Udwadia-Kalaba Approach for Three Link Manipulator Dynamics With Motion Constraints

Abstract

Aiming to dynamic modeling of a three-link manipulator subjected to motion constraints, a novel explicit approach to the dynamical equations based on Udwadia-Kalaba (UK) theory is established. The motion constraints on the three-link manipulator can be regarded as external constraints of the system. However, it is not easy to obtain explicit equations for the dynamic modeling of constrained systems. For a multibody system subjecting to motion constraints, it is common to introduce Lagrange multipliers, but obtaining an explicit dynamical equation using traditional Lagrange multipliers is difficult. In order to obtain such equations more simply, motion constraints are handled using the UK equation. Compared with the Lagrange method, the UK approach can simplify the analysis and solution of a constrained system, without the need to introduce additional auxiliary variables to solve the Lagrange equation. Based on a more real-life nominal system (whose parameters are known) model considering the uncertain environment, this paper develops a nonlinear controller that satisfies the required trajectory. This controller allows the nonlinear nominal system to track the desired trajectory exactly without linearizations or approximations. These continuous controllers compensate extra force to eliminate the errors caused by uncertainties. The controllers are based on a generalization of sliding surfaces. Error bounds on tracking caused by uncertainties are analytically obtained. The numerical results show the simplicity and efficacy of the proposed methodology, and the reliability of the error bounds.