Global Contact-Rich Planning with Sparsity-Rich Semidefinite Relaxations

TL;DR

This paper introduces a novel approach to contact-rich motion planning by exploiting sparsity in polynomial optimization, enabling fast, near-global solutions and demonstrating effectiveness in simulation and real-world robotics tasks.

Contribution

It develops high-order sparse semidefinite relaxations for contact-rich planning, leveraging robot kinematic sparsity, and releases the SPOT toolbox for automated sparsity exploitation.

Findings

SDP relaxations solve in seconds with small suboptimality

Effective in simulation and real-world robotics tasks

Demonstrates near-global optimality in complex planning problems

Abstract

We show that contact-rich motion planning is also sparsity-rich when viewed as polynomial optimization (POP). We can exploit not only the correlative and term sparsity patterns that are general to all POPs, but also specialized sparsity patterns from the robot kinematic structure and the separability of contact modes. Such sparsity enables the design of high-order but sparse semidefinite programming (SDPs) relaxations--building upon Lasserre's moment and sums of squares hierarchy--that (i) can be solved in seconds by off-the-shelf SDP solvers, and (ii) compute near globally optimal solutions to the nonconvex contact-rich planning problems with small certified suboptimality. Through extensive experiments both in simulation (Push Bot, Push Box, Push Box with Obstacles, and Planar Hand) and real world (Push T), we demonstrate the power of using convex SDP relaxations to generate global contact-rich motion plans. As a contribution of independent interest, we release the Sparse Polynomial Optimization Toolbox (SPOT)--implemented in C++ with interfaces to both Python and Matlab--that automates sparsity exploitation for robotics and beyond.