The Indirect Method for Generating Libraries of Optimal Periodic Trajectories and Its Application to Economical Bipedal Walking

Abstract

Trajectory optimization is an essential tool for generating efficient and dynamically consistent gaits in legged locomotion. This paper explores the indirect method of trajectory optimization, emphasizing its application in creating optimal periodic gaits for legged systems and contrasting it with the more commonly used direct method. While the direct method provides considerable flexibility in its implementation, it is limited by its input space parameterization. In contrast, the indirect method improves accuracy by defining control inputs as functions of the system's states and costates. We tackle the convergence challenges associated with indirect shooting methods, particularly through the systematic development of gait libraries by utilizing numerical continuation methods. Our contributions include: (1) the formalization of a general periodic trajectory optimization problem that extends existing first-order necessary conditions for a broader range of cost functions and operating conditions; (2) a methodology for efficiently generating libraries of optimal trajectories (gaits) utilizing a single shooting approach combined with numerical continuation methods, including a novel approach for reconstructing Lagrange multipliers and costates from passive gaits; and (3) a comparative analysis of the indirect and direct shooting methods using a compass-gait walker as a case study, demonstrating the former's superior accuracy in generating optimal gaits. The findings underscore the potential of the indirect method for generating families of optimal gaits, thereby advancing the field of trajectory optimization in legged robotics.

Figures (5)

• Figure 1: The compass-gait walker on a slope ($\gamma$). The robot's parameters are normalized by total mass $m$, gravity $g$ and leg length $l_\mathrm{o}$. Its proportions are defined as $a/b=1$ and $m_\mathrm{h}/m_\mathrm{l}=2$.
• Figure 2: Projections of the implicit curve $\boldsymbol{r}_\gamma^{-1}(\boldsymbol{0})$, computed using Algorithm \ref{['algo:OptimalContinuation']} with the continuation parameter $\sigma := \gamma$. The curve corresponds to a fixed average speed of $v_\text{avg} = 0.1~\sqrt{gl_\text{o}}$ and originates from the passive gait family with the shorter period time $T_\text{short}$. As the curve progresses towards level slope at $\gamma = 0^\circ$ (marked by $\ast$), it intersects the passive gait family with the longer period time $T_\text{long}$. The filled circles indicate points where the passive and optimal gaits coincide.
• Figure 3: Visualization of an optimal walking gait at an average velocity of $v_\text{avg} = 0.1 \sqrt{gl_\text{o}}$ on a level slope ($\gamma=0^\circ$). The keyframes depict the compass-gait walker, while the input and state trajectories (swing leg "sw" and stance leg "st") are shown for both the indirect (red) and direct (black) shooting methods. The black curves, representing the direct method, correspond to an input parameterization using a single cubic B-spline with $n_\xi=4$.
• Figure 4: Comparison of the direct ($\hat{\boldsymbol{r}}$) and indirect ($\boldsymbol{r}$) approaches in a MATLAB implementation on an Intel Core i5-8500 CPU @ 3.00GHz with 8GB RAM. In the direct method, the input space is parameterized with either Bézier polynomials or cubic B-Splines, introducing $n_\xi$ variables. The data in a) and b) corresponds to walking on a level slope ($\gamma = 0^\circ$) at an average speed of $v_\text{avg} = 0.1 \sqrt{gl_\text{o}}$. The correspondig (locally) optimal gaits were generated via continuation from a passive gait using Algorithm \ref{['algo:OptimalContinuation']}, as shown in Fig. \ref{['fig:passive2active']}.
• Figure 5: Projections of the implicit curves derived from the indirect method $\boldsymbol{r}_{v_\text{avg}}^{-1}(\boldsymbol{0})$ and the direct method $\hat{\boldsymbol{r}}_{v_\text{avg}}^{-1}(\boldsymbol{0})$, both computed using Algorithm \ref{['algo:OptimalContinuation']} with the continuation parameter $\sigma = v_\text{avg}$. These curves represent level-slope walking ($\gamma = 0^\circ$) and originate from an optimal gait at $v_\text{avg} = 0.1~\sqrt{gl_\text{o}}$ (denoted by $\ast$). The indirect method consistently yields gaits with lower costs, whereas the direct method (utilizing cubic B-splines with $n_\xi=4$) can distinguish local minima (solid black line) from saddle points (dotted black line) by applying second-order optimality conditions.

Theorems & Definitions (3)

• Remark 2.1
• Definition : Passive-Optimal Solution
• Remark 4.1