On the domain and error characterization in the singular perturbation modeling of closed kinematic chains

This paper addresses two important issues in modeling closed kinematic chains (CKC) as singularly perturbed systems, namely, the validity domain and the error characterization. The singular perturbation formulation (SPF) is obtained by replacing the algebraic constraint equation in an index-1 differential algebraic equations (DAE) model with an artificial fast dynamics. We first show that the SPF model has a larger validity domain than the DAE model, and boundaries of the domain are easy to determine. We then characterize the error between the SPF model and the DAE model by deriving explicit error bounds. Sufficient conditions that guarantee exponential convergence of the model error are established. We verify the analysis by simulating the dynamics of a CKC mechanism, the rice planar delta robot, and validating the simulation results with experimental data obtained on the real robot.