Page 207 -
P. 207

190                                       Computed-Torque Control

              There are some upper limits on the choice for   n  [Paul 1981]. Although the
            links of most industrial robots are massive, they may have some flexibility.
            Suppose that the frequency of the first flexible or resonant mode of link i is
                                                                      (4.4.23)
            with J the link inertia and k r  the link stiffness. Then, to avoid exciting the
            resonant mode, we should select   n <  r /2. Of course, the link inertia J changes
            with the arm configuration, so that its maximum value might be used in
            computing   r .
              Another upper bound on   n  is provided by considerations on actuator
            saturation. If the PD gains are too large, the torque τ(t) may reach its upper
            limits.
              Some more feeling for the choice of the PD gains is provided from error-
            boundedness considerations as follows. The transfer function of the closed-
            loop error system in (4.4.15) is
                                                                      (4.4.24)


            or if K v and K p are diagonal,

                                                                      (4.4.25)



                                                                      (4.4.26)

                                               -1
              We assume that the disturbance and M  are bounded (Table 3.3.1), so that
                                                                      (4.4.27)
                      —
                 —
            with m and d known for a given robot arm. Therefore,

                                                                      (4.4.28)


                                                                      (4.4.29)

              Now selecting the L 2—norm, the operator gain ||H(s)|| 2 is the maximum
            value of the Bode magnitude plot of H(s). For a critically damped system,

                                                                      (4.4.30)

            Therefore,




            Copyright © 2004 by Marcel Dekker, Inc.
   202   203   204   205   206   207   208   209   210   211   212