Geometric Gait Optimization for Kinodynamic Systems Using a Lie Group Integrator

TL;DR

This work tackles gait optimization for kinodynamic locomotion by forging a unified dynamic-kinematic model via Lagrangian reduction and Lie group integrators, enabling gradient-based optimization of periodic and transitional gaits. It introduces a four-tier motion-planning scheme—steady-state, acceleration, transitions, and turning—that leverages momentum dynamics and energy constraints to generate reusable gait primitives. The framework is demonstrated on three representative systems (roller racer, snakeboard, and intermediate-Reynolds-number swimmer) with both simulation and hardware validation on the roller racer, showing effective acceleration, steady-state cruising, and turning with manageable energy losses. The approach offers a scalable, geometry-preserving path to planning complex motions in kinodynamic locomotion, with potential for applying geometric insights to broader robotic and biological locomotion problems.

Abstract

This paper presents a gait optimization and motion planning framework for a class of locomoting systems with mixed kinematic and dynamic properties. Using Lagrangian reduction and differential geometry, we derive a general dynamic model that incorporates second-order dynamics and nonholonomic constraints, applicable to kinodynamic systems such as wheeled robots with nonholonomic constraints as well as swimming robots with nonisotropic fluid-added inertia and hydrodynamic drag. Building on Lie group integrators and group symmetries, we develop a variational gait optimization method for kinodynamic systems. By integrating multiple gaits and their transitions, we construct comprehensive motion plans that enable a wide range of motions for these systems. We evaluate our framework on three representative examples: roller racer, snakeboard, and swimmer. Simulation and hardware experiments demonstrate diverse motions, including acceleration, steady-state maintenance, gait transitions, and turning. The results highlight the effectiveness of the proposed method and its potential for generalization to other biological and robotic locomoting systems.

Figures (8)

• Figure 1: Execution of the comprehensive motion plan computed using the proposed method on the roller racer robot. The green curve depicts the robot's position trajectory, and the composite figure illustrates the robot's configuration at different times. The roller racer has only one actuated degree of freedom: the steering angle between the front and rear wheels, with all wheels being passive and driven by anti-slip constraints and steering angle changes. The motion plan incorporates multiple gaits and transitions, enabling the robot to: a) start from rest and nominal shape into the gait cycle, b) accelerate to the optimal steady-state motion, c) smoothly switch to the steady-state gait that maintains maximum speed, d) execute a turn with minimal energy loss and return to steady-state motion, and e) end the gait cycle and return to the nominal shape. Throughout the execution, the motion plan maintains the desired heading angle and meets the average power consumption constraints.
• Figure 2: Illustration and key characteristics of the three kinodynamic systems considered in this paper. The roller racer has four degrees of freedom (DOF): three DOFs for position, describing its pose in the world, and one DOF for shape (the heading angle between the two carts). The system includes two nonholonomic constraints, corresponding to the non-lateral slip constraints of the wheelsets on the two carts, resulting in one DOF of nonholonomic momentum. Linear viscous friction occurs in the driving direction of each wheelset. Similarly, the snakeboard has five DOFs: three DOFs for position and two DOFs for shape. The shape is defined by the rotor angle and the cart angle, with the assumption that the two carts rotate together in opposite directions. The two nonholonomic constraints, corresponding to the non-lateral slip constraints of the wheelsets, also result in one DOF of nonholonomic momentum. Linear viscous friction also acts in the driving direction of each wheelset. Finally, the intermediate Reynolds number swimmer has five DOFs: three DOFs for position and two DOFs for shape, defined by the joint angles between its three links. Unlike the other systems, it has no direct nonholonomic constraints, so its momentum corresponds to the three DOFs for position. Instead, fluid dynamics manifest as anisotropic fluid-added mass and hydrodynamic drag acting on each link.
• Figure 3: Demonstration of gradient computation for variational gait optimization based on Lie group integrator. For a principally kinematic system, $\accentset{\circ}{g} = 10u^{\intercal}$, the position trajectory $g\vert_{u=u_0}$ is represented by the solid line. The variation in the trajectory due to a control change $\Delta u$ is computed by summing the gradient contributions from the control change at each segment along the trajectory. The contribution of each segment is depicted as the dashed trajectory. Due to group symmetries, a control change at any segment does not affect the shape of the remaining trajectory, similar to rotating the joints of a robotic arm.
• Figure 4: Illustration of the optimal gaits and transitions for the roller racer (top) and their demonstration on the robot (middle: acceleration-transition-steady-state gait; bottom: steady-state-left-turn-steady-state and steady-state-right-turn-steady-state gaits). In the demonstration, the dashed line represents the robot’s position trajectory, with its color indicating the corresponding gait being executed at each point.
• Figure 5: Illustration of the optimal gaits and transitions for the snakeboard (top) and their demonstration on the robot (middle: acceleration-transition-steady-state gait; bottom: steady-state-left-turn-steady-state and steady-state-right-turn-steady-state gaits). The thickness of the gait cycles indicates the pacing of the gait, with thick lines representing slower shape changes and thin lines representing faster changes. Black arcs with arrows denote the direction of the gait. In the demonstration, the dashed line represents the robot’s position trajectory, with its color indicating the corresponding gait being executed at each point.