Control of Robot Manipulators in Joint Space R. Kelly, V. Santibanez and A. Loria

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.In principle, this has the consequence that large accelerations maybe reached for light links.Nevertheless, the presence of reduction mecha-nisms, such as gears and bands, may introduce undesired physical phenomenathat hamper the performance of the robot in its required task.Among thesephenomena we cite vibrations due to backlash among the teeth of the gears,positioning errors and energy waste caused by friction in the gears, positioningerrors 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 robotmanipulators.This has a positive impact on the tuning task of controllers, andmore particularly of PID controllers.Indeed, as has been shown in Chapter 3the complete dynamic of the robot with high-reduction-ratio transmissions isbasically characterized by the model of the actuators themselves, which areoften modeled by linear differential equations.Thus, in this scenario the differ-ential equation that governs the behavior of the closed-loop system becomeslinear 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 ac-tuators were ideal sources of torques and forces.Under this condition, thedynamic model of a robot of n DOF is given by (3.18), i.e.� � �M(q)q + C(q, q)q + g(q) =� , (9.2)9 PID Control 203where the vector of gravitational torques g(q) is clearly present.In this chap-ter, we assume that the joints of the robot are all revolute.We study the control system formed by the PID controller given by Equa-tion (9.1) and the robot model (9.2).This study is slightly more complex thanthat for the PID control of robots with high-reduction actuators.Specifically,we obtain a tuning procedure for PID control that guarantees achievement ofthe position control objective, locally.In other words, for the desired constant�position qd, tuning ensures that limt�!" q(t) =0, as long as the initial posi-� �tion error q(0) and the initial velocity error q(0) are sufficiently small.Froman analytic viewpoint this is done by proving local asymptotic stability of theorigin of the equation that describes the behavior of the closed-loop system.For this analysis, we use the following information drawn from Properties 4.1,4.2, and 4.3:1�" The matrix @(q) - C(q, q) is skew-symmetric.2" There exists a non-negative constant kC such that for all x, y, z " IRn,1we haveC(x, y)z d"kC y z.1" There exists a non-negative constant kg such that for all x, y " IRn, wehaveg(x) - g(y) d"kg x - y ," ( )where kg e" for all q " IRn."The integral action of the PID control law (9.1) introduces an additional��state variable that is denoted here by �, and whose time derivative is � = q.The PID control law may be expressed via the two following equations:�� = Kpq + Kvq + Ki� (9.3)�� = q.(9.4)The closed-loop equation is obtained by substituting the control action �from (9.3) in the robot model (9.2), i.e.�� M(q)q + C(q, q)q + g(q) =Kpq + Kvq + Ki�� = q,TT�� which may be written in terms of the state vector �T qT q , as�# �#�# �#�q��# �#�# �#�#d�# �#=�#��# �#.(9.5)�q�q�# �#�# �#dt �# �#�# �#�� - M(q)-1 Kpq + Kvq + Ki� - C(q, q)q - g(q)�qd � � � ��q204 9 PID ControlTT�� The equilibria of the equation above, if any, have the form �T qT q =T�"T 0T 0T where-1� � ��" = Ki [M(qd)qd + C(qd, qd)qd + g(qd)]must be a constant vector.Certainly, for �" to be a constant vector, if thedesired joint position qd is time-varying, it may not be arbitrary but shouldhave a very particular form.One way to obtain a qd for which �" is constant,is by solving the differential equations�# �# �# �# �# �#�qd qd qd(0)d�# �#=�# �# �# �#" IR2n,dt� � � �qd M(qd)-1 [� - C(qd, qd)qd - g(qd)] qd(0)(9.6)-1where � " IRn is a constant vector.This way �" = Ki �.In particular,0 0if � = 0 " IRn then the origin of the closed-loop Equation (9.5), is anequilibrium.Notice that the solution of (9.6) is simply the position q and�velocity q when one applies a constant torque � = � to the robot in question.0In general, it is not possible to obtain an expression in closed form for qd, soEquation (9.6) must be solved numerically.Nevertheless, the resulting desiredposition qd, may have a capricious form and therefore be of little utility.Thisis illustrated in the following example.Example 9.1.Consider the Pelican prototype robot studied in Chapter5, and shown in Figure 5.2.Considering � = 0 " IR2 and the initial condition [qd1 qd2 qd1�qd2]T = [-�/20 �/20 0 0]T ; the numerical solution of (9.6) for�qd(t), is shown in Figure 9.2.With qd(t), whose two components are shown in Figure 9.2, theTT�� origin �T qT q = 0 " IR6 is an equilibrium of the closed-loopequation formed by the PID control and the robot in question.f&In the case where the desired joint position qd is an arbitrary functionof time, and does not tend to a constant vector value, then the closed-loopequation has no equilibrium.In such cases, we cannot study the stability inthe sense of Lyapunov and in particular, one may not expect that the position�error q tend to zero.In the best case scenario, and under the hypothesis that� �the initial position and velocity errors q(0) and q(0) are small, the position�error q remains bounded.The formal proof of these claims is established inthe works cited at the end of the chapter.Let us come back to our discussion on the determination of an equilibriumfor the closed-loop system.We said that a sufficient condition for the existence9 PID Control 205[rad]1.00.75.qd2.50.qd1.25.00.-0.25.-0.50.-0.75.-1.0002468 10t [s]Figure 9.2.Desired joint positionsand unicity of the equilibrium for the closed-loop Equation (9.5) is that thedesired position qd(t) be constant.Denoting by qd such a constant vector theequilibrium is�# �# �# �#� Ki-1 g(qd)�# �# �# �#�# �# �# �#�q = 0 " IR3n.�# �# �# �#�# �# �# �#��q 0This equilibrium may of course, be translated to the origin via a suitablechange of variable, e.g.definingz = � - Ki-1g(qd).Then, the corresponding closed-loop equation may be expressed in terms ofT� �the state vector zT qT qT as�# �# �# �#�z q�# �# �# �#d�# �# �# �#�q = ��# �# �# -q �#.dt �# �# �# �#� � � � �q M(q)-1 [Kpq - Kvq + Kiz + g(qd) - C(q, q)q - g(q)](9.7)Notice that the previous equation is autonomous and its unique equilibriumT� �is the origin zT qT qT = 0 " IR3n.For the sequel, we find it convenient to adopt the following global changeof variables,206 9 PID Control�# �# �# �# �# �#w � I I 0 z�# �# �# �# �# �#� �q = 0 I 0 q (9.8)� �q 0 0 I qwith �>0.The closed-loop Equation (9.7) may be expressed as a function of the newvariables as�# �#w�# �#d�# �#�q =�# �#dt �# �#�q�# �#� �� q - q�# �#�# �#�# �#�-q.�# �#�# �#1 1� � � �M(q)-1 Kp - Ki q - Kvq + Kiw + g(qd) - C(q, q)q - g(q)� �(9.9)�These equations are autonomous and the origin of the state space, wT qTT�qT = 0 " IR3n is the unique equilibrium of the closed-loop system.Moreover, due to the globality of the change of variable (9.8), the sta-bility features of this equilibrium correspond to those for the equilibriumT� �zT qT qT = 0 " IR3n of Equation (9.7) [ Pobierz całość w formacie PDF ]

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