Towards Solutions of Manipulation Tasks via Optimal Control of Projected Dynamical Systems

TL;DR

This work presents a practical framework for planning manipulation tasks with contact and friction by modeling dynamics as a projected dynamical system and solving a discretized optimal control problem via a dynamical complementarity system. Contacts are encoded with signed distance functions, with ellipsoidal objects handled by a convex SDF formulation that guarantees a unique contact point, and friction is incorporated through a quasi‑static complementarity model. The discrete OCP is solved as a sequence of relaxed MPCCs using Finite Elements with Switch Detection (FESD) and Scholtes relaxation, demonstrated on planar pushing scenarios including collaboration between pushers. The approach demonstrates feasible, computationally tractable trajectories for frictionless and frictional planar manipulation, and lays the groundwork for extending to more shapes and three‑dimensional friction.

Abstract

We introduce a modeling framework for manipulation planning based on the formulation of the dynamics as a projected dynamical system. This method uses implicit signed distance functions and their gradients to formulate an equivalent gradient complementarity system. The optimal control problem is then solved via a direct method, discretized using finite-elements with switch detection. An extension to this approach is provided in the form of a friction formulation commonly used in quasi-static models. We show that this approach is able to generate trajectories for problems including multiple pushers, friction, and non-convex objects modeled as unions of convex ellipsoids with reasonable computational effort.

Figures (3)

• Figure 1: Several frames of the solution for the frictionless manipulation problem.
• Figure 2: Plots of solution to the frictionless pushing problem.
• Figure 3: Several frames of the solution for the manipulation problem with $\mu_f=0.5$.