Preferred Jacobian Differentiation and Direct Collocation Methods for an Efficient and Accurate Walker Gait Optimization

Abstract

Devices based on walker robotics research sometimes require a reference trajectory for the control systems of the devices to track. One valuable class of methods used to find an optimal reference trajectory is direct collocation, but even after selecting a method like direct collocation, several optimization design decisions remain. In order to determine the most desirable optimization settings, 600 optimizations were performed for the trajectory of a two degree-of-freedom (DOF) compass gait walker, and 200 optimizations were performed for a five-DOF link walker. These runs evaluated various combinations of optimization settings, including: numerical vs. symbolic vs. automatic differentiation; trapezoidal vs. Hermite Simpson collocation; numerical vs. symbolic calculation of joint accelerations; and inclusion or exclusion of joint accelerations in the decision variables. The different generated gaits were then compared in terms of computational efficiency and accuracy. The results showed that including joint accelerations as decision variables eliminated alternative gaits but increased computational complexity and variability. Symbolic acceleration evaluation was preferable when automatic differentiation was excluded. Automatic differentiation was shown to be significantly faster than the other two differentiation methods for both walking models. In addition, Hermite-Simpson collocation, although slower than trapezoidal, was the more accurate of the two approaches. These results can be applied to the derivation of optimal reference joint trajectories in future robotics applications.