Enhancing Robotic System Robustness via Lyapunov Exponent-Based Optimization

TL;DR

The paper tackles the challenge of quantifying and improving robustness in robotic systems by introducing a Lyapunov-exponent based metric, $L_{lambda}$, computed over long horizons via differentiable simulation. This approach provides a gradient-friendly means to analyze and co-design both hardware and control policies, even in the presence of contact-rich dynamics and limit cycles. Key contributions include a formalization of Lyapunov exponents for high-dimensional robotic systems, a differentiable computation framework, and empirical demonstrations on a planar manipulator, a 16-DOF spider, and a crawling quadruped showing improved robustness metrics and stability properties. The work offers a scalable methodology for robustness-driven design that can influence practical co-design workflows, while outlining limitations related to gradient quality over long simulations and the need for richer policies in future work.

Abstract

We present a novel approach to quantifying and optimizing stability in robotic systems based on the Lyapunov exponents addressing an open challenge in the field of robot analysis, design, and optimization. Our method leverages differentiable simulation over extended time horizons. The proposed metric offers several properties, including a natural extension to limit cycles commonly encountered in robotics tasks and locomotion. We showcase, with an ad-hoc JAX gradient-based optimization framework, remarkable power, and flexi-bility in tackling the robustness challenge. The effectiveness of our approach is tested through diverse scenarios of varying complexity, encompassing high-degree-of-freedom systems and contact-rich environments. The positive outcomes across these cases highlight the potential of our method in enhancing system robustness.

Figures (11)

• Figure 1: The transition map $\Phi$ maps an uncertainty of radius $\varepsilon$ around $x_i$ into an ellipsoid $\varepsilon^\top \text{d}\Phi^\top(x_i) \text{d}\Phi(x_i) \varepsilon$.
• Figure 2: Van der Pol oscillator $\mu=2$, on the left the Lyapunov spectrum is approximated for 100 different starting conditions. On the right, the trajectories in the state space are shown (starting points in blue). The shape of the attractor can be seen in light orange.
• Figure 3: Parallelized Lyapunov spectrum computation: first a forward simulation is performed and then the computation of the spectrum is parallelized with respect to the states using the simulator gradients.
• Figure 4: Framework for co-optimization of hardware and policy parameters with differentiable simulation.
• Figure 5: Robot manipulator scheme