Three-dimensional data-tracking dynamic optimization simulations of human locomotion generated by direct collocation
Introduction
Non-invasive measurement of muscle forces is infeasible thus computational modelling is needed to evaluate these quantities in vivo. Optimization theory is often used to address the mechanically redundant nature of the human locomotor system (Pedotti et al., 1978, Hardt, 1978, Zajac, 1993, Collins, 1995, Pandy, 2001). Computed muscle control (CMC) combines static optimization with linear feedback control to calculate muscle forces by tracking measurements of body-segmental motions (Thelen et al., 2003). CMC has been used in conjunction with OpenSim (Delp et al., 2007), an open-source musculoskeletal modelling and simulation environment, to generate dynamic simulations of walking (Arnold et al., 2007, Liu et al., 2008), running (Hamner et al., 2010) and landing from a jump (Mokhtarzadeh et al., 2014). However, this method is limited in several ways. First, static optimization solves a discrete set of optimization problems along the movement trajectory rather than evaluating the cost function over time (Crowninshield and Brand, 1981, Anderson and Pandy, 2001, Lin et al., 2012). Second, ground reaction forces obtained from experiment are applied directly to the model causing inconsistencies between the model-computed body-segmental motions and the measured ground reaction forces. These inconsistencies in the model dynamics require the application of artificial or residual forces and torques that potentially affect the calculated values of muscle forces. Third, predictive simulations of movement using dynamic optimization theory cannot be pursued because foot-ground contact is not simulated in the model. Finally, a large set of nonlinear differential equations are solved using explicit integration techniques that require small time steps and have poor convergence properties, particularly when the dynamic equations of motion are stiff; for example, when bodies with relatively small masses such as the patella are included in the model.
Muscle forces may also be calculated by using dynamic optimization to solve data-tracking problems in which the model-computed kinematics and kinetics are constrained to reproduce corresponding experimental data. Direct shooting is one of the most common techniques used to track experimental measurements within the framework of dynamic optimization theory. This method is implemented by discretizing the control variables (e.g., muscle excitations) while the state variables are found by integrating the system dynamic equations forward in time (Pandy et al., 1992). Unfortunately, convergence to an optimal solution may require several hours or even days of CPU time (Neptune et al., 2000, McLean et al., 2003). Menegaldo et al. (2006) combined a direct shooting method with inverse dynamics analysis to reproduce desired torque profiles at a lower computational cost. Although a significant advantage of applying dynamic optimization is that the cost function is evaluated over time, the equations for neuromusculoskeletal dynamics are still solved using explicit numerical integration when implementing direct shooting.
Implicit methods such as direct collocation convert the dynamic equations of motion into algebraic constraints by discretizing the state and control variables thus eliminating the need for explicit integration. Direct collocation has been used with relatively simple two-dimensional models to produce data-tracking dynamic optimization simulations of pedaling (Kaplan and Heegaard, 2001), walking (van den Bogert et al., 2011) and running (Miller and Hamill, 2015). De Groote et al. (2016) recently combined direct collocation with a three-dimensional (3D) musculoskeletal model to calculate muscle forces in normal walking, but foot-ground interaction was not explicitly simulated.
The overall goal of the present study was to assess the feasibility of generating accurate, computationally-efficient, data-tracking dynamic optimization solutions for human locomotion by combining the computational power of direct collocation with the flexibility and efficiency of OpenSim. Our specific aim was to implement direct collocation on a 3D neuromusculoskeletal model created in OpenSim to calculate lower-limb muscle forces for walking and running compatible with body-segmental motions and foot-ground forces obtained from experiment.
Section snippets
Human experiments
Gait data were collected from five healthy adult males (age: 26 ± 4 years; height: 178 ± 4 cm; weight: 70 ± 5 kg) at the Biomotion Laboratory of the University of Melbourne. The study was approved by the University’s Human Research Ethics Committee and informed written consent was obtained from each participant prior to data collection. Participants were asked to walk at their preferred speeds (1.4 ± 0.1 m/s) and to run at a prescribed speed of 2 m/s. Marker-trajectory and ground-reaction-force data were
Results
Direct collocation took on average 2.7 ± 1.0 h and 2.2 ± 1.6 h to compute the optimal solutions for walking and running, respectively. CPU time per iteration was similar for the two tasks (0.5 ± 0.0 and 0.5 ± 0.1 min for walking and running, respectively) but the total time taken to solve the walking problem was greater because more iterations were needed to converge to the optimal solution (320 ± 116 iterations for walking compared to 250 ± 171 iterations for running).
Model-predicted body-segmental
Discussion
The aim of this study was to assess the feasibility of generating accurate, computationally-efficient, data-tracking, dynamic optimization simulations of human locomotion by implementing direct collocation on a detailed 3D neuromusculoskeletal model created in OpenSim. Model-predicted body-segmental displacements, ground reaction forces and muscle excitation patterns were consistent with experimental gait data obtained for both walking and running gaits.
Direct collocation is potentially more
Conflict of interest
None of the authors have a conflict of interest in relation to the work reported here.
Acknowledgements
This work was supported by a Discovery Projects Grant from the Australian Research Council (DP160104366).
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