With PD control we are able to achieve the position control objective for robots whose models do not contain the gravitational term (i.e. g(q) = 0). In this case, the tuning procedure1 for PD control is trivial since it is enough to select the design matrices Kp and Kv to be symmetric and positive deﬁnite (see Chapter 6). In the case where the robot model contains the vector ofgravitational torques (i.e. g(q) = 0) and if in particular, g(q d ) = 0, where q d is the joint desired position, then the position control objective cannot be achieved by means of a simple PD control law. As a matter of fact it may happen that the ˜ position error q tends to a constant vector but which is always diﬀerent from the vector 0 ∈ IRn . Then, from an automatic control viewpoint and withthe aim of satisfying the position control objective, in this case it seems natural to introduce an Integral component to the PD control to drive the position error to zero. This reasoning justiﬁes the application of Proportional Integral Derivative (PID) control to robot manipulators. The PID control law is given by
˙ ˜ ˜ τ = Kp q + Kv q + Ki
˜ q (σ) dσ
where the designmatrices Kp , Kv , Ki ∈ IRn×n , which are respectively called “position, velocity and integral gains”, are symmetric positive deﬁnite and suitably selected. Figure 9.1 shows the block-diagram of the PID control for robot manipulators. Nowadays, most industrial robot manipulators are controlled by PID controllers. The wide use of robot manipulators in everyday applications, is testament to theperformance that can be achieved in a large variety of applications
By ‘tuning procedure’ the reader should interpret the process of determining the numerical values of the design parameters of the control law, which guarantee the achievement of the control objective.
9 PID Control
q ˙ q
Kv ˙ qd qd Σ
Figure 9.1. Block-diagram: PIDcontrol
when using PID control. However, in contrast to PD control, the tuning procedure for PID controllers, that is, the procedure to choose appropriate positive deﬁnite matrices Kp , Kv and Ki , is far from trivial. In practice, the tuning of PID controllers is easier for robots whose transmission system includes reduction mechanisms such as gears or bands. The use of these reductions eﬀectivelyincreases the torque or force produced by the actuators, and therefore, these are able to drive links of considerably large masses. In principle, this has the consequence that large accelerations may be reached for ‘light’ links. Nevertheless, the presence of reduction mechanisms, such as gears and bands, may introduce undesired physical phenomena that hamper the performance of the robot in itsrequired task. Among these phenomena we cite vibrations due to backlash among the teeth of the gears, positioning errors and energy waste caused by friction in the gears, positioning errors caused by vibrations and elasticity of the bands and by gear torsions. In spite of all these the use of reduction mechanisms is common in most robot manipulators. This has a positive impact on the tuning task ofcontrollers, and more particularly of PID controllers. Indeed, as has been shown in Chapter 3 the complete dynamic of the robot with high-reduction-ratio transmissions is basically characterized by the model of the actuators themselves, which are often modeled by linear diﬀerential equations. Thus, in this scenario the diﬀerential equation that governs the behavior of the closed-loop systembecomes linear and therefore, the tuning of the controller becomes relatively simple. This last topic is not treated here since it is well documented in the literature; the interested reader is invited to see the texts cited at the end of the chapter. Here, we consider the more general nonlinear case. In the introduction to Part II we assumed that the considered robot actuators were ideal sources of...