In this paper, we propose a novel approach to the control of closed kinematic chains (CKCs). This method is based on a recently developed singularly perturbed model for CKCs. Conventionally, the dynamics of CKCs are described by differential-algebraic equations (DAEs). Our approach transfers the control of the original DAE system to the control of an artificially created singularly perturbed system in which the slow dynamics corresponds to the original DAE when the perturbation parameter tends to zero. Compared to control schemes that rely on solving nonlinear algebraic constraint equations, the proposed method uses an ordinary differential equation (ODE) solver to obtain the dependent coordinates, hence, eliminates the need for Newton-type iterations and is amenable to real-time implementation. The composite Lyapunov function method is used to show that the closed-loop system, when controlled by typical open kinematic chain schemes, achieves asymptotic trajectory tracking. Simulations and experimental results on a parallel robot, the Rice planar Delta robot, are also presented to illustrate the efficacy of our method.
Asymptotic integration of singularly perturbed linear systems of differential-algebraic equations
