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268                            Robust Control of Robotic Manipulators

            of  q  for robots with prismatic joints, and that (5.2.10) is satisfied by
                                                                so that  a=(µ 2 -
            µ 1 )/(µ 2 +µ 1 )) [Spong and Vidyasagar 1987]. Finally, (5.2.10) is a result of the
            properties of the Coriolis and centripetal terms discussed in Section 3.3.
              We will give different representative designs of the feedback-linearization
            approach, starting with controllers whose behavior is studied using Lyapunov
            stability theory.


            Lyapunov Designs

            Static feedback compensators have been extensively used starting with the
            works of [Freund 1982] and [Tarn et al. 1984]. Consider the controller
            introduced in (4.4.13):


                                                                      (5.2.12)


            such that

                                                                      (5.2.13)


            It can be seen that by placing the poles of A c sufficiently far in the left half-
            plane, the robust stability of the closed- loop system in the presence of   is
            guaranteed. This was shown true in [Arimoto and Miyazaki 1985] for the
            case where                 as described in Theorem 4.4.1 and Example
            4.4.3. It was also shown true for the trajectory-following problem assuming
            that              in [Dawson et al. 1990] as described in Theorem 4.4.2.
            There are as many robust controllers designed using Lyapunov stability
            concepts as there are ways of choosing Lyapunov function candidates, and
            of designing the gain K to guarantee that the Lyapunov function candidate is
            decreasing along the trajectories of (5.2.13). To decrease the asymptotic
            trajectory error, however, excessively large gains may be required (see Example
            4.4.3). We therefore choose to use the passivity theorem and a choice of the
            gain matrix K that renders the linear part of the closed-loop system SPR. As
            described in Section 2.11, an output may be chosen to make the closed-loop
            system SPR; therefore allowing large passive uncertainties in the knowledge
            of  M(q). In fact, the state-feedback controller may be used to define an
            appropriate output Ke such that the input-output closed-loop linear systems
                      -1
            K(sI-A+BK) B is strictly positive real (SPR). Consider the following closed-
            loop linear system:





            Copyright © 2004 by Marcel Dekker, Inc.
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