Marcus G. Pandy

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.

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 optimal control model for maximum-height human jumping

To understand how intermuscular control, inertial interactions among body segments, and musculotendon dynamics coordinate human movement, we have chosen to study maximum-height jumping. Because this activity presents a relatively unambiguous performance criterion, it fits well into the framework of optimal control theory. The human body is modeled as a four-segment, planar, articulated linkage, with adjacent links joined together by frictionless revolutes. Driving the skeletal system are eight musculotendon actuators, each muscle modeled as a three-element, lumped-parameter entity, in series with tendon. Tendon is assumed to be elastic, and its properties are defined by a stress-strain curve. The mechanical behavior of muscle is described by a Hill-type contractile element, including both series and parallel elasticity. Driving the musculotendon model is a first-order representation of excitation-contraction (activation) dynamics. The optimal control problem is to maximize the height reached by the center of mass of the body subject to body-segmental, musculotendon, and activation dynamics, a zero vertical ground reaction force at lift-off, and constraints which limit the magnitude of the incoming neural control signals to lie between zero (no excitation) and one (full excitation). A computational solution to this problem was found on the basis of a Mayne-Polak dynamic optimization algorithm. Qualitative comparisons between the predictions of the model and previously reported experimental findings indicate that the model reproduces the major features of a maximum-height squat jump (i.e. limb-segmental angular displacements, vertical and horizontal ground reaction forces, sequence of muscular activity, overall jump height, and final lift-off time).

A Dynamic Optimization Solution for Vertical Jumping in Three Dimensions

A three-dimensional model of the human body is used to simulate a maximal vertical jump. The body is modeled as a 10-segment, 23 degree-of-freedom (dof), mechanical linkage, actuated by 54 muscles. Six generalized coordinates describe the position and orientation of the pelvis relative to the ground; the remaining nine segments branch in an open chain from the pelvis. The head, arms, and torso (HAT) are modeled as a single rigid body. The HAT articulates with the pelvis via a 3 dof ball-and-socket joint. Each hip is modeled as a 3 dof ball-and-socket joint, and each knee is modeled as a 1 dof hinge joint. Each foot is represented by a hindfoot and toes segment. The hindfoot articulates with the shank via a 2 dof universal joint, and the toes articulate with the hindfoot via a 1 dof hinge joint. Interaction of the feet with the ground is modeled using a series of spring-damper units placed under the sole of each foot. The path of each muscle is represented by either a series of straight lines or a combination of straight lines and space curves. Each actuator is modeled as a three-element, Hill-type muscle in series with tendon. A first-order process is assumed to model muscle excitation-contraction dynamics. Dynamic optimization theory is used to calculate the pattern of muscle excitations that produces a maximal vertical jump. Quantitative comparisons between model and experiment indicate that the model reproduces the kinematic, kinetic, and muscle-coordination patterns evident when humans jump to their maximum achievable heights.

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 muscles 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.

Optimal muscular coordination strategies for jumping

This paper presents a detailed analysis of an optimal control solution to a maximum height squat jump, based upon how muscles accelerate and contribute power to the body segments during the ground contact phase of jumping. Quantitative comparisons of model and experimental results expose a proximal-to-distal sequence of muscle activation (i.e. from hip to knee to ankle). We found that the contribution of muscles dominates both the angular acceleration and the instantaneous power of the segments. However, the contributions of gravity and segmental motion are insignificant, except the latter become important during the final 10% of the jump. Vasti and gluteus maximus muscles are the major energy producers of the lower extremity. These muscles are the prime movers of the lower extremity because they dominate the angular acceleration of the hip toward extension and the instantaneous power of the trunk. In contrast, the ankle plantarflexors (soleus, gastrocnemius, and the other plantarflexors) dominate the total energy of the thigh, though these muscles also contribute appreciably to trunk power during the final 20% of the jump. Therefore, the contribution of these muscles to overall jumping performance cannot be neglected. We found that the biarticular gastrocnemius increases jump height (i.e. the net vertical displacement of the center of mass of the body from standing) by as much as 25%. However, this increase is not due to any unique biarticular action (e.g. proximal-to-distal power transfer from the knee to the ankle), since jumping performance is similar when gastrocnemius is replaced with a uniarticular ankle plantarflexor.

Grand Challenge Competition to Predict In Vivo Knee Loads

Impairment of the human neuromusculoskeletal system can lead to significant mobility limitations and decreased quality of life. Computational models that accurately represent the musculoskeletal systems of individual patients could be used to explore different treatment options and optimize clinical outcome. The most significant barrier to model‐based treatment design is validation of model‐based estimates of in vivo contact and muscle forces. This paper introduces an annual “Grand Challenge Competition to Predict In Vivo Knee Loads” based on a series of comprehensive publicly available in vivo data sets for evaluating musculoskeletal model predictions of contact and muscle forces in the knee. The data sets come from patients implanted with force‐measuring tibial prostheses. Following a historical review of musculoskeletal modeling methods used for estimating knee muscle and contact forces, we describe the first two data sets used for the first two competitions and summarize four subsequent data sets to be used for future competitions. These data sets include tibial contact force, video motion, ground reaction, muscle EMG, muscle strength, static and dynamic imaging, and implant geometry data. Competition participants create musculoskeletal models to predict tibial contact forces without having access to the corresponding in vivo measurements. These blinded predictions provide an unbiased evaluation of the capabilities and limitations of musculoskeletal modeling methods. The paper concludes with a discussion of how these unique data sets can be used by the musculoskeletal modeling research community to improve the estimation of in vivo muscle and contact forces and ultimately to help make musculoskeletal models clinically useful.

Storage and utilization of elastic strain energy during jumping.

Based upon the optimal control solutions to a maximum-height countermovement jump (CMJ) and a maximum-height squat jump (SJ), this paper provides a quantitative description of how tendons and the elastic elements of muscle store and deliver energy during vertical jumping. After confirming the ability of the model to replicate the major features of each jump (i.e. muscle activation patterns, body-segmental motions, ground reaction forces, jump height, and total ground contact time), the time histories of the forces and shortening velocities of all the musculotendon actuators in the model were used to calculate the work done on the skeleton by tendons as well as the series-elastic elements, the parallel-elastic elements, and the contractile elements of muscle. We found that all the elastic tissues delivered nearly the same amount of energy to the skeleton during a CMJ and an SJ. The reason is twofold: first, nearly as much elastic strain energy was stored during the SJ as the CMJ; second, more stored elastic strain energy was lost as heat during the CMJ. There was also a difference in the way energy was stored during each jump. During the CMJ, strain energy stored in the elastic tissues came primarily from the gravitational potential energy of the skeleton as the more proximal extensor muscles were stretched during the downward phase of the jump. During the SJ, on the other hand, energy stored in the elastic tissues came primarily from the contractile elements as they did work to stretch the tendons and the series-elastic elements of the muscles. Increasing tendon compliance in the model led to an increase in elastic energy storage and utilization, but it also decreased the amount of energy delivered by the contractile elements to the skeleton. Jump height therefore remained almost the same for both jumps. These results suggest that elastic energy storage and utilization enhance jumping efficiency much more than overall jumping performance.

A Parameter Optimization Approach for the Optimal Control of Large-Scale Musculoskeletal Systems

This paper describes a computational method for solving optimal control problems involving large-scale, nonlinear, dynamical systems. Central to the approach is the idea that any optimal control problem can be converted into a standard nonlinear programming problem by parameterizing each control history using a set of nodal points, which then become the variables in the resulting parameter optimization problem. A key feature of the method is that it dispenses with the need to solve the two-point, boundary-value problem derived from the necessary conditions of optimal control theory. Gradient-based methods for solving such problems do not always converge due to computational errors introduced by the highly nonlinear characteristics of the costate variables. Instead, by converting the optimal control problem into a parameter optimization problem, any number of well-developed and proven nonlinear programming algorithms can be used to compute the near-optimal control trajectories. The utility of the parameter optimization approach for solving general optimal control problems for human movement is demonstrated by applying it to a detailed optimal control model for maximum-height human jumping. The validity of the near-optimal control solution is established by comparing it to a solution of the two-point, boundary-value problem derived on the basis of a bang-bang optimal control algorithm. Quantitative comparisons between model and experiment further show that the parameter optimization solution reproduces the major features of a maximum-height, countermovement jump (i.e., trajectories of body-segmental displacements, vertical and fore-aft ground reaction forces, displacement, velocity, and acceleration of the whole-body center of mass, pattern of lower-extremity muscular activity, jump height, and total ground contact time).

Dependence of Cruciate-Ligament Loading on Muscle Forces and External Load

A sagittal-plane model of the knee is used to predict and explain the relationships between the forces developed by the muscles, the external loads applied to the leg, and the forces induced in the cruciate ligaments during isometric exercises. The geometry of the model bones is adapted from cadaver data. Eleven elastic elements describe the geometric and mechanical properties of the cruciate ligaments, the collateral ligaments, and the posterior capsule. The model is actuated by 11 musculotendinous units, each unit represented as a three-element muscle in series with tendon. For isolated contractions of the quadriceps, ACL force increases as quadriceps force increases for all flexion angles between 0 and 80 degrees; the ACL is unloaded at flexion angles greater than 80 degrees. When quadriceps force is held constant, ACL force decreases monotonically as knee-flexion angle increases. The relationship between ACL force, quadriceps force, and knee-flexion angle is explained by the geometry of the knee-extensor mechanism and by the changing orientation of the ACL in the sagittal plane. For isolated contractions of the hamstrings, PCL force increases as hamstrings force increases for all flexion angles greater than 10 degrees; the PCL is unloaded at flexion angles less than 10 degrees. When hamstrings force is held constant, PCL force increases monotonically with increasing knee flexion. The relationship between PCL force, hamstrings force, and knee-flexion angle is explained by the geometry of the hamstrings and by the changing orientation of the PCL in the sagittal plane. At nearly all knee-flexion angles, hamstrings co-contraction is an effective means of reducing ACL force. Hamstrings co-contraction cannot protect the ACL near full extension of the knee because these muscles meet the tibia at small angles near full extension, and so cannot apply a sufficiently large posterior shear force to the leg. Moving the restraining force closer to the knee-flexion axis decreases ACL force; varying the orientation of the restraining force has only a small effect on cruciate-ligament loading.