Affine Geometric Heat Flow and Motion Planning for Dynamic Systems

Abstract

We present a new method for motion planning for control systems. The method aims to provide a natural computational framework in which a broad class of motion planning problems can be cast; including problems with holonomic and non-holonomic constraints, drift dynamics, obstacle constraints and constraints on the magnitudes of the applied controls. The method, which finds its inspiration in recent work on the so-called geometric heat flows and curve shortening flows, relies on a hereby introduced partial differential equation, which we call the affine geometric heat flow, which evolves an arbitrary differentiable path joining initial to final state in configuration space to a path that meets the constraints imposed on the problem. From this path, controls to be applied on the system can be extracted. We provide conditions guaranteeing that the controls extracted will drive the system arbitrarily close to the desired final state, while meeting the imposed constraints and illustrate the method on three canonical examples.

Geometric Heat Flow Method for Legged Locomotion Planning

“…The core of the planning algorithm is the affine geometric heat flow (AGHF) [5]. This flow starts from a curve x(t, 0) joining x init to x fin , shown by black line in Fig.…”

“…We then proceed to find such curves of minimal length via a homotopy initialized at an arbitrary curve joining x init to x fin . We refer the reader to [5] for a detailed presentation.…”

“…In this paper, we show that the geometric method we proposed for motion planning extends naturally to handle hybrid dynamics. Precisely, we show how the Ansatz developed in [4,5,6], contending that motion planning problems can be encoded into Riemannian metrics, can be applied and using fast solvers for parabolic partial differential equations (PDE), we can obtain efficiently natural motions.…”

“…The evolution of trajectory from arbitrary curve to planned motion is given by the geometric heat flow (GHF), whose solution is obtained via a PDE solver, instead of optimization solvers used in TO. These methods were extended in [5], in which the Affine Geometric Heat Flow (AGHF) is introduced to handle systems with drift. Motions for robot systems are generated using AGHF in [19,6].…”

“…The proposed algorithm can encode variety of constraints including contact constraints. Meanwhile, the convergence is guaranteed at given order, as explained in [5].…”