A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework

Abstract

The paper deals with controlled mechanical systems in which the number of control inputs is equal to the number of desired system outputs, and is smaller than the number of degrees of freedom of the system. The determination of control input strategy that force the underactuated system to complete the partly specified motion is a challenging problem. In the present formulation, the outputs (performance goals), expressed in terms of system states, are treated as constraints on the system—called control or program constraints as distinct from contact constraints in the classical sense, and a mathematical resemblance of the inverse control problem to the constrained system dynamics is exploited. However, while the reactions of contact constraints act in the directions orthogonal to the respective constraint manifold, the available control reactions may have arbitrary directions with respect to the program constraint manifold, and in the extreme may be tangent. A specific methodology must then be developed to find the solution of such “singular” problems, related to a class of control tracking problems such as position control of elastic joint robots, control of cranes, and aircraft control in prescribed trajectory flight. The governing equations of the problem arise as a set of differential-algebraic equations (DAEs), and an effective method for solving the DAEs, based on backward Euler method, is proposed. The open-loop control formulation obtained this way is then extended by a closed-loop control law to provide stable tracking of the required reference trajectories in the presence of perturbations. Some examples of applications of the theory and results of numerical simulations are reported.