In order to obtain the stabilizing feedback gains, an algebraic Riccati equation is solved relative to the discretized linear system. Furthermore, the quadratic performance index is exponentially weighted (see Chapter Four) so that a high degree of stability is achieved. For our design purposes, the degree of stability is chosen so that all poles of the closed-loop linearized system are guaranteed to lie within a circle of radius 0.7.
Solving an algebraic Riccati equation is a standard procedure used in selecting feedback gains for time-invariant linear systems. However, when the nominal trajectory is chosen to give translation of the hand, the linearized system becomes time-varying. In this case we assume that the slowly time-varying approach can be employed and hence, the feedback gains are selected to give stability to the family of frozen-time* systems. Here we employ a technique given in [24] which requires that an algebraic Riccati equation be solved for only several (6 in this case) frozen-time systems. Lagrange interpolation [34] is then used for calculating the gains at intermediate points.
Let us now consider the simulation procedure. Because the controller is a discrete-time system, updating the dynamics of the controller is easily accomplished using computer simulations. In order to update the dynamics of the plant (i.e., the manipulator) it is necessary to employ numerical integration techniques. Here we use a Runge-Kutta
numerical integration algorithm [19] which numerically integrates the nonlinear dynamic equation modeling the manipulator. To make the simulations realistic, the Runge-Kutta algorithm takes 10 steps to iterate
The term "frozen-time system" is used to denote the timeinvariant system obtained by "freezing" all time-varying parameters of a time-varying system.