A Basic Mechanical and Geometric Framework for Quasi-Static Manipulation

TL;DR

This paper introduces a geometric framework for analyzing quasi-static robotic manipulation using energy methods, force-space concepts, and control Hessians, demonstrated through an elastically-driven inverted pendulum example.

Contribution

It develops a novel geometric and energy-based approach to quasi-static manipulation, including a new metric and control Hessian analysis, with applications to robotic contact problems.

Findings

Derived a metric for quasi-static manipulation tasks.

Proposed the squared-Hessian for control stability analysis.

Applied methods to an elastically-driven inverted pendulum.

Abstract

In this work, we propose a geometric framework for analyzing mechanical manipulation, for instance, by a robotic agent. Under the assumption of conservative forces and quasi-static manipulation, we use energy methods to derive a metric. In the first part of the paper, we review how quasi-static mechanical manipulation tasks can be naturally described via the so-called force-space, i.e. the cotangent bundle of the configuration space, and its Lagrangian submanifolds. Then, via a second order analysis, we derive the control Hessian of total energy. As this is not necessarily positive-definite, from an optimal control perspective, we propose the use of the squared-Hessian, also motivated by insights derived from both mechanics (Gauss' Principle) and biology (Separation Principle). In the second part of the paper, we apply such methods to the problem of an elastically-driven, inverted pendulum. Despite its apparent simplicity, this example is representative of an important class of robotic manipulation problems for which we show how a smooth elastic potential can be derived by regularizing mechanical contact. We then show how graph theory can be used to connect each numerical solution to `nearby' ones, with weights derived from the very metric introduced in the first part of the paper.