Towards Geometric Motion Planning for High-Dimensional Systems: Gait-Based Coordinate Optimization and Local Metrics

TL;DR

A gait-based coordinate optimization method that overcomes the curse of dimensionality is proposed, and a unified geometric representation of locomotion is identified by generalizing various nonholonomic constraints into local metrics.

Abstract

Geometric motion planning offers effective and interpretable gait analysis and optimization tools for locomoting systems. However, due to the curse of dimensionality in coordinate optimization, a key component of geometric motion planning, it is almost infeasible to apply current geometric motion planning to high-dimensional systems. In this paper, we propose a gait-based coordinate optimization method that overcomes the curse of dimensionality. We also identify a unified geometric representation of locomotion by generalizing various nonholonomic constraints into local metrics. By combining these two approaches, we take a step towards geometric motion planning for high-dimensional systems. We test our method in two classes of high-dimensional systems - low Reynolds number swimmers and free-falling Cassie - with up to 11-dimensional shape variables. The resulting optimal gait in the high-dimensional system shows better efficiency compared to that of the reduced-order model. Furthermore, we provide a geometric optimality interpretation of the optimal gait.

Figures (6)

• Figure 1: Optimal gaits for the forward motion of a 12-link low Reynolds number swimmer and the pitch-rotational motion of a full model Cassie solved by the proposed high-dimensional geometric motion planning.
• Figure 2: The local connection (left) and CCF (right) of the forward ($x$) direction for a 3-link low Reynolds number swimmer in the optimal coordinates corresponding to the optimal gait solved by the proposed algorithm. The thickness of the gait corresponds to the pace, where a thicker pattern indicates slower movement.
• Figure 3: The displacements resulting from the optimal gaits of the 12-link low Reynold number swimmer and the full model Cassie, measured respectively in their original frames attached to the central joint and torso, and in the optimal coordinates solved using the proposed method. Rigid body rotations are converted to exponential coordinates $\xi$.
• Figure 4: Optimal gaits for each model of the low Reynolds number swimmer. The illustration shows the configuration of four quarter points within each gait's period. The displacement is scaled by the gait period to demonstrate efficiency. Each gait consumes unit power dissipation.
• Figure 5: Optimal gaits for the 12-link low Reynolds number swimmer (top) and the full model Cassie (bottom) in the 2D subspace with isometric 3D embedding. The contours show the CCF projected onto the subspace (left), and the CCF flux capture rate of the subspace relative to the maximum attainable CCF in the original shape space (right). Red, gray, and black represent positive, zero, and negative values of CCF, and represent 1, 0.5, and 0 for the capture rate. The thickness of the gait corresponds to the pace, where a thicker pattern indicates slower movement.