Constrained (closed-loop) robot simulation by local constraint propagation

Abstract

Robot mechanisms during many typical interactions with their environment have long posed a difficulty to computer simulation. This is due to the coupling of effects from the base and the tip (the closed loop problem), often complicated by mixtures of known and unknown endpoint forces and accelerations. This paper presents a linear-time (recursive) algorithm for the closed-loop forward dynamics of kinematic chains under very general environmental endpoint constraints. The statement of very complex systems of endpoint constraints is particularly easy, and the algorithm is conceptually simple and physically intuitive. The algorithm is then generalized to branching kinematic chains (trees), where every endpoint may be subject to environmental (closed-loop) constraints. The time complexity for constrained kinematic trees remains also linear in the number of joints. An appendix sketches an extension of the algorithm to general kinematic structures with one-dof joints, but containing arbitrary internal closed loops. The time complexity with internal loops is no worse than O(n) + O(l <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , for n joints and l internal loops; for many structures the algorithm attains O(n)+ O[l\log(l)].