Symbolic formulation of closed chain dynamics in independent coordinates

摘要

This article presents a symbolic formulation, in independent coordinates, of the equations of motion for multibody systems containing closed loops (closed chains). We first extend the recursive algorithm for open chain dynamics as derived in Ref.[18] to tree topology chains, and show how the corresponding recursive algorithm can be easily recast into an analytic set of closed-form equations. We then formulate the equations of motion for closed chains by applying D'Alembert's Principle to thereduced system ([15, 17]). Central to this approach is the evaluation of the transformation between the external generalized forces of the closed chain and its reduced system, and also its derivative. For general multibody systems these expressions become extremely complex, particularly the derivative of the transformation. We show that by formulating the equations of motion in a modern screw theoretic setting, and by taking advantage of some fundamental Lie algebraic identities, symbolic expressions forthese quantities can be obtained. The resulting formulation not only significantly reduces the symbolic complexity of the equations of motion, but also enables exact computation of the dynamics without resorting to numerical approximation. The formulation is illustrated with a three degrees of freedom (d.o.f.) parallel manipulator example.