Gait Switching and Enhanced Stabilization of Walking Robots with Deep Learning-based Reachability: A Case Study on Two-link Walker

TL;DR

The paper addresses stability assurance for learning-based legged locomotion by extending Hamilton-Jacobi reachability with deep learning to hybrid dynamics. It builds a parameterized RoA library across multiple gaits using a DeepReach-inspired neural network, and introduces a one-step predictive stabilizing controller along with a RoA-guided gait-switching strategy. Key findings show that the learned RoAs can improve stabilization success and enable safe gait transitions under perturbations, with a two-link walker case study illustrating superior performance to model-based controllers. This approach enhances explainability and reliability in learning-based locomotion, offering a path toward scalable, transparent stability guarantees for legged robots.

Abstract

Learning-based approaches have recently shown notable success in legged locomotion. However, these approaches often lack accountability, necessitating empirical tests to determine their effectiveness. In this work, we are interested in designing a learning-based locomotion controller whose stability can be examined and guaranteed. This can be achieved by verifying regions of attraction (RoAs) of legged robots to their stable walking gaits. This is a non-trivial problem for legged robots due to their hybrid dynamics. Although previous work has shown the utility of Hamilton-Jacobi (HJ) reachability to solve this problem, its practicality was limited by its poor scalability. The core contribution of our work is the employment of a deep learning-based HJ reachability solution to the hybrid legged robot dynamics, which overcomes the previous work's limitation. With the learned reachability solution, first, we can estimate a library of RoAs for various gaits. Second, we can design a one-step predictive controller that effectively stabilizes to an individual gait within the verified RoA. Finally, we can devise a strategy that switches gaits, in response to external perturbations, whose feasibility is guided by the RoA analysis. We demonstrate our method in a two-link walker simulation, whose mathematical model is well established. Our method achieves improved stability than previous model-based methods, while ensuring transparency that was not present in the existing learning-based approaches.

Figures (7)

• Figure 1: (a) Configuration of the two-link walker (grey: stance leg, blue: swing leg). (b) Hybrid limit cycle gaits in $q_1$-$q_2$ space with various walking step lengths. Black line indicates the switching surface. (c) $q_1$-$q_2$ slice of the numerical (top) and learned (bottom) target function $l(x;\beta)$ for $\beta=0.13$.
• Figure 2: Comparison between the value functions $V(T,x;\beta)$ for the gait with $\beta\!=\!0.13$, obtained by (a) the numerical method in mitchell2004toolboxchoi2022, and (b) (c) our method, where in (b), $V_\theta$ is not parametrized and $\beta$ is fixed as 0.13. The values are visualized in color contour map in $q_1-q_2$ slices along the gait. The zero-level sets (thick white line) represent the estimated BRTs.
• Figure 3: $q_1$-$q_2$ slices of gait BRTs (a) when angular velocities are 0, (b) values that lie on the gait with $\beta=0.13$ (* indicates where the slice is taken). Each BRT captures states from which the robot can be stabilized to the corresponding gait, and in the overlap region, pursuing any gait is feasible.
• Figure 4: Snapshots of the phase portrait of the trajectory, initialized at a perturbed state, stabilizing to the gait with $\beta=0.13$, under the OSP controller \ref{['eq:osp_problem']}. The trajectory evolves from yellow to blue while taking two walking steps, and we show the first walking step portion. Green dots represent the state where the snapshot is taken. The color contour map visualizes $V_\theta(t_i, \cdot;\beta)$ in \ref{['eq:osp_problem']}. [https://youtu.be/P7Vnr8jwSPc (https://youtu.be/P7Vnr8jwSPc)]
• Figure 5: Success rate of stabilization evaluated over a grid of 6,600 initial states. At each $(q_1, q_2) \in [-0.3, 0.3]\times[-1.0,1.0]$ value, we evaluate 25 combinations of $(\dot{q}_1, \dot{q}_2) \in \times[-1.0, 1.0]\times[-2.5, 2.5]$, and visualize the rate of the trajectories successfully converging to the gait ($\beta=0.13$).