  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.
