Comparison of Direct Collocation Optimal Control to Trajectory Optimization for Parameter Identification of an Ellipsoidal Foot–Ground Contact Model

Foot–ground contact models play an important role in the accuracy of predictive human gait simulations, and there is a need for a computationally-efficient dynamic contact model for predictive and evaluative studies. In this research, we generated symbolic dynamic equations for a 2D torque-driven 11-DOF human model with a 3D ellipsoidal volumetric foot–ground contact model. The main goal was to increase the prediction accuracy and decrease the computation time for human gait analyses compared to the previous studies that used numerical formulations and point foot–ground contact models.

A data-tracking optimization was developed to identify the contact parameters of the human gait model using two optimization approaches: trajectory optimization and optimal control. The first approach is developed with a global search algorithm based on inverse dynamics. In this algorithm, a local optimizer is repeatedly run from multiple potential start points to select the best start point while satisfying the constraints and reaching the lowest cost function value. The second approach is developed using direct collocation based on implicit dynamics. In this method, the optimization problem is solved using a variable-order adaptive orthogonal collocation method along with sparse nonlinear programming.

Optimal control was superior to trajectory optimization for identifying a large number of parameters; the simulated torques and ground reaction forces from the optimal control correlated better with the experimental data. For the optimal control, the root-mean-square errors of the resultant torques, tangential and normal ground reaction forces were 0.48 (N.m), 14.07 (N), and 26.44 (N), respectively. However, for the trajectory optimization, these errors were 15.19 (N.m), 36.51 (N), and 234.57 (N). Thus, the optimized contact model from the optimal control, which was developed symbolically and based on volumetric contact equations, is a suitable foot–ground contact model for predictive human gait simulations. Additionally, we demonstrated that optimal control could be used to predict the motion and torque for the metatarsal joints, which are not easily measurable in practice.