Control of dynamical systems governed by constrained coordinates

In this paper we develop feedback control laws for systems described by constrained generalized coordinates. The use of constrained coordinates leads to differentialalgebraic equations of motion. Previous research on controlling systems has concentrated primarily on using mathematical models in terms of independent coordinates. For several complex dynamical systems, it is more desirable to develop a mathematical model using a constrained coordinate formulation in terms of dependent coordinates. Examples include closed kinematic chains, vehicle dynamics and path tracking problems. Research in the last few decades has led to several advances in the treatment and in obtaining the solution of differential algebraic equations. We take advantage of these advances and introduce the constrained coordinate formulation to control. We design feedback control laws based on a pointwise-optimal formulation for systems described in terms of dependent generalized coordinates. We also use the constrained formulation in pathtracking control problems where one traditionally uses independent coordinates. Because the constrained formulation permits explicit minimization of the tracking error higher levels of accuracy are reached. Introduction With the advent of powerful digital computers, symbolic manipulation software, as well as simulation codes, it has become possible in recent years to develop the mathematical model and simulate the motion of complex dynamical systems. As systems become more complex, their modeling and control becomes more and more challenging, prompting one to look into additional methods of modeling and control. One such approach has been the use of constrained generalized coordinates. Describing motion in terms of constrained coordinates results in a set of differential-algebraic equations of motion [1]. The equations are differential with respect to the generalized coordinates and algebraic with respect to 1 Graduate Assistant. '* Associate Professor. Associate Fellow, AIAA. the Lagrange multiplier, because the derivative of the Lagrange multiplier does not appear in the formulation. The increased interest in constrained coordinates has been fueled by several advances that have made it possible to analyze and to solve for the response of systems described by differential algebraic equations [e.g.,1-15]. There is now powerful software like DASSL that has been developed to integrate differential-algebraic equations [4]. Systems that can be described by constrained coordinates include bodies subjected to nonholonomic constraints, such as vehicle motion, three-dimensional motion described in terms of Euler parameters, and problems involving friction. In addition, in many systems subjected to holonomic constraints, writing the equations of motion in terms of independent coordinates leads to lengthy and cumbersome expressions. A typical example to this is closed kinematic chains. In all examples considered above, the equations of motion have a much simpler form when expressed in terms of dependent coordinates. After one generates the constrained equations of motion one can eliminate the Lagrange multipliers and the redundant coordinates, but the resulting equations usually become very tedious. Such lengthy equations make it difficult to design a control law in terms of independent coordinates and to implement it on-line. Work on control of systems described by differential-algebraic equations has been restricted to low degree of freedom or application dependent systems, such as simple guidance systems or singular perturbation problems. In recent years, general results on existence of solutions and stability of control for differential-algebraic systems have been addressed in [17-19]. In this paper, we add actual feedback control design to those developments. We extend the pointwise-optimal control law developed in [20] to systems described by differential-algebraic equations. We implement the pointwise-optimal control law on a four bar linkage mechanism. We also use the constrained formulation in pathtracking problems where one traditionally uses independent coordinates. We show that because the constrained formulation perCopyright © 1996 by Haim Baruh. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. mils explicit minimization of the tracking error higher levels of accuracy can be reached. General Form of the System Equations Consider a system with n degrees of freedom. Lagrange's equations for a set of n independent coordinates are d 3L 3L (i) Consider now a system described by m generalized coordinates and acted upon by p constraints, expressed in the general velocity form m = 0. k= l,2,...,p;j=l,2, ..., m (2) which can also be written in vector form as AGI& + tea. t) = a (3) in which A is a matrix of order pxm, and b is a vector of order p . The vector a, = [qj q2 ... qm]. We now have n = m-p degrees of freedom. Introducing p Lagrange mulv tipliers Xj, X2, ... , Xp the equations of motion in the presence of constraints can be shown to be [16]