues concerning their use must be addressed. Do humans actually produce movements based on one given performance criterion, or does the performance objective change during the movement? If the optimized model results differ from those of a human performer, is it because the model is too simple or not constrained properly, or is the human not performing optimally? Finally, EMG data have illustrated various degrees of simultaneous antagonistic cocontraction during some movement sequences. Optimization models tend not to yield solutions predicting muscle antagonism around a joint, although some antagonism is predicted when models contain muscles that contribute to more than one joint moment. However, optimization models that seek to minimize or maximize specific cost functions will not predict antagonistic cocontraction associated with joint stability or stiffness. In the case of walking, for example, there is no single cost function that could simultaneously predict the relatively minimal muscle coactivation in healthy subjects and the substantial antagonism exhibited by patients with cerebral palsy. Despite these drawbacks, optimization models have increased the understanding of human biomechanics and will continue to do so in the future.
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The state space in the presentation of biomechanical data
Biomechanical data can be represented graphically in a variety of different ways.
The State Space
Biomechanical data can be represented graphically in a variety of different ways. A traditional and very informative type of graph is to display the changes in kinetic or kinematic parameters over time. Figure 13.4 presents an example of the changes in the left and right thigh angular displacements over a time period of 10 s while a subject is walking on a treadmill. Information about peak flexion and extension angles can be obtained from these graphs and quantified. A closer inspection of the graphs makes it quite evident that the coordination between these segments is primarily antiphase. However, especially during the stance phase there are more subtle changes in the patterns of coordination that are harder to obtain from these plots.
In the dynamical systems approach, the reconstruction of the so-called state space is essential in identifying the important features of the behavior of a system, such as its stability and ability to change and adapt to different environmental and task constraints. The state space is a representation of the relevant variables that will help identify these features. To help us understand and quantify coordination between joints or segments, it can be very useful to represent the system in a state space that is based on an angle-angle relative motion plot. Figure 13.5 presents a state space of the relative motion of the time series of the thigh and leg shown earlier in figure 13.4. This angle-angle plot can reveal regions where coordination changes take place as well as parts of the gait cycle where there is relative invariance in coordination patterns. These coordinative changes in the angle-angle plots can be further quantified by vector coding techniques that are discussed later.
Another form of state space is where the position and velocity of a joint or segment are plotted relative to each other. This state-space representation is also often referred to as the phase plane. An example of a position-velocity phase-plane plot is shown in figure 13.6, where the angular displacement of the thigh is plotted against the angular velocity. This is a higher-dimensional state space because the time derivative of position is used to identify the pattern. The phase-plane representation is a first and critical step in the quantification of coordination using continuous relative phase techniques.
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Attractors in State Space
In most forms of human movement, the dynamics in state space are limited to distinct regions, as can be seen in the phase-plane plot in figure 13.6, where the pattern in state space is limited to a fairly narrow, cyclical band. In dynamical systems, preferred regions in state space onto which the dynamics tend to settle are called attractors. Attractors can come in a variety of different forms. A seemingly simple kind of attractor is the point attractor. In this case, the dynamics in the system tend to converge onto one relatively fixed value in the state space. Figure 13.7, c and d, provides an example of a point attractor. The point attractor dynamics are shown in figure 13.7c by a relatively consistent value of discrete relative phase throughout the gait cycle. The discrete relative phase is based on the occurrence of peak flexion in the left and right thigh angular displacements, and the time series in 13.7c show that there is a consistent antiphase or 180° coordination pattern. This antiphase coordination can already be observed by comparing the individual time series in figure 13.4 but is more objectively quantified by the state-space plot in figure 13.7d, where so-called return maps (plotting the coordination at xn vs. xn+1, where n is the cycle number) identify a fixed-point attractor in state space. The relative phase dynamics are of the fixed-point type in this case, as a perturbation (sudden change in treadmill speed and return to the original speed; figure 13.7) results in a return to the original antiphase dynamics with a relative short latency. The perturbation here consisted of a brief (5 s) increase in treadmill speed from 1.2 m/s to 2.0 m/s, with a subsequent return to 1.2 m/s.
Whereas the coordination between the right and left legs can be identified in the form of a point attractor with a relatively fixed value of coordination from cycle to cycle, the dynamics of the individual limb segments show a very different pattern. These are clearly cyclical, as can be seen in the phase planes in figure 13.7, a and b. Attractors of this form are called limit-cycle attractors, and the dynamics converge onto a cyclical region in state space. These limit-cycle attractors are typically identified on the basis of a very narrow, overlapping band of the trajectories in the state space. However, the existence of the narrow band is a necessary but not sufficient condition to characterize the dynamics as a limit-cycle. An essential feature of the limit-cycle attractor is stability with respect to perturbation; to classify as a limit-cycle attractor, the system should also show resistance to perturbations. An example of this is given in figure 13.7, a and b. The regular cyclic pattern in the phase plane (solid line) represents the steady-state gait patterns while a subject was walking at a speed of 1.2 m/s. The dashed trajectories represent the perturbation phase of the trial when speed was suddenly increased to 2.0 m/s and the consequent return back to the original pattern. This return to the preperturbation dynamic is an essential feature of the limit-cycle attractor. The patterns in the phase plane can also serve as an energy plot. The convergence and divergence of trajectories in the phase plane can identify the loss and gain of energy in the system.
Higher-dimensional state spaces (three dimensions and up) can also reveal more complex types of attractors: quasiperiodic and chaotic attractors. The chaotic attractor demonstrates both stable attraction to a region in state space and variability. Figure 13.8 shows an example of the Lorenz attractor, a well-known chaotic attractor that emerges from dynamic interactions in fluid and air flow systems (Strogatz 1994). These dual features of stability and adaptability can be associated with a higher pattern complexity that is now commonly regarded as reflective of healthy and expert systems (see figure 13.3). As an example, increased heart rate variability is considered an important indicator of healthy heart function, reflecting a degree of complexity in organization in which disruptions can be compensated for more easily (Glass 2001; Lipsitz 2002).
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Using EMG to diagnose and treat low back pain
The diagnosis and treatment of low back pain are important issues.
EMG and Low Back Pain
The diagnosis and treatment of low back pain are important issues. In the United States, for example, back pain accounts for two-thirds of all workers compensation costs. Peripheral muscle problems may be one issue; atrophy of the multifidus is frequently observed in patients with low back pain (Hides et al. 1994). EMG is becoming an increasingly useful tool for the diagnosis and treatment of low back pain.
The trunk musculature plays an important role in tasks such as lifting and throwing. During these kinds of tasks as well as in other activities that might threaten the postural control of the individual and perturb balance, the nervous system implements strategies to activate trunk muscles while performing voluntary movements involving remote muscle groups. However, the prevalence of back pain in the general population, and the need to identify ergonomically efficient and safe ways to use back muscles, require that we understand the nature of activation in muscles controlling the trunk. Here, too, EMG is an important tool.
A number of issues render EMG recordings from the trunk musculature difficult to obtain and interpret. From an anatomical viewpoint, the architecture of the back muscles is complex. For example, it is generally accepted that injury or disorder of the multifidus frequently results in back pain. Both multifidus and erector spinae have distinct superficial and deep portions (Bustami 1986; Macintosh et al. 1986) with different histochemical and biomechanical characteristics (Bogduk et al. 1992; Dickx et al. 2010). In some respects, it is fortunate that most of the muscle mass of the multifidus lies in the more superficial portion, so that surface EMG has a greater possibility of characterizing the fibers that produce large amounts of force. However, the multifidus superficial portion has a different role than the deeper fibers. Fibers in the superficial portion cross numerous spinal segments and are thus in a better position to produce back extension. However, the deeper fibers are relatively short, crossing perhaps one or two segments, and thus protect segments of the lumbar spine from inappropriate shear or torsion torques (Macintosh and Valencia 1986).
In addition to issues concerning the activation of the back extensor muscles, a number of important questions require resolution by EMG analysis in which considerable electrocardiographic (ECG) artifact may be present. ECG artifact is particularly problematic during relatively low force contractions, exaggerating the ratio between the signal of interest and the interfering ECG signal. Recording EMG signals from abdominal, knee extensor, and other trunk muscles during normal human movements, for example, can result in the placement of electrodes well within recording range of the electrocardiogram. The relatively large ECG signal can exaggerate the amplitude of EMG activity, and the low-frequency characteristics of the ECG wave can also alter the frequency characteristics of the recorded EMG activity.
For example, the absence of sufficient trunk activity can produce instability resulting in low back pain (van Dieën et al. 2003). However, the amplitude of EMG activity required to ensure stability is small and thus can be affected by the ECG signal (Cholewicki et al. 1997). The best solution is to place the electrodes in a location from which no or minimal ECG artifact can be recorded. However, this is frequently difficult if not impossible. Consequently, numerous algorithms have been suggested to remove the ECG artifact.
One frequently implemented technique to remove this artifact is to compute a template of the ECG signal and subtract it from the electromyogram. This has been proved to be a reasonably successful procedure and has been implemented in studies recording from the diaphragm (Bartolo et al. 1996) as well as rectus abdominis (Hof 2009). Hof (2009) described a technique in which the ECG signal is recorded simultaneously with the EMG signal of interest, and template subtraction is then implemented (figure 8.15).
Digital filtering is another useful alternative for ECG artifact removal. Drake and Callaghan (2006) used a FIR (finite impulse response) filter with a hamming window, although they concluded that the most efficient filtering result could be obtained using a somewhat simpler fourth-order, 30 Hz high-pass cutoff Butterworth filter. Template subtraction improved extraction of the ECG signal but required a large amount of time.
Alternative ECG removal techniques include the use of adaptive filters (Lu et al. 2009; Marque et al. 2005), wavelet-independent component analysis (Taelman et al. 2007), and wavelet-based adaptive filters (Zhan et al. 2010). Irrespective of the technique used for ECG signal artifact removal, it is apparent that the resultant improvement in signal interpretation can be important. Hu and colleagues (2009), for example, found that the use of independent component analysis to remove artifact resulted in an improved ability to discriminate between patients with low back pain and normal subjects during both sitting and standing tasks.
Analysis of the EMG activity in back muscles has produced some interesting results, in part reflecting the unique anatomical issues discussed here. As long ago as 1962, Morris and colleagues noted that “the three muscles of the erector spinae group considered here . . .
do not always show parallel activity, and one may be active while the other two are inactive” (p. 519). This may suggest that the mere demonstration of greater or lesser EMG amplitude may not be indicative of abnormality in a particular muscle. Potentially more important than EMG amplitude is the timing of EMG activity. Deep and superficial portions of the multifidus are differentially active during arm movements (Moseley et al. 2002). One suggestion is that the manner in which the multifidus is differentially controlled in people with back pain compared with those without may be the source of chronic back pain (MacDonald et al. 2009).
Similarly, the analysis of EMG frequency characteristics in patients with back pain compared with control subjects has suggested that there may be differences between muscles and even differences within muscles. Patients with low back pain frequently exhibit alterations in the electromyographic response to fatigue in the back extensor muscles (Biedermann et al. 1991). This is one area in which spectral analysis of the EMG signals has been helpful. As in other muscles, median EMG frequency decreases in the trunk extensors with fatigue (Demoulin et al. 2007). It is interesting to note that Kramer and colleagues (2005) found that the magnitude of median frequency decrease was greater in healthy subjects than in individuals with back pain. Support for importance of electrode placement is obtained from the observations of Sung and colleagues (2009). Normal subjects and patients with back pain underwent an isometric contraction protocol that induced fatigue of the back extensors. EMG frequency measurements made in both the thoracic and the lumbar portions of the erector spinae indicated that the thoracic portion had a significantly lower median frequency than the lumbar portion in patients with low back pain. However, median frequency was lower in the lumbar portion than in the thoracic portion in control subjects. Thus, in the analysis of trunk musculature that might be involved in the development of back pain, EMG studies using wire electrodes frequently may be required to record from different portions of the muscle.
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Using musculoskeletal optimization models to generate control signals
A popular method for generating control signals in musculoskeletal models is to use some form of optimization.
Optimization Models
A popular method for generating control signals in musculoskeletal models is to use some form of optimization. In some cases, when the optimization criterion is derived from motor control principles, this is tantamount to developing a theoretical control model. In other cases, the optimization criterion may result in a coordinated movement pattern yet have little physiological basis. In general, the optimization approach is used in an effort to determine which set of model control signals will produce a result that optimizes (minimizes or maximizes) a given criterion measure. The criterion measure is known as a cost function, objective function, or performance criterion. The cost function can be relatively simple (e.g., find the solution that yields minimal muscle force) or complex (e.g., determine a nonlinear combination of maximal muscle force at minimal metabolic cost). It can be directly related to muscle function (e.g., minimize muscular work) or to some aspect of the motion under study (e.g., maximize vertical jump height). Alternatively, the cost function may be formulated to minimize the differences between model outputs (e.g., joint angles, ground reaction forces) and corresponding experimental measures. This latter case is often referred to as a tracking problem in that the goal is to find a solution that causes the model to follow, or track, the experimental data.
In all cases, the cost function serves as a guiding constraint that determines the selection of one particular set of optimal muscular controls from among many different possible solutions. Two major optimization approaches that have been used to control musculoskeletal models are referred to in the literature as static optimization and dynamic optimization. The meaning of static in this context is that the cost function is evaluated at each time step during a movement, independent of any prior or subsequent time steps. In contrast, dynamic optimization is dynamic in the sense that the entire movement sequence must be simulated to determine the value of the cost function.
Static Optimization
Initial attempts to predict individual muscle forces used static optimization models in conjunction with inverse dynamics analysis of an actual motor performance. The inverse dynamics analysis permits the calculation of net joint moments at incremental times throughout the movement (see chapter 5). In most applications, numerical optimization is used to find the set of muscle forces that balances these joint moments while also satisfying a selected cost function (figure 11.9). Thus, with this approach the muscle forces themselves, rather than neural signals, are the controls. The time-independent nature of static optimization allows solutions to be obtained with relatively little computational cost, but there are some drawbacks. One issue in early applications was sudden, nonphysiological switching on and off of muscle forces caused by the independence of solutions for sequential time increments (i.e., the optimization model balanced the joint moments separately for each time interval using very different sets of muscles). This problem can be avoided by careful selection of the initial guess in the optimization problem, such as by using the solution from one time step as the initial guess for the next time step. A stronger approach to address this issue is to use muscle models that invoke physiological realism through force-length and force-velocity relations and time-dependent stimulation-activation dynamics (chapter 9). When more detailed muscle models are included in a static optimization model, muscle activations become the control variables, rather than muscle forces.
Early studies using static optimization were plagued also by two other kinds of nonphysiological results. The first was the prediction of model forces that were too high for actual muscles to produce. This difficulty was easily addressed by defining physiologically valid maximal force constraints for each muscle in the musculoskeletal model. The second problem was that solutions would often select only one muscle to balance the net joint moment rather than choose a more realistic muscular synergy. Depending on the exact formulation, the muscle with the largest moment arm (if minimizing muscle force) or a favorable combination of moment arm and muscle strength (if minimizing muscle stress) would be selected to fully balance the joint moment, with zero force predicted in other synergist muscles. Mathematically, synergism can be produced by using nonlinear cost functions (e.g., minimizing the sum of squared or cubed muscle forces), although the physiological rationale for specific nonlinear cost functions is not always clear. A widely used nonlinear cost function proposed by Crowninshield and Brand (1981a) involves minimizing the sum of cubed muscle stresses. It was originally argued that this particular cost function would lead to solutions that maximize muscle endurance, making it appropriate for predicting muscle forces in submaximal tasks such as walking. However, the Crowninshield and Brand criterion has since been used to solve for muscle forces in a range of activities, some of which are unlikely candidates for maximizing muscle endurance (e.g., jumping, landing).
An additional issue with static optimization models is that they are essentially a decomposition of experimental joint moments into individual muscle forces. Therefore, the predicted muscle forces will be subject to any errors in the experimental joint moments in addition to any shortcomings associated with the approximation made in creating the musculoskeletal model. To complete this section, we present a static optimization example motivated by a classic review article by Crowninshield and Brand (1981b), which demonstrates the process of distributing an empirically determined elbow joint moment across a set of muscles spanning the elbow joint.
The net joint moment to be balanced in example 11.1 was a flexor moment, and only muscles with flexor moment arms were included in the model (figure 11.10). If an elbow extensor muscle had been included, there is no cost function of the type presented here that would predict force in an antagonist muscle. Any force in an elbow extensor would itself contribute to a higher cost function value and would also require greater elbow flexor forces to balance the target 10 N·m joint moment, further increasing the cost function value. The total lack of coactivation in this example is contrary to the common observation that during heavy activation of the elbow flexors, there is some activity in the triceps muscle group. The extensor coactivation likely helps stabilize the joint during heavy exertion but will not be predicted using traditional static optimization techniques in simple one DOF models. Although the example presented here focuses on obtaining numerical results for muscle forces, the interested reader is referred to the review by Crowninshield and Brand (1981b), which presents an interesting graphical interpretation of the solution to this static optimization problem.
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Dynamic Optimization
Ongoing work with static optimization models has led to the development of dynamic optimization or optimal control models, which are applied in conjunction with forward dynamics models of human motion. In contrast to the inverse dynamics approach, which uses experimental data from an actual performance to calculate net joint moments, forward dynamics analyses simulate the motion of the body from a given set of joint moments or muscle forces (see chapter 10). Dynamic optimization models are therefore often designed to find the muscle stimulation patterns that result in an optimal motion (figure 11.9). As mentioned earlier, the optimal motion may be one that maximizes (or minimizes) a performance criterion, such as maximizing jump height; or, the optimal motion may be defined as one that best reproduces a set of experimental data. Regardless, the variables that are optimized are usually the muscle stimulation patterns that control the motion of the musculoskeletal model. These stimulation patterns are used as the inputs to muscle models that predict individual muscle forces, which are then multiplied by the appropriate moment arms to compute the active muscle moments. The active muscle moments are combined with passive moments to compute the net joint moments, which actuate the skeletal model and produce movement. Thus, dynamic optimization leads to the synthesis of body segment kinematics associated with optimal performance, which is fundamentally different from the static optimization approach.
The optimal kinematics predicted from a dynamic optimization can be compared with experimental motion data; however, the experimental data are not required to obtain the solution, as they are with static optimization. This feature of dynamic optimization permits one to address questions for which no experimental data exist, such as testing the feasibility of possible forms of locomotion in extinct species (e.g., Nagano et al. 2005). Moreover, because dynamic optimization uses forward dynamics models that simulate the whole movement performance and provide complete muscle force time histories (rather than solutions at independent time increments), many of the problems associated with static optimization models are overcome. However, these advantages come at a computational cost, because dynamic optimization often requires a more detailed musculoskeletal model and the entire movement must be simulated for each possible solution. Thus, solving a dynamic optimization problem can easily require an order of magnitude more time than is required to solve a comparable static optimization problem. Although computational costs are difficult to compare in an objective manner, it is not uncommon for static optimization solutions to take seconds or minutes to obtain, whereas dynamic optimization solutions can take hours, days, or even weeks, when run on standard computer workstations.
In many cases, users of dynamic optimization must define the goal of the movement mathematically. This definition is easiest for movements in which the performance criterion to be optimized is clear in a mechanical sense. In vertical jumping, for example, the cost function can be stated as maximization of the vertical displacement of the body center of mass during the flight phase. If the model is constructed with appropriate constraints (realistic muscle properties and joint range of motion), jump heights and body segment motions approximating those of human jumpers can be attained (Pandy and Zajac 1991; van Soest et al. 1993). However, for many human movements the performance criterion is not as clear. In walking, for instance, the goal may seem to be getting from point A to point B in a certain amount of time. Unfortunately, this does not provide enough information to solve for a set of muscle stimulation patterns that will result in realistic walking. Successful simulations of walking have been generated using a cost function that minimizes the metabolic cost of transport (Anderson and Pandy 2001; Umberger 2010); however, the cost function formulation is considerably more complicated than for vertical jumping. Minimum-energy solutions also require the inclusion of an additional model for predicting the metabolic cost of muscle actions (e.g., Umberger et al. 2003). In cases where the performance criterion is not clear, dynamic optimization can serve as an effective approach for testing various theoretical criteria to see how well each can produce the desired movement patterns.
Movements for which the underlying criterion is difficult to define, such as walking or pedaling, have often been simulated by formulating and solving a tracking problem. This approach has a similar appeal to the EMG-based techniques described earlier, in that the resulting motion should closely approximate the way in which humans actually move. However, it is unclear which of several possible experimental measures (kinematics, kinetics, EMG) are the most important for the model to track. Also, because of errors in the experimental data and differences between the musculoskeletal model and the experimental subjects, it may be impossible for the model to track the data perfectly. This approach also limits much of the predictive ability of the dynamic optimization approach, as it is only possible to consider conditions for which experimental data are available.
The tracking approach has seen frequent use in hybrid solution algorithms, which seek to retain the advantages of forward dynamics and dynamic optimization while achieving the computational efficiency of static optimization. Two examples are computed muscle control (Thelen et al. 2003) and direct collocation (Kaplan and Heegaard 2001). Computed muscle control works by solving a static optimization problem at each time step within a single forward dynamics movement simulation. By using feedback on the kinematics and muscle activations at each time step of the numerical integration, computed muscle control can generate a forward dynamics simulation that optimally tracks a set of experimental data without solving a dynamic optimization problem that would require thousands of forward dynamics simulations. In contrast, direct collocation works by converting the differential equations of motion into a set of algebraic constraint equations and treating both the control variables (muscle stimulations) and state variables (positions and velocities) as unknowns in the optimization problem. Although they differ considerably in implementation, both techniques appear to be able to produce results that are similar to dynamic optimization tracking solutions, with a computational cost closer to that of static optimization.
Although static and dynamic optimization have both become popular means for controlling musculoskeletal models, several issues concerning their use must be addressed. Do humans actually produce movements based on one given performance criterion, or does the performance objective change during the movement? If the optimized model results differ from those of a human performer, is it because the model is too simple or not constrained properly, or is the human not performing optimally? Finally, EMG data have illustrated various degrees of simultaneous antagonistic cocontraction during some movement sequences. Optimization models tend not to yield solutions predicting muscle antagonism around a joint, although some antagonism is predicted when models contain muscles that contribute to more than one joint moment. However, optimization models that seek to minimize or maximize specific cost functions will not predict antagonistic cocontraction associated with joint stability or stiffness. In the case of walking, for example, there is no single cost function that could simultaneously predict the relatively minimal muscle coactivation in healthy subjects and the substantial antagonism exhibited by patients with cerebral palsy. Despite these drawbacks, optimization models have increased the understanding of human biomechanics and will continue to do so in the future.
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The state space in the presentation of biomechanical data
Biomechanical data can be represented graphically in a variety of different ways.
The State Space
Biomechanical data can be represented graphically in a variety of different ways. A traditional and very informative type of graph is to display the changes in kinetic or kinematic parameters over time. Figure 13.4 presents an example of the changes in the left and right thigh angular displacements over a time period of 10 s while a subject is walking on a treadmill. Information about peak flexion and extension angles can be obtained from these graphs and quantified. A closer inspection of the graphs makes it quite evident that the coordination between these segments is primarily antiphase. However, especially during the stance phase there are more subtle changes in the patterns of coordination that are harder to obtain from these plots.
In the dynamical systems approach, the reconstruction of the so-called state space is essential in identifying the important features of the behavior of a system, such as its stability and ability to change and adapt to different environmental and task constraints. The state space is a representation of the relevant variables that will help identify these features. To help us understand and quantify coordination between joints or segments, it can be very useful to represent the system in a state space that is based on an angle-angle relative motion plot. Figure 13.5 presents a state space of the relative motion of the time series of the thigh and leg shown earlier in figure 13.4. This angle-angle plot can reveal regions where coordination changes take place as well as parts of the gait cycle where there is relative invariance in coordination patterns. These coordinative changes in the angle-angle plots can be further quantified by vector coding techniques that are discussed later.
Another form of state space is where the position and velocity of a joint or segment are plotted relative to each other. This state-space representation is also often referred to as the phase plane. An example of a position-velocity phase-plane plot is shown in figure 13.6, where the angular displacement of the thigh is plotted against the angular velocity. This is a higher-dimensional state space because the time derivative of position is used to identify the pattern. The phase-plane representation is a first and critical step in the quantification of coordination using continuous relative phase techniques.
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Attractors in State Space
In most forms of human movement, the dynamics in state space are limited to distinct regions, as can be seen in the phase-plane plot in figure 13.6, where the pattern in state space is limited to a fairly narrow, cyclical band. In dynamical systems, preferred regions in state space onto which the dynamics tend to settle are called attractors. Attractors can come in a variety of different forms. A seemingly simple kind of attractor is the point attractor. In this case, the dynamics in the system tend to converge onto one relatively fixed value in the state space. Figure 13.7, c and d, provides an example of a point attractor. The point attractor dynamics are shown in figure 13.7c by a relatively consistent value of discrete relative phase throughout the gait cycle. The discrete relative phase is based on the occurrence of peak flexion in the left and right thigh angular displacements, and the time series in 13.7c show that there is a consistent antiphase or 180° coordination pattern. This antiphase coordination can already be observed by comparing the individual time series in figure 13.4 but is more objectively quantified by the state-space plot in figure 13.7d, where so-called return maps (plotting the coordination at xn vs. xn+1, where n is the cycle number) identify a fixed-point attractor in state space. The relative phase dynamics are of the fixed-point type in this case, as a perturbation (sudden change in treadmill speed and return to the original speed; figure 13.7) results in a return to the original antiphase dynamics with a relative short latency. The perturbation here consisted of a brief (5 s) increase in treadmill speed from 1.2 m/s to 2.0 m/s, with a subsequent return to 1.2 m/s.
Whereas the coordination between the right and left legs can be identified in the form of a point attractor with a relatively fixed value of coordination from cycle to cycle, the dynamics of the individual limb segments show a very different pattern. These are clearly cyclical, as can be seen in the phase planes in figure 13.7, a and b. Attractors of this form are called limit-cycle attractors, and the dynamics converge onto a cyclical region in state space. These limit-cycle attractors are typically identified on the basis of a very narrow, overlapping band of the trajectories in the state space. However, the existence of the narrow band is a necessary but not sufficient condition to characterize the dynamics as a limit-cycle. An essential feature of the limit-cycle attractor is stability with respect to perturbation; to classify as a limit-cycle attractor, the system should also show resistance to perturbations. An example of this is given in figure 13.7, a and b. The regular cyclic pattern in the phase plane (solid line) represents the steady-state gait patterns while a subject was walking at a speed of 1.2 m/s. The dashed trajectories represent the perturbation phase of the trial when speed was suddenly increased to 2.0 m/s and the consequent return back to the original pattern. This return to the preperturbation dynamic is an essential feature of the limit-cycle attractor. The patterns in the phase plane can also serve as an energy plot. The convergence and divergence of trajectories in the phase plane can identify the loss and gain of energy in the system.
Higher-dimensional state spaces (three dimensions and up) can also reveal more complex types of attractors: quasiperiodic and chaotic attractors. The chaotic attractor demonstrates both stable attraction to a region in state space and variability. Figure 13.8 shows an example of the Lorenz attractor, a well-known chaotic attractor that emerges from dynamic interactions in fluid and air flow systems (Strogatz 1994). These dual features of stability and adaptability can be associated with a higher pattern complexity that is now commonly regarded as reflective of healthy and expert systems (see figure 13.3). As an example, increased heart rate variability is considered an important indicator of healthy heart function, reflecting a degree of complexity in organization in which disruptions can be compensated for more easily (Glass 2001; Lipsitz 2002).
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Using EMG to diagnose and treat low back pain
The diagnosis and treatment of low back pain are important issues.
EMG and Low Back Pain
The diagnosis and treatment of low back pain are important issues. In the United States, for example, back pain accounts for two-thirds of all workers compensation costs. Peripheral muscle problems may be one issue; atrophy of the multifidus is frequently observed in patients with low back pain (Hides et al. 1994). EMG is becoming an increasingly useful tool for the diagnosis and treatment of low back pain.
The trunk musculature plays an important role in tasks such as lifting and throwing. During these kinds of tasks as well as in other activities that might threaten the postural control of the individual and perturb balance, the nervous system implements strategies to activate trunk muscles while performing voluntary movements involving remote muscle groups. However, the prevalence of back pain in the general population, and the need to identify ergonomically efficient and safe ways to use back muscles, require that we understand the nature of activation in muscles controlling the trunk. Here, too, EMG is an important tool.
A number of issues render EMG recordings from the trunk musculature difficult to obtain and interpret. From an anatomical viewpoint, the architecture of the back muscles is complex. For example, it is generally accepted that injury or disorder of the multifidus frequently results in back pain. Both multifidus and erector spinae have distinct superficial and deep portions (Bustami 1986; Macintosh et al. 1986) with different histochemical and biomechanical characteristics (Bogduk et al. 1992; Dickx et al. 2010). In some respects, it is fortunate that most of the muscle mass of the multifidus lies in the more superficial portion, so that surface EMG has a greater possibility of characterizing the fibers that produce large amounts of force. However, the multifidus superficial portion has a different role than the deeper fibers. Fibers in the superficial portion cross numerous spinal segments and are thus in a better position to produce back extension. However, the deeper fibers are relatively short, crossing perhaps one or two segments, and thus protect segments of the lumbar spine from inappropriate shear or torsion torques (Macintosh and Valencia 1986).
In addition to issues concerning the activation of the back extensor muscles, a number of important questions require resolution by EMG analysis in which considerable electrocardiographic (ECG) artifact may be present. ECG artifact is particularly problematic during relatively low force contractions, exaggerating the ratio between the signal of interest and the interfering ECG signal. Recording EMG signals from abdominal, knee extensor, and other trunk muscles during normal human movements, for example, can result in the placement of electrodes well within recording range of the electrocardiogram. The relatively large ECG signal can exaggerate the amplitude of EMG activity, and the low-frequency characteristics of the ECG wave can also alter the frequency characteristics of the recorded EMG activity.
For example, the absence of sufficient trunk activity can produce instability resulting in low back pain (van Dieën et al. 2003). However, the amplitude of EMG activity required to ensure stability is small and thus can be affected by the ECG signal (Cholewicki et al. 1997). The best solution is to place the electrodes in a location from which no or minimal ECG artifact can be recorded. However, this is frequently difficult if not impossible. Consequently, numerous algorithms have been suggested to remove the ECG artifact.
One frequently implemented technique to remove this artifact is to compute a template of the ECG signal and subtract it from the electromyogram. This has been proved to be a reasonably successful procedure and has been implemented in studies recording from the diaphragm (Bartolo et al. 1996) as well as rectus abdominis (Hof 2009). Hof (2009) described a technique in which the ECG signal is recorded simultaneously with the EMG signal of interest, and template subtraction is then implemented (figure 8.15).
Digital filtering is another useful alternative for ECG artifact removal. Drake and Callaghan (2006) used a FIR (finite impulse response) filter with a hamming window, although they concluded that the most efficient filtering result could be obtained using a somewhat simpler fourth-order, 30 Hz high-pass cutoff Butterworth filter. Template subtraction improved extraction of the ECG signal but required a large amount of time.
Alternative ECG removal techniques include the use of adaptive filters (Lu et al. 2009; Marque et al. 2005), wavelet-independent component analysis (Taelman et al. 2007), and wavelet-based adaptive filters (Zhan et al. 2010). Irrespective of the technique used for ECG signal artifact removal, it is apparent that the resultant improvement in signal interpretation can be important. Hu and colleagues (2009), for example, found that the use of independent component analysis to remove artifact resulted in an improved ability to discriminate between patients with low back pain and normal subjects during both sitting and standing tasks.
Analysis of the EMG activity in back muscles has produced some interesting results, in part reflecting the unique anatomical issues discussed here. As long ago as 1962, Morris and colleagues noted that “the three muscles of the erector spinae group considered here . . .
do not always show parallel activity, and one may be active while the other two are inactive” (p. 519). This may suggest that the mere demonstration of greater or lesser EMG amplitude may not be indicative of abnormality in a particular muscle. Potentially more important than EMG amplitude is the timing of EMG activity. Deep and superficial portions of the multifidus are differentially active during arm movements (Moseley et al. 2002). One suggestion is that the manner in which the multifidus is differentially controlled in people with back pain compared with those without may be the source of chronic back pain (MacDonald et al. 2009).
Similarly, the analysis of EMG frequency characteristics in patients with back pain compared with control subjects has suggested that there may be differences between muscles and even differences within muscles. Patients with low back pain frequently exhibit alterations in the electromyographic response to fatigue in the back extensor muscles (Biedermann et al. 1991). This is one area in which spectral analysis of the EMG signals has been helpful. As in other muscles, median EMG frequency decreases in the trunk extensors with fatigue (Demoulin et al. 2007). It is interesting to note that Kramer and colleagues (2005) found that the magnitude of median frequency decrease was greater in healthy subjects than in individuals with back pain. Support for importance of electrode placement is obtained from the observations of Sung and colleagues (2009). Normal subjects and patients with back pain underwent an isometric contraction protocol that induced fatigue of the back extensors. EMG frequency measurements made in both the thoracic and the lumbar portions of the erector spinae indicated that the thoracic portion had a significantly lower median frequency than the lumbar portion in patients with low back pain. However, median frequency was lower in the lumbar portion than in the thoracic portion in control subjects. Thus, in the analysis of trunk musculature that might be involved in the development of back pain, EMG studies using wire electrodes frequently may be required to record from different portions of the muscle.
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Using musculoskeletal optimization models to generate control signals
A popular method for generating control signals in musculoskeletal models is to use some form of optimization.
Optimization Models
A popular method for generating control signals in musculoskeletal models is to use some form of optimization. In some cases, when the optimization criterion is derived from motor control principles, this is tantamount to developing a theoretical control model. In other cases, the optimization criterion may result in a coordinated movement pattern yet have little physiological basis. In general, the optimization approach is used in an effort to determine which set of model control signals will produce a result that optimizes (minimizes or maximizes) a given criterion measure. The criterion measure is known as a cost function, objective function, or performance criterion. The cost function can be relatively simple (e.g., find the solution that yields minimal muscle force) or complex (e.g., determine a nonlinear combination of maximal muscle force at minimal metabolic cost). It can be directly related to muscle function (e.g., minimize muscular work) or to some aspect of the motion under study (e.g., maximize vertical jump height). Alternatively, the cost function may be formulated to minimize the differences between model outputs (e.g., joint angles, ground reaction forces) and corresponding experimental measures. This latter case is often referred to as a tracking problem in that the goal is to find a solution that causes the model to follow, or track, the experimental data.
In all cases, the cost function serves as a guiding constraint that determines the selection of one particular set of optimal muscular controls from among many different possible solutions. Two major optimization approaches that have been used to control musculoskeletal models are referred to in the literature as static optimization and dynamic optimization. The meaning of static in this context is that the cost function is evaluated at each time step during a movement, independent of any prior or subsequent time steps. In contrast, dynamic optimization is dynamic in the sense that the entire movement sequence must be simulated to determine the value of the cost function.
Static Optimization
Initial attempts to predict individual muscle forces used static optimization models in conjunction with inverse dynamics analysis of an actual motor performance. The inverse dynamics analysis permits the calculation of net joint moments at incremental times throughout the movement (see chapter 5). In most applications, numerical optimization is used to find the set of muscle forces that balances these joint moments while also satisfying a selected cost function (figure 11.9). Thus, with this approach the muscle forces themselves, rather than neural signals, are the controls. The time-independent nature of static optimization allows solutions to be obtained with relatively little computational cost, but there are some drawbacks. One issue in early applications was sudden, nonphysiological switching on and off of muscle forces caused by the independence of solutions for sequential time increments (i.e., the optimization model balanced the joint moments separately for each time interval using very different sets of muscles). This problem can be avoided by careful selection of the initial guess in the optimization problem, such as by using the solution from one time step as the initial guess for the next time step. A stronger approach to address this issue is to use muscle models that invoke physiological realism through force-length and force-velocity relations and time-dependent stimulation-activation dynamics (chapter 9). When more detailed muscle models are included in a static optimization model, muscle activations become the control variables, rather than muscle forces.
Early studies using static optimization were plagued also by two other kinds of nonphysiological results. The first was the prediction of model forces that were too high for actual muscles to produce. This difficulty was easily addressed by defining physiologically valid maximal force constraints for each muscle in the musculoskeletal model. The second problem was that solutions would often select only one muscle to balance the net joint moment rather than choose a more realistic muscular synergy. Depending on the exact formulation, the muscle with the largest moment arm (if minimizing muscle force) or a favorable combination of moment arm and muscle strength (if minimizing muscle stress) would be selected to fully balance the joint moment, with zero force predicted in other synergist muscles. Mathematically, synergism can be produced by using nonlinear cost functions (e.g., minimizing the sum of squared or cubed muscle forces), although the physiological rationale for specific nonlinear cost functions is not always clear. A widely used nonlinear cost function proposed by Crowninshield and Brand (1981a) involves minimizing the sum of cubed muscle stresses. It was originally argued that this particular cost function would lead to solutions that maximize muscle endurance, making it appropriate for predicting muscle forces in submaximal tasks such as walking. However, the Crowninshield and Brand criterion has since been used to solve for muscle forces in a range of activities, some of which are unlikely candidates for maximizing muscle endurance (e.g., jumping, landing).
An additional issue with static optimization models is that they are essentially a decomposition of experimental joint moments into individual muscle forces. Therefore, the predicted muscle forces will be subject to any errors in the experimental joint moments in addition to any shortcomings associated with the approximation made in creating the musculoskeletal model. To complete this section, we present a static optimization example motivated by a classic review article by Crowninshield and Brand (1981b), which demonstrates the process of distributing an empirically determined elbow joint moment across a set of muscles spanning the elbow joint.
The net joint moment to be balanced in example 11.1 was a flexor moment, and only muscles with flexor moment arms were included in the model (figure 11.10). If an elbow extensor muscle had been included, there is no cost function of the type presented here that would predict force in an antagonist muscle. Any force in an elbow extensor would itself contribute to a higher cost function value and would also require greater elbow flexor forces to balance the target 10 N·m joint moment, further increasing the cost function value. The total lack of coactivation in this example is contrary to the common observation that during heavy activation of the elbow flexors, there is some activity in the triceps muscle group. The extensor coactivation likely helps stabilize the joint during heavy exertion but will not be predicted using traditional static optimization techniques in simple one DOF models. Although the example presented here focuses on obtaining numerical results for muscle forces, the interested reader is referred to the review by Crowninshield and Brand (1981b), which presents an interesting graphical interpretation of the solution to this static optimization problem.
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Dynamic Optimization
Ongoing work with static optimization models has led to the development of dynamic optimization or optimal control models, which are applied in conjunction with forward dynamics models of human motion. In contrast to the inverse dynamics approach, which uses experimental data from an actual performance to calculate net joint moments, forward dynamics analyses simulate the motion of the body from a given set of joint moments or muscle forces (see chapter 10). Dynamic optimization models are therefore often designed to find the muscle stimulation patterns that result in an optimal motion (figure 11.9). As mentioned earlier, the optimal motion may be one that maximizes (or minimizes) a performance criterion, such as maximizing jump height; or, the optimal motion may be defined as one that best reproduces a set of experimental data. Regardless, the variables that are optimized are usually the muscle stimulation patterns that control the motion of the musculoskeletal model. These stimulation patterns are used as the inputs to muscle models that predict individual muscle forces, which are then multiplied by the appropriate moment arms to compute the active muscle moments. The active muscle moments are combined with passive moments to compute the net joint moments, which actuate the skeletal model and produce movement. Thus, dynamic optimization leads to the synthesis of body segment kinematics associated with optimal performance, which is fundamentally different from the static optimization approach.
The optimal kinematics predicted from a dynamic optimization can be compared with experimental motion data; however, the experimental data are not required to obtain the solution, as they are with static optimization. This feature of dynamic optimization permits one to address questions for which no experimental data exist, such as testing the feasibility of possible forms of locomotion in extinct species (e.g., Nagano et al. 2005). Moreover, because dynamic optimization uses forward dynamics models that simulate the whole movement performance and provide complete muscle force time histories (rather than solutions at independent time increments), many of the problems associated with static optimization models are overcome. However, these advantages come at a computational cost, because dynamic optimization often requires a more detailed musculoskeletal model and the entire movement must be simulated for each possible solution. Thus, solving a dynamic optimization problem can easily require an order of magnitude more time than is required to solve a comparable static optimization problem. Although computational costs are difficult to compare in an objective manner, it is not uncommon for static optimization solutions to take seconds or minutes to obtain, whereas dynamic optimization solutions can take hours, days, or even weeks, when run on standard computer workstations.
In many cases, users of dynamic optimization must define the goal of the movement mathematically. This definition is easiest for movements in which the performance criterion to be optimized is clear in a mechanical sense. In vertical jumping, for example, the cost function can be stated as maximization of the vertical displacement of the body center of mass during the flight phase. If the model is constructed with appropriate constraints (realistic muscle properties and joint range of motion), jump heights and body segment motions approximating those of human jumpers can be attained (Pandy and Zajac 1991; van Soest et al. 1993). However, for many human movements the performance criterion is not as clear. In walking, for instance, the goal may seem to be getting from point A to point B in a certain amount of time. Unfortunately, this does not provide enough information to solve for a set of muscle stimulation patterns that will result in realistic walking. Successful simulations of walking have been generated using a cost function that minimizes the metabolic cost of transport (Anderson and Pandy 2001; Umberger 2010); however, the cost function formulation is considerably more complicated than for vertical jumping. Minimum-energy solutions also require the inclusion of an additional model for predicting the metabolic cost of muscle actions (e.g., Umberger et al. 2003). In cases where the performance criterion is not clear, dynamic optimization can serve as an effective approach for testing various theoretical criteria to see how well each can produce the desired movement patterns.
Movements for which the underlying criterion is difficult to define, such as walking or pedaling, have often been simulated by formulating and solving a tracking problem. This approach has a similar appeal to the EMG-based techniques described earlier, in that the resulting motion should closely approximate the way in which humans actually move. However, it is unclear which of several possible experimental measures (kinematics, kinetics, EMG) are the most important for the model to track. Also, because of errors in the experimental data and differences between the musculoskeletal model and the experimental subjects, it may be impossible for the model to track the data perfectly. This approach also limits much of the predictive ability of the dynamic optimization approach, as it is only possible to consider conditions for which experimental data are available.
The tracking approach has seen frequent use in hybrid solution algorithms, which seek to retain the advantages of forward dynamics and dynamic optimization while achieving the computational efficiency of static optimization. Two examples are computed muscle control (Thelen et al. 2003) and direct collocation (Kaplan and Heegaard 2001). Computed muscle control works by solving a static optimization problem at each time step within a single forward dynamics movement simulation. By using feedback on the kinematics and muscle activations at each time step of the numerical integration, computed muscle control can generate a forward dynamics simulation that optimally tracks a set of experimental data without solving a dynamic optimization problem that would require thousands of forward dynamics simulations. In contrast, direct collocation works by converting the differential equations of motion into a set of algebraic constraint equations and treating both the control variables (muscle stimulations) and state variables (positions and velocities) as unknowns in the optimization problem. Although they differ considerably in implementation, both techniques appear to be able to produce results that are similar to dynamic optimization tracking solutions, with a computational cost closer to that of static optimization.
Although static and dynamic optimization have both become popular means for controlling musculoskeletal models, several issues concerning their use must be addressed. Do humans actually produce movements based on one given performance criterion, or does the performance objective change during the movement? If the optimized model results differ from those of a human performer, is it because the model is too simple or not constrained properly, or is the human not performing optimally? Finally, EMG data have illustrated various degrees of simultaneous antagonistic cocontraction during some movement sequences. Optimization models tend not to yield solutions predicting muscle antagonism around a joint, although some antagonism is predicted when models contain muscles that contribute to more than one joint moment. However, optimization models that seek to minimize or maximize specific cost functions will not predict antagonistic cocontraction associated with joint stability or stiffness. In the case of walking, for example, there is no single cost function that could simultaneously predict the relatively minimal muscle coactivation in healthy subjects and the substantial antagonism exhibited by patients with cerebral palsy. Despite these drawbacks, optimization models have increased the understanding of human biomechanics and will continue to do so in the future.
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The state space in the presentation of biomechanical data
Biomechanical data can be represented graphically in a variety of different ways.
The State Space
Biomechanical data can be represented graphically in a variety of different ways. A traditional and very informative type of graph is to display the changes in kinetic or kinematic parameters over time. Figure 13.4 presents an example of the changes in the left and right thigh angular displacements over a time period of 10 s while a subject is walking on a treadmill. Information about peak flexion and extension angles can be obtained from these graphs and quantified. A closer inspection of the graphs makes it quite evident that the coordination between these segments is primarily antiphase. However, especially during the stance phase there are more subtle changes in the patterns of coordination that are harder to obtain from these plots.
In the dynamical systems approach, the reconstruction of the so-called state space is essential in identifying the important features of the behavior of a system, such as its stability and ability to change and adapt to different environmental and task constraints. The state space is a representation of the relevant variables that will help identify these features. To help us understand and quantify coordination between joints or segments, it can be very useful to represent the system in a state space that is based on an angle-angle relative motion plot. Figure 13.5 presents a state space of the relative motion of the time series of the thigh and leg shown earlier in figure 13.4. This angle-angle plot can reveal regions where coordination changes take place as well as parts of the gait cycle where there is relative invariance in coordination patterns. These coordinative changes in the angle-angle plots can be further quantified by vector coding techniques that are discussed later.
Another form of state space is where the position and velocity of a joint or segment are plotted relative to each other. This state-space representation is also often referred to as the phase plane. An example of a position-velocity phase-plane plot is shown in figure 13.6, where the angular displacement of the thigh is plotted against the angular velocity. This is a higher-dimensional state space because the time derivative of position is used to identify the pattern. The phase-plane representation is a first and critical step in the quantification of coordination using continuous relative phase techniques.
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Attractors in State Space
In most forms of human movement, the dynamics in state space are limited to distinct regions, as can be seen in the phase-plane plot in figure 13.6, where the pattern in state space is limited to a fairly narrow, cyclical band. In dynamical systems, preferred regions in state space onto which the dynamics tend to settle are called attractors. Attractors can come in a variety of different forms. A seemingly simple kind of attractor is the point attractor. In this case, the dynamics in the system tend to converge onto one relatively fixed value in the state space. Figure 13.7, c and d, provides an example of a point attractor. The point attractor dynamics are shown in figure 13.7c by a relatively consistent value of discrete relative phase throughout the gait cycle. The discrete relative phase is based on the occurrence of peak flexion in the left and right thigh angular displacements, and the time series in 13.7c show that there is a consistent antiphase or 180° coordination pattern. This antiphase coordination can already be observed by comparing the individual time series in figure 13.4 but is more objectively quantified by the state-space plot in figure 13.7d, where so-called return maps (plotting the coordination at xn vs. xn+1, where n is the cycle number) identify a fixed-point attractor in state space. The relative phase dynamics are of the fixed-point type in this case, as a perturbation (sudden change in treadmill speed and return to the original speed; figure 13.7) results in a return to the original antiphase dynamics with a relative short latency. The perturbation here consisted of a brief (5 s) increase in treadmill speed from 1.2 m/s to 2.0 m/s, with a subsequent return to 1.2 m/s.
Whereas the coordination between the right and left legs can be identified in the form of a point attractor with a relatively fixed value of coordination from cycle to cycle, the dynamics of the individual limb segments show a very different pattern. These are clearly cyclical, as can be seen in the phase planes in figure 13.7, a and b. Attractors of this form are called limit-cycle attractors, and the dynamics converge onto a cyclical region in state space. These limit-cycle attractors are typically identified on the basis of a very narrow, overlapping band of the trajectories in the state space. However, the existence of the narrow band is a necessary but not sufficient condition to characterize the dynamics as a limit-cycle. An essential feature of the limit-cycle attractor is stability with respect to perturbation; to classify as a limit-cycle attractor, the system should also show resistance to perturbations. An example of this is given in figure 13.7, a and b. The regular cyclic pattern in the phase plane (solid line) represents the steady-state gait patterns while a subject was walking at a speed of 1.2 m/s. The dashed trajectories represent the perturbation phase of the trial when speed was suddenly increased to 2.0 m/s and the consequent return back to the original pattern. This return to the preperturbation dynamic is an essential feature of the limit-cycle attractor. The patterns in the phase plane can also serve as an energy plot. The convergence and divergence of trajectories in the phase plane can identify the loss and gain of energy in the system.
Higher-dimensional state spaces (three dimensions and up) can also reveal more complex types of attractors: quasiperiodic and chaotic attractors. The chaotic attractor demonstrates both stable attraction to a region in state space and variability. Figure 13.8 shows an example of the Lorenz attractor, a well-known chaotic attractor that emerges from dynamic interactions in fluid and air flow systems (Strogatz 1994). These dual features of stability and adaptability can be associated with a higher pattern complexity that is now commonly regarded as reflective of healthy and expert systems (see figure 13.3). As an example, increased heart rate variability is considered an important indicator of healthy heart function, reflecting a degree of complexity in organization in which disruptions can be compensated for more easily (Glass 2001; Lipsitz 2002).
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Using EMG to diagnose and treat low back pain
The diagnosis and treatment of low back pain are important issues.
EMG and Low Back Pain
The diagnosis and treatment of low back pain are important issues. In the United States, for example, back pain accounts for two-thirds of all workers compensation costs. Peripheral muscle problems may be one issue; atrophy of the multifidus is frequently observed in patients with low back pain (Hides et al. 1994). EMG is becoming an increasingly useful tool for the diagnosis and treatment of low back pain.
The trunk musculature plays an important role in tasks such as lifting and throwing. During these kinds of tasks as well as in other activities that might threaten the postural control of the individual and perturb balance, the nervous system implements strategies to activate trunk muscles while performing voluntary movements involving remote muscle groups. However, the prevalence of back pain in the general population, and the need to identify ergonomically efficient and safe ways to use back muscles, require that we understand the nature of activation in muscles controlling the trunk. Here, too, EMG is an important tool.
A number of issues render EMG recordings from the trunk musculature difficult to obtain and interpret. From an anatomical viewpoint, the architecture of the back muscles is complex. For example, it is generally accepted that injury or disorder of the multifidus frequently results in back pain. Both multifidus and erector spinae have distinct superficial and deep portions (Bustami 1986; Macintosh et al. 1986) with different histochemical and biomechanical characteristics (Bogduk et al. 1992; Dickx et al. 2010). In some respects, it is fortunate that most of the muscle mass of the multifidus lies in the more superficial portion, so that surface EMG has a greater possibility of characterizing the fibers that produce large amounts of force. However, the multifidus superficial portion has a different role than the deeper fibers. Fibers in the superficial portion cross numerous spinal segments and are thus in a better position to produce back extension. However, the deeper fibers are relatively short, crossing perhaps one or two segments, and thus protect segments of the lumbar spine from inappropriate shear or torsion torques (Macintosh and Valencia 1986).
In addition to issues concerning the activation of the back extensor muscles, a number of important questions require resolution by EMG analysis in which considerable electrocardiographic (ECG) artifact may be present. ECG artifact is particularly problematic during relatively low force contractions, exaggerating the ratio between the signal of interest and the interfering ECG signal. Recording EMG signals from abdominal, knee extensor, and other trunk muscles during normal human movements, for example, can result in the placement of electrodes well within recording range of the electrocardiogram. The relatively large ECG signal can exaggerate the amplitude of EMG activity, and the low-frequency characteristics of the ECG wave can also alter the frequency characteristics of the recorded EMG activity.
For example, the absence of sufficient trunk activity can produce instability resulting in low back pain (van Dieën et al. 2003). However, the amplitude of EMG activity required to ensure stability is small and thus can be affected by the ECG signal (Cholewicki et al. 1997). The best solution is to place the electrodes in a location from which no or minimal ECG artifact can be recorded. However, this is frequently difficult if not impossible. Consequently, numerous algorithms have been suggested to remove the ECG artifact.
One frequently implemented technique to remove this artifact is to compute a template of the ECG signal and subtract it from the electromyogram. This has been proved to be a reasonably successful procedure and has been implemented in studies recording from the diaphragm (Bartolo et al. 1996) as well as rectus abdominis (Hof 2009). Hof (2009) described a technique in which the ECG signal is recorded simultaneously with the EMG signal of interest, and template subtraction is then implemented (figure 8.15).
Digital filtering is another useful alternative for ECG artifact removal. Drake and Callaghan (2006) used a FIR (finite impulse response) filter with a hamming window, although they concluded that the most efficient filtering result could be obtained using a somewhat simpler fourth-order, 30 Hz high-pass cutoff Butterworth filter. Template subtraction improved extraction of the ECG signal but required a large amount of time.
Alternative ECG removal techniques include the use of adaptive filters (Lu et al. 2009; Marque et al. 2005), wavelet-independent component analysis (Taelman et al. 2007), and wavelet-based adaptive filters (Zhan et al. 2010). Irrespective of the technique used for ECG signal artifact removal, it is apparent that the resultant improvement in signal interpretation can be important. Hu and colleagues (2009), for example, found that the use of independent component analysis to remove artifact resulted in an improved ability to discriminate between patients with low back pain and normal subjects during both sitting and standing tasks.
Analysis of the EMG activity in back muscles has produced some interesting results, in part reflecting the unique anatomical issues discussed here. As long ago as 1962, Morris and colleagues noted that “the three muscles of the erector spinae group considered here . . .
do not always show parallel activity, and one may be active while the other two are inactive” (p. 519). This may suggest that the mere demonstration of greater or lesser EMG amplitude may not be indicative of abnormality in a particular muscle. Potentially more important than EMG amplitude is the timing of EMG activity. Deep and superficial portions of the multifidus are differentially active during arm movements (Moseley et al. 2002). One suggestion is that the manner in which the multifidus is differentially controlled in people with back pain compared with those without may be the source of chronic back pain (MacDonald et al. 2009).
Similarly, the analysis of EMG frequency characteristics in patients with back pain compared with control subjects has suggested that there may be differences between muscles and even differences within muscles. Patients with low back pain frequently exhibit alterations in the electromyographic response to fatigue in the back extensor muscles (Biedermann et al. 1991). This is one area in which spectral analysis of the EMG signals has been helpful. As in other muscles, median EMG frequency decreases in the trunk extensors with fatigue (Demoulin et al. 2007). It is interesting to note that Kramer and colleagues (2005) found that the magnitude of median frequency decrease was greater in healthy subjects than in individuals with back pain. Support for importance of electrode placement is obtained from the observations of Sung and colleagues (2009). Normal subjects and patients with back pain underwent an isometric contraction protocol that induced fatigue of the back extensors. EMG frequency measurements made in both the thoracic and the lumbar portions of the erector spinae indicated that the thoracic portion had a significantly lower median frequency than the lumbar portion in patients with low back pain. However, median frequency was lower in the lumbar portion than in the thoracic portion in control subjects. Thus, in the analysis of trunk musculature that might be involved in the development of back pain, EMG studies using wire electrodes frequently may be required to record from different portions of the muscle.
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Using musculoskeletal optimization models to generate control signals
A popular method for generating control signals in musculoskeletal models is to use some form of optimization.
Optimization Models
A popular method for generating control signals in musculoskeletal models is to use some form of optimization. In some cases, when the optimization criterion is derived from motor control principles, this is tantamount to developing a theoretical control model. In other cases, the optimization criterion may result in a coordinated movement pattern yet have little physiological basis. In general, the optimization approach is used in an effort to determine which set of model control signals will produce a result that optimizes (minimizes or maximizes) a given criterion measure. The criterion measure is known as a cost function, objective function, or performance criterion. The cost function can be relatively simple (e.g., find the solution that yields minimal muscle force) or complex (e.g., determine a nonlinear combination of maximal muscle force at minimal metabolic cost). It can be directly related to muscle function (e.g., minimize muscular work) or to some aspect of the motion under study (e.g., maximize vertical jump height). Alternatively, the cost function may be formulated to minimize the differences between model outputs (e.g., joint angles, ground reaction forces) and corresponding experimental measures. This latter case is often referred to as a tracking problem in that the goal is to find a solution that causes the model to follow, or track, the experimental data.
In all cases, the cost function serves as a guiding constraint that determines the selection of one particular set of optimal muscular controls from among many different possible solutions. Two major optimization approaches that have been used to control musculoskeletal models are referred to in the literature as static optimization and dynamic optimization. The meaning of static in this context is that the cost function is evaluated at each time step during a movement, independent of any prior or subsequent time steps. In contrast, dynamic optimization is dynamic in the sense that the entire movement sequence must be simulated to determine the value of the cost function.
Static Optimization
Initial attempts to predict individual muscle forces used static optimization models in conjunction with inverse dynamics analysis of an actual motor performance. The inverse dynamics analysis permits the calculation of net joint moments at incremental times throughout the movement (see chapter 5). In most applications, numerical optimization is used to find the set of muscle forces that balances these joint moments while also satisfying a selected cost function (figure 11.9). Thus, with this approach the muscle forces themselves, rather than neural signals, are the controls. The time-independent nature of static optimization allows solutions to be obtained with relatively little computational cost, but there are some drawbacks. One issue in early applications was sudden, nonphysiological switching on and off of muscle forces caused by the independence of solutions for sequential time increments (i.e., the optimization model balanced the joint moments separately for each time interval using very different sets of muscles). This problem can be avoided by careful selection of the initial guess in the optimization problem, such as by using the solution from one time step as the initial guess for the next time step. A stronger approach to address this issue is to use muscle models that invoke physiological realism through force-length and force-velocity relations and time-dependent stimulation-activation dynamics (chapter 9). When more detailed muscle models are included in a static optimization model, muscle activations become the control variables, rather than muscle forces.
Early studies using static optimization were plagued also by two other kinds of nonphysiological results. The first was the prediction of model forces that were too high for actual muscles to produce. This difficulty was easily addressed by defining physiologically valid maximal force constraints for each muscle in the musculoskeletal model. The second problem was that solutions would often select only one muscle to balance the net joint moment rather than choose a more realistic muscular synergy. Depending on the exact formulation, the muscle with the largest moment arm (if minimizing muscle force) or a favorable combination of moment arm and muscle strength (if minimizing muscle stress) would be selected to fully balance the joint moment, with zero force predicted in other synergist muscles. Mathematically, synergism can be produced by using nonlinear cost functions (e.g., minimizing the sum of squared or cubed muscle forces), although the physiological rationale for specific nonlinear cost functions is not always clear. A widely used nonlinear cost function proposed by Crowninshield and Brand (1981a) involves minimizing the sum of cubed muscle stresses. It was originally argued that this particular cost function would lead to solutions that maximize muscle endurance, making it appropriate for predicting muscle forces in submaximal tasks such as walking. However, the Crowninshield and Brand criterion has since been used to solve for muscle forces in a range of activities, some of which are unlikely candidates for maximizing muscle endurance (e.g., jumping, landing).
An additional issue with static optimization models is that they are essentially a decomposition of experimental joint moments into individual muscle forces. Therefore, the predicted muscle forces will be subject to any errors in the experimental joint moments in addition to any shortcomings associated with the approximation made in creating the musculoskeletal model. To complete this section, we present a static optimization example motivated by a classic review article by Crowninshield and Brand (1981b), which demonstrates the process of distributing an empirically determined elbow joint moment across a set of muscles spanning the elbow joint.
The net joint moment to be balanced in example 11.1 was a flexor moment, and only muscles with flexor moment arms were included in the model (figure 11.10). If an elbow extensor muscle had been included, there is no cost function of the type presented here that would predict force in an antagonist muscle. Any force in an elbow extensor would itself contribute to a higher cost function value and would also require greater elbow flexor forces to balance the target 10 N·m joint moment, further increasing the cost function value. The total lack of coactivation in this example is contrary to the common observation that during heavy activation of the elbow flexors, there is some activity in the triceps muscle group. The extensor coactivation likely helps stabilize the joint during heavy exertion but will not be predicted using traditional static optimization techniques in simple one DOF models. Although the example presented here focuses on obtaining numerical results for muscle forces, the interested reader is referred to the review by Crowninshield and Brand (1981b), which presents an interesting graphical interpretation of the solution to this static optimization problem.
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Dynamic Optimization
Ongoing work with static optimization models has led to the development of dynamic optimization or optimal control models, which are applied in conjunction with forward dynamics models of human motion. In contrast to the inverse dynamics approach, which uses experimental data from an actual performance to calculate net joint moments, forward dynamics analyses simulate the motion of the body from a given set of joint moments or muscle forces (see chapter 10). Dynamic optimization models are therefore often designed to find the muscle stimulation patterns that result in an optimal motion (figure 11.9). As mentioned earlier, the optimal motion may be one that maximizes (or minimizes) a performance criterion, such as maximizing jump height; or, the optimal motion may be defined as one that best reproduces a set of experimental data. Regardless, the variables that are optimized are usually the muscle stimulation patterns that control the motion of the musculoskeletal model. These stimulation patterns are used as the inputs to muscle models that predict individual muscle forces, which are then multiplied by the appropriate moment arms to compute the active muscle moments. The active muscle moments are combined with passive moments to compute the net joint moments, which actuate the skeletal model and produce movement. Thus, dynamic optimization leads to the synthesis of body segment kinematics associated with optimal performance, which is fundamentally different from the static optimization approach.
The optimal kinematics predicted from a dynamic optimization can be compared with experimental motion data; however, the experimental data are not required to obtain the solution, as they are with static optimization. This feature of dynamic optimization permits one to address questions for which no experimental data exist, such as testing the feasibility of possible forms of locomotion in extinct species (e.g., Nagano et al. 2005). Moreover, because dynamic optimization uses forward dynamics models that simulate the whole movement performance and provide complete muscle force time histories (rather than solutions at independent time increments), many of the problems associated with static optimization models are overcome. However, these advantages come at a computational cost, because dynamic optimization often requires a more detailed musculoskeletal model and the entire movement must be simulated for each possible solution. Thus, solving a dynamic optimization problem can easily require an order of magnitude more time than is required to solve a comparable static optimization problem. Although computational costs are difficult to compare in an objective manner, it is not uncommon for static optimization solutions to take seconds or minutes to obtain, whereas dynamic optimization solutions can take hours, days, or even weeks, when run on standard computer workstations.
In many cases, users of dynamic optimization must define the goal of the movement mathematically. This definition is easiest for movements in which the performance criterion to be optimized is clear in a mechanical sense. In vertical jumping, for example, the cost function can be stated as maximization of the vertical displacement of the body center of mass during the flight phase. If the model is constructed with appropriate constraints (realistic muscle properties and joint range of motion), jump heights and body segment motions approximating those of human jumpers can be attained (Pandy and Zajac 1991; van Soest et al. 1993). However, for many human movements the performance criterion is not as clear. In walking, for instance, the goal may seem to be getting from point A to point B in a certain amount of time. Unfortunately, this does not provide enough information to solve for a set of muscle stimulation patterns that will result in realistic walking. Successful simulations of walking have been generated using a cost function that minimizes the metabolic cost of transport (Anderson and Pandy 2001; Umberger 2010); however, the cost function formulation is considerably more complicated than for vertical jumping. Minimum-energy solutions also require the inclusion of an additional model for predicting the metabolic cost of muscle actions (e.g., Umberger et al. 2003). In cases where the performance criterion is not clear, dynamic optimization can serve as an effective approach for testing various theoretical criteria to see how well each can produce the desired movement patterns.
Movements for which the underlying criterion is difficult to define, such as walking or pedaling, have often been simulated by formulating and solving a tracking problem. This approach has a similar appeal to the EMG-based techniques described earlier, in that the resulting motion should closely approximate the way in which humans actually move. However, it is unclear which of several possible experimental measures (kinematics, kinetics, EMG) are the most important for the model to track. Also, because of errors in the experimental data and differences between the musculoskeletal model and the experimental subjects, it may be impossible for the model to track the data perfectly. This approach also limits much of the predictive ability of the dynamic optimization approach, as it is only possible to consider conditions for which experimental data are available.
The tracking approach has seen frequent use in hybrid solution algorithms, which seek to retain the advantages of forward dynamics and dynamic optimization while achieving the computational efficiency of static optimization. Two examples are computed muscle control (Thelen et al. 2003) and direct collocation (Kaplan and Heegaard 2001). Computed muscle control works by solving a static optimization problem at each time step within a single forward dynamics movement simulation. By using feedback on the kinematics and muscle activations at each time step of the numerical integration, computed muscle control can generate a forward dynamics simulation that optimally tracks a set of experimental data without solving a dynamic optimization problem that would require thousands of forward dynamics simulations. In contrast, direct collocation works by converting the differential equations of motion into a set of algebraic constraint equations and treating both the control variables (muscle stimulations) and state variables (positions and velocities) as unknowns in the optimization problem. Although they differ considerably in implementation, both techniques appear to be able to produce results that are similar to dynamic optimization tracking solutions, with a computational cost closer to that of static optimization.
Although static and dynamic optimization have both become popular means for controlling musculoskeletal models, several issues concerning their use must be addressed. Do humans actually produce movements based on one given performance criterion, or does the performance objective change during the movement? If the optimized model results differ from those of a human performer, is it because the model is too simple or not constrained properly, or is the human not performing optimally? Finally, EMG data have illustrated various degrees of simultaneous antagonistic cocontraction during some movement sequences. Optimization models tend not to yield solutions predicting muscle antagonism around a joint, although some antagonism is predicted when models contain muscles that contribute to more than one joint moment. However, optimization models that seek to minimize or maximize specific cost functions will not predict antagonistic cocontraction associated with joint stability or stiffness. In the case of walking, for example, there is no single cost function that could simultaneously predict the relatively minimal muscle coactivation in healthy subjects and the substantial antagonism exhibited by patients with cerebral palsy. Despite these drawbacks, optimization models have increased the understanding of human biomechanics and will continue to do so in the future.
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The state space in the presentation of biomechanical data
Biomechanical data can be represented graphically in a variety of different ways.
The State Space
Biomechanical data can be represented graphically in a variety of different ways. A traditional and very informative type of graph is to display the changes in kinetic or kinematic parameters over time. Figure 13.4 presents an example of the changes in the left and right thigh angular displacements over a time period of 10 s while a subject is walking on a treadmill. Information about peak flexion and extension angles can be obtained from these graphs and quantified. A closer inspection of the graphs makes it quite evident that the coordination between these segments is primarily antiphase. However, especially during the stance phase there are more subtle changes in the patterns of coordination that are harder to obtain from these plots.
In the dynamical systems approach, the reconstruction of the so-called state space is essential in identifying the important features of the behavior of a system, such as its stability and ability to change and adapt to different environmental and task constraints. The state space is a representation of the relevant variables that will help identify these features. To help us understand and quantify coordination between joints or segments, it can be very useful to represent the system in a state space that is based on an angle-angle relative motion plot. Figure 13.5 presents a state space of the relative motion of the time series of the thigh and leg shown earlier in figure 13.4. This angle-angle plot can reveal regions where coordination changes take place as well as parts of the gait cycle where there is relative invariance in coordination patterns. These coordinative changes in the angle-angle plots can be further quantified by vector coding techniques that are discussed later.
Another form of state space is where the position and velocity of a joint or segment are plotted relative to each other. This state-space representation is also often referred to as the phase plane. An example of a position-velocity phase-plane plot is shown in figure 13.6, where the angular displacement of the thigh is plotted against the angular velocity. This is a higher-dimensional state space because the time derivative of position is used to identify the pattern. The phase-plane representation is a first and critical step in the quantification of coordination using continuous relative phase techniques.
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Attractors in State Space
In most forms of human movement, the dynamics in state space are limited to distinct regions, as can be seen in the phase-plane plot in figure 13.6, where the pattern in state space is limited to a fairly narrow, cyclical band. In dynamical systems, preferred regions in state space onto which the dynamics tend to settle are called attractors. Attractors can come in a variety of different forms. A seemingly simple kind of attractor is the point attractor. In this case, the dynamics in the system tend to converge onto one relatively fixed value in the state space. Figure 13.7, c and d, provides an example of a point attractor. The point attractor dynamics are shown in figure 13.7c by a relatively consistent value of discrete relative phase throughout the gait cycle. The discrete relative phase is based on the occurrence of peak flexion in the left and right thigh angular displacements, and the time series in 13.7c show that there is a consistent antiphase or 180° coordination pattern. This antiphase coordination can already be observed by comparing the individual time series in figure 13.4 but is more objectively quantified by the state-space plot in figure 13.7d, where so-called return maps (plotting the coordination at xn vs. xn+1, where n is the cycle number) identify a fixed-point attractor in state space. The relative phase dynamics are of the fixed-point type in this case, as a perturbation (sudden change in treadmill speed and return to the original speed; figure 13.7) results in a return to the original antiphase dynamics with a relative short latency. The perturbation here consisted of a brief (5 s) increase in treadmill speed from 1.2 m/s to 2.0 m/s, with a subsequent return to 1.2 m/s.
Whereas the coordination between the right and left legs can be identified in the form of a point attractor with a relatively fixed value of coordination from cycle to cycle, the dynamics of the individual limb segments show a very different pattern. These are clearly cyclical, as can be seen in the phase planes in figure 13.7, a and b. Attractors of this form are called limit-cycle attractors, and the dynamics converge onto a cyclical region in state space. These limit-cycle attractors are typically identified on the basis of a very narrow, overlapping band of the trajectories in the state space. However, the existence of the narrow band is a necessary but not sufficient condition to characterize the dynamics as a limit-cycle. An essential feature of the limit-cycle attractor is stability with respect to perturbation; to classify as a limit-cycle attractor, the system should also show resistance to perturbations. An example of this is given in figure 13.7, a and b. The regular cyclic pattern in the phase plane (solid line) represents the steady-state gait patterns while a subject was walking at a speed of 1.2 m/s. The dashed trajectories represent the perturbation phase of the trial when speed was suddenly increased to 2.0 m/s and the consequent return back to the original pattern. This return to the preperturbation dynamic is an essential feature of the limit-cycle attractor. The patterns in the phase plane can also serve as an energy plot. The convergence and divergence of trajectories in the phase plane can identify the loss and gain of energy in the system.
Higher-dimensional state spaces (three dimensions and up) can also reveal more complex types of attractors: quasiperiodic and chaotic attractors. The chaotic attractor demonstrates both stable attraction to a region in state space and variability. Figure 13.8 shows an example of the Lorenz attractor, a well-known chaotic attractor that emerges from dynamic interactions in fluid and air flow systems (Strogatz 1994). These dual features of stability and adaptability can be associated with a higher pattern complexity that is now commonly regarded as reflective of healthy and expert systems (see figure 13.3). As an example, increased heart rate variability is considered an important indicator of healthy heart function, reflecting a degree of complexity in organization in which disruptions can be compensated for more easily (Glass 2001; Lipsitz 2002).
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Using EMG to diagnose and treat low back pain
The diagnosis and treatment of low back pain are important issues.
EMG and Low Back Pain
The diagnosis and treatment of low back pain are important issues. In the United States, for example, back pain accounts for two-thirds of all workers compensation costs. Peripheral muscle problems may be one issue; atrophy of the multifidus is frequently observed in patients with low back pain (Hides et al. 1994). EMG is becoming an increasingly useful tool for the diagnosis and treatment of low back pain.
The trunk musculature plays an important role in tasks such as lifting and throwing. During these kinds of tasks as well as in other activities that might threaten the postural control of the individual and perturb balance, the nervous system implements strategies to activate trunk muscles while performing voluntary movements involving remote muscle groups. However, the prevalence of back pain in the general population, and the need to identify ergonomically efficient and safe ways to use back muscles, require that we understand the nature of activation in muscles controlling the trunk. Here, too, EMG is an important tool.
A number of issues render EMG recordings from the trunk musculature difficult to obtain and interpret. From an anatomical viewpoint, the architecture of the back muscles is complex. For example, it is generally accepted that injury or disorder of the multifidus frequently results in back pain. Both multifidus and erector spinae have distinct superficial and deep portions (Bustami 1986; Macintosh et al. 1986) with different histochemical and biomechanical characteristics (Bogduk et al. 1992; Dickx et al. 2010). In some respects, it is fortunate that most of the muscle mass of the multifidus lies in the more superficial portion, so that surface EMG has a greater possibility of characterizing the fibers that produce large amounts of force. However, the multifidus superficial portion has a different role than the deeper fibers. Fibers in the superficial portion cross numerous spinal segments and are thus in a better position to produce back extension. However, the deeper fibers are relatively short, crossing perhaps one or two segments, and thus protect segments of the lumbar spine from inappropriate shear or torsion torques (Macintosh and Valencia 1986).
In addition to issues concerning the activation of the back extensor muscles, a number of important questions require resolution by EMG analysis in which considerable electrocardiographic (ECG) artifact may be present. ECG artifact is particularly problematic during relatively low force contractions, exaggerating the ratio between the signal of interest and the interfering ECG signal. Recording EMG signals from abdominal, knee extensor, and other trunk muscles during normal human movements, for example, can result in the placement of electrodes well within recording range of the electrocardiogram. The relatively large ECG signal can exaggerate the amplitude of EMG activity, and the low-frequency characteristics of the ECG wave can also alter the frequency characteristics of the recorded EMG activity.
For example, the absence of sufficient trunk activity can produce instability resulting in low back pain (van Dieën et al. 2003). However, the amplitude of EMG activity required to ensure stability is small and thus can be affected by the ECG signal (Cholewicki et al. 1997). The best solution is to place the electrodes in a location from which no or minimal ECG artifact can be recorded. However, this is frequently difficult if not impossible. Consequently, numerous algorithms have been suggested to remove the ECG artifact.
One frequently implemented technique to remove this artifact is to compute a template of the ECG signal and subtract it from the electromyogram. This has been proved to be a reasonably successful procedure and has been implemented in studies recording from the diaphragm (Bartolo et al. 1996) as well as rectus abdominis (Hof 2009). Hof (2009) described a technique in which the ECG signal is recorded simultaneously with the EMG signal of interest, and template subtraction is then implemented (figure 8.15).
Digital filtering is another useful alternative for ECG artifact removal. Drake and Callaghan (2006) used a FIR (finite impulse response) filter with a hamming window, although they concluded that the most efficient filtering result could be obtained using a somewhat simpler fourth-order, 30 Hz high-pass cutoff Butterworth filter. Template subtraction improved extraction of the ECG signal but required a large amount of time.
Alternative ECG removal techniques include the use of adaptive filters (Lu et al. 2009; Marque et al. 2005), wavelet-independent component analysis (Taelman et al. 2007), and wavelet-based adaptive filters (Zhan et al. 2010). Irrespective of the technique used for ECG signal artifact removal, it is apparent that the resultant improvement in signal interpretation can be important. Hu and colleagues (2009), for example, found that the use of independent component analysis to remove artifact resulted in an improved ability to discriminate between patients with low back pain and normal subjects during both sitting and standing tasks.
Analysis of the EMG activity in back muscles has produced some interesting results, in part reflecting the unique anatomical issues discussed here. As long ago as 1962, Morris and colleagues noted that “the three muscles of the erector spinae group considered here . . .
do not always show parallel activity, and one may be active while the other two are inactive” (p. 519). This may suggest that the mere demonstration of greater or lesser EMG amplitude may not be indicative of abnormality in a particular muscle. Potentially more important than EMG amplitude is the timing of EMG activity. Deep and superficial portions of the multifidus are differentially active during arm movements (Moseley et al. 2002). One suggestion is that the manner in which the multifidus is differentially controlled in people with back pain compared with those without may be the source of chronic back pain (MacDonald et al. 2009).
Similarly, the analysis of EMG frequency characteristics in patients with back pain compared with control subjects has suggested that there may be differences between muscles and even differences within muscles. Patients with low back pain frequently exhibit alterations in the electromyographic response to fatigue in the back extensor muscles (Biedermann et al. 1991). This is one area in which spectral analysis of the EMG signals has been helpful. As in other muscles, median EMG frequency decreases in the trunk extensors with fatigue (Demoulin et al. 2007). It is interesting to note that Kramer and colleagues (2005) found that the magnitude of median frequency decrease was greater in healthy subjects than in individuals with back pain. Support for importance of electrode placement is obtained from the observations of Sung and colleagues (2009). Normal subjects and patients with back pain underwent an isometric contraction protocol that induced fatigue of the back extensors. EMG frequency measurements made in both the thoracic and the lumbar portions of the erector spinae indicated that the thoracic portion had a significantly lower median frequency than the lumbar portion in patients with low back pain. However, median frequency was lower in the lumbar portion than in the thoracic portion in control subjects. Thus, in the analysis of trunk musculature that might be involved in the development of back pain, EMG studies using wire electrodes frequently may be required to record from different portions of the muscle.
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Using musculoskeletal optimization models to generate control signals
A popular method for generating control signals in musculoskeletal models is to use some form of optimization.
Optimization Models
A popular method for generating control signals in musculoskeletal models is to use some form of optimization. In some cases, when the optimization criterion is derived from motor control principles, this is tantamount to developing a theoretical control model. In other cases, the optimization criterion may result in a coordinated movement pattern yet have little physiological basis. In general, the optimization approach is used in an effort to determine which set of model control signals will produce a result that optimizes (minimizes or maximizes) a given criterion measure. The criterion measure is known as a cost function, objective function, or performance criterion. The cost function can be relatively simple (e.g., find the solution that yields minimal muscle force) or complex (e.g., determine a nonlinear combination of maximal muscle force at minimal metabolic cost). It can be directly related to muscle function (e.g., minimize muscular work) or to some aspect of the motion under study (e.g., maximize vertical jump height). Alternatively, the cost function may be formulated to minimize the differences between model outputs (e.g., joint angles, ground reaction forces) and corresponding experimental measures. This latter case is often referred to as a tracking problem in that the goal is to find a solution that causes the model to follow, or track, the experimental data.
In all cases, the cost function serves as a guiding constraint that determines the selection of one particular set of optimal muscular controls from among many different possible solutions. Two major optimization approaches that have been used to control musculoskeletal models are referred to in the literature as static optimization and dynamic optimization. The meaning of static in this context is that the cost function is evaluated at each time step during a movement, independent of any prior or subsequent time steps. In contrast, dynamic optimization is dynamic in the sense that the entire movement sequence must be simulated to determine the value of the cost function.
Static Optimization
Initial attempts to predict individual muscle forces used static optimization models in conjunction with inverse dynamics analysis of an actual motor performance. The inverse dynamics analysis permits the calculation of net joint moments at incremental times throughout the movement (see chapter 5). In most applications, numerical optimization is used to find the set of muscle forces that balances these joint moments while also satisfying a selected cost function (figure 11.9). Thus, with this approach the muscle forces themselves, rather than neural signals, are the controls. The time-independent nature of static optimization allows solutions to be obtained with relatively little computational cost, but there are some drawbacks. One issue in early applications was sudden, nonphysiological switching on and off of muscle forces caused by the independence of solutions for sequential time increments (i.e., the optimization model balanced the joint moments separately for each time interval using very different sets of muscles). This problem can be avoided by careful selection of the initial guess in the optimization problem, such as by using the solution from one time step as the initial guess for the next time step. A stronger approach to address this issue is to use muscle models that invoke physiological realism through force-length and force-velocity relations and time-dependent stimulation-activation dynamics (chapter 9). When more detailed muscle models are included in a static optimization model, muscle activations become the control variables, rather than muscle forces.
Early studies using static optimization were plagued also by two other kinds of nonphysiological results. The first was the prediction of model forces that were too high for actual muscles to produce. This difficulty was easily addressed by defining physiologically valid maximal force constraints for each muscle in the musculoskeletal model. The second problem was that solutions would often select only one muscle to balance the net joint moment rather than choose a more realistic muscular synergy. Depending on the exact formulation, the muscle with the largest moment arm (if minimizing muscle force) or a favorable combination of moment arm and muscle strength (if minimizing muscle stress) would be selected to fully balance the joint moment, with zero force predicted in other synergist muscles. Mathematically, synergism can be produced by using nonlinear cost functions (e.g., minimizing the sum of squared or cubed muscle forces), although the physiological rationale for specific nonlinear cost functions is not always clear. A widely used nonlinear cost function proposed by Crowninshield and Brand (1981a) involves minimizing the sum of cubed muscle stresses. It was originally argued that this particular cost function would lead to solutions that maximize muscle endurance, making it appropriate for predicting muscle forces in submaximal tasks such as walking. However, the Crowninshield and Brand criterion has since been used to solve for muscle forces in a range of activities, some of which are unlikely candidates for maximizing muscle endurance (e.g., jumping, landing).
An additional issue with static optimization models is that they are essentially a decomposition of experimental joint moments into individual muscle forces. Therefore, the predicted muscle forces will be subject to any errors in the experimental joint moments in addition to any shortcomings associated with the approximation made in creating the musculoskeletal model. To complete this section, we present a static optimization example motivated by a classic review article by Crowninshield and Brand (1981b), which demonstrates the process of distributing an empirically determined elbow joint moment across a set of muscles spanning the elbow joint.
The net joint moment to be balanced in example 11.1 was a flexor moment, and only muscles with flexor moment arms were included in the model (figure 11.10). If an elbow extensor muscle had been included, there is no cost function of the type presented here that would predict force in an antagonist muscle. Any force in an elbow extensor would itself contribute to a higher cost function value and would also require greater elbow flexor forces to balance the target 10 N·m joint moment, further increasing the cost function value. The total lack of coactivation in this example is contrary to the common observation that during heavy activation of the elbow flexors, there is some activity in the triceps muscle group. The extensor coactivation likely helps stabilize the joint during heavy exertion but will not be predicted using traditional static optimization techniques in simple one DOF models. Although the example presented here focuses on obtaining numerical results for muscle forces, the interested reader is referred to the review by Crowninshield and Brand (1981b), which presents an interesting graphical interpretation of the solution to this static optimization problem.
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Dynamic Optimization
Ongoing work with static optimization models has led to the development of dynamic optimization or optimal control models, which are applied in conjunction with forward dynamics models of human motion. In contrast to the inverse dynamics approach, which uses experimental data from an actual performance to calculate net joint moments, forward dynamics analyses simulate the motion of the body from a given set of joint moments or muscle forces (see chapter 10). Dynamic optimization models are therefore often designed to find the muscle stimulation patterns that result in an optimal motion (figure 11.9). As mentioned earlier, the optimal motion may be one that maximizes (or minimizes) a performance criterion, such as maximizing jump height; or, the optimal motion may be defined as one that best reproduces a set of experimental data. Regardless, the variables that are optimized are usually the muscle stimulation patterns that control the motion of the musculoskeletal model. These stimulation patterns are used as the inputs to muscle models that predict individual muscle forces, which are then multiplied by the appropriate moment arms to compute the active muscle moments. The active muscle moments are combined with passive moments to compute the net joint moments, which actuate the skeletal model and produce movement. Thus, dynamic optimization leads to the synthesis of body segment kinematics associated with optimal performance, which is fundamentally different from the static optimization approach.
The optimal kinematics predicted from a dynamic optimization can be compared with experimental motion data; however, the experimental data are not required to obtain the solution, as they are with static optimization. This feature of dynamic optimization permits one to address questions for which no experimental data exist, such as testing the feasibility of possible forms of locomotion in extinct species (e.g., Nagano et al. 2005). Moreover, because dynamic optimization uses forward dynamics models that simulate the whole movement performance and provide complete muscle force time histories (rather than solutions at independent time increments), many of the problems associated with static optimization models are overcome. However, these advantages come at a computational cost, because dynamic optimization often requires a more detailed musculoskeletal model and the entire movement must be simulated for each possible solution. Thus, solving a dynamic optimization problem can easily require an order of magnitude more time than is required to solve a comparable static optimization problem. Although computational costs are difficult to compare in an objective manner, it is not uncommon for static optimization solutions to take seconds or minutes to obtain, whereas dynamic optimization solutions can take hours, days, or even weeks, when run on standard computer workstations.
In many cases, users of dynamic optimization must define the goal of the movement mathematically. This definition is easiest for movements in which the performance criterion to be optimized is clear in a mechanical sense. In vertical jumping, for example, the cost function can be stated as maximization of the vertical displacement of the body center of mass during the flight phase. If the model is constructed with appropriate constraints (realistic muscle properties and joint range of motion), jump heights and body segment motions approximating those of human jumpers can be attained (Pandy and Zajac 1991; van Soest et al. 1993). However, for many human movements the performance criterion is not as clear. In walking, for instance, the goal may seem to be getting from point A to point B in a certain amount of time. Unfortunately, this does not provide enough information to solve for a set of muscle stimulation patterns that will result in realistic walking. Successful simulations of walking have been generated using a cost function that minimizes the metabolic cost of transport (Anderson and Pandy 2001; Umberger 2010); however, the cost function formulation is considerably more complicated than for vertical jumping. Minimum-energy solutions also require the inclusion of an additional model for predicting the metabolic cost of muscle actions (e.g., Umberger et al. 2003). In cases where the performance criterion is not clear, dynamic optimization can serve as an effective approach for testing various theoretical criteria to see how well each can produce the desired movement patterns.
Movements for which the underlying criterion is difficult to define, such as walking or pedaling, have often been simulated by formulating and solving a tracking problem. This approach has a similar appeal to the EMG-based techniques described earlier, in that the resulting motion should closely approximate the way in which humans actually move. However, it is unclear which of several possible experimental measures (kinematics, kinetics, EMG) are the most important for the model to track. Also, because of errors in the experimental data and differences between the musculoskeletal model and the experimental subjects, it may be impossible for the model to track the data perfectly. This approach also limits much of the predictive ability of the dynamic optimization approach, as it is only possible to consider conditions for which experimental data are available.
The tracking approach has seen frequent use in hybrid solution algorithms, which seek to retain the advantages of forward dynamics and dynamic optimization while achieving the computational efficiency of static optimization. Two examples are computed muscle control (Thelen et al. 2003) and direct collocation (Kaplan and Heegaard 2001). Computed muscle control works by solving a static optimization problem at each time step within a single forward dynamics movement simulation. By using feedback on the kinematics and muscle activations at each time step of the numerical integration, computed muscle control can generate a forward dynamics simulation that optimally tracks a set of experimental data without solving a dynamic optimization problem that would require thousands of forward dynamics simulations. In contrast, direct collocation works by converting the differential equations of motion into a set of algebraic constraint equations and treating both the control variables (muscle stimulations) and state variables (positions and velocities) as unknowns in the optimization problem. Although they differ considerably in implementation, both techniques appear to be able to produce results that are similar to dynamic optimization tracking solutions, with a computational cost closer to that of static optimization.
Although static and dynamic optimization have both become popular means for controlling musculoskeletal models, several issues concerning their use must be addressed. Do humans actually produce movements based on one given performance criterion, or does the performance objective change during the movement? If the optimized model results differ from those of a human performer, is it because the model is too simple or not constrained properly, or is the human not performing optimally? Finally, EMG data have illustrated various degrees of simultaneous antagonistic cocontraction during some movement sequences. Optimization models tend not to yield solutions predicting muscle antagonism around a joint, although some antagonism is predicted when models contain muscles that contribute to more than one joint moment. However, optimization models that seek to minimize or maximize specific cost functions will not predict antagonistic cocontraction associated with joint stability or stiffness. In the case of walking, for example, there is no single cost function that could simultaneously predict the relatively minimal muscle coactivation in healthy subjects and the substantial antagonism exhibited by patients with cerebral palsy. Despite these drawbacks, optimization models have increased the understanding of human biomechanics and will continue to do so in the future.
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The state space in the presentation of biomechanical data
Biomechanical data can be represented graphically in a variety of different ways.
The State Space
Biomechanical data can be represented graphically in a variety of different ways. A traditional and very informative type of graph is to display the changes in kinetic or kinematic parameters over time. Figure 13.4 presents an example of the changes in the left and right thigh angular displacements over a time period of 10 s while a subject is walking on a treadmill. Information about peak flexion and extension angles can be obtained from these graphs and quantified. A closer inspection of the graphs makes it quite evident that the coordination between these segments is primarily antiphase. However, especially during the stance phase there are more subtle changes in the patterns of coordination that are harder to obtain from these plots.
In the dynamical systems approach, the reconstruction of the so-called state space is essential in identifying the important features of the behavior of a system, such as its stability and ability to change and adapt to different environmental and task constraints. The state space is a representation of the relevant variables that will help identify these features. To help us understand and quantify coordination between joints or segments, it can be very useful to represent the system in a state space that is based on an angle-angle relative motion plot. Figure 13.5 presents a state space of the relative motion of the time series of the thigh and leg shown earlier in figure 13.4. This angle-angle plot can reveal regions where coordination changes take place as well as parts of the gait cycle where there is relative invariance in coordination patterns. These coordinative changes in the angle-angle plots can be further quantified by vector coding techniques that are discussed later.
Another form of state space is where the position and velocity of a joint or segment are plotted relative to each other. This state-space representation is also often referred to as the phase plane. An example of a position-velocity phase-plane plot is shown in figure 13.6, where the angular displacement of the thigh is plotted against the angular velocity. This is a higher-dimensional state space because the time derivative of position is used to identify the pattern. The phase-plane representation is a first and critical step in the quantification of coordination using continuous relative phase techniques.
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Attractors in State Space
In most forms of human movement, the dynamics in state space are limited to distinct regions, as can be seen in the phase-plane plot in figure 13.6, where the pattern in state space is limited to a fairly narrow, cyclical band. In dynamical systems, preferred regions in state space onto which the dynamics tend to settle are called attractors. Attractors can come in a variety of different forms. A seemingly simple kind of attractor is the point attractor. In this case, the dynamics in the system tend to converge onto one relatively fixed value in the state space. Figure 13.7, c and d, provides an example of a point attractor. The point attractor dynamics are shown in figure 13.7c by a relatively consistent value of discrete relative phase throughout the gait cycle. The discrete relative phase is based on the occurrence of peak flexion in the left and right thigh angular displacements, and the time series in 13.7c show that there is a consistent antiphase or 180° coordination pattern. This antiphase coordination can already be observed by comparing the individual time series in figure 13.4 but is more objectively quantified by the state-space plot in figure 13.7d, where so-called return maps (plotting the coordination at xn vs. xn+1, where n is the cycle number) identify a fixed-point attractor in state space. The relative phase dynamics are of the fixed-point type in this case, as a perturbation (sudden change in treadmill speed and return to the original speed; figure 13.7) results in a return to the original antiphase dynamics with a relative short latency. The perturbation here consisted of a brief (5 s) increase in treadmill speed from 1.2 m/s to 2.0 m/s, with a subsequent return to 1.2 m/s.
Whereas the coordination between the right and left legs can be identified in the form of a point attractor with a relatively fixed value of coordination from cycle to cycle, the dynamics of the individual limb segments show a very different pattern. These are clearly cyclical, as can be seen in the phase planes in figure 13.7, a and b. Attractors of this form are called limit-cycle attractors, and the dynamics converge onto a cyclical region in state space. These limit-cycle attractors are typically identified on the basis of a very narrow, overlapping band of the trajectories in the state space. However, the existence of the narrow band is a necessary but not sufficient condition to characterize the dynamics as a limit-cycle. An essential feature of the limit-cycle attractor is stability with respect to perturbation; to classify as a limit-cycle attractor, the system should also show resistance to perturbations. An example of this is given in figure 13.7, a and b. The regular cyclic pattern in the phase plane (solid line) represents the steady-state gait patterns while a subject was walking at a speed of 1.2 m/s. The dashed trajectories represent the perturbation phase of the trial when speed was suddenly increased to 2.0 m/s and the consequent return back to the original pattern. This return to the preperturbation dynamic is an essential feature of the limit-cycle attractor. The patterns in the phase plane can also serve as an energy plot. The convergence and divergence of trajectories in the phase plane can identify the loss and gain of energy in the system.
Higher-dimensional state spaces (three dimensions and up) can also reveal more complex types of attractors: quasiperiodic and chaotic attractors. The chaotic attractor demonstrates both stable attraction to a region in state space and variability. Figure 13.8 shows an example of the Lorenz attractor, a well-known chaotic attractor that emerges from dynamic interactions in fluid and air flow systems (Strogatz 1994). These dual features of stability and adaptability can be associated with a higher pattern complexity that is now commonly regarded as reflective of healthy and expert systems (see figure 13.3). As an example, increased heart rate variability is considered an important indicator of healthy heart function, reflecting a degree of complexity in organization in which disruptions can be compensated for more easily (Glass 2001; Lipsitz 2002).
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Using EMG to diagnose and treat low back pain
The diagnosis and treatment of low back pain are important issues.
EMG and Low Back Pain
The diagnosis and treatment of low back pain are important issues. In the United States, for example, back pain accounts for two-thirds of all workers compensation costs. Peripheral muscle problems may be one issue; atrophy of the multifidus is frequently observed in patients with low back pain (Hides et al. 1994). EMG is becoming an increasingly useful tool for the diagnosis and treatment of low back pain.
The trunk musculature plays an important role in tasks such as lifting and throwing. During these kinds of tasks as well as in other activities that might threaten the postural control of the individual and perturb balance, the nervous system implements strategies to activate trunk muscles while performing voluntary movements involving remote muscle groups. However, the prevalence of back pain in the general population, and the need to identify ergonomically efficient and safe ways to use back muscles, require that we understand the nature of activation in muscles controlling the trunk. Here, too, EMG is an important tool.
A number of issues render EMG recordings from the trunk musculature difficult to obtain and interpret. From an anatomical viewpoint, the architecture of the back muscles is complex. For example, it is generally accepted that injury or disorder of the multifidus frequently results in back pain. Both multifidus and erector spinae have distinct superficial and deep portions (Bustami 1986; Macintosh et al. 1986) with different histochemical and biomechanical characteristics (Bogduk et al. 1992; Dickx et al. 2010). In some respects, it is fortunate that most of the muscle mass of the multifidus lies in the more superficial portion, so that surface EMG has a greater possibility of characterizing the fibers that produce large amounts of force. However, the multifidus superficial portion has a different role than the deeper fibers. Fibers in the superficial portion cross numerous spinal segments and are thus in a better position to produce back extension. However, the deeper fibers are relatively short, crossing perhaps one or two segments, and thus protect segments of the lumbar spine from inappropriate shear or torsion torques (Macintosh and Valencia 1986).
In addition to issues concerning the activation of the back extensor muscles, a number of important questions require resolution by EMG analysis in which considerable electrocardiographic (ECG) artifact may be present. ECG artifact is particularly problematic during relatively low force contractions, exaggerating the ratio between the signal of interest and the interfering ECG signal. Recording EMG signals from abdominal, knee extensor, and other trunk muscles during normal human movements, for example, can result in the placement of electrodes well within recording range of the electrocardiogram. The relatively large ECG signal can exaggerate the amplitude of EMG activity, and the low-frequency characteristics of the ECG wave can also alter the frequency characteristics of the recorded EMG activity.
For example, the absence of sufficient trunk activity can produce instability resulting in low back pain (van Dieën et al. 2003). However, the amplitude of EMG activity required to ensure stability is small and thus can be affected by the ECG signal (Cholewicki et al. 1997). The best solution is to place the electrodes in a location from which no or minimal ECG artifact can be recorded. However, this is frequently difficult if not impossible. Consequently, numerous algorithms have been suggested to remove the ECG artifact.
One frequently implemented technique to remove this artifact is to compute a template of the ECG signal and subtract it from the electromyogram. This has been proved to be a reasonably successful procedure and has been implemented in studies recording from the diaphragm (Bartolo et al. 1996) as well as rectus abdominis (Hof 2009). Hof (2009) described a technique in which the ECG signal is recorded simultaneously with the EMG signal of interest, and template subtraction is then implemented (figure 8.15).
Digital filtering is another useful alternative for ECG artifact removal. Drake and Callaghan (2006) used a FIR (finite impulse response) filter with a hamming window, although they concluded that the most efficient filtering result could be obtained using a somewhat simpler fourth-order, 30 Hz high-pass cutoff Butterworth filter. Template subtraction improved extraction of the ECG signal but required a large amount of time.
Alternative ECG removal techniques include the use of adaptive filters (Lu et al. 2009; Marque et al. 2005), wavelet-independent component analysis (Taelman et al. 2007), and wavelet-based adaptive filters (Zhan et al. 2010). Irrespective of the technique used for ECG signal artifact removal, it is apparent that the resultant improvement in signal interpretation can be important. Hu and colleagues (2009), for example, found that the use of independent component analysis to remove artifact resulted in an improved ability to discriminate between patients with low back pain and normal subjects during both sitting and standing tasks.
Analysis of the EMG activity in back muscles has produced some interesting results, in part reflecting the unique anatomical issues discussed here. As long ago as 1962, Morris and colleagues noted that “the three muscles of the erector spinae group considered here . . .
do not always show parallel activity, and one may be active while the other two are inactive” (p. 519). This may suggest that the mere demonstration of greater or lesser EMG amplitude may not be indicative of abnormality in a particular muscle. Potentially more important than EMG amplitude is the timing of EMG activity. Deep and superficial portions of the multifidus are differentially active during arm movements (Moseley et al. 2002). One suggestion is that the manner in which the multifidus is differentially controlled in people with back pain compared with those without may be the source of chronic back pain (MacDonald et al. 2009).
Similarly, the analysis of EMG frequency characteristics in patients with back pain compared with control subjects has suggested that there may be differences between muscles and even differences within muscles. Patients with low back pain frequently exhibit alterations in the electromyographic response to fatigue in the back extensor muscles (Biedermann et al. 1991). This is one area in which spectral analysis of the EMG signals has been helpful. As in other muscles, median EMG frequency decreases in the trunk extensors with fatigue (Demoulin et al. 2007). It is interesting to note that Kramer and colleagues (2005) found that the magnitude of median frequency decrease was greater in healthy subjects than in individuals with back pain. Support for importance of electrode placement is obtained from the observations of Sung and colleagues (2009). Normal subjects and patients with back pain underwent an isometric contraction protocol that induced fatigue of the back extensors. EMG frequency measurements made in both the thoracic and the lumbar portions of the erector spinae indicated that the thoracic portion had a significantly lower median frequency than the lumbar portion in patients with low back pain. However, median frequency was lower in the lumbar portion than in the thoracic portion in control subjects. Thus, in the analysis of trunk musculature that might be involved in the development of back pain, EMG studies using wire electrodes frequently may be required to record from different portions of the muscle.
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Using musculoskeletal optimization models to generate control signals
A popular method for generating control signals in musculoskeletal models is to use some form of optimization.
Optimization Models
A popular method for generating control signals in musculoskeletal models is to use some form of optimization. In some cases, when the optimization criterion is derived from motor control principles, this is tantamount to developing a theoretical control model. In other cases, the optimization criterion may result in a coordinated movement pattern yet have little physiological basis. In general, the optimization approach is used in an effort to determine which set of model control signals will produce a result that optimizes (minimizes or maximizes) a given criterion measure. The criterion measure is known as a cost function, objective function, or performance criterion. The cost function can be relatively simple (e.g., find the solution that yields minimal muscle force) or complex (e.g., determine a nonlinear combination of maximal muscle force at minimal metabolic cost). It can be directly related to muscle function (e.g., minimize muscular work) or to some aspect of the motion under study (e.g., maximize vertical jump height). Alternatively, the cost function may be formulated to minimize the differences between model outputs (e.g., joint angles, ground reaction forces) and corresponding experimental measures. This latter case is often referred to as a tracking problem in that the goal is to find a solution that causes the model to follow, or track, the experimental data.
In all cases, the cost function serves as a guiding constraint that determines the selection of one particular set of optimal muscular controls from among many different possible solutions. Two major optimization approaches that have been used to control musculoskeletal models are referred to in the literature as static optimization and dynamic optimization. The meaning of static in this context is that the cost function is evaluated at each time step during a movement, independent of any prior or subsequent time steps. In contrast, dynamic optimization is dynamic in the sense that the entire movement sequence must be simulated to determine the value of the cost function.
Static Optimization
Initial attempts to predict individual muscle forces used static optimization models in conjunction with inverse dynamics analysis of an actual motor performance. The inverse dynamics analysis permits the calculation of net joint moments at incremental times throughout the movement (see chapter 5). In most applications, numerical optimization is used to find the set of muscle forces that balances these joint moments while also satisfying a selected cost function (figure 11.9). Thus, with this approach the muscle forces themselves, rather than neural signals, are the controls. The time-independent nature of static optimization allows solutions to be obtained with relatively little computational cost, but there are some drawbacks. One issue in early applications was sudden, nonphysiological switching on and off of muscle forces caused by the independence of solutions for sequential time increments (i.e., the optimization model balanced the joint moments separately for each time interval using very different sets of muscles). This problem can be avoided by careful selection of the initial guess in the optimization problem, such as by using the solution from one time step as the initial guess for the next time step. A stronger approach to address this issue is to use muscle models that invoke physiological realism through force-length and force-velocity relations and time-dependent stimulation-activation dynamics (chapter 9). When more detailed muscle models are included in a static optimization model, muscle activations become the control variables, rather than muscle forces.
Early studies using static optimization were plagued also by two other kinds of nonphysiological results. The first was the prediction of model forces that were too high for actual muscles to produce. This difficulty was easily addressed by defining physiologically valid maximal force constraints for each muscle in the musculoskeletal model. The second problem was that solutions would often select only one muscle to balance the net joint moment rather than choose a more realistic muscular synergy. Depending on the exact formulation, the muscle with the largest moment arm (if minimizing muscle force) or a favorable combination of moment arm and muscle strength (if minimizing muscle stress) would be selected to fully balance the joint moment, with zero force predicted in other synergist muscles. Mathematically, synergism can be produced by using nonlinear cost functions (e.g., minimizing the sum of squared or cubed muscle forces), although the physiological rationale for specific nonlinear cost functions is not always clear. A widely used nonlinear cost function proposed by Crowninshield and Brand (1981a) involves minimizing the sum of cubed muscle stresses. It was originally argued that this particular cost function would lead to solutions that maximize muscle endurance, making it appropriate for predicting muscle forces in submaximal tasks such as walking. However, the Crowninshield and Brand criterion has since been used to solve for muscle forces in a range of activities, some of which are unlikely candidates for maximizing muscle endurance (e.g., jumping, landing).
An additional issue with static optimization models is that they are essentially a decomposition of experimental joint moments into individual muscle forces. Therefore, the predicted muscle forces will be subject to any errors in the experimental joint moments in addition to any shortcomings associated with the approximation made in creating the musculoskeletal model. To complete this section, we present a static optimization example motivated by a classic review article by Crowninshield and Brand (1981b), which demonstrates the process of distributing an empirically determined elbow joint moment across a set of muscles spanning the elbow joint.
The net joint moment to be balanced in example 11.1 was a flexor moment, and only muscles with flexor moment arms were included in the model (figure 11.10). If an elbow extensor muscle had been included, there is no cost function of the type presented here that would predict force in an antagonist muscle. Any force in an elbow extensor would itself contribute to a higher cost function value and would also require greater elbow flexor forces to balance the target 10 N·m joint moment, further increasing the cost function value. The total lack of coactivation in this example is contrary to the common observation that during heavy activation of the elbow flexors, there is some activity in the triceps muscle group. The extensor coactivation likely helps stabilize the joint during heavy exertion but will not be predicted using traditional static optimization techniques in simple one DOF models. Although the example presented here focuses on obtaining numerical results for muscle forces, the interested reader is referred to the review by Crowninshield and Brand (1981b), which presents an interesting graphical interpretation of the solution to this static optimization problem.
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Dynamic Optimization
Ongoing work with static optimization models has led to the development of dynamic optimization or optimal control models, which are applied in conjunction with forward dynamics models of human motion. In contrast to the inverse dynamics approach, which uses experimental data from an actual performance to calculate net joint moments, forward dynamics analyses simulate the motion of the body from a given set of joint moments or muscle forces (see chapter 10). Dynamic optimization models are therefore often designed to find the muscle stimulation patterns that result in an optimal motion (figure 11.9). As mentioned earlier, the optimal motion may be one that maximizes (or minimizes) a performance criterion, such as maximizing jump height; or, the optimal motion may be defined as one that best reproduces a set of experimental data. Regardless, the variables that are optimized are usually the muscle stimulation patterns that control the motion of the musculoskeletal model. These stimulation patterns are used as the inputs to muscle models that predict individual muscle forces, which are then multiplied by the appropriate moment arms to compute the active muscle moments. The active muscle moments are combined with passive moments to compute the net joint moments, which actuate the skeletal model and produce movement. Thus, dynamic optimization leads to the synthesis of body segment kinematics associated with optimal performance, which is fundamentally different from the static optimization approach.
The optimal kinematics predicted from a dynamic optimization can be compared with experimental motion data; however, the experimental data are not required to obtain the solution, as they are with static optimization. This feature of dynamic optimization permits one to address questions for which no experimental data exist, such as testing the feasibility of possible forms of locomotion in extinct species (e.g., Nagano et al. 2005). Moreover, because dynamic optimization uses forward dynamics models that simulate the whole movement performance and provide complete muscle force time histories (rather than solutions at independent time increments), many of the problems associated with static optimization models are overcome. However, these advantages come at a computational cost, because dynamic optimization often requires a more detailed musculoskeletal model and the entire movement must be simulated for each possible solution. Thus, solving a dynamic optimization problem can easily require an order of magnitude more time than is required to solve a comparable static optimization problem. Although computational costs are difficult to compare in an objective manner, it is not uncommon for static optimization solutions to take seconds or minutes to obtain, whereas dynamic optimization solutions can take hours, days, or even weeks, when run on standard computer workstations.
In many cases, users of dynamic optimization must define the goal of the movement mathematically. This definition is easiest for movements in which the performance criterion to be optimized is clear in a mechanical sense. In vertical jumping, for example, the cost function can be stated as maximization of the vertical displacement of the body center of mass during the flight phase. If the model is constructed with appropriate constraints (realistic muscle properties and joint range of motion), jump heights and body segment motions approximating those of human jumpers can be attained (Pandy and Zajac 1991; van Soest et al. 1993). However, for many human movements the performance criterion is not as clear. In walking, for instance, the goal may seem to be getting from point A to point B in a certain amount of time. Unfortunately, this does not provide enough information to solve for a set of muscle stimulation patterns that will result in realistic walking. Successful simulations of walking have been generated using a cost function that minimizes the metabolic cost of transport (Anderson and Pandy 2001; Umberger 2010); however, the cost function formulation is considerably more complicated than for vertical jumping. Minimum-energy solutions also require the inclusion of an additional model for predicting the metabolic cost of muscle actions (e.g., Umberger et al. 2003). In cases where the performance criterion is not clear, dynamic optimization can serve as an effective approach for testing various theoretical criteria to see how well each can produce the desired movement patterns.
Movements for which the underlying criterion is difficult to define, such as walking or pedaling, have often been simulated by formulating and solving a tracking problem. This approach has a similar appeal to the EMG-based techniques described earlier, in that the resulting motion should closely approximate the way in which humans actually move. However, it is unclear which of several possible experimental measures (kinematics, kinetics, EMG) are the most important for the model to track. Also, because of errors in the experimental data and differences between the musculoskeletal model and the experimental subjects, it may be impossible for the model to track the data perfectly. This approach also limits much of the predictive ability of the dynamic optimization approach, as it is only possible to consider conditions for which experimental data are available.
The tracking approach has seen frequent use in hybrid solution algorithms, which seek to retain the advantages of forward dynamics and dynamic optimization while achieving the computational efficiency of static optimization. Two examples are computed muscle control (Thelen et al. 2003) and direct collocation (Kaplan and Heegaard 2001). Computed muscle control works by solving a static optimization problem at each time step within a single forward dynamics movement simulation. By using feedback on the kinematics and muscle activations at each time step of the numerical integration, computed muscle control can generate a forward dynamics simulation that optimally tracks a set of experimental data without solving a dynamic optimization problem that would require thousands of forward dynamics simulations. In contrast, direct collocation works by converting the differential equations of motion into a set of algebraic constraint equations and treating both the control variables (muscle stimulations) and state variables (positions and velocities) as unknowns in the optimization problem. Although they differ considerably in implementation, both techniques appear to be able to produce results that are similar to dynamic optimization tracking solutions, with a computational cost closer to that of static optimization.
Although static and dynamic optimization have both become popular means for controlling musculoskeletal models, several issues concerning their use must be addressed. Do humans actually produce movements based on one given performance criterion, or does the performance objective change during the movement? If the optimized model results differ from those of a human performer, is it because the model is too simple or not constrained properly, or is the human not performing optimally? Finally, EMG data have illustrated various degrees of simultaneous antagonistic cocontraction during some movement sequences. Optimization models tend not to yield solutions predicting muscle antagonism around a joint, although some antagonism is predicted when models contain muscles that contribute to more than one joint moment. However, optimization models that seek to minimize or maximize specific cost functions will not predict antagonistic cocontraction associated with joint stability or stiffness. In the case of walking, for example, there is no single cost function that could simultaneously predict the relatively minimal muscle coactivation in healthy subjects and the substantial antagonism exhibited by patients with cerebral palsy. Despite these drawbacks, optimization models have increased the understanding of human biomechanics and will continue to do so in the future.
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The state space in the presentation of biomechanical data
Biomechanical data can be represented graphically in a variety of different ways.
The State Space
Biomechanical data can be represented graphically in a variety of different ways. A traditional and very informative type of graph is to display the changes in kinetic or kinematic parameters over time. Figure 13.4 presents an example of the changes in the left and right thigh angular displacements over a time period of 10 s while a subject is walking on a treadmill. Information about peak flexion and extension angles can be obtained from these graphs and quantified. A closer inspection of the graphs makes it quite evident that the coordination between these segments is primarily antiphase. However, especially during the stance phase there are more subtle changes in the patterns of coordination that are harder to obtain from these plots.
In the dynamical systems approach, the reconstruction of the so-called state space is essential in identifying the important features of the behavior of a system, such as its stability and ability to change and adapt to different environmental and task constraints. The state space is a representation of the relevant variables that will help identify these features. To help us understand and quantify coordination between joints or segments, it can be very useful to represent the system in a state space that is based on an angle-angle relative motion plot. Figure 13.5 presents a state space of the relative motion of the time series of the thigh and leg shown earlier in figure 13.4. This angle-angle plot can reveal regions where coordination changes take place as well as parts of the gait cycle where there is relative invariance in coordination patterns. These coordinative changes in the angle-angle plots can be further quantified by vector coding techniques that are discussed later.
Another form of state space is where the position and velocity of a joint or segment are plotted relative to each other. This state-space representation is also often referred to as the phase plane. An example of a position-velocity phase-plane plot is shown in figure 13.6, where the angular displacement of the thigh is plotted against the angular velocity. This is a higher-dimensional state space because the time derivative of position is used to identify the pattern. The phase-plane representation is a first and critical step in the quantification of coordination using continuous relative phase techniques.
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Attractors in State Space
In most forms of human movement, the dynamics in state space are limited to distinct regions, as can be seen in the phase-plane plot in figure 13.6, where the pattern in state space is limited to a fairly narrow, cyclical band. In dynamical systems, preferred regions in state space onto which the dynamics tend to settle are called attractors. Attractors can come in a variety of different forms. A seemingly simple kind of attractor is the point attractor. In this case, the dynamics in the system tend to converge onto one relatively fixed value in the state space. Figure 13.7, c and d, provides an example of a point attractor. The point attractor dynamics are shown in figure 13.7c by a relatively consistent value of discrete relative phase throughout the gait cycle. The discrete relative phase is based on the occurrence of peak flexion in the left and right thigh angular displacements, and the time series in 13.7c show that there is a consistent antiphase or 180° coordination pattern. This antiphase coordination can already be observed by comparing the individual time series in figure 13.4 but is more objectively quantified by the state-space plot in figure 13.7d, where so-called return maps (plotting the coordination at xn vs. xn+1, where n is the cycle number) identify a fixed-point attractor in state space. The relative phase dynamics are of the fixed-point type in this case, as a perturbation (sudden change in treadmill speed and return to the original speed; figure 13.7) results in a return to the original antiphase dynamics with a relative short latency. The perturbation here consisted of a brief (5 s) increase in treadmill speed from 1.2 m/s to 2.0 m/s, with a subsequent return to 1.2 m/s.
Whereas the coordination between the right and left legs can be identified in the form of a point attractor with a relatively fixed value of coordination from cycle to cycle, the dynamics of the individual limb segments show a very different pattern. These are clearly cyclical, as can be seen in the phase planes in figure 13.7, a and b. Attractors of this form are called limit-cycle attractors, and the dynamics converge onto a cyclical region in state space. These limit-cycle attractors are typically identified on the basis of a very narrow, overlapping band of the trajectories in the state space. However, the existence of the narrow band is a necessary but not sufficient condition to characterize the dynamics as a limit-cycle. An essential feature of the limit-cycle attractor is stability with respect to perturbation; to classify as a limit-cycle attractor, the system should also show resistance to perturbations. An example of this is given in figure 13.7, a and b. The regular cyclic pattern in the phase plane (solid line) represents the steady-state gait patterns while a subject was walking at a speed of 1.2 m/s. The dashed trajectories represent the perturbation phase of the trial when speed was suddenly increased to 2.0 m/s and the consequent return back to the original pattern. This return to the preperturbation dynamic is an essential feature of the limit-cycle attractor. The patterns in the phase plane can also serve as an energy plot. The convergence and divergence of trajectories in the phase plane can identify the loss and gain of energy in the system.
Higher-dimensional state spaces (three dimensions and up) can also reveal more complex types of attractors: quasiperiodic and chaotic attractors. The chaotic attractor demonstrates both stable attraction to a region in state space and variability. Figure 13.8 shows an example of the Lorenz attractor, a well-known chaotic attractor that emerges from dynamic interactions in fluid and air flow systems (Strogatz 1994). These dual features of stability and adaptability can be associated with a higher pattern complexity that is now commonly regarded as reflective of healthy and expert systems (see figure 13.3). As an example, increased heart rate variability is considered an important indicator of healthy heart function, reflecting a degree of complexity in organization in which disruptions can be compensated for more easily (Glass 2001; Lipsitz 2002).
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Using EMG to diagnose and treat low back pain
The diagnosis and treatment of low back pain are important issues.
EMG and Low Back Pain
The diagnosis and treatment of low back pain are important issues. In the United States, for example, back pain accounts for two-thirds of all workers compensation costs. Peripheral muscle problems may be one issue; atrophy of the multifidus is frequently observed in patients with low back pain (Hides et al. 1994). EMG is becoming an increasingly useful tool for the diagnosis and treatment of low back pain.
The trunk musculature plays an important role in tasks such as lifting and throwing. During these kinds of tasks as well as in other activities that might threaten the postural control of the individual and perturb balance, the nervous system implements strategies to activate trunk muscles while performing voluntary movements involving remote muscle groups. However, the prevalence of back pain in the general population, and the need to identify ergonomically efficient and safe ways to use back muscles, require that we understand the nature of activation in muscles controlling the trunk. Here, too, EMG is an important tool.
A number of issues render EMG recordings from the trunk musculature difficult to obtain and interpret. From an anatomical viewpoint, the architecture of the back muscles is complex. For example, it is generally accepted that injury or disorder of the multifidus frequently results in back pain. Both multifidus and erector spinae have distinct superficial and deep portions (Bustami 1986; Macintosh et al. 1986) with different histochemical and biomechanical characteristics (Bogduk et al. 1992; Dickx et al. 2010). In some respects, it is fortunate that most of the muscle mass of the multifidus lies in the more superficial portion, so that surface EMG has a greater possibility of characterizing the fibers that produce large amounts of force. However, the multifidus superficial portion has a different role than the deeper fibers. Fibers in the superficial portion cross numerous spinal segments and are thus in a better position to produce back extension. However, the deeper fibers are relatively short, crossing perhaps one or two segments, and thus protect segments of the lumbar spine from inappropriate shear or torsion torques (Macintosh and Valencia 1986).
In addition to issues concerning the activation of the back extensor muscles, a number of important questions require resolution by EMG analysis in which considerable electrocardiographic (ECG) artifact may be present. ECG artifact is particularly problematic during relatively low force contractions, exaggerating the ratio between the signal of interest and the interfering ECG signal. Recording EMG signals from abdominal, knee extensor, and other trunk muscles during normal human movements, for example, can result in the placement of electrodes well within recording range of the electrocardiogram. The relatively large ECG signal can exaggerate the amplitude of EMG activity, and the low-frequency characteristics of the ECG wave can also alter the frequency characteristics of the recorded EMG activity.
For example, the absence of sufficient trunk activity can produce instability resulting in low back pain (van Dieën et al. 2003). However, the amplitude of EMG activity required to ensure stability is small and thus can be affected by the ECG signal (Cholewicki et al. 1997). The best solution is to place the electrodes in a location from which no or minimal ECG artifact can be recorded. However, this is frequently difficult if not impossible. Consequently, numerous algorithms have been suggested to remove the ECG artifact.
One frequently implemented technique to remove this artifact is to compute a template of the ECG signal and subtract it from the electromyogram. This has been proved to be a reasonably successful procedure and has been implemented in studies recording from the diaphragm (Bartolo et al. 1996) as well as rectus abdominis (Hof 2009). Hof (2009) described a technique in which the ECG signal is recorded simultaneously with the EMG signal of interest, and template subtraction is then implemented (figure 8.15).
Digital filtering is another useful alternative for ECG artifact removal. Drake and Callaghan (2006) used a FIR (finite impulse response) filter with a hamming window, although they concluded that the most efficient filtering result could be obtained using a somewhat simpler fourth-order, 30 Hz high-pass cutoff Butterworth filter. Template subtraction improved extraction of the ECG signal but required a large amount of time.
Alternative ECG removal techniques include the use of adaptive filters (Lu et al. 2009; Marque et al. 2005), wavelet-independent component analysis (Taelman et al. 2007), and wavelet-based adaptive filters (Zhan et al. 2010). Irrespective of the technique used for ECG signal artifact removal, it is apparent that the resultant improvement in signal interpretation can be important. Hu and colleagues (2009), for example, found that the use of independent component analysis to remove artifact resulted in an improved ability to discriminate between patients with low back pain and normal subjects during both sitting and standing tasks.
Analysis of the EMG activity in back muscles has produced some interesting results, in part reflecting the unique anatomical issues discussed here. As long ago as 1962, Morris and colleagues noted that “the three muscles of the erector spinae group considered here . . .
do not always show parallel activity, and one may be active while the other two are inactive” (p. 519). This may suggest that the mere demonstration of greater or lesser EMG amplitude may not be indicative of abnormality in a particular muscle. Potentially more important than EMG amplitude is the timing of EMG activity. Deep and superficial portions of the multifidus are differentially active during arm movements (Moseley et al. 2002). One suggestion is that the manner in which the multifidus is differentially controlled in people with back pain compared with those without may be the source of chronic back pain (MacDonald et al. 2009).
Similarly, the analysis of EMG frequency characteristics in patients with back pain compared with control subjects has suggested that there may be differences between muscles and even differences within muscles. Patients with low back pain frequently exhibit alterations in the electromyographic response to fatigue in the back extensor muscles (Biedermann et al. 1991). This is one area in which spectral analysis of the EMG signals has been helpful. As in other muscles, median EMG frequency decreases in the trunk extensors with fatigue (Demoulin et al. 2007). It is interesting to note that Kramer and colleagues (2005) found that the magnitude of median frequency decrease was greater in healthy subjects than in individuals with back pain. Support for importance of electrode placement is obtained from the observations of Sung and colleagues (2009). Normal subjects and patients with back pain underwent an isometric contraction protocol that induced fatigue of the back extensors. EMG frequency measurements made in both the thoracic and the lumbar portions of the erector spinae indicated that the thoracic portion had a significantly lower median frequency than the lumbar portion in patients with low back pain. However, median frequency was lower in the lumbar portion than in the thoracic portion in control subjects. Thus, in the analysis of trunk musculature that might be involved in the development of back pain, EMG studies using wire electrodes frequently may be required to record from different portions of the muscle.
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Using musculoskeletal optimization models to generate control signals
A popular method for generating control signals in musculoskeletal models is to use some form of optimization.
Optimization Models
A popular method for generating control signals in musculoskeletal models is to use some form of optimization. In some cases, when the optimization criterion is derived from motor control principles, this is tantamount to developing a theoretical control model. In other cases, the optimization criterion may result in a coordinated movement pattern yet have little physiological basis. In general, the optimization approach is used in an effort to determine which set of model control signals will produce a result that optimizes (minimizes or maximizes) a given criterion measure. The criterion measure is known as a cost function, objective function, or performance criterion. The cost function can be relatively simple (e.g., find the solution that yields minimal muscle force) or complex (e.g., determine a nonlinear combination of maximal muscle force at minimal metabolic cost). It can be directly related to muscle function (e.g., minimize muscular work) or to some aspect of the motion under study (e.g., maximize vertical jump height). Alternatively, the cost function may be formulated to minimize the differences between model outputs (e.g., joint angles, ground reaction forces) and corresponding experimental measures. This latter case is often referred to as a tracking problem in that the goal is to find a solution that causes the model to follow, or track, the experimental data.
In all cases, the cost function serves as a guiding constraint that determines the selection of one particular set of optimal muscular controls from among many different possible solutions. Two major optimization approaches that have been used to control musculoskeletal models are referred to in the literature as static optimization and dynamic optimization. The meaning of static in this context is that the cost function is evaluated at each time step during a movement, independent of any prior or subsequent time steps. In contrast, dynamic optimization is dynamic in the sense that the entire movement sequence must be simulated to determine the value of the cost function.
Static Optimization
Initial attempts to predict individual muscle forces used static optimization models in conjunction with inverse dynamics analysis of an actual motor performance. The inverse dynamics analysis permits the calculation of net joint moments at incremental times throughout the movement (see chapter 5). In most applications, numerical optimization is used to find the set of muscle forces that balances these joint moments while also satisfying a selected cost function (figure 11.9). Thus, with this approach the muscle forces themselves, rather than neural signals, are the controls. The time-independent nature of static optimization allows solutions to be obtained with relatively little computational cost, but there are some drawbacks. One issue in early applications was sudden, nonphysiological switching on and off of muscle forces caused by the independence of solutions for sequential time increments (i.e., the optimization model balanced the joint moments separately for each time interval using very different sets of muscles). This problem can be avoided by careful selection of the initial guess in the optimization problem, such as by using the solution from one time step as the initial guess for the next time step. A stronger approach to address this issue is to use muscle models that invoke physiological realism through force-length and force-velocity relations and time-dependent stimulation-activation dynamics (chapter 9). When more detailed muscle models are included in a static optimization model, muscle activations become the control variables, rather than muscle forces.
Early studies using static optimization were plagued also by two other kinds of nonphysiological results. The first was the prediction of model forces that were too high for actual muscles to produce. This difficulty was easily addressed by defining physiologically valid maximal force constraints for each muscle in the musculoskeletal model. The second problem was that solutions would often select only one muscle to balance the net joint moment rather than choose a more realistic muscular synergy. Depending on the exact formulation, the muscle with the largest moment arm (if minimizing muscle force) or a favorable combination of moment arm and muscle strength (if minimizing muscle stress) would be selected to fully balance the joint moment, with zero force predicted in other synergist muscles. Mathematically, synergism can be produced by using nonlinear cost functions (e.g., minimizing the sum of squared or cubed muscle forces), although the physiological rationale for specific nonlinear cost functions is not always clear. A widely used nonlinear cost function proposed by Crowninshield and Brand (1981a) involves minimizing the sum of cubed muscle stresses. It was originally argued that this particular cost function would lead to solutions that maximize muscle endurance, making it appropriate for predicting muscle forces in submaximal tasks such as walking. However, the Crowninshield and Brand criterion has since been used to solve for muscle forces in a range of activities, some of which are unlikely candidates for maximizing muscle endurance (e.g., jumping, landing).
An additional issue with static optimization models is that they are essentially a decomposition of experimental joint moments into individual muscle forces. Therefore, the predicted muscle forces will be subject to any errors in the experimental joint moments in addition to any shortcomings associated with the approximation made in creating the musculoskeletal model. To complete this section, we present a static optimization example motivated by a classic review article by Crowninshield and Brand (1981b), which demonstrates the process of distributing an empirically determined elbow joint moment across a set of muscles spanning the elbow joint.
The net joint moment to be balanced in example 11.1 was a flexor moment, and only muscles with flexor moment arms were included in the model (figure 11.10). If an elbow extensor muscle had been included, there is no cost function of the type presented here that would predict force in an antagonist muscle. Any force in an elbow extensor would itself contribute to a higher cost function value and would also require greater elbow flexor forces to balance the target 10 N·m joint moment, further increasing the cost function value. The total lack of coactivation in this example is contrary to the common observation that during heavy activation of the elbow flexors, there is some activity in the triceps muscle group. The extensor coactivation likely helps stabilize the joint during heavy exertion but will not be predicted using traditional static optimization techniques in simple one DOF models. Although the example presented here focuses on obtaining numerical results for muscle forces, the interested reader is referred to the review by Crowninshield and Brand (1981b), which presents an interesting graphical interpretation of the solution to this static optimization problem.
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Dynamic Optimization
Ongoing work with static optimization models has led to the development of dynamic optimization or optimal control models, which are applied in conjunction with forward dynamics models of human motion. In contrast to the inverse dynamics approach, which uses experimental data from an actual performance to calculate net joint moments, forward dynamics analyses simulate the motion of the body from a given set of joint moments or muscle forces (see chapter 10). Dynamic optimization models are therefore often designed to find the muscle stimulation patterns that result in an optimal motion (figure 11.9). As mentioned earlier, the optimal motion may be one that maximizes (or minimizes) a performance criterion, such as maximizing jump height; or, the optimal motion may be defined as one that best reproduces a set of experimental data. Regardless, the variables that are optimized are usually the muscle stimulation patterns that control the motion of the musculoskeletal model. These stimulation patterns are used as the inputs to muscle models that predict individual muscle forces, which are then multiplied by the appropriate moment arms to compute the active muscle moments. The active muscle moments are combined with passive moments to compute the net joint moments, which actuate the skeletal model and produce movement. Thus, dynamic optimization leads to the synthesis of body segment kinematics associated with optimal performance, which is fundamentally different from the static optimization approach.
The optimal kinematics predicted from a dynamic optimization can be compared with experimental motion data; however, the experimental data are not required to obtain the solution, as they are with static optimization. This feature of dynamic optimization permits one to address questions for which no experimental data exist, such as testing the feasibility of possible forms of locomotion in extinct species (e.g., Nagano et al. 2005). Moreover, because dynamic optimization uses forward dynamics models that simulate the whole movement performance and provide complete muscle force time histories (rather than solutions at independent time increments), many of the problems associated with static optimization models are overcome. However, these advantages come at a computational cost, because dynamic optimization often requires a more detailed musculoskeletal model and the entire movement must be simulated for each possible solution. Thus, solving a dynamic optimization problem can easily require an order of magnitude more time than is required to solve a comparable static optimization problem. Although computational costs are difficult to compare in an objective manner, it is not uncommon for static optimization solutions to take seconds or minutes to obtain, whereas dynamic optimization solutions can take hours, days, or even weeks, when run on standard computer workstations.
In many cases, users of dynamic optimization must define the goal of the movement mathematically. This definition is easiest for movements in which the performance criterion to be optimized is clear in a mechanical sense. In vertical jumping, for example, the cost function can be stated as maximization of the vertical displacement of the body center of mass during the flight phase. If the model is constructed with appropriate constraints (realistic muscle properties and joint range of motion), jump heights and body segment motions approximating those of human jumpers can be attained (Pandy and Zajac 1991; van Soest et al. 1993). However, for many human movements the performance criterion is not as clear. In walking, for instance, the goal may seem to be getting from point A to point B in a certain amount of time. Unfortunately, this does not provide enough information to solve for a set of muscle stimulation patterns that will result in realistic walking. Successful simulations of walking have been generated using a cost function that minimizes the metabolic cost of transport (Anderson and Pandy 2001; Umberger 2010); however, the cost function formulation is considerably more complicated than for vertical jumping. Minimum-energy solutions also require the inclusion of an additional model for predicting the metabolic cost of muscle actions (e.g., Umberger et al. 2003). In cases where the performance criterion is not clear, dynamic optimization can serve as an effective approach for testing various theoretical criteria to see how well each can produce the desired movement patterns.
Movements for which the underlying criterion is difficult to define, such as walking or pedaling, have often been simulated by formulating and solving a tracking problem. This approach has a similar appeal to the EMG-based techniques described earlier, in that the resulting motion should closely approximate the way in which humans actually move. However, it is unclear which of several possible experimental measures (kinematics, kinetics, EMG) are the most important for the model to track. Also, because of errors in the experimental data and differences between the musculoskeletal model and the experimental subjects, it may be impossible for the model to track the data perfectly. This approach also limits much of the predictive ability of the dynamic optimization approach, as it is only possible to consider conditions for which experimental data are available.
The tracking approach has seen frequent use in hybrid solution algorithms, which seek to retain the advantages of forward dynamics and dynamic optimization while achieving the computational efficiency of static optimization. Two examples are computed muscle control (Thelen et al. 2003) and direct collocation (Kaplan and Heegaard 2001). Computed muscle control works by solving a static optimization problem at each time step within a single forward dynamics movement simulation. By using feedback on the kinematics and muscle activations at each time step of the numerical integration, computed muscle control can generate a forward dynamics simulation that optimally tracks a set of experimental data without solving a dynamic optimization problem that would require thousands of forward dynamics simulations. In contrast, direct collocation works by converting the differential equations of motion into a set of algebraic constraint equations and treating both the control variables (muscle stimulations) and state variables (positions and velocities) as unknowns in the optimization problem. Although they differ considerably in implementation, both techniques appear to be able to produce results that are similar to dynamic optimization tracking solutions, with a computational cost closer to that of static optimization.
Although static and dynamic optimization have both become popular means for controlling musculoskeletal models, several issues concerning their use must be addressed. Do humans actually produce movements based on one given performance criterion, or does the performance objective change during the movement? If the optimized model results differ from those of a human performer, is it because the model is too simple or not constrained properly, or is the human not performing optimally? Finally, EMG data have illustrated various degrees of simultaneous antagonistic cocontraction during some movement sequences. Optimization models tend not to yield solutions predicting muscle antagonism around a joint, although some antagonism is predicted when models contain muscles that contribute to more than one joint moment. However, optimization models that seek to minimize or maximize specific cost functions will not predict antagonistic cocontraction associated with joint stability or stiffness. In the case of walking, for example, there is no single cost function that could simultaneously predict the relatively minimal muscle coactivation in healthy subjects and the substantial antagonism exhibited by patients with cerebral palsy. Despite these drawbacks, optimization models have increased the understanding of human biomechanics and will continue to do so in the future.
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The state space in the presentation of biomechanical data
Biomechanical data can be represented graphically in a variety of different ways.
The State Space
Biomechanical data can be represented graphically in a variety of different ways. A traditional and very informative type of graph is to display the changes in kinetic or kinematic parameters over time. Figure 13.4 presents an example of the changes in the left and right thigh angular displacements over a time period of 10 s while a subject is walking on a treadmill. Information about peak flexion and extension angles can be obtained from these graphs and quantified. A closer inspection of the graphs makes it quite evident that the coordination between these segments is primarily antiphase. However, especially during the stance phase there are more subtle changes in the patterns of coordination that are harder to obtain from these plots.
In the dynamical systems approach, the reconstruction of the so-called state space is essential in identifying the important features of the behavior of a system, such as its stability and ability to change and adapt to different environmental and task constraints. The state space is a representation of the relevant variables that will help identify these features. To help us understand and quantify coordination between joints or segments, it can be very useful to represent the system in a state space that is based on an angle-angle relative motion plot. Figure 13.5 presents a state space of the relative motion of the time series of the thigh and leg shown earlier in figure 13.4. This angle-angle plot can reveal regions where coordination changes take place as well as parts of the gait cycle where there is relative invariance in coordination patterns. These coordinative changes in the angle-angle plots can be further quantified by vector coding techniques that are discussed later.
Another form of state space is where the position and velocity of a joint or segment are plotted relative to each other. This state-space representation is also often referred to as the phase plane. An example of a position-velocity phase-plane plot is shown in figure 13.6, where the angular displacement of the thigh is plotted against the angular velocity. This is a higher-dimensional state space because the time derivative of position is used to identify the pattern. The phase-plane representation is a first and critical step in the quantification of coordination using continuous relative phase techniques.
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Attractors in State Space
In most forms of human movement, the dynamics in state space are limited to distinct regions, as can be seen in the phase-plane plot in figure 13.6, where the pattern in state space is limited to a fairly narrow, cyclical band. In dynamical systems, preferred regions in state space onto which the dynamics tend to settle are called attractors. Attractors can come in a variety of different forms. A seemingly simple kind of attractor is the point attractor. In this case, the dynamics in the system tend to converge onto one relatively fixed value in the state space. Figure 13.7, c and d, provides an example of a point attractor. The point attractor dynamics are shown in figure 13.7c by a relatively consistent value of discrete relative phase throughout the gait cycle. The discrete relative phase is based on the occurrence of peak flexion in the left and right thigh angular displacements, and the time series in 13.7c show that there is a consistent antiphase or 180° coordination pattern. This antiphase coordination can already be observed by comparing the individual time series in figure 13.4 but is more objectively quantified by the state-space plot in figure 13.7d, where so-called return maps (plotting the coordination at xn vs. xn+1, where n is the cycle number) identify a fixed-point attractor in state space. The relative phase dynamics are of the fixed-point type in this case, as a perturbation (sudden change in treadmill speed and return to the original speed; figure 13.7) results in a return to the original antiphase dynamics with a relative short latency. The perturbation here consisted of a brief (5 s) increase in treadmill speed from 1.2 m/s to 2.0 m/s, with a subsequent return to 1.2 m/s.
Whereas the coordination between the right and left legs can be identified in the form of a point attractor with a relatively fixed value of coordination from cycle to cycle, the dynamics of the individual limb segments show a very different pattern. These are clearly cyclical, as can be seen in the phase planes in figure 13.7, a and b. Attractors of this form are called limit-cycle attractors, and the dynamics converge onto a cyclical region in state space. These limit-cycle attractors are typically identified on the basis of a very narrow, overlapping band of the trajectories in the state space. However, the existence of the narrow band is a necessary but not sufficient condition to characterize the dynamics as a limit-cycle. An essential feature of the limit-cycle attractor is stability with respect to perturbation; to classify as a limit-cycle attractor, the system should also show resistance to perturbations. An example of this is given in figure 13.7, a and b. The regular cyclic pattern in the phase plane (solid line) represents the steady-state gait patterns while a subject was walking at a speed of 1.2 m/s. The dashed trajectories represent the perturbation phase of the trial when speed was suddenly increased to 2.0 m/s and the consequent return back to the original pattern. This return to the preperturbation dynamic is an essential feature of the limit-cycle attractor. The patterns in the phase plane can also serve as an energy plot. The convergence and divergence of trajectories in the phase plane can identify the loss and gain of energy in the system.
Higher-dimensional state spaces (three dimensions and up) can also reveal more complex types of attractors: quasiperiodic and chaotic attractors. The chaotic attractor demonstrates both stable attraction to a region in state space and variability. Figure 13.8 shows an example of the Lorenz attractor, a well-known chaotic attractor that emerges from dynamic interactions in fluid and air flow systems (Strogatz 1994). These dual features of stability and adaptability can be associated with a higher pattern complexity that is now commonly regarded as reflective of healthy and expert systems (see figure 13.3). As an example, increased heart rate variability is considered an important indicator of healthy heart function, reflecting a degree of complexity in organization in which disruptions can be compensated for more easily (Glass 2001; Lipsitz 2002).
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