Towards Tight Convex Relaxations for Contact-Rich Manipulation

TL;DR

This paper tackles global motion planning for contact-rich manipulation by formulating the problem as a shortest path in a graph of convex sets (GCS), where each vertex encodes a convex relaxation (via semidefinite programming) of the nonconvex, bilinear dynamics within a fixed contact mode. By combining tight per-mode SDP relaxations with the GCS framework, the authors obtain a single convex program whose solution can be rounded to a feasible contact-rich trajectory, while providing an upper bound on optimality. The experimental results in planar pushing show near-global optimality (average gap around 10%) and higher success rates than a state-of-the-art contact-implicit method, including successful hardware demonstrations on a real robot. This approach enables efficient global reasoning over discrete mode sequences and continuous dynamics, and generalizes to more complex multi-contact scenarios beyond planar pushing.

Abstract

We present a novel method for global motion planning of robotic systems that interact with the environment through contacts. Our method directly handles the hybrid nature of such tasks using tools from convex optimization. We formulate the motion-planning problem as a shortest-path problem in a graph of convex sets, where a path in the graph corresponds to a contact sequence and a convex set models the quasi-static dynamics within a fixed contact mode. For each contact mode, we use semidefinite programming to relax the nonconvex dynamics that results from the simultaneous optimization of the object's pose, contact locations, and contact forces. The result is a tight convex relaxation of the overall planning problem, that can be efficiently solved and quickly rounded to find a feasible contact-rich trajectory. As an initial application for evaluating our method, we apply it on the task of planar pushing. Exhaustive experiments show that our convex-optimization method generates plans that are consistently within a small percentage of the global optimum, without relying on an initial guess, and that our method succeeds in finding trajectories where a state-of-the-art baseline for contact-rich planning usually fails. We demonstrate the quality of these plans on a real robotic system.

Figures (6)

• Figure 1: The experimental planar pushing setup. A cylindrical finger is attached to a robotic arm that is pushing a T-shaped object on the table into its target configuration.
• Figure 2: a) The slider-pusher kinematic quantities. b) The contact point and the contact forces.
• Figure 3: a) An example of a configuration-space partitioning $\mathcal{Q}_1, \ldots, \mathcal{Q}_4$ and the linear approximations $\phi_1, \ldots, \phi_4$ for a slider with convex planar geometry. b) The graph of non-contact modes that is added between every pair of vertices corresponding to contact modes $\mathcal{C}_i$ and $\mathcal{C}_j$. Corresponding modes are written in text next to the vertices.
• Figure 4: Our planner simultaneously reasons about both discrete mode switches and continuous motion. Here, an example of a planar pushing plan with multiple mode switches is shown for a T-shaped slider geometry.
• Figure 5: Our method is able to generate close-to globally optimal plans for pushing tasks with collision-free motion planning between contact modes. Here, two different pushing trajectories for a T-shaped slider are shown, stabilized with a feedback controller on a real robotic system.