Elastic Stability of Cosserat Rods and Parallel Continuum Robots

Abstract

Classic theories in nonlinear elasticity have increasingly been used to obtain accurate and efficient models for continuum robots and other elastic structures. Numerically computed solutions of these models typically satisfy the first-order conditions necessary for equilibrium, but do not provide any information about the elastic stability of the solution. The inability to detect or avoid physically unstable model solutions poses a major hindrance to reliable model-based simulation, planning, design, and control. In this paper, we adapt results from optimal control to determine the stability of Kirchhoff rods and Cosserat rods subject to general end constraints, including coupled multirod models which describe parallel continuum robots. We formulate a sufficient condition for the stability of a solution, a numerical test for evaluating this condition, and a heuristic stability metric. We verify that our numerical stability test agrees with the classical results for the buckling of single columns with various end constraints and for multicolumn frames. We then validate our approach experimentally on a six degree-of-freedom parallel continuum robot.