Optimal Control Approach for Gait Transition with Riemannian Splines

TL;DR

This work addresses the challenge of smooth, energy-efficient gait transitions in robotic locomotion by formulating a geometric optimal control framework on curved shape spaces. It recasts gait transitions as boundary-value problems derived from a variational principle, accommodating drag-dominated and inertia-dominated regimes with distinct cost metrics. The authors demonstrate path-, acceleration-, and torque-optimal gait transitions for two three-link swimmers in viscous and perfect-fluid environments, using an indirect shooting method to solve the resulting boundary-value problems. The findings highlight how the choice of cost shapes the transition trajectories and point toward a broad, geometry-based pathway for designing gait-based controllers in complex locomotion systems.

Abstract

Robotic locomotion often relies on sequenced gaits to efficiently convert control input into desired motion. Despite extensive studies on gait optimization, achieving smooth and efficient gait transitions remains challenging. In this paper, we propose a general solver based on geometric optimal control methods, leveraging insights from previous works on gait efficiency. Building upon our previous work, we express the effort to execute the trajectory as distinct geometric objects, transforming the optimization problems into boundary value problems. To validate our approach, we generate gait transition trajectories for three-link swimmers across various fluid environments. This work provides insights into optimal trajectory geometries and mechanical considerations for robotic locomotion.

Figures (4)

• Figure 1: The optimal gait transition for inertia-dominated systems from the gait generating the forward motion to the gait generating the turning motion. (a) The three different gait-switching trajectories starting from the same point in metric-weighted space. $\alpha_1$ and $\alpha_2$ are the joint angles of the swimmer. The red curve denotes the torque-optimal trajectory, the blue curve denotes the acceleration-optimal trajectory, and the black curve denotes the geodesic trajectory. The geodesic trajectory, the shortest curve between points, nonsmoothly connects to each gait. The torque-optimal and acceleration-optimal trajectories smoothly connect to each gait. (b) The body trajectory generated by the gait switching scenario: a sequence consisting of the forward gait, the torque-optimal gait transition, and the turning gait. The center of mass markers and black arrows denote where the swimmer's head is pointing at each time step. The first gray, red, and the second gray curve respectively indicate the body trajectory generated by the forward gait, the torque-optimal gait transition, and the turning gait.
• Figure 2: (a) Schematic of a three-link viscous swimmer. The shape of the system is described by two joint angles and is a two-dimensional shape space. $\alpha$ denotes the joint angles. $x$-$y^b$ denotes the body coordinate of the system. (b) Visualization of the metric field of viscous three-link swimmers. Infinitesimal circles at each point indicate how much the shape space distorts in each direction in comparison to the actual space. Physically, it means how much effort is required to produce shape changes in a given direction at each point. Longer ellipse axes denote directions in which shape changes are cheaper. (c) The projection of the actual space onto the flat 2D plane that minimizes distortion. Because the actual space is curved, the approximation must distort when forced into the flat plane. there are still some elliptically of the infinitesimal circles. The pathlength in this manifold can approximate the actual effort to change from one shape to another shape.
• Figure 3: The variational curves (a) in the spherical parameter space (b) and the spherical space, respectively. The spherical manifold is parameterized by the polar angle $\theta \in [-\pi/2,\pi/2]$ and the azimuthal angle $\phi \in [-\pi,\pi]$. Suppose that the variation family $\Gamma(s,t)$ is parameterized by one parameter $s$, and the curves in the family have the same initial and final points. The black curves are in the same variational family. The black solid curve is a geodesic. The shortest path among curves in the family is a geodesic curve, and the covariant acceleration is zero along the geodesic curve. The red arrow indicates the variation field, $V(s,t)$, which tells the infinitesimal difference between the neighborhood of the geodesic.
• Figure 4: The gait switching trajectories from the forward gait to the turning gait. The results of columns respectively are distinguished by path-optimal trajectories for viscous swimmers, acceleration-optimal trajectories for perfect fluid swimmers, and torque-optimal trajectories for perfect fluid swimmers in order. The gait-switching scenario includes a one-cycle and extra execution of forward gait (the first gray curves), gait transition (red curves), and a one-cycle execution of turning gait (the second gray curves). Especially, the lower-left and upper-right gray closed curve in the first row of figures denotes the forward gait and turning gait, respectively. The velocity directions of both gaits are clockwise. The figures in the first row represent the shape trajectory. The second row shows the changes in forward net displacement for a single scenario, while the third row illustrates the changes in turning net displacement. Finally, the fourth row displays the changes in accumulated cost. The length unit in the figure is the link length of the swimmers. The red curves denote the gait-switching trajectories. Each marker in the first row on the forward gait denotes where the initial points are. The circle marker in the first row denotes that the solver generates the corresponding trajectory, and the diagonal cross marker in the first row indicates that the solver fails to generate the corresponding trajectory. The gait-switching trajectories start from the forward gait and join into the target gait smoothly. From the second row of figures, the numbers in the figure distinguish the trajectories based on different initial points. The red star marker tells when the execution of the gait-switching trajectory is ended. The gray circle marker tells when the execution of the forward gait or the turning gait is ended.