Joint space control
Traditionally, control design in robot manipulators is understood as the simple fact of tuning a PD (Proportional and Derivative) compensator at the level of each motor driving the manipulator joints. Fundamentally, a PD controller is a position and a vel
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J oint space control Traditionally, control design in robot manipulators is understood as the simple fact of tuning a PD (Proportional and Derivative) compensator at the level of each motor driving the manipulator joints. Fundamentally, a PD controller is a position and a velocity feedback that has good closed-loop properties when applied to a double integrator. This controller provides a natural way to stabilize double integrators since it can be understood as an additional mechanical (active) spring and damper which reduces oscillations. To this extent, the control of an n-joint manipulator can be interpreted as the control of n independent chains of double integrators for which a PD controller can be designed. In reality, the manipulator dynamics is much more complex than a simple decoupled second-order linear system. It includes coupling terms and nonlinear components such as gravity, Coriolis and centrifugal forces and friction. The first experiments conducted with real robots were performed with simple PD compensators and have been, in general, satisfactory as far as stability and middle range performance are concerned. There is some explanation for this. First, as it was understood in the beginning of the 80's, PD controllers can stabilize not only a double integrator structure but also the complete Lagrange robot manipulator dynamics (in the sense of Lyapunov stability) due to the passivity properties enjoyed by the manipulator model. But before this was known, the main reason why the experiments were successful is because the complete robot manipulator dynamics is locally equivalent to a linear model described by a set of double integrators and, as such, is locally stabilizable. This local domain enlarges as the nonlinearities and the coupling terms become less important, or equivalently as the reduction ratio of the gear boxes (or the harmonic drives) becomes larger. In addition, this region of attraction can be enlarged by increasing
C. C. de Wit et al. (eds.), Theory of Robot Control © Springer-Verlag London Limited 1996
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CHAPTER 2. JOINT SPACE CONTROL
the controller gains. It is instructive for comparative purposes to classify the control objectives into the following two classes: • Regulation which sometimes is also called point-to-point control. A fixed configuration in the joint space is specified; the objective is to regulate the joint variables about the desired position in spite of torque disturbances and independently of the initial conditions. The behaviour of transients and overshooting are, in general, not guaranteed . • Tracking control consists of following a time-varying joint reference trajectory specified within the manipulator workspace. In general, this desired trajectory is assumed to comply with the actuators' capacity. In other words, the joint velocity and acceleration associated with the desired trajectory should respect the maximum velocity and acceleration limits of the manipulator. The control objective is thus to asymptotically track the desired trajectory in spite of d
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