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212                                       Computed-Torque Control































                              Figure 4.4.10: (Cont) (c) w n =50 rad/s.



            robot controls designer to understand. See [Franklin et al. 1986] for a reference
            on classical control theory.
              A simplified dynamical model of a robot arm with electric actuators may
            be written as (Section 3.6)


                                                                      (4.4.52)


            with J m  the actuator motor inertia, B m  the rotor damping constant, k m  the
            torque constant, k b  the back emf constant, R the armature resistance, and r i
            the gear ratio for joint i. The motor angle is denoted θ i (t). The constant portions
            of the diagonal elements of M(q) are denoted m ii . The time-varying portions
            of these elements, as well as the off-diagonal elements of M(q), the nonlinear
            terms N(q,  ), and any disturbances   d  are all lumped into the disturbance
            d i (t). Thus d i (t) contains the effects on joint i of all the other joints. The control
            input is the motor armature voltage v i (t).
              Note that predominantly motor parameters appear in this equation. In
            fact, if the gear ratio is small, even m ii  may be neglected. For this reason, if
            the gear ratio is small, the robot arm control problem virtually reduces to the
            problem of controlling the actuator motors.


            Copyright © 2004 by Marcel Dekker, Inc.
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