Robust Control of Robotic Manipulators308 (5.3.11) A representative of this class was developed in [Spong et al. 1987] and is given as follows. THEOREM 5.3–4: The trajectory error e is uniformly ultimately bounded (UUB) with the controller Figure 5.3.8: (a) errors of joint 1; (b) errors of joint 1; (c) torques of joints 1 and 2 Copyright © 2004 by Marcel Dekker, Inc. 309 (1) where (2) Figure 5.3.9: (a) errors of joint 1; (b) errors of joint 1; (c) torques of joints 1 and 2 5.3 Nonlinear Controllers Copyright © 2004 by Marcel Dekker, Inc. Robust Control of Robotic Manipulators310 and (3) (4) (5) Proof: Choose the Lyapunov function and proceed as in Theorem 2.10.4. See [Spong et al. 1987] for details. Note that in the equations above, the matrix B is defined as in (5.2.3), the  i ’s are defined as in (5.2.10), and the matrix P is the symmetric, positive- definite solution of the lyapunov equation (5.3.12), where Q is symmetric and positive-definite matrix and A c is given in (5.2.14). (5.3.12) In particular, the choice of Q (5.3.13) leads to the following P: (5.3.14) The expression of P in (5.3.14) may therefore be used in the expression of v r in (2). This design is summarized in Table 5.3.4. Copyright © 2004 by Marcel Dekker, Inc. Robust Control of Robotic Manipulators312 Upon closer examination of Spong’s controller in Theorem 5.3.4, it becomes clear that v r depends on the servo gains K p and K v through p. This might obscure the effect of adjusting the servo gains and may be avoided The same trajectory is followed by the two-link robot as shown in Figure 5.3.10. Note that although the trajectory errors seem to be diverging, they are indeed ultimately bounded and may be shown to be so by running the simulation for a long time. Figure 5.3.10: (a) errors of joint 1; (b) errors of joint 1; (c) torques of joints 1 and 2 Copyright © 2004 by Marcel Dekker, Inc. 313 as described in [Dawson et al. 1990]. THEOREM 5.3–5: The trajectory error e is uniformly ultimately bounded (UUB) with the controller (1) where Figure 5.3.11: (a) errors of joint 1; (b) errors of joint 1; (c) torques of joints 1 and 2 5.3 Nonlinear Controllers Copyright © 2004 by Marcel Dekker, Inc. Robust Control of Robotic Manipulators314 and (2) where ␥ i ’s are positive scalars. Proof: We again choose and and proceed as in Theorem 2.10.4. See [Dawson et al. 1990] for details. Note that p no longer contains the servo gains and, as such, one may adjust K p and K v without tampering with the auxiliary control v r . As was also shown in [Dawson et al. 1990], if the initial error e(0)=0 and by choosing K v =2K p =k v I, the tracking error may be bounded by the following, which shows the direct effect of the control parameters on the tracking error: (5.3.15) Finally, note that if e(0)=e . (0)=0, the uniform boundedness of e(t) may be deduced. This controller is given in Table 5.3.5. EXAMPLE 5.3–6: Saturation Controller 2 In this example, let Copyright © 2004 by Marcel Dekker, Inc. Robust Control of Robotic Manipulators316 5.4 Dynamics Redesign In this section we present two other approaches to design robust controllers. The first starts with the mechanical design of the robot and proposes to design robots such that their dynamics are simple and decoupled. It then solves the robust controller problem by eliminating its causes. The second approach may be recast into one of the approaches discussed previously, but it presents such a novel way to looking at the problem that we decided to include it separately. Decoupled Designs It was shown throughout the previous chapters that the controller complexity is directly dependent on that of the robot dynamics. Thus it would make sense to design robots such that they have simple dynamics making their control much easier. This approach is advocated in [Asada and Youcef-Toumi 1987]. In fact, it is shown that certain robotic structures will have a decoupled dynamical structures resulting in a decoupled set of n SISO nonlinear systems which are easier controlled than the one MIMO nonlinear system. The decoupling is achieved by modifying the dimensions and mass properties of the arm to cancel out the velocity-dependent terms and decouple the inertia matrix. An illustrative example of such robots is given in the next example. EXAMPLE 5.4–1: Decoupled Design Consider the robot described in Figure 5.4.1. This mechanism is known as the five-bar linkage and its dynamics are described when the following condition holds: by Note that the inertia matrix is decoupled and position independent. The controller given by Copyright © 2004 by Marcel Dekker, Inc. Robust Control of Robotic Manipulators318 Some standard robotic structures may also be decoupled by design. Studies have been carried out to partially or totally decouple robots up to six links. The interested reader is referred to [Yang and Tzeng 1986], [Asada and Youcef-Toumi 1987], and [Kazerooni 1989] for good discussions of this topic. Imaginary Robot Concept The decoupled design alternative is very useful if the control engineer has access to, or can modify, the robot design at an early stage. It is more reasonable, however, to assume that the robot has already been constructed to satisfy may mechanical requirements before the control law is actually implemented. Thus a dynamics redesign is difficult if not impossible. The imaginary robot concept is presented as an alternative robust design methodology [Gu and Loh 1988]. The development of this approach is described next. Consider an output function of the robot given by (5.4.1) so that (5.4.2) and (5.4.3) The generalized output y may denote the coordinates of the end effector of the robot or the trajectory joint error q d -q. The imaginary robot concept attempts to simplify the design of the control law for the physical robot, by controlling an “imaginary” robot that is close to the actual robot. This choice of the controller is shown to achieve the global stability of an imaginary robot whose joint positions are described by the components of the vector y. The methodology starts by decomposing M(q) as follows: (5.4.4) and then using the controller (5.4.5) (5.4.6) Copyright © 2004 by Marcel Dekker, Inc. 319 Since M(q) is unknown, however, the actual M ~ is not available. The resulting controller is then simpler and may be applied to the physical robot to lead acceptable, if not optimal behavior. The following theorem illustrates a controller to guarantee the boundedness of the error. THEOREM 5.4–1: Let (1) (2) The ∞ stability of the closed-loop system is guaranteed if (3) Proof: This is an immediate result of Theorem 5.2.2. See [Gu and Loh 1988] for more detail and for illustrative examples. The controller is shown in Table 5.4.1. Table 5.4.1: Computed-Torque-Like Robot Controllers. 5.4 Dynamics Redesign Copyright © 2004 by Marcel Dekker, Inc. [...]... 198 8 [Canudas de Wit and Fixot 199 1] C.Canudas de Wit and N.Fixot Robot Control via Robust Estimated State Feedback” IEEE Trans Autom Control, vol 36, number 12, pp 1 497 –1501, December, 199 1 [Chen 198 9] Y.Chen “Replacing a PID Controller by a Lag-Lead Compensator for a Robot: A Frequency Response Approach” IEEE Trans Robot Autom vol 5, number 2, pp 174–182, April, 198 9 [Chen et al 199 0] Y-F Chen and. .. Control for Robot Manipulators of Sensory Capability” Proc Third Int Symp Robot Res., Gouvieux, France July, 198 5 [Asada and Youcef-Toumi 198 7] H.Asada and K.Youcef-Toumi “Direct-Drive Robots: Theory and Practice MIT Press, Cambridge, MA, 198 7 [Becker and Grimm 198 8] N.Becker and W.M.Grimm “On L2 and L8 Stability Approaches for the Robust Control of Robot Manipulators” IEEE Trans Autom Control, vol 33,... Robust Control for Rigid Robots” IEEE Control Syst Mag., vol 11, number 2, pp 24–30, 199 1 [Anderson 198 9] R.J.Anderson “A Network Approach to Force Control in Robotics and Teleoperation” Ph.D Thesis, Departemnt of Electrical & Computer Engineering University of Illinois at Urbana-Champaign, 198 9 [Arimoto and Miyazaki 198 5] S.Arimoto and F.Miyazaki “Stabiliy and Robustness of PID Feedback Control for Robot. .. Y-F Chen and T.Mita and S.Wahui “A New and simple Algorithm for Sliding Mode Control of Robot Arms” IEEE Trans Autom Control, vol 35, number 7, pp 828–8 29, 199 0 [Corless 198 9] M.Corless “Tracking Controllers for Uncertain Systems: Application to a Manutec R3 Robot J Dyn Syst Meas Control vol 111, pp 6 09 618, December, 198 9 [Craig 198 8] J.J.Craig “Adaptive Control of Mechanical Manipulators” Addis... bounded and that M(q) is a positive-definite matrix (see Chapter 2), we can state that r and are bounded From the definition of r given in (6.3.8), we can use standard linear control arguments to state that e and (and hence q and ) are bounded Since e, , r, and are bounded, we can use (6.3.10) and (6.3.14) to show that (and hence Copyright © 2004 by Marcel Dekker, Inc 346 Adaptive Control of Robotic Manipulators... method [Slotine 198 8] of motivating a PD plus gravity controller [Arimoto and Miyazaki 198 6] is to write the manipulator dynamics in the conservationof-energy form (6.3.1) where the left-hand side of (6.3.1) is the derivative of the manipulator kinetic energy, and the right-hand side of (6.3.1) represents the power supplied from the actuators minus the power dissipated due to gravity and friction Note... Reading, MA, 198 8 321 Copyright © 2004 by Marcel Dekker, Inc REFERENCES 323 [Vidyasagar 198 5] M.Vidyasagar Control Systems Synthesis: A Factorization Approach” MIT Press, Cambridge, MA 198 5 [Yang and Tzeng 198 6] D.C-H.Yang and S.W.Tzeng “Simplification and Linearization of Manipulator Dynamics by the Design of Inertia Distribution” Int J Rob Res vol 5, number 3, pp 120–128, 198 6 [Yeung and Chen 198 8] K.S.Yeung... design adaptive controllers for robotic manipulators EXAMPLE 6.3–1: Adaptive Inertia-Related Controller We wish to design and simulate the adaptive inertia-related controller given in Table 6.3.1 for the two-link arm given in Figure 6.2.1 Assuming that the friction is negligible and the link lengths are exactly known, the adaptive inertia-related torque controller can be written as (1) and Copyright... the parameter estimates, Kv and Kp are control gain matrices, qd is used to denote the desired trajectory, and the tracking error e is defined by e=qd-q We now illustrate the approximate computed-torque controller by examining an example EXAMPLE 6.2–1: Approximate Computed-Torque Controller We wish to design and simulate an approximate computed-torque controller for the two-link arm given in Figure 6.2.1... [Yeung and Chen 198 8] K.S.Yeung and Y.P.Chen “A New Controller Design for Manipulators Using the Theory of Variable Structure Systems” IEEE Trans Autom Control, vol 33, number 2, pp 200–206, February, 198 8 [Young 197 8] K-K.D.Young “Controller Design for a Manipulator Using theory of Variale Structure Systems” IEEE Trans Syst Man Cybern vol 8, number 2, pp 210–218, February, 197 8 Copyright © 2004 by Marcel . immediate result of Theorem 5.2.2. See [Gu and Loh 198 8] for more detail and for illustrative examples. The controller is shown in Table 5.4.1. Table 5.4.1: Computed-Torque-Like Robot Controllers. 5.4. vol. 5, number 2, pp. 174–182, April, 198 9. [Chen et al. 199 0] Y-F Chen and T.Mita and S.Wahui. “A New and simple Algorithm for Sliding Mode Control of Robot Arms”. IEEE Trans. Autom. Control, . simplify the design of the control law for the physical robot, by controlling an “imaginary” robot that is close to the actual robot. This choice of the controller is shown to achieve the global