An inverse dynamics approach based on the fundamental equations of constrained motion and on the theory of optimal control

In this investigation, a new algorithm for the nonlinear control of mechanical systems is developed. The method proposed in this paper can be used for the forward and inverse dynamics of nonlinear mechanical systems. For this purpose, the Udwadia-Kalaba equations, also known as the fundamental equations of constrained motion, are combined with the feedback control strategy resulting from the theory of optimal control applied to linear time-invariant dynamical systems. In forward dynamics problems, the fundamental equations of constrained motion allow for explicitly calculating the generalized constraint forces associated with a nonlinear set of kinematic constraints. Furthermore, in inverse dynamic problems, the Udwadia-Kalaba equations can be effectively used for computing the generalized control forces that impose a prescribed dynamic behavior to the mechanical system under consideration. In this dual case, the desired dynamic behavior is described in terms of nonlinear algebraic equations that play the role of the kinematic joints encountered in the direct problem of forward dynamics. Conversely, it is shown in this work that the mathematical tool of the optimal control theory can be employed for the practical design of an effective compensation controller that improves the performance of the nonlinear control laws devised by using the fundamental equations of constrained motion. Employing the new approach proposed in this paper, the compensation controller designed by using the theory of optimal control is fully integrated into the nonlinear set of control laws obtained considering the general form of the Udwadia-Kalaba equations. The method developed in this paper is tested by means of numerical experiments. For this purpose, the nonlinear dynamic equations of a physical pendulum are used in order to exemplify the analytical developments carried out in this work and for assessing in a straightforward manner the performance of the proposed methodology.