Control of closed kinematic chains using a singularly perturbed dynamic model

In this paper, we propose a new method to the control of closed kinematic chains (CKC). This method is based on a recently developed singularly perturbed model for CKC. Conventionally, the dynamics of CKC are described by differential-algebraic equations (DAE). 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 small perturbation parameter tends to zero. Compared to control schemes which rely on the solution of nonlinear algebraic constraint equations, the proposed method uses an 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 local 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.