A Dynamic Optimization Solution for Vertical Jumping in Three Dimensions

A computational framework to predict post-treatment outcome for gait-related disorders

Clinicians often use intuitive models based on clinical experience or regression models based on population studies to plan treatment of gait-related disorders. Because such models are constructed using data collected from previous patients, the predicted clinical outcome for a particular patient may not be reliable. We propose a new approach that uses computational models based on engineering mechanics to predict post-treatment outcome from pre-treatment movement data. The approach utilizes a four-phase optimization process built around a dynamic, patient-specific gait model. The first three phases calibrate the model's joint, inertial, and control parameters, respectively, where the control parameters are weights in an optimization cost function that tracks the patient's pre-treatment gait motion and loads. The last phase predicts the patient's post-treatment gait pattern by performing a tracking optimization with the calibrated model modified to simulate the selected treatment. We demonstrate the approach by simulating how two treatments for knee osteoarthritis (OA)--gait modification and high tibial osteotomy (HTO) surgery--alter the external knee adduction torque for a specific patient. By performing multiple tracking optimizations, we calibrated the model's parameter values to reproduce the patient's knee adduction torque curve for a toe out gait motion. When we performed a tracking optimization with the calibrated model using a modified footpath to simulate an increased stance width, the predicted reduction in both adduction torque peaks matched experimental results to within 4.8% error. When we performed a tracking optimization with the same model using modified leg geometry to simulate HTO surgery, the predicted reductions were consistent with published data. The approach requires further evaluation with a larger number of patients to determine its effectiveness for planning the treatment of gait-related disorders on a patient-specific basis.

Muscular coordination of knee motion during the terminal-swing phase of normal gait

Children with cerebral palsy often walk with diminished knee extension during the terminal-swing phase, resulting in a troublesome "crouched" posture at initial contact and a shortened stride. Treatment of this gait abnormality is challenging because the factors that extend the knee during normal walking are not well understood, and because the potential of individual muscles to limit terminal-swing knee extension is unknown. This study analyzed a series of three-dimensional, muscle-driven dynamic simulations to quantify the angular accelerations of the knee induced by muscles and other factors during swing. Simulations were generated that reproduced the measured gait dynamics and muscle excitation patterns of six typically developing children walking at self-selected speeds. The knee was accelerated toward extension in the simulations by velocity-related forces (i.e., Coriolis and centrifugal forces) and by a number of muscles, notably the vasti in mid-swing (passive), the hip extensors in terminal swing, and the stance-limb hip abductors, which accelerated the pelvis upward. Knee extension was slowed in terminal swing by the stance-limb hip flexors, which accelerated the pelvis backward. The hamstrings decelerated the forward motion of the swing-limb shank, but did not contribute substantially to angular motions of the knee. Based on these data, we hypothesize that the diminished knee extension in terminal swing exhibited by children with cerebral palsy may, in part, be caused by weak hip extensors or by impaired hip muscles on the stance limb that result in abnormal accelerations of the pelvis.

Are patient-specific joint and inertial parameters necessary for accurate inverse dynamics analyses of gait?

Variations in joint parameter (JP) values (axis positions and orientations in body segments) and inertial parameter (IP) values (segment masses, mass centers, and moments of inertia) as well as kinematic noise alter the results of inverse dynamics analyses of gait. Three-dimensional linkage models with joint constraints have been proposed as one way to minimize the effects of noisy kinematic data. Such models can also be used to perform gait optimizations to predict post-treatment function given pre-treatment gait data. This study evaluates whether accurate patient-specific JP and IP values are needed in three-dimensional linkage models to produce accurate inverse dynamics results for gait. The study was performed in two stages. First, we used optimization analyses to evaluate whether patient-specific JP and IP values can be calibrated accurately from noisy kinematic data, and second, we used Monte Carlo analyses to evaluate how errors in JP and IP values affect inverse dynamics calculations. Both stages were performed using a dynamic, 27 degrees-of-freedom, full-body linkage model and synthetic (i.e., computer generated) gait data corresponding to a nominal experimental gait motion. In general, JP but not IP values could be found accurately from noisy kinematic data. Root-mean-square (RMS) errors were 3 degrees and 4 mm for JP values and 1 kg, 22 mm, and 74 500 kg * mm2 for IP values. Furthermore, errors in JP but not IP values had a significant effect on calculated lower-extremity inverse dynamics joint torques. The worst RMS torque error averaged 4% bodyweight * height (BW * H) due to JP variations but less than 0.25% (BW * H) due to IP variations. These results suggest that inverse dynamics analyses of gait utilizing linkage models with joint constraints should calibrate the model's JP values to obtain accurate joint torques.

Optimal coordination of maximal-effort horizontal and vertical jump motions--a computer simulation study

BACKGROUND

The purpose of this study was to investigate the coordination strategy of maximal-effort horizontal jumping in comparison with vertical jumping, using the methodology of computer simulation.

METHODS

A skeletal model that has nine rigid body segments and twenty degrees of freedom was developed. Thirty-two Hill-type lower limb muscles were attached to the model. The excitation-contraction dynamics of the contractile element, the tissues around the joints to limit the joint range of motion, as well as the foot-ground interaction were implemented. Simulations were initiated from an identical standing posture for both motions. Optimal pattern of the activation input signal was searched through numerical optimization. For the horizontal jumping, the goal was to maximize the horizontal distance traveled by the body's center of mass. For the vertical jumping, the goal was to maximize the height reached by the body's center of mass.

RESULTS

As a result, it was found that the hip joint was utilized more vigorously in the horizontal jumping than in the vertical jumping. The muscles that have a function of joint flexion such as the m. iliopsoas, m. rectus femoris and m. tibialis anterior were activated to a greater level during the countermovement in the horizontal jumping with an effect of moving the body's center of mass in the forward direction. Muscular work was transferred to the mechanical energy of the body's center of mass more effectively in the horizontal jump, which resulted in a greater energy gain of the body's center of mass throughout the motion.

CONCLUSION

These differences in the optimal coordination strategy seem to be caused from the requirement that the body's center of mass needs to be located above the feet in a vertical jumping, whereas this requirement is not so strict in a horizontal jumping.

Dynamic analysis of above-knee amputee gait

BACKGROUND

It is important to understand the characteristics of amputee gait to develop more functional prostheses. The aim of this study is to quantitatively evaluate amputee gait by dynamic analysis of the musculoskeletal system during level walking and stair climbing.

METHODS

Dynamic analysis using gait analysis, electromyography and musculoskeletal modeling for above-knee amputees (n=8) and healthy adults (n=10) was performed to evaluate the muscle balance, muscle force, and moment of each major muscle in each ambulatory task. Time-distance parameters and the kinematic parameter of gait analysis were calculated, and a root mean square electromyogram of major muscles and hamstring and tibialis anterior coactivity was measured using electromyography. Lastly, dynamic analyses of above-knee amputee gaits were performed using musculoskeletal models with scaled bones and redefined muscles for each subject.

FINDINGS

Most kinematic parameters showed statistically no difference among the tasks, excluding pelvic tilt, pelvic obliquity, and hip abduction. Major muscle activities and coactivities of the hamstring and tibialis anterior showed that the stair ascent task needed more muscle activity than the stair descent task and level walking. The muscle activity and coactivity of amputees were greater than those of healthy subjects, excluding the hamstring coactivity during stair ascent (P<0.05). Lastly, dynamic analysis showed that weakened abductor and excessive adductor and then inadequate torque during all tasks were quantitatively observed.

INTERPRETATION

Dynamic analysis of amputee gait enabled us to quantify the contribution of major muscles at the hip and knee joint mainly in daily ambulatory tasks of above-knee amputees and may be helpful in designing functional prostheses.

Sensitivity of muscle force estimates to variations in muscle–tendon properties

The aim of this study was to determine the sensitivity of muscle force estimates to changes in some of the parameters which are commonly used to describe models of muscle-tendon actuation. The sensitivity analysis was performed on three parameters: optimal muscle-fiber length, muscle physiological cross-sectional area (PCSA), and tendon rest length. The muscles selected for the analysis were posterior gluteus medius/minimus, vasti, soleus, and sartorius. Each parameter was perturbed from its nominal value, and an optimization problem was solved to determine the relative influence of each parameter on the calculated values of muscle force. Muscle forces were calculated for a simulated cycle of normal walking. Parameter sensitivity was quantified using two new metrics: an integrated sensitivity ratio, which quantified the effect of changing a single parameter for any muscle on the time history of force developed by that muscle; and a summed cross-sensitivity ratio, which quantified the effect of changing one parameter for any muscle on the time histories of forces developed by all of the other muscles. The results showed that muscle force estimates for walking are most sensitive to changes in tendon rest length and least sensitive to changes in muscle PCSA. For soleus, for example, the integrated sensitivity ratios for tendon rest length were an order of magnitude greater than those for muscle-fiber length and PCSA. For vasti, the integrated sensitivity ratios for tendon rest length were twice as large as those for muscle-fiber length and nearly an order of magnitude greater than those for PCSA. Overall, changes in the tendon rest lengths of vasti and soleus and changes in the fiber length of vasti were most critical to model estimates of muscle force. Our results emphasize the importance of obtaining accurate estimates of tendon rest length and muscle-fiber length, particularly for those actuators that function as prime movers during locomotion (gluteus maximus, gluteus medius/minimus, vasti, soleus, and gastrocnemius).

Model-based estimation of muscle forces exerted during movements

Estimation of individual muscle forces during human movement can provide insight into neural control and tissue loading and can thus contribute to improved diagnosis and management of both neurological and orthopaedic conditions. Direct measurement of muscle forces is generally not feasible in a clinical setting, and non-invasive methods based on musculoskeletal modeling should therefore be considered. The current state of the art in clinical movement analysis is that resultant joint torques can be reliably estimated from motion data and external forces (inverse dynamic analysis). Static optimization methods to transform joint torques into estimates of individual muscle forces using musculoskeletal models, have been known for several decades. To date however, none of these methods have been successfully translated into clinical practice. The main obstacles are the lack of studies reporting successful validation of muscle force estimates, and the lack of user-friendly and efficient computer software. Recent advances in forward dynamics methods have opened up new opportunities. Forward dynamic optimization can be performed such that solutions are less dependent on measured kinematics and ground reaction forces, and are consistent with additional knowledge, such as the force-length-velocity-activation relationships of the muscles, and with observed electromyography signals during movement. We conclude that clinical applications of current research should be encouraged, supported by further development of computational tools and research into new algorithms for muscle force estimation and their validation.

Musculotendon Parameters and Musculoskeletal Pathways Within the Human Foot

In order to create a flexible model of the foot for dynamic musculoskeletal models, anthropometric data combined with geometric information describing the intrinsic musculature are needed. In this study, the left feet of two male and two female cadavers were dissected to expose the intrinsic musculotendon pathways. Three-dimensional coordinates of bony landmarks, tendon origins, insertions, and via points were digitized to submillimeter accuracy. Muscle architectural parameters were also measured, including volume, weight, and pennation angle and sarcomere, fascicle, and free tendon lengths. Optimal muscle fascicle lengths, pennation angles at optimal length, physiological cross-sectional areas (PCSA), and tendon slack lengths were calculated from the directly measured values. Fascicle length and pennation angle varied greatly within each subject. Average fascicle lengths normalized by optimal fascicle length varied between 0.73 and 1.25, with 75% of the formalin-preserved muscles being found in a shortened state. The muscle volume and PCSA also had a large variability within subjects but less variation between subjects. The ratio of tendon slack length to optimal fascicle length was found to vary between 1.05 and 9.56. Using this data, a deformable model of the foot can now be created. It is envisioned that deformable feet will significantly improve

A neuromusculoskeletal tracking method for estimating individual muscle forces in human movement

A neuromusculoskeletal tracking (NMT) method was developed to estimate muscle forces from observed motion data. The NMT method combines skeletal motion tracking and optimal neuromuscular tracking to produce forward simulations of human movement quickly and accurately. The skeletal motion tracker calculates the joint torques needed to actuate a skeletal model and track observed segment angles and ground forces in a forward simulation of the motor task. The optimal neuromuscular tracker resolves the muscle redundancy problem dynamically and finds the muscle excitations (and muscle forces) needed to produce the joint torques calculated by the skeletal motion tracker. To evaluate the accuracy of the NMT method, kinematics and ground forces obtained from an optimal control (parameter optimization) solution for maximum-height jumping were contaminated with both random and systematic noise. These data served as input observations to the NMT method as well as an inverse dynamics analysis. The NMT solution was compared to the input observations, the original optimal solution, and a simulation driven by the inverse dynamics torques. The results show that, in contrast to inverse dynamics, the NMT method is able to produce an accurate forward simulation consistent with the optimal control solution. The NMT method also requires 3 orders-of-magnitude less CPU time than parameter optimization. The speed and accuracy of the NMT method make it a promising new tool for estimating muscle forces using experimentally obtained kinematics and ground force data.

The effect of speed and influence of individual muscles on hamstring mechanics during the swing phase of sprinting

The purpose of this study was to characterize the effect of speed and influence of individual muscles on hamstring stretch, loading, and work during the swing phase of sprinting. We measured three-dimensional kinematics and electromyography (EMG) activities of 19 athletes sprinting on a treadmill at speeds ranging from 80% to 100% of maximum speed. We then generated muscle-actuated forward dynamic simulations of swing and double float phases of the sprinting gait cycle. Simulated lower extremity joint angles and model predicted excitations were similar to measured quantities. Swing phase simulations were used to characterize the effects of speed on the peak stretch, maximum force, and negative work of the biceps femoris long head (BF), the most often injured hamstring muscle. Perturbations of the double float simulations were used to assess the influence of individual muscles on BF stretch. Peak hamstring musculotendon stretch occurred at approximately 90% of the gait cycle (late swing) and was independent of speed. Peak hamstring force and negative musculotendon work increased significantly with speed (p<0.05). Muscles in the lumbo-pelvic region had greater influence on hamstring stretch than muscles acting about the knee and ankle. In particular, the hip flexors were found to induce substantial hamstring stretch in the opposite limb, with that influence increasing with running speed. We conclude that hamstring strain injury during sprinting may be related to the performance of large amounts of negative work over repeated strides and/or resulting from a perturbation in pelvic muscle coordination that induces excessive hamstring stretch in a single stride.

Contributions of muscles, ligaments, and the ground-reaction force to tibiofemoral joint loading during normal gait

The aim of this study was twofold: first, to determine which muscles and ligaments resist the adduction moment at the knee during normal walking; and second, to describe and explain the contributions of muscles, ligaments, and the ground reaction force to medial and lateral compartment loading. Muscle forces, ground reaction forces, and joint motions obtained from a dynamic optimization solution for normal walking were used as input to a three-dimensional model of the lower limb. A static equilibrium problem was solved at each instant of the gait cycle to determine tibiofemoral joint loading at the knee. Medial compartment loading was determined mainly by the orientation of the ground reaction force. Because this force vector passed medial to the knee, it applied an adduction moment about the joint during stance. In contrast, all of the force transmitted by the lateral compartment was due to muscle and ligament action. The muscles that contributed most to support and forward propulsion during normal walking (quadriceps and gastrocnemius) also contributed most to knee stability in the frontal plane. The knee ligaments, particularly those of the posterior lateral corner, provided stability to the knee at certain periods of the stance phase, when activity of the important stabilizing muscles was low.

Parallel asynchronous particle swarm optimization

The high computational cost of complex engineering optimization problems has motivated the development of parallel optimization algorithms. A recent example is the parallel particle swarm optimization (PSO) algorithm, which is valuable due to its global search capabilities. Unfortunately, because existing parallel implementations are synchronous (PSPSO), they do not make efficient use of computational resources when a load imbalance exists. In this study, we introduce a parallel asynchronous PSO (PAPSO) algorithm to enhance computational efficiency. The performance of the PAPSO algorithm was compared to that of a PSPSO algorithm in homogeneous and heterogeneous computing environments for small- to medium-scale analytical test problems and a medium-scale biomechanical test problem. For all problems, the robustness and convergence rate of PAPSO were comparable to those of PSPSO. However, the parallel performance of PAPSO was significantly better than that of PSPSO for heterogeneous computing environments or heterogeneous computational tasks. For example, PAPSO was 3.5 times faster than was PSPSO for the biomechanical test problem executed on a heterogeneous cluster with 20 processors. Overall, PAPSO exhibits excellent parallel performance when a large number of processors (more than about 15) is utilized and either (1) heterogeneity exists in the computational task or environment, or (2) the computation-to-communication time ratio is relatively small.

Predicted threshold against backward balance loss in gait

The purpose of this study was to determine the minimum forward center of mass (COM) velocity required to prevent backward loss of balance in gait as function of the initial COM position. We hypothesized that these threshold values would be different from those previously published for standing because of the postural differences between gait and standing. To investigate this issue, we constructed a seven-link, nine-degree-of-freedom biomechanical model and employed dynamic optimization to estimate these threshold values under two initial postural conditions: (1) the posture at the beginning of swing phase (i.e., at toe-off), and (2) symmetric bipedal standing. In particular, for a range of possible COM positions posterior to the base of support (BOS), simulated annealing was used to search for the minimum velocity that could carry the COM into the BOS and avoid backward loss of balance. We found that the stability boundary against backward balance loss in walking had a similar overall trend as that previously published for standing. In general, the minimal COM velocity necessary to prevent a backward loss of balance in walking was greater than that in symmetric bipedal standing, and the difference could approach 30% or more when the COM started 0.5 and 1.0 foot-lengths behind the BOS. These discrepancies suggest that simpler biomechanical models, while being more efficient and easier to employ, may not always be adequate for exploring stability limits of humans.

Integrating modelling and experiments to assess dynamic musculoskeletal function in humans

Magnetic resonance imaging, bi-plane X-ray fluoroscopy and biomechanical modelling are enabling technologies for the non-invasive evaluation of muscle, ligament and joint function during dynamic activity. This paper reviews these various technologies in the context of their application to the study of human movement. We describe how three-dimensional, subject-specific computer models of the muscles, ligaments, cartilage and bones can be developed from high-resolution magnetic resonance images; how X-ray fluoroscopy can be used to measure the relative movements of the bones at a joint in three dimensions with submillimetre accuracy; how complex 3-D dynamic simulations of movement can be performed using new computational methods based on non-linear control theory; and how musculoskeletal forces derived from such simulations can be used as inputs to elaborate finite-element models of a joint to calculate contact stress distributions on a subject-specific basis. A hierarchical modelling approach is highlighted that links rigid-body models of limb segments with detailed finite-element models of the joints. A framework is proposed that integrates subject-specific musculoskeletal computer models with highly accurate in vivo experimental data.

Muscle, ligament, and joint-contact forces at the knee during walking

PURPOSE

In vivo measurement of the forces and strains in human tissues is currently impracticable. Computer modeling and simulation allows estimates of these quantities to be obtained noninvasively. This paper reviews our recent work on muscle, ligament, and joint loading at the knee during gait.

METHODS

Muscle and ground-reaction forces obtained from a sophisticated computer simulation of walking were input into a detailed model of the lower limb to obtain ligament and joint-contact loading at the knee for one full cycle of gait.

RESULTS

Peak anterior cruciate ligament (ACL) force occurred in early stance and was mainly determined by the anterior pull of the patellar tendon on the tibia. The medial collateral ligament was the primary restraint to anterior tibial translation (ATT) in the ACL-deficient knee. ATT in the ACL-deficient knee can be reduced to the level calculated for the intact knee by increasing hamstrings muscle force. Reducing quadriceps force was insufficient to restore ATT to the level calculated for the intact knee. For both normal and ACL-deficient walking, the resultant force acting between the femur and tibia remained mainly on the medial side of the knee. The knee adductor moment was resisted by a combination of muscle and ligament forces.

CONCLUSION

Knee-ligament loading during the stance phase of gait is explained by the pattern of anterior shear force applied to the leg. The distribution of force at the tibiofemoral joint is determined by the variation in the external adductor moment applied at the knee. The forces acting at the tibiofemoral and patellofemoral joints are similar during normal and ACL-deficient gait. Hamstrings facilitation is more effective than quadriceps avoidance in reducing ATT during ACL-deficient gait.

A ‘cheap’ optimal control approach to estimate muscle forces in musculoskeletal systems

This paper shows a new method to estimate the muscle forces in musculoskeletal systems based on the inverse dynamics of a multi-body system associated optimal control. The redundant actuator problem is solved by minimizing a time-integral cost function, augmented with a torque-tracking error function, and muscle dynamics is considered through differential constraints. The method is compared to a previously implemented human posture control problem, solved using a Forward Dynamics Optimal Control approach and to classical static optimization, with two different objective functions. The new method provides very similar muscle force patterns when compared to the forward dynamics solution, but the computational cost is much smaller and the numerical robustness is increased. The results achieved suggest that this method is more accurate for the muscle force predictions when compared to static optimization, and can be used as a numerically 'cheap' alternative to the forward dynamics and optimal control in some applications.

Optimal control simulations reveal mechanisms by which arm movement improves standing long jump performance

Optimal control simulations of the standing long jump were developed to gain insight into the mechanisms of enhanced performance due to arm motion. The activations that maximize standing long jump distance of a joint torque actuated model were determined for jumps with free and restricted arm movement. The simulated jump distance was 40 cm greater when arm movement was free (2.00 m) than when it was restricted (1.60 m). The majority of the performance improvement in the free arm jump was due to the 15% increase (3.30 vs. 2.86 m/s) in the take-off velocity of the center of gravity. Some of the performance improvement in the free arm jump was attributable to the ability of the jumper to swing the arms backwards during the flight phase to alleviate excessive forward rotation and position the body segments properly for landing. In restricted arm jumps, the excessive forward rotation was avoided by "holding back" during the propulsive phase and reducing the activation levels of the ankle, knee, and hip joint torque actuators. In addition, swinging the arm segments allowed the lower body joint torque actuators to perform 26 J more work in the free arm jump. However, the most significant contribution to developing greater take-off velocity came from the additional 80 J work done by the shoulder actuator in the jump with free arm movement.

Combined finite-element and rigid-body analysis of human jaw joint dynamics

The jaw joint plays a crucial role in human mastication. It acts as a guidance for jaw movements and as a fulcrum for force generation. The joint is subjected to loading which causes tensions and deformations in its cartilaginous structures. These are assumed to be a major determinant for development, maintenance and also degeneration of the joint. To analyze the distribution of tensions and deformations in the cartilaginous structures of the jaw joint during jaw movement, a dynamical model of the human masticatory system has been constructed. Its movements are controlled by muscle activation. The articular cartilage layers and articular disc were included as finite-element (FE) models. As this combination of rigid-body and FE modeling had not been applied to musculoskeletal systems yet, its benefits and limitations were assessed by simulating both unloaded and loaded jaw movements. It was demonstrated that joint loads increase with muscle activation, irrespective of the external loads. With increasing joint load, the size of the stressed area of the articular surfaces was enlarged, whereas the peak stresses were much less affected. The results suggest that the articular disc enables distribution of local contact stresses over a much wider area of the very incongruent articular surfaces by transforming compressive principal stress into shear stress.

Evaluation of a particle swarm algorithm for biomechanical optimization

Optimization is frequently employed in biomechanics research to solve system identification problems, predict human movement, or estimate muscle or other internal forces that cannot be measured directly. Unfortunately, biomechanical optimization problems often possess multiple local minima, making it difficult to find the best solution. Furthermore, convergence in gradient-based algorithms can be affected by scaling to account for design variables with different length scales or units. In this study we evaluate a recently-developed version of the particle swarm optimization (PSO) algorithm to address these problems. The algorithm's global search capabilities were investigated using a suite of difficult analytical test problems, while its scale-independent nature was proven mathematically and verified using a biomechanical test problem. For comparison, all test problems were also solved with three off-the-shelf optimization algorithms--a global genetic algorithm (GA) and multistart gradient-based sequential quadratic programming (SQP) and quasi-Newton (BFGS) algorithms. For the analytical test problems, only the PSO algorithm was successful on the majority of the problems. When compared to previously published results for the same problems, PSO was more robust than a global simulated annealing algorithm but less robust than a different, more complex genetic algorithm. For the biomechanical test problem, only the PSO algorithm was insensitive to design variable scaling, with the GA algorithm being mildly sensitive and the SQP and BFGS algorithms being highly sensitive. The proposed PSO algorithm provides a new off-the-shelf global optimization option for difficult biomechanical problems, especially those utilizing design variables with different length scales or units.

Contribution of non-extensor muscles of the leg to maximal-effort countermovement jumping

BACKGROUND

The purpose of this study was to determine the effects of non-extensor muscles of the leg (i.e., muscles whose primary function is not leg extension) on the kinematics and kinetics of human maximal-effort countermovement jumping. Although it is difficult to address this type of question through experimental procedures, the methodology of computer simulation can be a powerful tool.

METHODS

A skeletal model that has nine rigid body segments and twenty degrees of freedom was developed. Two sets of muscle models were attached to this skeletal model: all (most of) major muscles in the leg ("All Muscles" model) and major extensor muscles in the leg (i.e., muscles whose primary function is leg extension; "Extensors Only" model). Neural activation input signal was represented by a series of step functions with a step duration of 0.05 s. Simulations were started from an identical upright standing posture. The optimal pattern of the activation input signal was searched through extensive random-search numerical optimization with a goal of maximizing the height reached by the mass centre of the body after jumping up.

RESULTS

The simulated kinematics was almost two-dimensional, suggesting the validity of two-dimensional analyses when evaluating net mechanical outputs around the joints using inverse dynamics. A greater jumping height was obtained for the "All Muscles" model (0.386 m) than for the "Extensors Only" model (0.301 m). For the "All Muscles" model, flexor muscles developed force in the beginning of the countermovement. For the "All Muscles" model, the sum of the work outputs from non-extensor muscles was 47.0 J, which was 13% of the total amount (359.9 J). The quantitative distribution of the work outputs from individual muscles was markedly different between these two models.

CONCLUSION

It was suggested that the contribution of non-extensor muscles in maximal-effort countermovement jumping is substantial. The use of a computer simulation model that includes non-extensor muscles seems to be more desirable for the assessment of muscular outputs during jumping.

Force, work and power output of lower limb muscles during human maximal-effort countermovement jumping

The purpose of this study was to simulate human maximal-effort countermovement jumping with a three-dimensional neuromusculoskeletal model. The specific aim was to investigate muscle force, work and power output of major lower limb muscles during the motion. A neuromusculoskeletal model that has nine rigid body segments, 20 degrees of freedom, 32 Hill-type lower limb muscles was developed. The neural activation input signal was represented by a series of step functions with step duration of 0.05 s. The excitation-contraction dynamics of the contractile element, the tissues around the joints to limit the joint range of motion, as well as the foot-ground interaction were implemented. A simulation was started from a standing posture. Optimal pattern of the activation input signal was searched through numerical optimization with a goal of maximizing the height reached by the mass center of body after jumping up. As a result, feasible kinematics, ground reaction force profile and muscle excitation profile were generated. It was found that monoarticular muscles had major contributions of mechanical work and power output, whereas biarticular muscles had minor contributions. Hip adductors, abductors and external rotator muscles were vigorously activated, although their mechanical work and power output was minor because of their limited length change during the motion. Joint flexor muscles such as m. iliopsoas, m. biceps femoris short head and m. tibialis anterior were activated in the beginning of the motion with an effect of facilitating the generation of a countermovement.

Using computed muscle control to generate forward dynamic simulations of human walking from experimental data

The objective of this study was to develop an efficient methodology for generating muscle-actuated simulations of human walking that closely reproduce experimental measures of kinematics and ground reaction forces. We first introduce a residual elimination algorithm (REA) to compute pelvis and low back kinematic trajectories that ensure consistency between whole-body dynamics and measured ground reactions. We then use a computed muscle control (CMC) algorithm to vary muscle excitations to track experimental joint kinematics within a forward dynamic simulation. CMC explicitly accounts for delays in muscle force production resulting from activation and contraction dynamics while using a general static optimization framework to resolve muscle redundancy. CMC was used to compute muscle excitation patterns that drove a 21-degrees-of-freedom, 92 muscle model to track experimental gait data of 10 healthy young adults. Simulated joint kinematics closely tracked experimental quantities (mean root-mean-squared errors generally less than 1 degrees), and the time histories of muscle activations were similar to electromyographic recordings. A simulation of a half-cycle of gait could be generated using approximately 30 min of computer processing time. The speed and accuracy of REA and CMC make it practical to generate subject-specific simulations of gait.

Evaluation of theories of complex movement planning in different levels of gravity

Due to high redundancy of degrees of freedom in the human body, we can perform any movement, from the simplest to the most complex, in many different ways. Several studies are still trying to identify the motor strategies that master this redundancy and generate the movements whose characteristics are highly stereotyped. The aim of this work is to build a simulator that is able to evaluate different motor planning hypotheses. The most interesting applications of this tool occur in studies of the motor strategy in microgravity conditions. The comparison between simulated movements and kinematics data recorded both on Earth, and during a 5-month mission on board the Mir station shows that for a complex whole-body movement (such as trunk bending) a single planning criterion cannot explain all movement aspects. However, the simulator allows an understanding of the motor planning adaptation of astronauts. In space, the lack of equilibrium constraint (which on Earth brings about the center of mass control) leads to a new motor strategy that minimizes dynamic interactions with the floor.

Multibody dynamic simulation of knee contact mechanics

Multibody dynamic musculoskeletal models capable of predicting muscle forces and joint contact pressures simultaneously would be valuable for studying clinical issues related to knee joint degeneration and restoration. Current three-dimensional multibody knee models are either quasi-static with deformable contact or dynamic with rigid contact. This study proposes a computationally efficient methodology for combining multibody dynamic simulation methods with a deformable contact knee model. The methodology requires preparation of the articular surface geometry, development of efficient methods to calculate distances between contact surfaces, implementation of an efficient contact solver that accounts for the unique characteristics of human joints, and specification of an application programming interface for integration with any multibody dynamic simulation environment. The current implementation accommodates natural or artificial tibiofemoral joint models, small or large strain contact models, and linear or nonlinear material models. Applications are presented for static analysis (via dynamic simulation) of a natural knee model created from MRI and CT data and dynamic simulation of an artificial knee model produced from manufacturer's CAD data. Small and large strain natural knee static analyses required 1 min of CPU time and predicted similar contact conditions except for peak pressure, which was higher for the large strain model. Linear and nonlinear artificial knee dynamic simulations required 10 min of CPU time and predicted similar contact force and torque but different contact pressures, which were lower for the nonlinear model due to increased contact area. This methodology provides an important step toward the realization of dynamic musculoskeletal models that can predict in vivo knee joint motion and loading simultaneously.

Neuromusculoskeletal computer modeling and simulation of upright, straight-legged, bipedal locomotion of Australopithecus afarensis (A.L. 288-1)

The skeleton of Australopithecus afarensis (A.L. 288-1, better known as "Lucy") is by far the most complete record of locomotor morphology of early hominids currently available. Even though researchers agree that the postcranial skeleton of Lucy shows morphological features indicative of bipedality, only a few studies have investigated Lucy's bipedal locomotion itself. Lucy's energy expenditure during locomotion has been the topic of much speculation, but has not been investigated, except for several estimates derived from experimental data collected on other animals. To gain further insights into how Lucy may have walked, we generated a full three-dimensional (3D) reconstruction and forward-dynamic simulation of upright bipedal locomotion of this ancient human ancestor. Laser-scanned 3D bone geometries were combined with state-of-the-art neuromusculoskeletal modeling and simulation techniques from computational biomechanics. A detailed full 3D neuromusculoskeletal model was developed that encompassed all major bones, joints (10), and muscles (52) of the lower extremity. A model of muscle force and heat production was used to actuate the musculoskeletal system, and to estimate total energy expenditure during locomotion. Neural activation profiles for each of the 52 muscles that produced a single step of locomotion, while at the same time minimizing the energy consumed per meter traveled, were searched through numerical optimization. The numerical optimization resulted in smooth locomotor kinematics, and the predicted energy expenditure was appropriate for upright bipedal walking in an individual of Lucy's body size.

Dynamic simulation of FES-cycling: influence of individual parameters

Cycling by means of functional electrical simulation (FES) is an attractive training method for spinal cord injured (SCI) subjects. FES-cycling performance is influenced by a number of parameters like seating position, physiological parameters, conditions of surface stimulation, and pedaling rate. The objective of this paper was the determination of the influence of the most important parameters on optimal muscle stimulation patterns and power output of FES-cycling on a noncircular pedal path. The rider-cycle system was modeled as a planar articulated rigid body linkage on which the muscle forces are applied via joint moments and implemented into a forward dynamic simulation of FES-cycling. For model validation, the generated drive torques that are predicted by the simulation were compared to measurements with an individual paraplegic subject. Then, a sensitivity analysis was carried out to determine the influences of the most important parameters for surface stimulation of gluteus maximus, quadriceps, hamstrings, and peroneus reflex. The results show how optimal stimulation patterns and the expected mean active power output can be estimated based on measured individual parameters and adjusted geometry and stimulation parameters for a particular SCI-subject. This can considerably improve FES-cycling performance and relieve the patients by shortening the time that is necessary for experimental adaptation of the stimulation patterns.

Evaluation of parallel decomposition methods for biomechanical optimizations

As the complexity of musculoskeletal models continues to increase, so will the computational demands of biomechanical optimizations. For this reason, parallel biomechanical optimizations are becoming more common. Most implementations parallelize the optimizer. In this study, an alternate approach is investigated that parallelizes the analysis function (i.e., a kinematic or dynamic simulation) called repeatedly by the optimizer to calculate the cost function and constraints. To evaluate this approach, a system identification problem involving a kinematic ankle joint model was solved using a gradient-based optimizer and three parallel decomposition methods: gradient calculation decomposition, analysis function decomposition, or both methods combined. For a given number of processors, analysis function decomposition exhibited the best performance despite the highest communication and synchronization overhead, while gradient calculation decomposition demonstrated the worst performance due to the fact that the necessary line searches were not performed in parallel. These findings suggest that the method of parallelization most commonly used for biomechanical optimizations may not be the most efficient, depending on the optimization algorithm used. In many applications, the best computational strategy may be to focus on parallelizing the analysis function.

Muscular contributions to hip and knee extension during the single limb stance phase of normal gait: a framework for investigating the causes of crouch gait

Crouch gait, a troublesome movement abnormality among persons with cerebral palsy, is characterized by excessive flexion of the hips and knees during stance. Treatment of crouch gait is challenging, at present, because the factors that contribute to hip and knee extension during normal gait are not well understood, and because the potential of individual muscles to produce flexion or extension of the joints during stance is unknown. This study analyzed a three-dimensional, muscle-actuated dynamic simulation of walking to quantify the angular accelerations of the hip and knee induced by muscles during normal gait, and to rank the potential of the muscles to alter motions of these joints. Examination of the muscle actions during single limb stance showed that the gluteus maximus, vasti, and soleus make substantial contributions to hip and knee extension during normal gait. Per unit force, the gluteus maximus had greater potential than the vasti to accelerate the knee toward extension. These data suggest that weak hip extensors, knee extensors, or ankle plantar flexors may contribute to crouch gait, and strengthening these muscles--particularly gluteus maximus--may improve hip and knee extension. Abnormal forces generated by the iliopsoas or adductors may also contribute to crouch gait, as our analysis showed that these muscles have the potential to accelerate the hip and knee toward flexion. This work emphasizes the need to consider how muscular forces contribute to multijoint movements when attempting to identify the causes of abnormal gait.

Model Prediction of Anterior Cruciate Ligament Force during Drop-Landings

PURPOSE

The aim of this study was to calculate and explain the pattern of force transmitted to the anterior cruciate ligament during soft-style drop-landings. We hypothesized that peak ACL loading is due to the anterior pull of the quadriceps on the tibia, as these muscles develop large eccentric forces upon impact.

METHODS

A three-dimensional model of the body was used to simulate drop-landing. The simulation was performed by entering into the model muscle excitation patterns based on experimental EMG. The input excitation patterns were modified to create a performance response of the model that matched experimental data. Joint angles, ground reaction forces, and muscle forces obtained from the landing simulation were then applied to a model of the lower limb that incorporated a three-dimensional model of the knee.

RESULTS

The model ACL was loaded only in the first 25% of the landing phase. Peak ACL force (approximately 0.4 BW) resulted from a complex interaction between the patellar tendon force, the compressive force acting at the tibiofemoral joint, and the force applied by the ground to the lower leg. The patellar tendon force and tibiofemoral contact force both applied significant anterior shear forces to the shank throughout the landing phase. These effects were modulated by another significant posterior shear force applied by the ground reaction, which served to limit the maximum force transmitted to the ACL.

CONCLUSION

The pattern of ACL force in drop-landing cannot be explained by the anterior pull of the quadriceps force alone.

Contributions of muscle forces and toe-off kinematics to peak knee flexion during the swing phase of normal gait: an induced position analysis

A three-dimensional dynamic simulation of walking was used together with induced position analysis to determine how kinematic conditions at toe-off and muscle forces following toe-off affect peak knee flexion during the swing phase of normal gait. The flexion velocity of the swing-limb knee at toe-off contributed 30 degrees to the peak knee flexion angle; this was larger than any contribution from an individual muscle or joint moment. Swing-limb muscles individually made large contributions to knee angle (i.e., as large as 22 degrees), but their actions tended to balance one another, so that the combined contribution from all swing-limb muscles was small (i.e., less than 3 degrees of flexion). The uniarticular muscles of the swing limb made contributions to knee flexion that were an order of magnitude larger than the biarticular muscles of the swing limb. The results of the induced position analysis make clear the importance of knee flexion velocity at toe-off relative to the effects of muscle forces exerted after toe-off in generating peak knee flexion angle. In addition to improving our understanding of normal gait, this study provides a basis for analyzing stiff-knee gait, a movement abnormality in which knee flexion in swing is diminished.

Muscles that influence knee flexion velocity in double support: implications for stiff-knee gait

Adequate knee flexion velocity at toe-off is important for achieving normal swing-phase knee flexion during gait. Consequently, insufficient knee flexion velocity at toe-off can contribute to stiff-knee gait, a movement abnormality in which swing-phase knee flexion is diminished. This work aims to identify the muscles that contribute to knee flexion velocity during double support in normal gait and the muscles that have the most potential to alter this velocity. This objective was achieved by perturbing the forces generated by individual muscles during double support in a forward dynamic simulation of normal gait and observing the effects of the perturbations on peak knee flexion velocity. Iliopsoas and gastrocnemius were identified as the muscles that contribute most to increasing knee flexion velocity during double support. Increased forces in vasti, rectus femoris, and soleus were found to decrease knee flexion velocity. Vasti, rectus femoris, gastrocnemius, and iliopsoas were all found to have large potentials to influence peak knee flexion velocity during double support. The results of this work indicate which muscles likely contribute to the diminished knee flexion velocity at toe-off observed in stiff-knee gait, and identify the treatment strategies that have the most potential to increase this velocity in persons with stiff-knee gait.

Pattern of anterior cruciate ligament force in normal walking

The goal of this study was to calculate and explain the pattern of anterior cruciate ligament (ACL) loading during normal level walking. Knee-ligament forces were obtained by a two-step procedure. First, a three-dimensional (3D) model of the whole body was used together with dynamic optimization theory to calculate body-segmental motions, ground reaction forces, and leg-muscle forces for one cycle of gait. Joint angles, ground reaction forces, and muscle forces obtained from the gait simulation were then input into a musculoskeletal model of the lower limb that incorporated a 3D model of the knee. The relative positions of the femur, tibia, and patella and the forces induced in the knee ligaments were found by solving a static equilibrium problem at each instant during the simulated gait cycle. The model simulation predicted that the ACL bears load throughout stance. Peak force in the ACL (303 N) occurred at the beginning of single-leg stance (i.e., contralateral toe off). The pattern of ACL force was explained by the shear forces acting at the knee. The balance of muscle forces, ground reaction forces, and joint contact forces applied to the leg determined the magnitude and direction of the total shear force acting at the knee. The ACL was loaded whenever the total shear force pointed anteriorly. In early stance, the anterior shear force from the patellar tendon dominated the total shear force applied to the leg, and so maximum force was transmitted to the ACL at this time. ACL force was small in late stance because the anterior shear forces supplied by the patellar tendon, gastrocnemius, and tibiofemoral contact were nearly balanced by the posterior component of the ground reaction.

Changes in Foot Loading Following Plantar Fasciotomy: A Computer Modeling Study

Forward dynamic simulations of a toe-rise task were developed to explore the outcomes of plantar fasciotomy, a surgery commonly performed to relieve heel pain. The specific objectives of this study were to develop such a simulation, validate its predictions, and simulate rising on toes using a model from which the plantar fascia had been removed. Root-mean squared differences between the intact model and measurements of healthy subjects were found to be 0.009 body weights (BW) and 0.055 BW for the horizontal and vertical ground reaction forces and 7.1 mm, 11.3 mm, and 0.48 deg for the horizontal, vertical and rotational positions of the pelvis. Simulated plantar fasciotomy increased passive arch torques by 7.4%, increased metatarsal head contact forces by 18%, and resulted in greater toe flexor activity. These simulations may explain the mechanisms behind plantar fasciotomy complications when patients perform activities that require loading of the plantarflexors and the longitudinal arch.

Comparison of shear forces and ligament loading in the healthy and ACL-deficient knee during gait

The purpose of this study was to predict and explain the pattern of shear force and ligament loading in the ACL-deficient knee during walking, and to compare these results to similar calculations for the healthy knee. Musculoskeletal modeling and computer simulation were combined to calculate ligament forces in the ACL-deficient knee during walking. Joint angles, ground-reaction forces, and the corresponding lower-extremity muscle forces obtained from a whole-body dynamic optimization simulation of walking were input into a second three-dimensional model of the lower extremity that represented the knee as a six degree-of-freedom spatial joint. Anterior tibial translation (ATT) increased throughout the stance phase of gait when the model ACL was removed. The medial collateral ligament (MCL) was the primary restraint to ATT in the ACL-deficient knee. Peak force in the MCL was three times greater in the ACL-deficient knee than in the ACL-intact knee; however, peak force sustained by the MCL in the ACL-deficient knee was limited by the magnitude of the total anterior shear force applied to the tibia. A decrease in anterior tibial shear force was brought about by a decrease in the patellar tendon angle resulting from the increase in ATT. These results suggest that while the MCL acts as the primary restraint to ATT in the ACL-deficient knee, changes in patellar tendon angle reduce total anterior shear force at the knee.

Simple and complex models for studying muscle function in walking

While simple models can be helpful in identifying basic features of muscle function, more complex models are needed to discern the functional roles of specific muscles in movement. In this paper, two very different models of walking, one simple and one complex, are used to study how muscle forces, gravitational forces and centrifugal forces (i.e. forces arising from motion of the joints) combine to produce the pattern of force exerted on the ground. Both the simple model and the complex one predict that muscles contribute significantly to the ground force pattern generated in walking; indeed, both models show that muscle action is responsible for the appearance of the two peaks in the vertical force. The simple model, an inverted double pendulum, suggests further that the first and second peaks are due to net extensor muscle moments exerted about the knee and ankle, respectively. Analyses based on a much more complex, muscle-actuated simulation of walking are in general agreement with these results; however, the more detailed model also reveals that both the hip extensor and hip abductor muscles contribute significantly to vertical motion of the centre of mass, and therefore to the appearance of the first peak in the vertical ground force, in early single-leg stance. This discrepancy in the model predictions is most probably explained by the difference in model complexity. First, movements of the upper body in the sagittal plane are not represented properly in the double-pendulum model, which may explain the anomalous result obtained for the contribution of a hip-extensor torque to the vertical ground force. Second, the double-pendulum model incorporates only three of the six major elements of walking, whereas the complex model is fully 3D and incorporates all six gait determinants. In particular, pelvic list occurs primarily in the frontal plane, so there is the potential for this mechanism to contribute significantly to the vertical ground force, especially during early single-leg stance when the hip abductors are activated with considerable force.

Biomechanical modeling and optimal control of human posture

The present work describes the biomechanical modeling of human postural mechanics in the saggital plane and the use of optimal control to generate open-loop raising-up movements from a squatting position. The biomechanical model comprises 10 equivalent musculotendon actuators, based on a 40 muscles model, and three links (shank, thigh and HAT-Head, Arms and Trunk). Optimal control solutions are achieved through algorithms based on the Consistent Approximations Theory (Schwartz and Polak, 1996), where the continuous non-linear dynamics is represented in a discrete space by means of a Runge-Kutta integration and the control signals in a spline-coefficient functional space. This leads to non-linear programming problems solved by a sequential quadratic programming (SQP) method. Due to the highly non-linear and unstable nature of the posture dynamics, numerical convergence is difficult, and specific strategies must be implemented in order to allow convergence. Results for control (muscular excitations) and angular trajectories are shown using two final simulation times, as well as specific control strategies are discussed.

A Computational Technique for Determining the Ground Reaction Forces in Human Bipedal Stance

Our long-term goal is to use a musculoskeletal modeling approach for developing controller algorithms to restore standing balance to individuals with lower extremity paralysis using functional electrical stimulation. This paper describes a technique that facilitates this approach by avoiding the numerical problems associated with modeling the closed kinematic chain formed by the two lower extremities and the ground while standing. Specifically, we propose an optimization technique to estimate the magnitude and origin of the ground reaction force (GRF) vector on one of the feet, resulting in an equivalent open-chain formulation. Using this technique, we performed a series of inverse dynamic computations to determine the GRF and center of pressure (COP) values for five standing postures: neutral, neutral with forward lean, neutral with backward lean, wide, and tandem. The optimization procedure elicited force results that satisfy equilibrium and result in COP locations that are consistent and physically reasonable.

Experimental/analytical analysis of human locomotion using bondgraphs

A new vectorial bondgraph approach for modeling and simulation of human locomotion is introduced. The vectorial bondgraph is applied to an eight-segment gait model to derive the equations of motion for studying ground reaction forces (GRFs) and centers of pressure (COPs) in single and double support phases of ground and treadmill walking. A phase detection technique and accompanying transition equation is proposed with which the GRFs and COPs may be calculated for the transitions from double-to-single and single-to-double support phases. Good agreement is found between model predictions and experimental data obtained from force plate measurements. The bondgraph modeling approach is shown to be both informative and adaptable, in the sense that the model resembles the human body structure, and that modeled body segments can be easily added or removed.

Individual muscle contributions to support in normal walking

The purpose of this study was to quantify the contributions made by individual muscles to support of the whole body during normal gait. A muscle's contribution to support was described by its contribution to the time history of the vertical force exerted by the ground. The analysis was based on a three-dimensional, muscle-actuated model of the body and a dynamic optimization solution for normal walking. The results showed that, in early stance, before the foot was placed flat on the ground, support was provided mainly by the ankle dorsiflexors. After foot-flat, but before contralateral toe-off, support was generated primarily by gluteus maximus, vasti, and posterior gluteus medius/minimus; these muscles were responsible for the first peak seen in the vertical ground-reaction force. The majority of support in midstance was provided by gluteus medius/minimus, with gravity assisting significantly as well. The ankle plantarflexors generated nearly all support in late stance; these muscles were responsible for the second peak in the vertical ground-reaction force. The results showed also that centrifugal forces act to decrease the vertical ground-reaction force, but only by minor amounts, and that resistance of the skeleton to the force of gravity is no larger than 1/2 body weight throughout the gait cycle.

Generating dynamic simulations of movement using computed muscle control

Computation of muscle excitation patterns that produce coordinated movements of muscle-actuated dynamic models is an important and challenging problem. Using dynamic optimization to compute excitation patterns comes at a large computational cost, which has limited the use of muscle-actuated simulations. This paper introduces a new algorithm, which we call computed muscle control, that uses static optimization along with feedforward and feedback controls to drive the kinematic trajectory of a musculoskeletal model toward a set of desired kinematics. We illustrate the algorithm by computing a set of muscle excitations that drive a 30-muscle, 3-degree-of-freedom model of pedaling to track measured pedaling kinematics and forces. Only 10 min of computer time were required to compute muscle excitations that reproduced the measured pedaling dynamics, which is over two orders of magnitude faster than conventional dynamic optimization techniques. Simulated kinematics were within 1 degrees of experimental values, simulated pedal forces were within one standard deviation of measured pedal forces for nearly all of the crank cycle, and computed muscle excitations were similar in timing to measured electromyographic patterns. The speed and accuracy of this new algorithm improves the feasibility of using detailed musculoskeletal models to simulate and analyze movement.

A dynamic model of the knee and lower limb for simulating rising movements

A two-dimensional dynamical model of the human body was developed and used to simulate muscle and knee-ligament loading during a fast rising movement. The hip, ankle, and toes were each modeled as a simple hinge joint. Relative movements of the femur, tibia, and patella in the sagittal plane were described using a more detailed representation of the knee. The geometry of the model bones was adapted from cadaver data. Eleven elastic elements described the geometric and mechanical properties of the knee ligaments and joint capsule. The patella was assumed to be massless. Smooth hypersurfaces were constructed and used to calculate the position and orientation of the patella during a forward integration of the model. Each hypersurface was formed by applying the principle of static equilibrium to approximate patellofemoral mechanics during the simulation. The model was actuated by 22 musculotendinous units, each unit represented as a three-element muscle in series with tendon. A first-order process was assumed to model muscle excitation-contraction dynamics. Dynamic optimization theory was used to calculate the pattern of muscle excitations that produces a coordinated rising movement from an initial squatting position in minimum time. The calculations support the contention that squatting is a relatively safe exercise for rehabilitation following ACL reconstruction. ACL forces remain less than 20 N for the duration of the task.

Computer modeling and simulation of human movement

Recent interest in using modeling and simulation to study movement is driven by the belief that this approach can provide insight into how the nervous system and muscles interact to produce coordinated motion of the body parts. With the computational resources available today, large-scale models of the body can be used to produce realistic simulations of movement that are an order of magnitude more complex than those produced just 10 years ago. This chapter reviews how the structure of the neuromusculoskeletal system is commonly represented in a multijoint model of movement, how modeling may be combined with optimization theory to simulate the dynamics of a motor task, and how model output can be analyzed to describe and explain muscle function. Some results obtained from simulations of jumping, pedaling, and walking are also reviewed to illustrate the approach.

Dynamic optimization of human walking

A three-dimensional, neuromusculoskeletal model of the body was combined with dynamic optimization theory to simulate normal walking on level ground. The body was modeled as a 23 degree-of-freedom mechanical linkage, actuated by 54 muscles. The dynamic optimization problem was to calculate the muscle excitation histories, muscle forces, and limb motions subject to minimum metabolic energy expenditure per unit distance traveled. Muscle metabolic energy was calculated by slimming five terms: the basal or resting heat, activation heat, maintenance heat, shortening heat, and the mechanical work done by all the muscles in the model. The gait cycle was assumed to be symmetric; that is, the muscle excitations for the right and left legs and the initial and terminal states in the model were assumed to be equal. Importantly, a tracking problem was not solved. Rather only a set of terminal constraints was placed on the states of the model to enforce repeatability of the gait cycle. Quantitative comparisons of the model predictions with patterns of body-segmental displacements, ground-reaction forces, and muscle activations obtained from experiment show that the simulation reproduces the salient features of normal gait. The simulation results suggest that minimum metabolic energy per unit distance traveled is a valid measure of walking performance.

Muscle recruitment by the min/max criterion -- a comparative numerical study

This paper introduces the min/max criterion for simulation of muscle recruitment in multiple muscle systems. The criterion is introduced and justified by comparison to two known criterion types: the polynomial criterion and the soft saturation criterion. The comparison is performed on a planar three-muscle elbow model performing a dumbbell curl. A generalized form of the soft saturation criterion is introduced, and it is shown that the min/max criterion can be interpreted as the limit of the classical criteria when the exponents in their mathematical expressions approach infinity. We finally show how the min/max criterion can be cast into a form that allows for efficient and robust numerical solution by linear programming. It is concluded that the min/max criterion possesses a number of attractive physiological as well as algorithmic advantages.

Static and dynamic optimization solutions for gait are practically equivalent

The proposition that dynamic optimization provides better estimates of muscle forces during gait than static optimization is examined by comparing a dynamic solution with two static solutions. A 23-degree-of-freedom musculoskeletal model actuated by 54 Hill-type musculotendon units was used to simulate one cycle of normal gait. The dynamic problem was to find the muscle excitations which minimized metabolic energy per unit distance traveled, and which produced a repeatable gait cycle. In the dynamic problem, activation dynamics was described by a first-order differential equation. The joint moments predicted by the dynamic solution were used as input to the static problems. In each static problem, the problem was to find the muscle activations which minimized the sum of muscle activations squared, and which generated the joint moments input from the dynamic solution. In the first static problem, muscles were treated as ideal force generators; in the second, they were constrained by their force-length-velocity properties; and in both, activation dynamics was neglected. In terms of predicted muscle forces and joint contact forces, the dynamic and static solutions were remarkably similar. Also, activation dynamics and the force-length-velocity properties of muscle had little influence on the static solutions. Thus, for normal gait, if one can accurately solve the inverse dynamics problem and if one seeks only to estimate muscle forces, the use of dynamic optimization rather than static optimization is currently not justified. Scenarios in which the use of dynamic optimization is justified are suggested.

An efficient simulation method for discrete-value controlled large-scale neuromyoskeletal system models

An efficient Euler-Adams hybrid integration scheme for simulating on the computer discrete-value controlled large-scale neuromyoskeletal system models is presented. If, as discussed in the model, the differential equations describing the recruitment and excitation dynamics of the muscular subsystem are independent of the corresponding contraction-dynamical state variables, they can be integrated separately over certain time intervals by a modified Euler routine that handles discontinuous right-hand sides efficiently. The resulting myostates can then be stored and used as continuous input values for the subsequent integration by an Adams predictor-corrector algorithm of the remaining contraction-dynamical and skeletomechanical state differential equations. With such an Euler-Adams hybrid integration routine one avoids the detrimental effects and efficiency losses associated with frequent stop-restart cycles of otherwise efficient Adams-type algorithms, which cycles are forced by discontinuities on the right-hand side of the myostate equations. In the example presented, a reduction in the execution time by a factor of about 5 could be achieved by implementing the proposed technique.

Dynamic simulation of human movement using large-scale models of the body

A three-dimensional model of the body was used to simulate two different motor tasks: vertical jumping and normal walking on level ground. The pattern of muscle excitations, body motions, and ground-reaction forces for each task were calculated using dynamic optimization theory. For jumping, the performance criterion was to maximize the height reached by the center of mass of the body; for walking, the measure of performance was metabolic energy consumed per meter walked. Quantitative comparisons of the simulation results with experimental data obtained from people indicate that the model reproduces the salient features of maximum-height jumping and normal walking on the level. Analyses of the model solutions will allow detailed explanations to be given about the actions of specific muscles during each of these tasks.