SDRS: Shape-Differentiable Robot Simulator

TL;DR

SDRS addresses the challenge of differentiable robot simulation when robot shapes undergo large geometric and topological changes. It achieves this by representing each robot link as a union of convex polyhedra and by formulating contact with a separating plane that acts as a zero-mass auxiliary body, yielding a barrier energy that remains differentiable. The framework provides position-level dynamics, differentiable convex-contact and friction models, and an adjoint-based gradient flow enabling end-to-end co-design of shape and control with proven differentiability properties. Practical benchmarks demonstrate improved co-design performance and the ability to deform shapes and topology to achieve robust grasping and locomotion, highlighting SDRS’s potential for scalable, gradient-based robot design.

Abstract

Robot simulators are indispensable tools across many fields, and recent research has significantly improved their functionality by incorporating additional gradient information. However, existing differentiable robot simulators suffer from non-differentiable singularities, when robots undergo substantial shape changes. To address this, we present the Shape-Differentiable Robot Simulator (SDRS), designed to be differentiable under significant robot shape changes. The core innovation of SDRS lies in its representation of robot shapes using a set of convex polyhedrons. This approach allows us to generalize smooth, penalty-based contact mechanics for interactions between any pair of convex polyhedrons. Using the separating hyperplane theorem, SDRS introduces a separating plane for each pair of contacting convex polyhedrons. This separating plane functions as a zero-mass auxiliary entity, with its state determined by the principle of least action. This setup ensures global differentiability, even as robot shapes undergo significant geometric and topological changes. To demonstrate the practical value of SDRS, we provide examples of robot co-design scenarios, where both robot shapes and control movements are optimized simultaneously.

Figures (19)

• Figure 1: Starting from an initial guess (left), we optimize the shape of gripper to firmly grasp the Stanford bunny (right).
• Figure 2: We illustrate the co-design problem of a robot arm trying to reach a target apple, with the robot's configuration space being the 3 joint angles (red). We first represent each robot link shape as a set of (possibly overlapping) convex polyhedrons (middle), e.g. using V-HACD, where we use different colors for different rigid bodies. A connectivity constraint denoted as $x_{ijk}$ is introduced for each pair of overlapping convex polyhedrons (red). Our SDRS is a differentiable robot simulator tailored to the convex-polyhedron-based representation, where we can differentiate with respect to the polyhedron vertices. Such representation allows both geometry and topology changes by continuously modifying the vertices, e.g. to drill a hole in the middle of the brown link (right).
• Figure 3: The constraint to ensure that the attachment point $x_i^{\lambda(i)}$ overlaps both links.
• Figure 4: The separating hyperplane (dashed) between a pair of convex polyhedrons with normal $n$ (red). The closest distance (red) between them is realized by $x_{ij}$ and $x_{i'j'}$.
• Figure 5: We assume the separating plane is moving at an in-plane angular speed of $\omega$ and linear speed of $u$. Each vertex $T_i(d,\theta^t)x_{ij}^m$ is projected onto the plane as $B^{ij,i'j'}T_i(d,\theta^t)x_{ij}^m$, to compute the relative velocity.

Theorems & Definitions (32)

• Lemma 7.1
• proof
• Lemma 7.2
• proof
• Lemma 7.3
• proof
• Lemma 7.4
• proof
• Lemma 7.5
• proof