CRC Press - Robotics and Automation Handbook Episode 2 Part 3 pot

17 -18 Robotics and Automation Handbook the latter term following since θ is constant, i.e., ˜ θ = ˆ θ. Using the parameter update law (17.102) gives ˙ V =−e T Qe (17.105) From this it follows that the position tracking errors converge to zero asymptotically and the parameter estimation errors remain bounded. Furthermore, it can be shown that the estimated parameters converge to the true parameters provided the reference trajectory satisfies the condition of persistencyof excitation, αI ≤  t 0 +T t 0 Y T (q d , ˙ q d , ¨ q d )Y(q d , ˙ q d , ¨ q d )dt ≤ β I (17.106) for all t 0 ,whereα, β, and T are positive constants. In order to implement this adaptive feedback linearization scheme, however, one notes that the accel- eration ¨ q is needed in the parameter update law and that ˆ M must be guaranteed to be invertible, possibly by the use of projection in the parameter space. Later work was devoted to overcome these two drawbacks to this scheme by using so-called indirect approaches based on a (filtered) prediction error. 17.3.7 Passivity-Based Approaches One may also exploit the passivity of the rigid robot dynamics to derive robust and adaptive control algorithms for manipulators. These methods are qualitatively different from the previous methods which were based on feedback linearization. In the passivity-based approach, we modify the inner loop control as τ = ˆ M(q)a + ˆ C(q, ˙ q)v + ˆ g(q) − Kr (17.107) where v, a, and r are given as v = ˙ q d −   q a = ˙ v = ¨ q d −  ˙  q r = ˙ q d − v = ˙  q +  q with K , diagonal matrices of positive gains. In terms of thelinear parametrization of the robot dynamics, the control (17.107) becomes τ = Y(q, ˙ q, a, v) ˆ θ − Kr (17.108) and the combination of Equation (17.107) with Equation (17.41) yields M(q) ˙ r + C(q, ˙ q)r + Kr = Y ˜ θ (17.109) Note that, unlike the inverse dynamics control, the modified inner loop control (17.41) does not achieve a linear, decoupled system, even in the known parameter case ˆ θ = θ. However, in this formulation the regressor Y in Equation (17.109) does not contain the acceleration ¨ q nor is the inverse of the estimated inertia matrix required. These represent distinct advantages over the feedback linearization based schemes. 17.3.8 Passivity-Based Robust Control In the robust passivity-based approach of [36], the term ˆ θ in Equation (17.108) is chosen as ˆ θ = θ 0 + u (17.110) where θ 0 is a fixed nominal parameter vector and u is an additional control term. The system (17.109) then becomes M(q) ˙ r + C(q, ˙ q)r + Kr = Y(a,v, q, ˙ q)( ˜ θ +u) (17.111) Copyright © 2005 by CRC Press LLC Robust and Adaptive Motion Control of Manipulators 17 -19 where ˜ θ = θ 0 − θ is a constant vector and represents the parametric uncertainty in the system. If the uncertainty can be bounded by finding a nonnegative constant, ρ ≥ 0, such that  ˜ θ=θ 0 − θ≤ρ (17.112) then the additional term u can be designed according to the expression, u =    −ρ Y T r ||Y T r || ,if ||Y T r || > − ρ  Y T r,if||Y T r || ≤  (17.113) The Lyapunov function V = 1 2 r T M(q)r +  q T K  q (17.114) is used to show uniform ultimate boundedness of the tracking error. Note that ˜ θ is constant and so is not a state vector as in adaptive control. Calculating ˙ V yields ˙ V =r T M ˙ r + 1 2 r T ˙ Mr + 2  q T K ˙  q (17.115) =−r T Kr +2  q T K ˙  q + 1 2 r T ( ˙ M −2C)r +r T Y( ˜ θ +u) (17.116) Using the passivity property and the definition of r , this reduces to ˙ V =−  q T  T K   q − ˙  q T K ˙  q +r T Y( ˜ θ +u) (17.117) Uniform ultimate boundedness of the tracking error follows with the control u from (17.113). See [36] for details. Comparing this approach with the approach in thesection (17.3.5),we see that finding a constant bound ρ for the constant vector ˜ θ is much simpler than finding a time-varying bound for η in Equation (17.44). The bound ρ in this case depends only on the inertia parameters of the manipulator, while ρ(x,t)in Equation (17.69) depends on the manipulator state vector and the reference trajectory and, in addition, requires some assumptions on the estimated inertia matrix ˆ M(q). 17.3.9 Passivity-Based Adaptive Control In the adaptive approach the vector ˆ θ in Equation (17.109) is now taken to be a time-varying estimate of the true parameter vector θ. Combining the control law (17.107) with (17.41) yields M(q) ˙ r + C(q, ˙ q)r + Kr = Y ˜ θ (17.118) The parameter estimate ˆ θ may be computed using standard methods such as gradient or least squares. For example, using the gradient update law ˙  θ =− −1 Y T (q, ˙ q, a, v)r (17.119) together with the Lyapunov function V = 1 2 r T M(q)r +  q T K  q + 1 2 ˜ θ T  ˜ θ (17.120) results in global convergence of the tracking errors to zero and boundedness of the parameter estimates since ˙ V =−  q T  T K   q − ˙  q T K ˙  q + ˜ θ T { ˙ ˆ θ +Y T r } (17.121) See [38] for details. Copyright © 2005 by CRC Press LLC 17 -20 Robotics and Automation Handbook PERFORMANCE MEASURE + + + CONTROL ROBOT MODEL N MODEL 2 MODEL 1 – – – min J i J 1 J 2 J N e t2 e t N e t1 t ^ N t ^ 2 t ^ 1 ( q , q . ) q ^ t • • • FIGURE 17.9 Multiple-model-based hybrid control architecture. 17.3.10 Hybrid Control A Hybrid System is one that has both continuous-time and discrete-event or logic-based dynamics. Supervisory Control, Logic-Based Switching Control, and Multiple-Model Control are typical control architectures in this context. In the robotics context, hybrid schemes can be combined with robust and adaptive control methods to further improve robustness. In particular, because the preceeding robust and adaptivecontrolmethods provideonlyasymptotic (i.e., as t →∞)errorbounds,thetransient performance may not be acceptable. Hybrid control methods have been shown to improve transient performance over fixed robust and adaptive controllers. The use of the term Hybrid Control in this context should not be confused with the notion of Hybrid Position/Force Control [41]. The latter is a familiar approach to force control of manipulators in which the term hybrid refers to the combination of pure force control and pure motion control. Figure 17.9 shows the Multiple-Model approach of [12], which has been applied to the adaptive control of manipulators. In this architecture, the multiple models have the same structure but may have different nominal parameters in case a robust control scheme is used, or different initial parameter estimates if an adaptive control scheme is used. Because all models have the same inputs and desired outputs, the identification errors e I j are available at each instant for the j th model. The idea is then to define a performance measure, for example, J (e I j (t)) = γ e 2 I j (t) + β  t 0 e 2 I j (σ )dσ with γ, β>0 (17.122) and switch into the closed loop the control input that results in the smallest value of J at each instant. 17.4 Conclusions We have given a brief overview of the basic results in robust and adaptive control of robot manipulators. In most cases, we have given only the simplest forms of the algorithms, both for ease of exposition and for reasons of space. An extensive literature is available that contains numerous extensions of these basic results. The attached list of references is by no means an exhaustive one. The book [10] is an excellent and Copyright © 2005 by CRC Press LLC Robust and Adaptive Motion Control of Manipulators 17 -21 highly detailed treatment of the subject and a good starting point for further reading. Also, the reprint book [37] contains several of the original sources of material surveyed here. In addition, the two survey papers [1] and [28] provide additional details on the robust and adaptive control outlined here. Several important areas of interest have been omitted for space reasons including output feedback control, learning control, fuzzy control, neural networks, and visual servoing, and control of flexible robots. The reader should consult the references at the end for background on these and other subjects. References [1] Abdallah, C. et al., Survey of robust control for rigid robots, IEEE Control Systems Magazine, Vol. 11, No.2,pp.24–30, Feb. 1991. [2] Amestegui, M., Ortega, R., and Ibarra, J.M., Adaptive linearizing-decoupling robot control: A com- parative study, Proc. 5th Yale Workshop on Applications of Adaptive Systems Theory, NewHaven,CT, 1987. [3] Balestrino, A., De Maria, G., and Sciavicco, L., An adaptive model following control for robotic manipulators, ASME J. Dynamic Syst., Meas., Contr., Vol. 105, pp. 143–151, 1983. [4] Bayard, D.S. and Wen, J.T., A new class of control laws for robotic manipulators-Part 2. Adaptive case, Int. J. Contr., Vol. 47, No. 5, pp. 1387–1406, 1988. [5] Becker, N. and Grimm, W.M., On L 2 and L ∞ -stability approaches for the robust control of robot manipulators, IEEE Trans. Automat. Contr., Vol. 33, No. 1, pp. 118–122, Jan. 1988. [6] Berghuis, H. and Nijmeijer, H., Global regulation of robots using only position measurements, Syst. Contr. Lett., Vol. 1, pp. 289–293, 1993. [7] Berghuis, H., Ortega, R., and Nijmeier, H., A robust adaptive robot controller, IEEE Trans. Robotics Automat., Vol. 9, No. 6, pp.825–830, Dec. 1993. [8] Brogliato, B., Landau, I.D., and Lozano, R., Adaptive motion control of robot manipulators: A unified approach based on passivity, Int. J. Robust Nonlinear Contr., Vol. 1, pp. 187–202, 1991. [9] Campion, G. and Bastin, G., Analysis of an adaptive controller for manipulators: Robustness versus flexibility, Syst. Contr. Lett., Vol. 12, pp. 251–258, 1989. [10] Canudas de Wit, C. et al., Theory of Robot Control, Springer-Verlag, London, 1996. [11] Canudas de Wit, C. and Fixot, N., Adaptive control of robot manipulators via velocity estimated state feedback, IEEE Trans. Automatic Contr., Vol. 37, pp. 1234–1237, 1992. [12] Ciliz, M.K. and Narendra, K.S., Intelligent control of robotic manipulators: A multiple model based approach, IEEE Conf. on Decision and Control, New Orleans, LA, pp. 422–427, December 1995. [13] Corless, M. and Leitmann, G., Continuous state feedback guaranteeing uniform ultimate bound- edness for uncertain dynamic systems, IEEE Trans. Automatic Contr., Vol. 26, pp. 1139–1144, 1981. [14] Craig, J.J., Adaptive Control of Mechanical Manipulators, Addison-Wesley, Reading, MA, 1988. [15] Craig, J.J., Hsu, P., and Sastry, S., Adaptive control of mechanical manipulators, Proc. IEEE Int. Conf. Robotics Automation, San Francisco, CA, March 1986. [16] De Luca, A., Dynamic control of robots with joint elasticity, Proc. IEEE Conf. on Robotics and Automation, Philadelphia, PA, pp. 152–158, 1988. [17] Desoer, C.A. and Vidyasagar, M., Feedback Systems: Input-Output Properties, Academic Press, New York, 1975. [18] Filippov, A.F., Differential equations with discontinuous right-hand side, Amer. Math. Soc. Transl., Vol. 42, pp. 199–231, 1964. [19] Hager,G.D., A modular system for robust positioning using feedback from stereo vision, IEEE Trans. Robotics Automat., Vol. 13, No. 4, pp. 582–595, August 1997. [20] Ioannou, P.A. and Kokotovi ´ c, P.V., Instability analysis and improvement of robustness of adaptive control, Automatica, Vol. 20, No. 5, pp. 583–594, 1984. [21] Khatib, O., A unified approach for motion and force control of robot manipulators: the operational space formulation, IEEE J. Robotics Automat., Vol. RA–3,No.1,pp.43–53, Feb. 1987. Copyright © 2005 by CRC Press LLC 17 -22 Robotics and Automation Handbook [22] Kim, Y.H. and Lewis, F.L., Optimal design of CMAC neural-network controller for robot manipu- lators, IEEE Trans. on Sys. Man and Cybernetics, Vol. 30, No. 1, pp. 22–31, Feb. 2000. [23] Koditschek, D., Natural motion of robot arms, Proc. IEEE Conf. on Decision and Control, Las Vegas, NV, pp. 733–735, 1984. [24] Kreutz, K., On manipulator control by exact linearization, IEEE Trans. Automat. Contr., Vol. 34, No.7,pp.763–767, July 1989. [25] Latombe, J.C., Robot Motion Planning, Kluwer, Boston, MA, 1990. [26] Luh, J., Walker, M., and Paul, R., Resolved-acceleration control of mechanical manipulators, IEEE Trans. Automat. Contr., Vol. AC–25, pp. 468–474, 1980. [27] Middleton, R.H. and Goodwin, G.C., Adaptive computed torque control for rigid link manipulators, Syst. Contr. Lett., Vol. 10, pp. 9–16, 1988. [28] Ortega, R. and Spong, M.W., Adaptive control of rigid robots: a tutorial, Proc. IEEE Conf. on Decision and Control, Austin, TX, pp. 1575–1584, 1988. [29] Paul, R.C., Modeling, trajectory calculation, and servoing of a computer controlled arm, Stanford A.I. Lab, A.I. Memo 177, Stanford, CA, Nov. 1972. [30] Dahleh, M.A. and Pearson, J.B., L 1 -optimal compensators for continuous-time systems, IEEE Trans. Automat. Contr., Vol. AC-32, No. 10, pp. 889–895, Oct. 1987. [31] Porter, D.W. and Michel, A.N., Input-output stability of time varying nonlinear multiloop feedback systems, IEEE Trans. Automat. Contr., Vol. AC-19, No. 4, pp. 422–427, Aug. 1974. [32] Sadegh, N. and Horowitz, R., An exponentially stable adaptive control law for robotic manipulators, Proc. American Control Conf., San Diego, pp. 2771–2777, May 1990. [33] Schwartz, H.M. and Warshaw, G., On the richness condition for robot adaptive control, ASME Winter Annual Meeting, DSC-Vol. 14, pp. 43–49, Dec. 1989. [34] Slotine, J J.E. and Li, W., On the adaptive control of robot manipulators, Int. J. Robotics Res., Vol. 6, No.3,pp.49–59, 1987. [35] Spong, M.W., Modeling and control of elastic joint manipulators, J. Dyn. Sys., Meas. Contr., Vol. 109, pp. 310–319, 1987. [36] Spong, M.W., On the robust control of robot manipulators, IEEE Trans. Automat. Contr., Vol. 37, pp. 1782–1786, Nov. 1992. [37] Spong, M.W., Lewis, F., and Abdallah, C., Robot Control: Dynamics, Motion Planning, and Analysis, IEEE Press, 1992. [38] Spong, M.W., Ortega, R., and Kelly, R., Comments on ‘Adaptive manipulator control’, IEEE Trans. Automat. Contr., Vol. AC–35, No. 6, pp. 761–762, 1990. [39] Spong, M.W. and Vidyasagar, M., Robust nonlinear control of robot manipulators, Proc. 24th IEEE Conf. Decision and Contr., Fort Lauderdale, FL, pp. 1767–1772, Dec. 1985. [40] Spong, M.W. and Vidyasagar, M., Robust linear compensator design for nonlinear robotic control, IEEE J. Robotics Automation, Vol. RA-3, No. 4, pp. 345–350, Aug. 1987. [41] Spong, M.W. and Vidyasagar, M., Robot Dynamics and Control, John Wiley & Sons, New York, 1989. [42] Su, C Y., Leung, T.P., and Zhou, Q J., A novel variable structure control scheme for robot trajectory control, IFAC World Congress, Vol. 9, pp. 121–124, Tallin, Estonia, August 1990. [43] Vidyasagar, M., Control Systems Synthesis: A Factorization Approach, MIT Press, Cambridge, MA, 1985. [44] Vidyasagar, M., Optimal rejection of persistent bounded disturbances, IEEE Trans. Automat. Contr., Vol. AC–31, No. 6, pp. 527–534, June 1986. [45] Yaz, E., Comments on On the robust control of robot manipulators by M.W. Spong, IEEE Trans. Automatic Control, Vol. 38, No. 3, pp. 511–512, Mar. 1993. [46] Yoo, B.K. and Ham, W.C., Adaptive control of robot manipulator using fuzzy compensator, IEEE Trans. on Fuzzy Systems, Vol. 8, No. 2, pp. 186–199, Apr. 2000. [47] Yoshikawa, T., Foundations of Robotics: Analysis and Control, MIT Press, Cambridge, MA, 1990. Copyright © 2005 by CRC Press LLC Robust and Adaptive Motion Control of Manipulators 17 -23 [48] Youla,D.C.,Jabr, H.A.,and Bongiorno,J.J., Modern Wiener-Hopf design ofoptimal controllers–Part 2: the multivariable case, IEEE Trans. Automatic Control, Vol. AC-21, pp. 319–338, June 1976. [49] Zhang, F., Dawson, D.M., deQueiroz, M.S., and Dixon, W.E., Global adaptive output feedback tracking control of robot manipulators, IEEE Trans. Automat. Contr., Vol. AC–45, No. 6, pp. 1203– 1208, June 2000. Copyright © 2005 by CRC Press LLC 18 Sliding Mode Control of Robotic Manipulators Hector M. Gutierrez Florida Institute of Technology 18.1 Sliding Mode Controller Design–An Overview 18.2 The Sliding Mode Formulation of the Robot Manipulator Motion Control Problem 18.3 Equivalent Control and Chatter Free Sliding Control 18.4 Control of Robotic Manipulators by Continuous Sliding Mode Laws Sliding Mode Formulation of the Robotic Manipulator Motion Control Problem • Sliding Mode Controller Design 18.5 Conclusions 18.1 Sliding Mode Controller Design An Overview Sliding mode design [1–6] has several features that make it an attractive technique to solve tracking problemsin motioncontrolof robotic manipulators, the most important being its robustness to parametric uncertainties and unmodelled dynamics, and the computational simplicity of the algorithm. One way of looking at sliding mode controller design is to think of the design process as a two-step procedure. First, a region of the state space where the system behaves as desired (sliding surface) is defined. Then, a control action that takes the system into such surface and keeps it there is to be determined. Robustness is usually achieved based on a switching control law. The design of the control action can be based on different strategies, a straightforward one being to define a condition that makes the sliding surface an attractive region for the state vector trajectories. Consider a nonlinear affine system of the form: x (n i ) i = f i (x) + m  j=1 b ij (x)u j , i = 1, , m, j = 1, , m (18.1) where u = [u 1 , , u m ] T is the vector of m control inputs, and the state x is composed of the x i coordinates to be tracked and their first (n i − 1) derivatives. Such systems are called square systems because they have as many control inputs u j as outputs to be controlled x i [3]. The motion control problem to be addressed is the one of making the state vector x track a desired trajectory r. Consider first the time-varying manifoldσ givenbythe intersectionof the surfacess i (x, t) =0, i =1, , m,specified bythe componentsof Copyright © 2005 by CRC Press LLC Sliding Mode Control of Robotic Manipulators 18 -3 J 1 q 1 r 1 J 2 m 1 m 2 q 2 r 2 y c x c constrain surface q FIGURE 18.1 Rigid-link rigid-joint robot interacting with a constrain surface. therefore either a position tracking problem in the q-state space or a force control problem or a hybrid of both. Example 18.1 Consider the force control problem of the two-link robot depicted in Figure 18.1 [7]. A control scheme to track a desired contact force F d is shown in Figure 18.2, where J c is the robot’s Jacobian matrix relative to the coordinate system fixed at the point of contact x c , y c , T is the n × n diagonal selection matrix with elements equal to zero in the force controlled directions, F is the measured contact force, τ ff is the output of the feed-forward controller, and τ sm the output of the sliding mode controller (Figure 18.2). The matrices used to estimate the torque vector are H(q) =  (m 1 + m 2 )r 2 1 + m 2 r 2 2 + 2m 2 r 1 r 2 C 2 + J 1 m 2 r 2 2 + m 2 r 1 r 2 C 2 m 2 r 2 2 + m 2 r 1 r 2 C 2 m 2 r 2 2 + J 2  (18.8) c(q, ˙ q) =  −m 2 r 1 r 2 S 2 ˙ q 2 2 − 2m 2 r 1 r 2 S 2 ˙ q 2 1 ˙ q 2 2 m 2 r 1 r 2 S 2 ˙ q 2 1  (18.9) F d F I-T J c T force controller robot and constrain surface I-T J c T ++ + – τ ff τ sm τ es FIGURE 18.2 Force controller with feed-forward compensation. Copyright © 2005 by CRC Press LLC Sliding Mode Control of Robotic Manipulators 18 -5 where >0 is the boundary layer thickness. ε = /λ n−1 is the corresponding boundary layer width, whereλ isthe design parameter of the sliding surface (18.2) for the case λ 1 = λ 2 =···=λ n . The switching control law (18.5) remains the same outside the boundary layer, and inside b L (t) the control action is interpolated by using, e.g., u sw = s/ , providing an overall piece-wise continuous control law. This creates an upper bound in tracking error given by λ (as opposed to perfect tracking): ∀t ≥ 0, |x (i) (t) −r (i) (t)|≤ (2λ) i ε; i = 0, , n − 1, for all trajectories starting inside b L (t). The boundary layer thickness  can be made time-varying to tune up the control law to exploit the control bandwidth available, and in that case the sliding condition (18.4) becomes |s(x, t)|≥ ⇒ 1 2 d dt s 2 ≤ ( ˙  − η)|s(x, t)| (18.13) A simple switching term that satisfies (18.13) is u sw = (k(x, t) − ˙ ) sat  s   , sat(z) =  z, |z|≤1 sgn(z), otherwise (18.14) which replaces Equation (18.5). This method eliminates control chatter provided that high-frequency unmodelled dynamics are not excited [3] and that the corresponding trade-off in tracking accuracy is acceptable. The equivalent control (u eq ) [3, 5, 9] is the continuous control law that would maintain dS/dt = 0 if the dynamics were exactly known. Consider the nonlinear affine system (18.15) and (18.16) with the associated sliding surface (18.17): ˙ x 1 =  f 1 (x 1 , x 2 ) (18.15) ˙ x 2 =  f 2 (x 1 , x 2 ) + B 2 (x)u + B 2 (x)  d(t) (18.16)  S = { x : ϕ(t) −s a (x) =s(x, t) = 0 } (18.17) where u is the m ×1 vector of control inputs,  d is the m ×1 vector of input disturbances, x 2 is the vector of m states to be controlled, x 1 is the (n −m) vector of autonomous states,  f is the n ×1vectorofsystem’s equations, B 2 (x)isam × m input gain matrix, s a (x)isam × 1 continuous function of the states to be tracked, and ϕ(t) is the m × 1 vector of desired trajectories. The equivalent control is obtained from d dt (ϕ(t) −s a (x, u =u eq )) = 0 (18.18) by calculating dS/dt =0 from Equation (18.18), replacing the system’s Equation (18.15) and Equa- tion (18.16) in the resulting expression, and finally solving for u eq . This yields u eq =−  d +(G 2 B 2 ) −1  d ϕ dt − G 2  f 2 − G 1  f 1  (18.19) where ds a /dt = G 1 x 1 + G 2 x 2 and G 1 , G 2 are defined as [∂s a /∂ x 1 ] = G 1 ,[∂s a /∂ x 2 ] = G 2 . The continuous control (18.19) is a nonrobust control law because it assumes the plant model to be exact. Several different techniques have been proposed to achieve robustness against parametric variations and unmodelled dynamics (disturbances) [6,9] by a continuous control law (as opposed to a switching control action such as Equation (18.5)), which is obviously essential in systems where the control inputs are continuous functions and hence Equation (18.5) cannot be realized. One such technique [9] is based on r Given the Lyapunov function candidate v = S T S/2, if the Lyapunov stability criteria are satisfied, the solution ϕ(t) − s a (x) = 0 is stable for all possible trajectories of systems (18.15) and (18.16). r The problem then becomes that of finding the control u that satisfies the Lyapunov condition dv dt =−S T DS ≤ 0 for some positive definite matrix D. Copyright © 2005 by CRC Press LLC Sliding Mode Control of Robotic Manipulators 18 -9 [12] Tso, S.K., Xu, Y., and Shum, H.Y., Variable structure model reference adaptive control of robot manipulators, Proc. 1991 IEEE International Conference on Robotics and Automation, Sacramento, CA, Vol. 3, pp. 2148–2153, April 1991. [13] Arai, F., Fukuda,T., Shoi, W., and Wada, H., Models of mechanical systems for controller design, Proc. IFAC Workshop on Motion Control for Intelligent Automation, Perugia, Italy, pp. 13–19, 1992. Copyright © 2005 by CRC Press LLC [...]... between the position of the mass at the actuator x1 and a reference r is applied to control motion A proportional controller acting on force fed back from the environment is applied to control force and improve interactive behavior The control law Copyright © 20 05 by CRC Press LLC 1 9-4 Robotics and Automation Handbook x1 x2 k Fa Fe m2 m1 b3 b1 b2 FIGURE 19 .2 Model including a single structural resonance... Copyright © 20 05 by CRC Press LLC Impedance and Interaction Control 1 9-1 1 These requirements ensure that Z(s ) is a positive real function and lead to the following interesting and useful extensions: 1 If Z(s )is positive real, so is its inverse, the admittance function Y (s ) = Z −1 (s ), and Y (s ) has the same properties 2 If equality is restricted from condition 3, the system is dissipative and is called... below Copyright © 20 05 by CRC Press LLC 1 9 -3 Impedance and Interaction Control to actuator forces F a and environmental forces F e A single damper b connected to ground represents frictional losses The Laplace-transformed equation of motion for this simple model is as follows: (ms 2 + bs )X = F a + F e (19.1) where X is the Laplace transform of the mass position x A proportional-integral motion controller... proportional-integral motion controller is applied F a = K p (R − X) + KI (R − X) s (19 .2) where R is the Laplace transform of the reference position and K p and K I are proportional and integral gains, respectively In isolation, F e = 0 and the closed-loop transfer function is K ps + K I X = 3 + bs 2 + K s + K R ms P I (19 .3) From the Routh-Hurwitz stability criterion, a condition for isolated stability is the following... Z(s ) and Y (s ) each lie wholly within the closed right half-plane b Z(s ) and Y (s ) each have phase in the closed interval between −90◦ and +90◦ 3 If condition 3 is met in equality, the system is passive (but not strictly passive) a The Nyquist contours of Z(s ) and Y (s ) each lie wholly within the open right half-plane b Z(s ) and Y (s ) each have phase in the open interval between −90◦ and +90◦... represented by the model shown in Figure 19 .2 Two masses m1 and m2 are connected by a spring of stiffness k Frictional losses are represented by dampers b1 and b2 connected from the masses to ground and damper b3 in parallel with the spring One mass is driven by the actuator force F a and the other is subject to F e , an interaction force with the environment A proportional-derivative (PD) controller acting... parameters and controller gains If the system is not isolated but instead is connected to a spring to ground, such that F e = −ke x2 , the closed loop characteristic polynomial is changed It is now easy to find parameters such that this polynomial has right-half-plane roots For example, if m1 = m2 = 10, b1 = b2 = b3 = 1, k = 100, K = 10, B = 1, K f = 10, and ke = 100, the closed loop poles are at 2. 78 ±... actuator and assumed motion sensor are co-located .3 However, the addition of a force feedback loop renders the robot control system vulnerable to coupled instability In part this is because the actuator and force sensor are not co-located Given the difficulty of designing a robot with no significant dynamics interposed between its actuators and the points at which it contacts its environment (see Tilley and. .. proven another useful result via this argument, particularly helpful in testing for coupled stability of systems As can easily be determined from Table 19.1, an ideal spring in admittance causality produces +90◦ of phase, and an ideal mass produces −90◦ , both for all frequencies, making each passive Copyright © 20 05 by CRC Press LLC 1 9-1 3 Impedance and Interaction Control but not strictly passive... coupled in the power-continuous way described above and illustrated in Figure 19.4, the phase property of passive systems constrains the total phase of the “open-loop” transfer function to between −180◦ and +180◦ (the phase of each of the two port functions is between −90◦ and +90◦ , and the two are summed) Because the phase never crosses these bounds, the coupled system is never unstable and is at worst . 2m 2 r 1 r 2 C 2 + J 1 m 2 r 2 2 + m 2 r 1 r 2 C 2 m 2 r 2 2 + m 2 r 1 r 2 C 2 m 2 r 2 2 + J 2  (18.8) c(q, ˙ q) =  −m 2 r 1 r 2 S 2 ˙ q 2 2 − 2m 2 r 1 r 2 S 2 ˙ q 2 1 ˙ q 2 2 m 2 r 1 r 2 S 2 ˙ q 2 1  (18.9) F d F I-T. Robotics Automat., Vol. RA 3, No.1,pp. 43 53, Feb. 1987. Copyright © 20 05 by CRC Press LLC 17 -2 2 Robotics and Automation Handbook [22 ] Kim, Y.H. and Lewis, F.L., Optimal design of CMAC neural-network. (17. 121 ) See [38 ] for details. Copyright © 20 05 by CRC Press LLC 17 -2 0 Robotics and Automation Handbook PERFORMANCE MEASURE + + + CONTROL ROBOT MODEL N MODEL 2 MODEL 1 – – – min J i J 1 J 2 J N e t2 e t N e t1 t ^ N t ^ 2 t ^ 1 ( q ,