Integrated Design and PD Control of High-Speed Closed-loop Mechanisms

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1 F. X. Wu Division of Biomedical Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada W. J. Zhang* Department of Mechanical Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada chris Q. Li School of Mechanical and Production Engineering, Nanyang Technological University, 50 Nanyang Ave., Singapore P. R. Ouyang Department of Mechanical Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada Integrated Design and PD Control of High-Speed Closed-loop Mechanisms The performance of an electromechanical system not only depends on its controller design, but also on the design of its mechanical structure. In order to achieve the excellent performance of the four-bar-link mechanism by employing the simple PD control, we redesign the structure of the four-bar-link mechanism by a mass-redistribution scheme to simplify the dynamic model. Theoretically, we analyze the stability of the closed-loop system consisting of the PD controller and several kinds of four-bar-link mechanisms, and discuss the relations between the performance of the PD controller and its gains and the mechanical design. The obtained results show that the performance of the PD controller may be significantly improved by using the methodology of Design For Control (DFC). The effectiveness of the proposed methodology has also been verified by some simulation studies. DOI: / Keywords: DFC, Four-Bar-Link Mechanism, Simple PD Control, Performance 1 Introduction For serial photos or open-chain mechanisms, a wellestablished formulation of the equations of motion 1 exists and a wealth of control results 5 have been developed during the last two decades. However, open-chain mechanisms possess some inherent disadvantages, for example, the position accuracy at the endpoint of the long robot arm is considerably low; a small amount of error at each revolution joint is magnified at the endpoint of the arm as its length gets longer; most importantly, the mechanical stiffness of the open-chain construction is inherently poor. As a result, the accuracy of the motion tracking performance can be deteriorated. The research trend in modern machinery development therefore shifts toward the design of a new generation of mechanism, i.e., the closed-chain mechanisms for the position and the trajectory tracking purpose. Comparing with the open-chain mechanism, the dynamic equations of the closed-chain mechanism include more system parameters and are of more complexity under the same degrees of freedom dof of the system. It thus renders difficulties for control engineers to control a closed-chain mechanism to follow a trajectory precisely and quickly. Several methods reported in the literature were proposed to handle these difficulties 6 8. As suggested by Lin and Chen 6, a very complex control structure that is composed of several sub-control algorithms, such as a model reference adaptive control, a disturbance compensation loop, and a modified switching controller plus some feedback loops was proposed to control the four-bar-link mechanism to follow a presimplified trajectory. Ghorbel 7,8 presented PD control with simple gravity compensation for a two-dof closed-chain mechanism to position the endpoint. Although Ghorbel s controller is the same in form as the controller used to the open-chain robot, the complexity of its computation is by far more. In general, intensive computation can result in the difficulty in physical realization of a controller for high-speed performance. There is indeed a trend to apply parallel computation/processing techniques 9 for controlling closed-loop mechanism systems with multi-dof. In Toumi s works 10,11, a different design strategy was *Author to whom all correspondence should be sent Contributed by the Dynamic Systems and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the ASME Dynamic Systems and Control Division, Nov. 000; final revision, April 00. Associate Editor: C. Rahn. adopted. With the aim to simplify the dynamic model of their concerned manipulator system, a parallelogram closed-loop mechanism structure is added into an open-loop robot structure; as such, the dynamic decoupling of generalized conditions actuator variables is achieved. High motion tracking performance can, thus, be achieved by applying a relatively simple control algorithm. Following Toumi s design strategy, Diken 1 improved the motion tracking performance of an open-loop robot manipulator by applying a mass redistribution scheme. In his study, the structure of a robot arm is first reduced into dynamically equivalent point masses so as to eliminate the gravitational term in the dynamic model. A simple algorithm is then applied to control the system and satisfactory performance trajectory tracking can be obtained. A more general concept called Design For Control DFC was proposed by the authors elsewhere 1,14. The essence of the concept suggests designing the mechanical structure of a programmable machine by fully exploring the physical understanding of the overall system with consideration of the facilitation of controller design as well as the execution of control action with the least hardware restriction. An intuitive way to realize this objective is to design an appropriate structure for the mechanical part so that it can result in a simple dynamic model and thus a more predictable dynamic response. In our previous work 1,14, we demonstrated the benefits to control a mechanical system by judicious selection of mass distribution, which results in the force balancing. In this paper, we further consider simplifying the dynamical model in terms of making the inertia term configuration-invariant or partial configuration-invariant. This paper is organized as follows. Section of this paper recalls the dynamic model of a four-bar linkage system. In Section, synthesis of the mass redistribution design is presented to derive configuration-invariant qualities. Section 4 analytically proves the stability of the PD controller and discusses the relation between the performance of the PD controller and the mechanical structure design. In Section 5, a conclusion is drawn and future work is discussed. Description of the Four-Bar-Link Mechanism For the clarity of description, the dynamic model of the four-bar linkage is rewritten here. The details of the derivation could be found in references 15,16. Figure 1 illustrates the configuration 5 Õ Vol. 14, DECEMBER 00 Copyright 00 by ASME Transactions of the ASME

2 of the four-bar linkage under study. For link i, the location of the center of mass which is denoted by a darkened circle shown in this figure is described by variables r i and i. Furthermore, m i and L i denote the mass and the length of the link, respectively, and J i is the moment of inertia with respect to the centroid of link i. The Lagrange s equation is applied to derive the dynamic model of the linkage, i.e., d K K P (1) dt where K denoting the kinetic energy, P the potential energy, and the external input torque. In Eq. 1, input crack angle 1 is specified as the generalized coordinate to describe the motion of the four-bar-link mechanism. The kinetic energy of the linkage system can be expressed by () K i1 1 m iv ix V iy 1 J i i with i representing the angular velocity of link i, and V ix and V iy representing the x and y axis velocity components of the mass center of link k. i, V ix, and V iy can be further expressed by V ixu i 1 V iy v i 1 i i 1 The detailed expression of u i, v i, and i can be found in Appendix A. By substituting Eq. into Eq., we obtain with () K 1 A 1 1 (4) A 1 i1 m i u i v i J i i (5) The potential energy of the mechanism can be expressed by P g m 1 r 1 sin 1 1 m L 1 sin 1 r sin m L 4 sin 4 r sin g (6) where,, 4, 1,, and are illustrated in Fig. 1, and g is the gravity constant. Substituting Eqs. 5 and 6 into Eq. 1 and using the relations defined by Eq., we have A da 1 1 G (7) d 1 where A( 1 ) is the non-constant generalized inertia coefficient term, (1/) da( 1 )/d 1 is the Coriolis/centripetal coefficient term, and G( 1 ) is the gravity term. Their full expansions are described in Appendix B. From Appendix B, we can obtain the following properties for the four-bar-link mechanism: Property 1. For any kinematically valid input crack angle 1, A( 1 ), and (1/) da( 1 )/d 1 are bounded, and A( 1 ) is positive. That is, there exist three positive constants a 1, a, and a such that a 1 A 1 a, 1 da 1 a (8) d 1 Property. For any kinematically valid input crack angle 1, G( 1 ) is bounded. That is, there exists a positive constant g 1 such that G 1 g 1 (9) Redesign of the Four-Bar-Link Mechanism As shown in Eq. 7, the dynamic model of the four-bar linkage is quite complicated. To design a control algorithm for this system to achieve high performance is not a simple task. Following the DFC concept, this section will present the modification of the mass distribution of the four-bar-link mechanism, with the aim to find out some configuration-invariant mechanism, which may simplify the dynamic model of the mechanism so as to facilitate the controller design..1 Configuration-Invariance of the Potential Energy CIPE. CIPE means that the potential energy of a system does not change with respect to a different configuration. This requires that the global center of mass GCM of a mechanism stays stationary during the operation of the system. The mass distribution is represented by m i, r i, i, as shown in Fig. 1. The GCM of a four-bar-link mechanism can be expressed by r c 1 m m i r ci (10) i1 where m i1 m i ; r c denotes the position vector of the GCM, r ci (i1,,) the position vector of the center of mass of link i. The position vector r ci are expressed by r c1 r 1 e i 1 1 r c L 1 e i 1r e i r c L 4 e i 4r e i Substitution of the above equations into Eq. 10 leads to: Fig. 1 Structure of a four-bar-link mechanism mr c m 1 r 1 e i 1m L 1 e i 1m r e i e i m r e i e i m L 4 e i 4 (11) The unit vector e i 1, e i, e i are constrained by the kinematics loop equation, i.e., L 1 e i 1L e i L e i L 4 e i 40 Substitution of the vector e i, solved out of the above equation, into Eq. 11 leads to: L mr c m 1 r 1 e i 1 1m L 1 m r e i L e 1 i m r e i L e i 4 m L 4 m e i 4 m L L r e i L r e i Journal of Dynamic Systems, Measurement, and Control DECEMBER 00, Vol. 14 Õ 5

3 In the above equation, in order to keep the stationary GCM, the coefficients of the vectors e i 1 and e i must vanish, i.e., m 1 r 1 e i 1m L 1 m L 1 L r e i 0, m r e i m L L r e i 0 From Fig. 1, the following relationship holds r e i L r e i (1) Substitution of the above equation into the first equation in 1 yields: L m 1 r 1 e i 1 1m r e i 0 (1) L Therefore, from the second equation in 1 and Eq. 1, the conditions for the configuration-invariance of the potential energy in the four-bar-link mechanism are L 1 m1r1m r, 1 L (14) L m r m r, L From Eq. 14, it can be seen that whenever the mass and the location of the center of the mass of one of the links are given, the mass distribution of the remaining two links can then be determined. Furthermore, Eq. 14 can also be applied to determine the size and location of counterweights or negative masses that may need to be added to the mechanism for the configurationinvariance of the potential energy. When Eq. 14 holds, the potential energy of the four-bar-link mechanism is configuration invariant, so P 0 (15) 1 After the modification schemes are applied, the dynamic model of the linkage given in Eq. 7 is simplified as: A da 1 1 (16) d 1 From literature 1, Eq. 14 implies that the shaking force is completely balanced.. Configuration-Invariance of the Generalized Inertia CIGI. The work reported in 10,11 showed a parallel drive five-bar-link mechanism, in which the distance between the two motors is zero and the two pairs of opposite links are parallel. Under the certain conditions see the literature 10,11 for details, the generalized inertia of the parallel drive five-bar-link mechanism is configuration-invariant. In the care of the four-barlink mechanism, the parallel structure requires the following condition, i.e., L 1 L, L L 4 (17) This then leads to the following kinematics motion behavior, i.e., 1, 0 and, 1 1, 0 Substitution of the above equations into Eq. 5 causes A 1 C 0 C 1 cons tan ta That is to say, a parallel four-bar-link mechanism holds the CIGI property. Furthermore, it is possible to prove that in fact the parallel structure is necessary and sufficient for a four-bar-link mechanism to have the CIGI property. If both Eqs. 14 and 17 hold simultaneously, the system holds both CIPE and CIGI properties. In this case, the dynamic model is reduced to: A 1 (18). Partial Configuration-Invariance of the Generalized Inertia PCIGI. The four-bar-link mechanism with a parallel structure has, however, a limited application scope. It should be interesting to consider partial CIGI for a four-bar-link mechanism with non-parallel structure. In particular, the impact of a partial CIGI on control design is to be encountered. Through some empirical studies, we have found a special situation that can lead to a simple dynamic model with a partial CIGI. Let the mass center of the second link be placed on the end of the input link. For the system to be of CIPE, we obtain from Eq. 14: r 0, m 1 r 1 m L 1, 1, r 0 (19) In this situation, the generalized inertia is reduced to: A 1 C 0 J J m r m r m r L 1 cos 1 C 0 J J (0) That is, the expression in in Eq. 0 is a constant zero when the condition 19 holds. The remainder of the discussion is largely on the impact of this partial CIGI PCIGI for short on control system design. 4 Control Algorithm Design and Stability Analysis If the dynamics of a four-bar-link mechanism are exactly known, the computer-torque controller is known to be asymptotically stable in tracking a desired trajectory. However, in practice, the dynamics are in most cases not perfectly available. The PD control method still enjoys its competitive role in real industrial control applications. Although the PD control methods do not hold asymptotic stability for trajectory tracking task, the trajectory tracking error can be bounded. In the following, we first give a theorem for stability analysis for the PD control methods for the four-bar-link mechanism, and then apply this theorem to discuss the stationary error of PD controller for various situations of configuration-invariance. The proof of this theorem is provided in Appendix C. 4.1 PD Controller and its Stability Analysis. Consider the following PD controller: tk p e tk d et (1) where (t) is the driving torque generated by the controller, K p and K d are the proportional and derivative gains, respectively, e (t) d 1 (t) 1 (t) represents the angular trajectory error appeared in the input crank with d 1 (t) and 1 (t) as the desired and actual angular displacements, respectively, and e(t) the angular velocity error of the input crank. Assume that the desired velocity d(t) and acceleration d(t) are bounded, that is, there exist two positive constants b 1 and b such that db 1, db () From Eqs. 16 and 1, the tracking error equation can be obtained: 54 Õ Vol. 14, DECEMBER 00 Transactions of the ASME

4 A 1 e 1 da 1 1 ek p e K d e 1 da 1 d 1 d e d 1 A 1 d 1 da 1 d G 1 () d 1 For simplicity, let w(t)a( 1 ) d(1/) da( 1 )/d 1 d G( 1 ), then wta b a b 1 g 1 (4) By employing the Lyapunov approach, we obtain the following theorem: Theorem. If the gains of the PD controller K d and K p and the constant are chosen such that 0 K pk d a b 1 (5) K d K p K p a K p max a 4a c 1, a 1 (6) then, the trajectory tracking error can be bounded, and this bound is given by lim x 4c 1 t K p 1K p /a (7) where x e e T, and the constants c 1 and is defined in Appendix C. To give an impression of the reliability of this theorem, we use this theorem to verify the asymptotic stability of a PD controller for point-to-point control. In this case, the desired trajectory can be viewed as a fixed point, thus d d0. From Eq., we know b 1 b 0. Further, if the gravity term is cancelled, then c 1 0. According to the theorem, we obtain lim t x0, and this implies the closed-loop is asymptotically stable as stated in the existing literature 1,7,8. 4. Impact of CIPE and CIGI on Control System Design Performance of the PD controller in the case of CIPE. If the potential energy of the mechanism is configurationinvariant, then g 1 0. According to the theorem, we have lim x 41 a b a b 1 t K p 1K p /a (8) Comparing Eqs. 8 and 7, we can conclude that the relative reduction of the stationary trajectory tracking error the case without CIPE versus the case with CIPE is in proportion to the term of g 1 /(a b a b 1 g ). In other words, the control performance improvement in terms of the stationary trajectory tracking error reduction is in the percent of this term. The larger this term the more improvement is the control performance achieved. 4.. Performance of the PD controller in the case of both CIPE and CIGI. In this case, the dynamics equation is reduced to Eq. 18, thus a 0. According to the theorem, we have lim x 41 a b t K p 1K p /a (9) With the same discussion above, we can conclude that the control performance improvement in terms of the stationary trajectory tracking error reduction is in the percent of g 1 a b 1 /(a b a b 1 g 1 ) term. It is noted that the relative reduction here is larger than that in the case of CIPE. This shows the significance in Table 1 Mechanical parameters of two cases: case 1 CIPE; case CIPE and PCIGI Parameters Case 1 Case L 1 m L m L m L 4 m r 1 m r m r m m 1 kg m kg m kg J 1 kg.m J kg.m J kg.m rad rad 0 0 rad 4 rad 0 0 further improvement of the control performance when the fourbar-link mechanism system possesses the property of CIGI on the top of possessing the CIPE property. Furthermore, if the desired velocity is constant as discussed in the literature 17, thus b 0. From inequality 7, the trajectory error satisfies lim t x0. Therefore, the PD controller may result in the four-bar-link mechanism with both CIPE and CIGI to track asymptotically the desired constant velocity. 4.. Performance of the PD controller in the case of both CIPE and PCIGI. In this case, although the resultant dynamic equation is the same as Eq. 16, but some terms that produce the fluctuation in both A( 1 ) and (1/) da( 1 )/d 1 have been cancelled. It is possible that the magnitudes of constants a 1 /a and a are reduced. According to the theorem, this implies that employing the DFC design method may significantly decrease the trajectory tracking error. The following simulation of cases will also show this point. 4. Simulation. To investigate the effectiveness of PCIGI, simulation studies were carried out for the four-bar-link mechanism of two cases, one of which is only CIPE, and another of which is CIPE and PCIGI. The parameters of the four-bar-link mechanism under different situations are recorded in Table I. Case 1 describes the mechanism only with CIPE, and case describes the parameters of the modified mechanism with both CIPE and PCIGI. In the simulation, the input crank was required to rotate at a high-speed constant velocity of 0 rad/s, and the controller gains were selected to be K p 15 and K d 5, where minimal performance indexes are obtained for case 1 1. Comparing the simulation results, it is observed that, after applying the DFC method such that the four-bar-link mechanism possess both CIPE and PCIGI, the motion tracking performance of the system is improved significantly. As shown in Fig., the angular displacement tracking performance is improved in case. Figure b shows the angular velocity tracking errors are reduced from 0.14 rad/s in case 1 to 0.06 rad/s in case, and the improved magnitude exceeds 50%. Figure c shows the simulated result of the angular velocity. From the control torque profiles shown in Fig. (d), it is shown that less control energy is consumed in case, and the maximum of torque in case is only a half of that in case 1. 5 Conclusion The four-bar-link mechanism can be designed to possess both the CIPE and CIGI properties. The control performance of a PD controller for trajectory tracking for this case can be further improved over the case where the mechanism only possess the CIPE property. The mechanism with the CIGI property renders itself to Journal of Dynamic Systems, Measurement, and Control DECEMBER 00, Vol. 14 Õ 55

5 be a parallel structure which restricts its application. Therefore, the case where the mechanism possesses the PCIGI property is worth study. It is shown that indeed the control performance can be improved with the PCIGI property in addition to the CIPE property. The above conclusion is not only supported by the simulation study but also by the proposed theorem. This theorem describes the relationship among the stationary trajectory tracking error, control gains, and the properties of the physical structure of the four-bar-link mechanism. Therefore, this theorem can be used as a tool for designing PD controllers for the four-bar-link mechanism to achieve a desired performance for trajectory tracking. Although the study presented here is for the four-bar-link mechanism, these conclusions could most likely be extended to more general closed-loop mechanical systems driven by servomotors. The extended work for a five-bar-link mechanism driven by two servomotors is well under way. Appendix A The terms u i, v i, and i can be described as follows 1: 1 i L 1sin j 1 1 L i sin i j, with i and j being any cyclic permutation of and ; (A1) u i v i u R v R y ir x ir i, (A) with R representing the Point A, B, and D of Link 1,, and, respectively. The detailed expression of u i, v i, and i are given as 1: Fig. Tracking performance of the four-bar-link mechanism with the PD controller under two cases: a Profiles of angular displacement tracking errors, b profiles of angular velocity tracking errors, c responses of angular velocities, and d profiles of torque dotted line: for case 1, only with CIPE; solid line: for case, with both CIPE and PCIGI 56 Õ Vol. 14, DECEMBER 00 Transactions of the ASME

6 1 1 L 1sin 1 L sin L 1sin 1 L sin u 1 v 1S 1 1 C 1 1 C 1 1 S 1 u v L 1S 1 L 1 C 1 S C C S u v S C L S C S L C (A) (A4) where, C i cos i, S i sin i. Note i, i and the local coordinate of each link are shown in Fig.. Furthermore, the coordinates of point P i on link i can be determined with reference to the local coordinate system: x ir x i x R i cos i i sin i Appendix B y ir y i y R i sin i i cos i Term A( 1 ) in Eq. 7 can be expressed in a more compact form A 1 C 0 C 1 C C cos 1 (B1) where C i (i0,1,,) are coefficients containing the parameters for mass distribution. The coefficients C i (i0,1,,) are given as 1 C 0 J 1 m 1 r 1 m L 1 C 1 J m r C J m r C m r L 1 The expression of da( 1 )/d 1 in Eq. 7 can be written as: (B) da 1 d d C 1 C C d 1 d 1 d 1 d cos 1 d 1 sin 1 1 (B) where d L 1D 1 D d 1 L sin d L 1D D 4 d 1 L sin D 1 1sin cos 1 D sin 1 cos D 1sin cos 1 D 4 sin 1 cos (B4) The expression of G( 1 ) in Eq. 7 can be written as: G 1 m 1 r 1 cos 1 1 m L 1 cos 1 m r cos m r cos g (B5) Appendix C In this Appendix, we prove theorem in Section 4. Construct the candidate Lyapunov function as follows: It may be estimated as: V 1 A 1 e 1 K p e A 1 e e xq 1 x 1 a 1 e 1 K p e a e e V 1 a e 1 K p e a e e xq x (C1) (C) where x e e T and Q 1 1 a 1 a, Q a K 1 p a a a K p Since the matrix Q 1 is positive definite when K p a /a 1, the function V in Eq. C1 is positive, and 1 x V x (C) where 1 and are positive constants defined by the minimum eigenvalue of the matrix Q 1 and the maximum eigenvalue of the matrix Q, respectively, i.e., 1 a 1K p a 1 K p a 0, (C4) a K p a K p a 0 Differentiating the Lyapunov function C1 along the trajectory of Eq. yields to V A 1 e e 1 A 1 1 e K p e e A 1 1e e e 1 A 1 e ea 1 e K d e 1 A 1 d e A 1 e 1 K p e K d e e ew e 1 K d a b 1 a e K p e K d e e 1 a b a b 1 g 1 xa e e A 1 1 e e xq xc 1 xc x c 1 xc x a x where c 1 and c are positive constants defined as: (C5) c 1 1 a b a b 1 g 1 (C6) and c is the minimum eigenvalue of the matrix below K Q p 0.5K d 0.5K d K d a b 1 a (C7) If the condition 5 holds, c K pk d a b 1 a K d a b 1 a K p K d K p 0 (C8) Journal of Dynamic Systems, Measurement, and Control DECEMBER 00, Vol. 14 Õ 57

7 From Eq. C5, V 0, if 1 x (C9) where 1 and are two constants defined as: 1 c c 4a c 1, c c 4a c 1 (C10) a a If the following inequality 6 holds, 1 and are positive. Thus, if the inequalities 5 and 6 hold, the inequalities C and C9 is true. Therefore, similar to Theorem 1 in the literature 18, we can obtain when the initial error x(0) satisfies x(0) lim xte / 1 1. t (C11) In order to analyze the effects of the parameters of mechanism on the tracking error, by using Eqs. C4 and C5, we obtain E 1 1 a K p a K p a a 1 K p a 1 K p a 4a c 1 a K p K p a a a 1 K p K p a 1 a a K p a 1 a c 1 c Substituting C6 and C8 into C1 yields to c c 4a c 1 a a c c 4a c 1 (C1) E 4c 1 K p 1K p /a (C1) where a 1 /a. So the result in the theorem is concluded from Eqs. C11 and C1. References 1 Craig, J. J., 1989, Introduction to Robotics: Mechanics and Control, nd edition, Addison Wesley, Reading, MA. Murray, R. M., Li, Z., and Sastry, S. S., 1994, A Mathematical Introduction to Robotic Manipulator, CRC Press Inc. Craig, J. J., 1988, Adaptive Control of Mechanical Manipulator, Addison- Wesley Publishing Co. 4 Slotine, J. E., and Li, W., 1991, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, NJ. 5 Ortega, R., and Spong, M. W., 1989, Adaptive Motion Control of Rigid Robots: A Tutorial, Automatica, 56, pp Lin, M. C., and Chen, J. S., 1996, Experiment Toward MARC Design for Linkage System, Mechatronics, 66, pp Ghorbel, F., 1997, PD Control of Closed-Chain Mechanical Systems: An Experimental Study, Proc. of 5th IFAC Symp. on Robot Control SYROCO 97, Nantes, France, 1, pp Ghorbel, F., 1997, A Validation Study of PD Control of Closed-Chain Mechanical Systems, Proc. of IEEE 6th Conf. on Decision and Control, San Diego, pp Gosselin, C. M., 1996, Parallel Computational Algorithms for the Kinematics and Dynamics of Planar and Spatial Parallel Manipulators, ASME J. Dyn. Syst., Meas., Control, 1181, pp Asada, H., and Youcef-Toumi, K., 1987, Direct-Drive Robots: Theory and Practice, MIT Press. 11 Youcef-Toumi, K., and Kuo, A. T. Y., 199, High-Speed Trajectory Control of a Direct-Drive Manipulator, IEEE Trans. Rob. Autom., 91, pp Diken, H., 1997, Trajectory Control of Mass Balanced Manipulator, Mech. Mach. Theory,, pp Zhang, W. J., Li, Q., and Guo, L. S., 1999, Integrated Design of Mechanical Structure and Control Algorithm for a Programmable Four-Bar-Linkage, Mechatronics, 44, pp Zhang, W. J., and Li, Q., 1999, Design for Control: A New Principle for Technical Systems Development, 1th Int. Conf. on Engineering Design, Aug., Munich, Germany. 15 Doughty, S., 1988, Mechanics of Machines, John Wiley & Sons, Inc. 16 Burton, P., 1979, Kinematics and Dynamics of Planar Machinery, Prentice- Hall, Inc., Englewood Cliffs, NJ. 17 Tao, J., and Sadler, J. P., 1995, Constant Speed Control of a Motor Driven Mechanism System, Mech. Mach. Theory, 05, pp Dawson, D. M., Qu, Z., Lewis, F. L., and Dorsey, J. F., 1990, Robust Control for the Tracking of Robot Motion, Int. J. Control, 5, pp Õ Vol. 14, DECEMBER 00 Transactions of the ASME