This study evaluated whether a global particle swarm optimization algorithm could solve two large-scale human movement problems possessing 660 design variables. Both problems utilized a three-dimensional, 27 DOF, full-body gait model whose joint and inertial parameters were calibrated to gait data collected from a single patient with knee OA. The cost function for both problems sought to reduce the peak knee adduction torque either by increasing the foot progression angle (i.e., the toe out gait optimization) or by minimizing the adduction torque directly without any changes in foot path (i.e., the modified gait optimization). Penalty terms were included to ensure that the predicted gait motions were physically realistic. The PAPSO algorithm achieved an improved solution only for the toe out gait optimization problem, whereas a gradient-based NLS algorithm achieved significant and realistic improvements for both problems. These findings suggest that global optimization algorithms proven to work well for small- to medium-scale problems may not work well for large-scale human movement problems. Researchers should not assume that an optimization solution is the global minimum simply because a global algorithm was used to generate it, and they should exercise caution when extrapolating the performance of global optimizers to problems involving hundreds of design variables.
There are at least three reasons why the PAPSO algorithm performed worse than did the NLS algorithm on these two benchmark problems. First, no tuning was performed of PAPSO algorithm parameters. It is possible that our PAPSO algorithm could have performed better had time-consuming algorithm parameter tuning been performed. Such tuning was not performed in the present study since few researchers take the time to do it (i.e., researchers want to find algorithms that work well “out of the box”), and since studies that evaluate optimization algorithm performance normally do not tune the algorithm parameters to the specific problems being investigated [13
]. Second, the shape of the design space made it difficult to locate the global minimum without the use of gradient information. Though the design space was smooth for both sample problems, the minimum was located in a small deep hole in design space (), making it extremely difficult for a particle to migrate into this hole using only stochastic updates. Third, only a single set of penalty weights was used for both optimizations problems rather than ramping up the weights slowly over a sequence of optimization runs [33
]. A sequence of optimizations with increasing penalty weights was not performed because of the excessive computation time required by this approach. Furthermore, the selected weights posed no problem for the NLS algorithm, suggesting that the PSO algorithm may have convergence difficulties for problems involving penalty terms. Differences in optimization algorithm performance cannot be attributed to differences in problem formulation, since both algorithms used exactly the same formulations. In fact, the PSO algorithm possesses the advantage of insensitivity to design variable scaling, which gradient-based algorithms with finite-difference gradients do not possess [24
Visualization of the design space for the modified gait optimization problem when two design variables are varied within their initial bounds and all other design variables are held constant.
Since the global optimization results reported in this study were based on a single optimization algorithm, it is not clear that similar poor performance would be observed for other global algorithms. However, this will likely be the case based on previous benchmark problems solved with a synchronous version of our PSO algorithm and several other global algorithms [24
]. At a minimum, researchers who desire to use a global optimization algorithm on a large-scale human movement problem should compare their results with those from a gradient-based algorithm as a check.
The best result for both optimizations was consistent with recent experimental observations. The toe out gait optimization predicted that increasing the foot progression angle by 15 deg would reduce the second but not the first peak of knee adduction torque curve. When Guo et al
] increased the experimental toe out angle by 15 deg, they found little change in the first adduction torque peak and roughly a 40% reduction in the second one. When the subject in our study increased his toe out angle by approximately 15 deg, the experimental reduction in the second peak was approximately 30% with little change in the first one [8
]. By comparison, our NLS toe out gait optimization predicted a 32.2% reduction in the second peak and only a slight change in the first one, with both peaks being within ± 1 standard deviation of the mean values measured from our subject over three trials. The modified gait optimization predicted that internally rotating the hips so as to medialize the knees would reduce both peaks of the knee adduction torque curve. When Davis et al
] recently trained a healthy subject to walk with the hips internally rotated and the knees medialized, they observed a 28% reduction in the first knee adduction torque peak. Our NLS modified gait optimization predicted a comparable reduction of 32.4%.
In summary, this study evaluated the ability of a global particle swarm optimization algorithm to solve large-scale human movement problems. Though the PSO algorithm used in our study has performed well on small- to medium-scale benchmark and biomechanical optimization problems [26
], it did not perform well on the two large-scale human movement problems investigated in this study, with a gradient-based nonlinear least squares algorithm performing better on both problems. Significant algorithm parameter tuning or use of a global-local hybrid algorithm may be necessary for PSO and other global optimizers to solve large-scale human movement problems. Based on the results of this study, the PSO algorithm is not recommended for solving large-scale human movement optimization problems possessing constraints or competing terms in the cost function. Researchers should exercise caution when evaluating the results of large-scale human movement optimization problems solved using global algorithms.