Robot Manipulator Control Theory and Practice - Frank L.Lewis Part 8 pptx

Robust Control of Robotic Manipulators268 of q for robots with prismatic joints, and that (5.2.10) is satisfied by so that a=(µ 2 - µ 1 )/(µ 2 +µ 1 )) [Spong and Vidyasagar 1987]. Finally, (5.2.10) is a result of the properties of the Coriolis and centripetal terms discussed in Section 3.3. We will give different representative designs of the feedback-linearization approach, starting with controllers whose behavior is studied using Lyapunov stability theory. Lyapunov Designs Static feedback compensators have been extensively used starting with the works of [Freund 1982] and [Tarn et al. 1984]. Consider the controller introduced in (4.4.13): (5.2.12) such that (5.2.13) It can be seen that by placing the poles of A c sufficiently far in the left half- plane, the robust stability of the closed- loop system in the presence of ␩ is guaranteed. This was shown true in [Arimoto and Miyazaki 1985] for the case where as described in Theorem 4.4.1 and Example 4.4.3. It was also shown true for the trajectory-following problem assuming that in [Dawson et al. 1990] as described in Theorem 4.4.2. There are as many robust controllers designed using Lyapunov stability concepts as there are ways of choosing Lyapunov function candidates, and of designing the gain K to guarantee that the Lyapunov function candidate is decreasing along the trajectories of (5.2.13). To decrease the asymptotic trajectory error, however, excessively large gains may be required (see Example 4.4.3). We therefore choose to use the passivity theorem and a choice of the gain matrix K that renders the linear part of the closed-loop system SPR. As described in Section 2.11, an output may be chosen to make the closed-loop system SPR; therefore allowing large passive uncertainties in the knowledge of M(q). In fact, the state-feedback controller may be used to define an appropriate output Ke such that the input-output closed-loop linear systems K(sI-A+BK) -1 B is strictly positive real (SPR). Consider the following closed- loop linear system: Copyright © 2004 by Marcel Dekker, Inc. 269 (5.2.14) It may then be shown using Theorem 2.11.5 that this system is SPR if (5.2.15) with the choice of (5.2.16) such that (5.2.17) is the positive-definite solution to the Lyapunov equation (5.2.18) and (5.2.19) The next theorem presents sufficient conditions for the uniform boundedness of the trajectory error. THEOREM 5.2–1: The closed-loop system given by (5.2.13) will be uniformly bounded if and where K v =2aI and K p =4aI. Proof: 5.2 Feedback-Linearization Controllers Copyright © 2004 by Marcel Dekker, Inc. Robust Control of Robotic Manipulators270 Consider the closed-loop system given by (5.2.8), with the controller (5.2.12), and choose the following Lyapunov function candidate: (1) where is the Lyapunov function corresponding to the SPR system (5.2.14). Then if ⌬≥0, we have that V>0. This condition is satisfied for ≥µ 2 I. Then differentiate to find (2) To guarantee that V < 0 recall the bounds (5.2.8)–(5.2.11), and write (3) where . Note that ||e|| may be factored out of (3) without affecting the sign definiteness of the equation. The uniform boundedness of the error is then guaranteed using Lemma 2.10.3 and Theorem 2.10.3 if (4) which is guaranteed if (5) or (6) The error will be bounded by a term that goes to zero as a increases (see Theorem 2.10.3 and its proof in [Dawson et al. 1990] for details). This analysis then allows ⌬ to be arbitrarily large as long as ≥µ 2 I, as shown in the next example. In fact, if N were known, global asymptotic stability is assured from the passivity theorem since in that case ␦=0. The controller is It is instructive to study (6) and try to understand the contribution of each of its terms. The following choices will help satisfy (6). 1. Large gains K p and K v which correspond to a large a. ᭿ Copyright © 2004 by Marcel Dekker, Inc. summarized in Table 5.2.1. 271 2. A good knowledge of N which translates into small ␤ i ’s. 3. A large µ 1 or a large inertia matrix M(q). 4. A trajectory with a small c, this a small desired acceleration d . Figure 5.2.2: K p =50, K v =25 (a) errors of joint 1; (b) errors of joint 2; (c) torques of joints 1 and 2. The following example illustrates the sufficiency of condition (6) and of the effects of larger gains K p and K v . 5.2 Feedback-Linearization Controllers Copyright © 2004 by Marcel Dekker, Inc. Robust Control of Robotic Manipulators272 (1) where EXAMPLE 5.2–1: Static Controller (Lyapunov Design) In all our examples in this chapter we use the two-link revolute-joint robot first described in Chapter 3, Example 3.2.2, whose dynamics are repeated here: (2) The parameters m 1 =1kg, m 2 =1kg, a 1 =1m, a 2 =1m, and g=9.8 m/s 2 are given. Let the desired trajectory used in all examples throughout this chapter be described by Then . It may then be shown that Let =0 then or that Then use =6I and a=172 to satisfy (6). In fact, these values will lead to a larger controller gains than are actually needed. Suppose instead that we let 6 ^ I, =0, and that Copyright © 2004 by Marcel Dekker, Inc. 273 (3) Note that this is basically a computed-torque-like PD controller. A simulation of the robot’s trajectory is shown in Figure 5.2.2. We also start our simulation at =0. The effect of increasing the gains is shown in Figure 5.2.3, which corresponds to the controller (4) Note that at least initially, more torque is required for the higher-gains case (compare Figs. 5.2.2c and 5.2.3c) but that the errors magnitude is greatly reduced by expanding more effort. There are other proofs of the uniform boundedness of these static controllers. In particular, the results in [Dawson et al. 1990] provide an explicit expression for the bound on e in terms of the controller gains. In the interest of brevity and to present different designs, we choose to limit our development to one controller in this section. As discussed in Section 4.4, a residual stead-state error may be present even when using an exact computed-torque controller if disturbances are present. A common cure for this problem (and one that will eliminate constant disturbances) is to introduce integral feedback as done in Section 4.4. Such a controller may again be used within a robust controller framework and will lead to similar improvements if the integrator windup problem is avoided (see Section 4.4). In the next section we show the stability of static controllers similar to the ones designed here and use input-output stability methods to design more general dynamic compensators. Input-Output Designs In this section we group designs that show the stability of the trajectory error using input-output methods. In particular, we present controllers that show ∞ and 2 stability of the error. We divide this section into a subsection that deals with static controllers such as the ones described previously, ᭿ 5.2 Feedback-Linearization Controllers Copyright © 2004 by Marcel Dekker, Inc. Robust Control of Robotic Manipulators276 the error signals was shown using a static controller. The norms used in (5.2.8)–(5.2.10) are then ∞ norms. The development of this controller starts with assumptions (5.2.8), (5.2.9), and a modification of (5.2.10) to (5.2.20) This assumption is justified by the fact that N is composed of gravity and velocity-dependent terms which may be bounded independent from the position error e [see (5.1.1)]. We shall also assume that =0. Let us then choose the state-feedback controller (5.2.12) repeated here for convenience: (5.2.21) The corresponding input-output differential equation (5.2.22) A block diagram description of this equation is given in Figure 5.2.4. Consider now the transfer function from η (taken as an independent input) to e: (5.2.23) or (5.2.24) It can be seen that K v and K p are both diagonal, with , a critically damped response it achieved at every joint [see (4.4.22), (4.4.30), and (3.3.32)]. The infinity operator gains of P 11 (s) and P 12 (s) are (see Lemma 2.5.2 and Example 2.5.8) (5.2.25) where (5.2.26) Consider then the following inequalities: Copyright © 2004 by Marcel Dekker, Inc. 277 and using (5.2.8)–(5.2.11), we have that (5.2.27) The following theorem presents sufficient conditions for the boundedness of the error that parallel those of Theorem 5.2.1. THEOREM 5.2–2: Suppose that and (1) Then the ∞ stability of the error is guaranteed if (2) Proof: The condition above results from applying the small-gain theorem to the closed-loop system, under the assumption that e(0)=0 so that the quadratic term ||e|| 2 is small. See [Craig 1988] for details. Note that (2) reduces to and further to (5.2.28) Let us study the inequality above to determine the effect of each term. The following observations are made to help satisfy (5.2.28). 1. A large k v will help satisfy the stability condition. Note: That will also imply a large k p . 5.2 Feedback-Linearization Controllers Copyright © 2004 by Marcel Dekker, Inc. Robust Control of Robotic Manipulators278 2. A good knowledge of N, which will translate into small ␤ i ’s. 3. A large µ 1 or a large inertia matrix M(q). 4. A trajectory with a small d . 5. Robots whose inertia matrix M(q) does not vary greatly throughout its workspace (i.e. µ 1 ≈µ 2 )), so that a is small. Note that a small a is needed to guarantee that at least < 1 in (5.2.28). This will translate into the severe requirement that the matrix M should be close to the inertia matrix M(q) in all configurations of the robot. The controller is summarized in Table 5.2.2. These observations are similar to those made after inequality (6) and are illustrated in the next example. EXAMPLE 5.2–2: Static Controller (Input-Output Design) Consider the nonlinear controller (5.2.6), where (1) Therefore, (2) Condition (1) is then satisfied if k v >720. This of course is a large bound that can be improved by choosing a better . A simulation of the closed-loop behavior for k p =225 and k v =30 is shown in Figure 5.2.5. The errors magnitudes are much smaller than those achieved with the PD controllers of Example 5.2.1 with a comparable control effort. This improvement came with the expense of knowing the inertia matrix M(q) as seen in (1). Dynamic Controllers The controllers discussed so far are static controllers in that they do not have a mechanism of storing previous state information. In Chapter 4 and in this chapter, these controllers could operate only on the current position and velocity errors. In this section we present three approaches to show the robustness of dynamic controllers based on the feedback-linearization ᭿ Copyright © 2004 by Marcel Dekker, Inc. [...]... if ␥1=max ␥11, ␥12 and ␥2k=␥11kp+␥12kv On the other hand, assuming that d=0 and ␤2=0, the 2 stability of e was shown in [Becker and Grimm 1 988 ] if (5.2.36) where as given in Lemma 2.5.2 This controller is summarized in Table 5.2.4 Copyright © 2004 by Marcel Dekker, Inc  288 Robust Control of Robotic Manipulators is due to the fact that the velocity terms are truly negligent in this particular application... 2; (c) torques of joints 1 and 2 Two-Degree-of-Freedom Design It is well known that the two-DOF structure is the most general linear controller structure The two-DOF design allows us simultaneously to specify the desired response to a command input and guarantee the robustness of Copyright © 2004 by Marcel Dekker, Inc 5.2 Feedback-Linearization Controllers 289 the closed-loop system This design was... Robust Control of Robotic Manipulators the other hand, if one can show the passivity of the system, which maps ␶ to a new vector r which is a filtered version of e, a controller that closes the loop between -r and ␶ will guarantee the asymptotic stability of both e and This indirect use of the passivity property was illustrated in [Ortega and Spong 1 988 ] and will be discussed first Let the controller... this theory to robot control seems to be in [Young 19 78] , where the set-point regulation problem ( d=0)was solved using the following controller: (5.3.4) where i=1,…, n for an n-link robot, and ri are the switching planes, (5.3.5) It is then shown, using the hierarchy of the sliding surfaces r1, r2,…, rn and given bounds on the uncertainties in the manipulators model, that one can find ␶ + and ␶ - in... associated with variable-structure controllers We shall address the second issue later in this section, and answer the first by admitting that although initial applications of variable-structure theory did indeed gloss over the physics of robots, later designs (such as the Copyright © 2004 by Marcel Dekker, Inc 2 98 Robust Control of Robotic Manipulators ones discussed here) by [Slotine 1 985 ] and [Chen et al... asymptotically stable [Slotine 1 988 ] This approach was used in the adaptive control literature to design passive controllers [Ortega and Spong 1 988 ], but its modification in the design of robust controllers when M, Vm and G are not exactly known is not immediately obvious Such modifications will be given in the variable-structure designs, but first, we present a simple controller to illustrate the robustness... needed in order to find Kp and Kv We will not discuss this particular design and refer the interested reader to [Anderson 1 989 ] Variable-Structure Controllers In this section we group designs that use variable-structure (VSS) controllers The VSS theory has been applied to the control of many nonlinear processes [DeCarlo et al 1 988 ] One of the main features of this approach is that one only needs to drive... because it relies on classical frequency-domain SISO concepts The general structure is shown in Figure 5.2 .8 A two-DOF robust controller was designed and simulated in [Sugie et al 1 988 ] and will be presented next Let the plant be given by (5.2.5) and consider the following factorization: G(s)=N(s)D-1(s), where D(s)=s2, N(s)=I (5.2.37) The following result presents a two-DOF compensator which will robustly... 2004 by Marcel Dekker, Inc  286 Robust Control of Robotic Manipulators stability result of [Spong and Vidyasagar 1 987 ] It was shown in [Becker and Grimm 1 988 ], however, that the 2 stability of the error cannot be guaranteed unless the problem is reformulated and more assumptions are made It effect, the error will be bounded, but it may or may not have a finite energy In particular, noisy measurements... two-DOF structure of Figure 5.2 .8 Let K1(S) be a stable system and K2(s) be a compensator to stabilize G(s) Then the controller u=s2K1v+K2(K1v-q) (1) will lead to the closed-loop system q=K1v and the closed-loop error system (5.2.13) will be (2) ∞ stable Proof: With simple block-diagram manipulations, it may be shown that the closedloop system is q=K1v The actual robustness analysis is involved and . Then the controller u=s 2 K 1 v+K 2 (K 1 v-q) (1) will lead to the closed-loop system q=K1v (2) and the closed-loop error system (5.2.13) will be ∞ stable. Proof: With simple block-diagram manipulations,. by Marcel Dekker, Inc. 293 5.3 Nonlinear Controllers There is a class of robot controllers that are not computed-torque-like controllers. These controllers are obtained directly from the robot. the stability condition. Note: That will also imply a large k p . 5.2 Feedback-Linearization Controllers Copyright © 2004 by Marcel Dekker, Inc. Robust Control of Robotic Manipulators2 78 2. A good