Algorithmic differentiation improves the computational efficiency of OpenSim-based trajectory optimization of human movement

Abstract

Algorithmic differentiation (AD) is an alternative to finite differences (FD) for evaluating function derivatives. The primary aim of this study was to demonstrate the computational benefits of using AD instead of FD in OpenSim-based trajectory optimization of human movement. The secondary aim was to evaluate computational choices including different AD tools, different linear solvers, and the use of first-or second-order derivatives. First, we enabled the use of AD in OpenSim through a custom source code transformation tool and through the operator overloading tool ADOL-C. Second, we developed an interface between OpenSim and CasADi to solve trajectory optimization problems. Third, we evaluated computational choices through simulations of perturbed balance, two-dimensional predictive simulations of walking, and three-dimensional tracking simulations of walking. We performed all simulations using direct collocation and implicit differential equations. Using AD through our custom tool was between 1.8 ± 0.1 and 17.8 ± 4.9 times faster than using FD, and between 3.6 ± 0.3 and 12.3 ± 1.3 times faster than using AD through ADOL-C. The linear solver efficiency was problem-dependent and no solver was consistently more efficient. Using second-order derivatives was more efficient for balance simulations but less efficient for walking simulations. The walking simulations were physiologically realistic. These results highlight how the use of AD drastically decreases computational time of trajectory optimization problems as compared to more common FD. Overall, combining AD with direct collocation and implicit differential equations decreases the computational burden of trajectory optimization of human movement, which will facilitate their use for biomechanical applications requiring the use of detailed models of the musculoskeletal system. Algorithmic differentiation speeds up trajectory optimization of human movement PLOS ONE | https://doi.Fig 2.Flowchart depicting the optimal control framework. We developed two approaches (AD-ADOLC and AD-Recorder) to make an OpenSim function F and its forward (F fwd) and reverse (F rev) directional derivatives available within the CasADi environment for use by the NLP solver during the optimization. In the AD-ADOLC approach (top), ADOL-C's algorithms are used in a C++ code to provide F fwd and F rev. In the AD-Recorder approach (bottom), Recorder provides the expression graph of F as MATLAB source code from which CasADi's C-code generator generates C-code containing F, F fwd, and F rev. The AD-Recorder approach combines operator overloading, when generating the expression graph, and source code transformation, when processing the expression graph to generate C-code for F, F fwd, and F rev. In both approaches, the code comprising F, F fwd, and F rev is compiled as a Dynamic-link Library (DLL), which is imported as an external function within the CasADi environment. In our application, F represents the multi-body dynamics and is called when formulating the optimal control problem. The latte...

Efficient trajectory optimization for curved running using a 3D musculoskeletal model with implicit dynamics

“…Future studies are needed to compare the efficiency of algorithmic and analytical derivatives in optimal control simulations. Previously, it was shown that using algorithmic derivatives is faster than finite differences which are used by OpenSim 43 . Machine learning approaches were recently investigated in the field of computer graphics to speed up musculoskeletal simulation 44 , 45 .…”

Physics-Based Simulations to Predict the Differential Effects of Motor Control and Musculoskeletal Deficits on Gait Dysfunction in Cerebral Palsy: A Retrospective Case Study

“…We formulated our problems in MATLAB using CasADi (Andersson et al, 2019), applied direct collocation using a third order Radau quadrature collocation scheme with 150 mesh intervals per gait cycle, and solved the resulting nonlinear programming problems with the solver IPOPT (Wächter and Biegler, 2006). We applied algorithmic differentiation to compute derivatives (Falisse et al, 2019a). We started each optimization from multiple initial guesses and selected the result with the lowest optimal cost.…”

OpenSim Moco: Musculoskeletal optimal control

“…Supporting mesh refinement would allow the solver to increase the number of mesh intervals in time ranges with fast dynamics, thereby improving accuracy. Computing the nonlinear program derivatives with algorithmic differentiation instead of finite differences would vastly improve the speed of Moco, but would require substantial changes to the source code of OpenSim and its dependencies [40].…”

“…More recent versions of CasADi support finite differences and CasADi understands the structure of the nonlinear program objective and constraint functions, allowing for potentially more efficient finite difference calculations than with Tropter, which treats the nonlinear program objective and constraints as blackbox functions [63]. If OpenSim provides exact derivatives in the future, we can exploit the algorithmic differentiation modes in Tropter and CasADi [40]. Those distributing Moco as a dependency of closed-source software may prefer distributing Moco without CasADi, as CasADi's "weak copyleft" GNU Lesser General Public License 3.0 places requirements on how CasADi is redistributed.…”

“…The advantages of direct collocation have led biomechanists to use the method for tracking motions [16,23], predicting motions [24][25][26][27][28][29][30][31][32][33], fitting muscle properties [34], and optimizing design parameters [35]. Researchers have made key methodological advances, including efficiently handling multibody and muscle dynamics via implicit formulations [36,37], minimizing energy consumption [38,39], and employing algorithmic differentiation to simulate complex models more rapidly compared to using finite differences [40]. Along with these methodological advances, researchers have discovered that minimizing an energy-related cost produces non-physiological motions during walking [24], skipping is the most efficient gait on our moon [41], and unilateral amputees can improve gait symmetry with only a minor increase in effort [42].…”