Dynamic Models for Control System Design of Integrated Robot and Drive Systems

Transcription

1 M. C. Good L. M. Sweet K. L. Strobel General Electric Company, Corporate Research and Development, Schenectady, N.Y Dynamic Models for Control System Design of Integrated Robot and Drive Systems The design of high performance motion controls for industrial robots is based on accurate models for the robot arm and drive systems. This paper presents analytical models and experimental data to show that interactions between electromechanical drives coupled with compliant linkages to arm link drive points are of fundamental importance to robot control system design. Flexibility in harmonic drives produces resonances in the 5 Hz to 8 Hz range. Flexibility in the robot linkages and joints connecting essentially rigid arm members produces higher frequency modes at 4 Hz and 40 Hz. The nonlinear characteristics of the drive system are modeled, and compared to experimental data. The models presented have been validated over the frequency range 0 to 50 Hz. The paper concludes with a brief discussion of the influence of model characteristics on motion control design. Introduction The successful design of any high-performance control system is based on accurate knowledge of the dynamics of the physical plant. This knowledge is particularly important when the "unmodeled dynamics", physical effects not included in the model, produce poles near the imaginary axis. If the control design does not account for these lightly-damped poles, there is great risk that the closed-loop system will become unstable or exhibit very poor robustness to variations in plant parameters. The design of multivariable controllers for industrial robots is a subject of great current interest. Nearly all papers in the literature on the subject of robot motion control are based on models for the arm that, based on the results of research presented in this paper, are inadequate for a large class of contemporary arm mechanical designs. Since there are major differences in the dynamic behavior of real robot arms from the idealized models found in the literature, it is difficult to assess the benefits to be obtained through implementation of advanced control schemes. The principal limitation of published models for industrial robots is the assumption that their dynamic behavior is represented adequately by interconnected rigid bodies driven by actuators modeled as pure torque sources or as first-order lags. It is noteworthy that there is very limited published experimental evidence to substantiate this assumption. In contrast, in this paper the importance of dynamic interactions between the multiple link arm and its electromechanical drive systems is demonstrated through analysis and experimentation. Specifically, this paper addresses the following aspects of model development for the purposes of control system design for real industrial robots: Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received at ASME Headquarters, January 29, 985. (A) Identification of resonant behavior induced through dynamic interactions of drive motors and the robot arm, coupled by harmonic drives and linkages with series compliance. (B) Characterization of structural modes of the robot arm, including an assessment of the relative importance of flexibility in the drive system versus the arm links themselves. (C) Characterization of nonlinearities in the drive system, due to motor current limiters, Coulomb friction, and stiffening spring characteristics in the harmonic drive. The paper concludes with a brief discussion of the influence of the integrated robot and drive system models on controller design. Limitations of Rigid Body Robot Models Nearly all models for robot dynamics presented in the literature are based on the assumption that the arm is a linkage of connected rigid bodies, as exemplified by the widely-cited references in [-5]. Using the Newton-Euler, Lagrange, Kane, or other approaches, the kinematics and dynamics of multiple link robot arms are derived and reduced to the following form: H(.q)q + C{q,q) + G{q) + K{qYM=T () In this equation q is a vector of robot joint angles, H is a moment of inertia matrix, C is a matrix specifying centrifugal and Coriolis effects, G is a vector specifying gravitational effects, K\s a Jacobian matrix relating joint torques to M, the vector of forces and moments applied to the end effector, and Tis the vector of torques applied by the actuators at the drive points on each arm link. The characteristics of equation () as applied to multiple Journal of Dynamic Systems, Measurement, and Control MARCH 985, Vol. 07/53 Copyright 985 by ASME

2 FORE ARM ROTATION Fig. Geometry of industrial robot used for analysis and experimental studies link robot arms have been exhaustively discussed in the literature, with particular attention paid to: (A) Dependence of effective inertia, reflected at the actuator drive point, on robot arm configuration and end-effector pay load. (B) Cross-coupling of the set of second-order differential equations due to Coriolis, centripetal and inertia forces. (C) Nonlinear behavior due to arm geometry and crosscoupling effects. In much of the literature, the actuators providing the drive torques are modelled as pure torque sources, or as first-order lags. This assumption is the Achilles' heel of the class of robot dynamic models represented by equation (). References [6, 7] are among the few studies to recognize the significance of drive system and mechanical compliance. As a vehicle for demonstrating the importance of drive system compliance (as opposed to arm link flexibility), in the present paper we select the robot shown in Fig. as representative of a large class of contemporary industrial arms. The robot is powered by electro-mechanical drives, consisting of d-c or a-c motors in series with a harmonic drive used for speed reduction. (A "harmonic drive" is a compact, high-torque, high-ratio, in-line gear mechanism incorporating a rigid "circular spline", an elliptical "wave generator", and a nonrigid "flexspline".) The motors and harmonic drives are mounted on or near the base, with drive torques transmitted to the drive points on the arm links via mechanical links such as bars or chains. Mounting of drive motors in this manner, in contrast with mounting at the joints, can been shown (using the methods of [4]) to provide superior dynamic characteristics. The dynamic response of the integrated system consisting of the drive motor, harmonic drive, linkages, and arm may be measured experimentally by recording the motor current, motor velocity (tachometer feedback signal), and arm motion (integrated accelerometer signal) as the servo is excited with random or sinusoidal input signals. For a given (small) signal level, the drive system may be characterized by a linearized frequency response function. An example of such a frequency response function, computed through FFT analysis of signals recorded during random excitation testing, is shown in Fig. 2. The strong anti-resonance/resonance phenomenon exhibited in Fig. 2 is typical of electro-mechanical drives with compliance in series with the load, and is grossly divergent from the torque source or first-order lag representation of the drive system cited earlier. Although this behavior is well-known by the industrial controls technical community, it has been ignored in much robotics research. 54/Vol. 07, MARCH 985 FREQUENCY. Hi Fig. 2 Comparison of linearized model and measured transfer functions for resonant robot joint servodrive system Phase F, Mag //I Frequency (Hz) Fig. 3 Results of frequency response testing using impact hammer, showing drive system resonance (9 Hz) and away and yaw structural modes (4 Hz and 40 Hz) The existence of resonant behavior inside the robot motion control loop has a dramatic impact oh control system design, providing the motivation for development of the models presented in the following sections. For example, one approach to improving robot motion control performance is to decouple the dynamics of the robot links through nonlinear control [8, 9]. The decoupling action performed by the controller is particularly significant in the same frequency range as the above resonant behavior. Decoupling controllers that do not take this strong resonance into account have little chance of successful implementation. Similarly, adaptive control is an alternative approach proposed frequently [0-2]; however, from adaptive control theory it is known that "unmodeled dynamics" such as this drive system resonance lead to severe stability and robustness problems [3]. The existence of drive system interactions does not preclude future use of nonlinear or decoupling control strategies; rather it demands that realistic robot and drive system models be used in development of implementable algorithms. A Transactions of the ASME /

3 TEST k z, Cj - TORSIONAL k«,c x -LATERAL C ^ TEST k,c - TORSIONAL Fig. 4 Four degree of freedom dynamic model for robot drive system flexibility. Model shown here is for base rotation drive, but also applies to fore-arm and upper-arm drives. Linearized Models for Electromechanical Drives In this section models are presented for the robot electromechanical drives valid for small motions about a given arm position with fixed payload. These models are useful for design of linear controllers, a point of departure for the more general nonlinear case. The range of variation of model parameters with varying payload and arm position is indicated at the end of this section, information which may be used in studies of control system robustness and adaptive control strategies. Extensions of the linear models to include nonlinear effects are presented in a later section. Resonant Joint Control Models. For the robot shown in Fig., the drives for base rotation, forearm, and upper arm have resonant responses to motor torque inputs similar to those shown in Fig. 2. The discussion in this section pertains to the base rotations specifically, with the concepts directly extendable to the other two drives. As well as its response to servo motor inputs, exemplified by Fig. 2, the response of the arm to external torques and forces (e.g., inertial loadings, tool forces) is of interest. Figure 3 shows a frequency response function for the lateral acceleration of the wrist in response to a lateral force applied at the wrist, measured by an impulse hammer technique. Three resonances below 50 Hz can be observed in this response. Note that the characteristics of the velocity servo will have some influence on these results. The dynamic characteristics which are displayed in Figs. 2 and 3 for frequencies up to 50 Hz may be accounted for by a four degree-of-freedom (DOF) model of the drive system coupled to the robot arm, as illustrated in Fig. 4. The four degrees of freedom are rotations of motor and base inertias, plus yaw and sway rigid body modes of the robot forearm. Note that the four degrees of freedom for the dynamic model of this figure should not be confused with the conventional kinematic "degrees of freedom" associated with the five controlled axes of the robot. The main source of flexibility and energy dissipation is the harmonic drive. Additional flexibility in the upper-arm links allows lateral and rotational motions of the forearm. The dynamic modes of this system consist of an aperiodic "rigid-body" mode, a "torsional" mode in which the motor inertia oscillates against the combined inertia of the base and arm, a "sway" mode in which the predominant motion is a lateral oscillation of the forearm, and a "yaw" mode in which the forearm oscillates about a vertical axis. The torsional mode is responsible for the resonance at 9.6 Hz in Fig. 2; the torsional sway and yaw modes give rise to the resonances at 9, 4, and 40 Hz in Fig FREQUENCY. Hi Fig. 5 Comparison of linearized model and measured transfer functions for non-resonant robot joint servodrive system The sway and yaw modes are not significantly excited by motor inputs, so that for the frequency bandwidth of interest for position control (0 Hz, say), perhaps only the rigid-body and torsional modes are of concern, and a simplified twoinertia model could be used for control design purposes. However, it should be noted that both of the higher frequency "structural" modes could be excited by forces arising from tool/workpiece contact or from inertia forces generated by wrist motions. In some configurations the sway mode could be excited by Coriolis forces arising from upper-arm and forearm motions. The mechanics of the "structural" modes are considered further in Section 4. Assuming linear elements for the model of Fig. 4, and combining the base and arm inertias into one inertia /, (referred to the motor shaft speed), a transfer function model may be derived that accounts for the magnitude and phase characteristics presented in Fig. 2. For a high current servo loop gain and small armature inductance, the transfer function between motor torque and current command is essentially a pure gain K t. Then, the motor velocity W m and the load velocity W i in response to current commands I c and load torque disturbances 7", are given by: W m K i [J l s 2 + (B i +c)s+k] Kj(cs + k) D" J m s 2 cs+k + {B m +c)s + k where D" = J m J x s } + [J, (5, + c) + J, (B, + c)]s 2 + [k(j, + J l ) + B, B l +c(b, +,)]s + 0B m +5,). Using component data or experimentally identified parameters for the motor and drive system, the desired drive system model to be used in control system design may be developed for various motion amplitude levels. Using the shorthand notation: K(a)[z,w] to represent: K{s/a + l)[(s/w) 2 + 2z(s/w) + ], (2) Journal of Dynamic Systems, Measurement, and Control MARCH 985, Vol. 07/55

4 0 & 3.,0 <J * to - W no- - \ In 0' n ir \.0 ' TEST fi A Y i.test 0 \ \L NORMALIZED FREQUENCY Ulo i a A - \ NORMALIZED FREQUENCY Wo Fig. 6 Comparison of linearized model and measured transfer functions for sway and yaw structural modes the transfer functions appropriate to the data in Fig. 2 become: W m W, 26.7[0.0,44.7] 236(222) (20.7)[0.29,60.] 66.7(222) 5336[0.52,44.7] (3) Evaluating the transfer function for W m /I c gives the "model" Bode plot in Fig. 2, which compares favorably with the experimental data. Similar, close agreement is obtained for load velocity in response to current command, with the experimental data measured as velocity at the robot wrist. The dynamic modes of the open-loop servo displayed in equation (3) consist of the aperiodic "rigid-body" mode, with a time constant of approximately /20.7 = 48 ms, and the "torsional" mode with a natural frequency of about 60 rad/s = 9.6 Hz, and a damping ratio of about 0.3. Nonlinearities in the robot arm and drive systems produce changes in the linearized control model gains and pole locations. Configuration and payload changes can each produce two-fold variations in the referred load inertia J t. From equation (2) it can be seen that the natural frequency and damping ratio of the zero in the W { /I c transfer function will correspondingly vary inversely with J t. The natural frequency of the torsional mode depends in the same way on the effective inertia J <, f{ = J m J l /(J m +J ), rather than /,. A large part of the energy dissipation, represented by the effective viscous damping coefficients in equation (2), arises in fact from Coulomb friction, so that both the DC gain (which is inversely proportional to the total external damping B, +B X ) and the damping ratios of the poles and zeroes vary widely with motion amplitude. Gain variations of 300 percent and damping ratios as low as 5 percent of critical have been observed. The variation of natural frequency with amplitude due to the nonlinear hardening spring characteristic can be as much as 40 percent. Non-Resonant Joint Control Models. The twist and bend axes of the robot (located at the wrist) do not exhibit the previously described resonant behavior in the frequency range below 50 Hz. The harmonic drive output of the wrist bend motor is coupled to the bend axis by a chain-rod arrangement. The chains add considerable absolute damping to the "load" in a two-inertia model for the bend axis. Also, the load inertia referred to the motor shaft is considerably smaller than the motor inertia, in contrast to the rotation axis case where these two inertias are of similar magnitude. The result of these effects is to increase both the anti-resonance and resonance Ic Fig. 7 Comparison of linearized model and measured structural mode shapes for sway and yaw structural modes frequencies and bring them very close together so that, with the increased damping, no resonance is observed. The absence of resonant behavior permits reduction in the order of the model needed to represent the drive. Following the same procedure as before yields the transfer function model: K, (4) Ir (J m +J i )s+(b m +B i ) An example frequency response measurement for the wrist twist axis is shown in Fig. 5. The corresponding equation (4) model: W 3 44 ZJS. = ±ZL (5) V Ic (74.4) is also plotted in Fig. 5. Again, the nonlinear friction causes substantial changes in the gain and bandwidth of this system as the motion amplitude changes: the effective viscous damping may vary by a factor of five or more. Structural Modes There has been considerable discussion in the literature of the extension of robot motion controls to include control of structural modes in the arm links themselves [5]. In this discussion we define "structural modes" to include motions resulting from flexibility in the robot mechanical system that would not be predicted from a rigid body model for the arm. There are two principal sources of mechanical flexibility in the arm: (A) Bending and torsion of the arm members, as described in [5]. (B) Flexibility in the joints and mechanical linkages connecting the drive systems to the arm members, with the members modeled as rigid bodies. Experimental results obtained for the robot modeled in this paper showed that motions produced by joint and linkage flexibility were dominant. The models presented in this section are useful in design of motion controls in the frequency range where the modes produced by this type of flexibility is important. While many robots available today have similar characteristics, future robot designs may include greater beam flexibility to increase the ratio of payload to arm mass. In this latter case, models of the form given in [5] will be necessary. To investigate the arm dynamic modes, standard structural dynamics analysis techniques were used: accelerometers were mounted at several locations along the robot forearm and the robot was excited either by applying random signal inputs to the robot velocity servo, or by applying impulsive forces at various locations along the forearm with an instrumented hammer. For simplicity in analysis, the structural modes were investigated analytically for the two-dof sub-system consisting of the forearm and its compliant support, which is excited by motion of the base, 9;,, and which has degrees of freedom in 56/Vol. 07, MARCH 985 Transactions of the ASME

5 Ettsv Fig. 8 Nonlinear model for robot drive system, including stiffening spring for harmonic drive, viscous plus Coulomb friction damping on both motor and load inertias, and motor torque limiting sway, x, and yaw, 9 (see Fig. 4). The natural frequencies and mode shapes for the structural modes of the more complete model of Fig. 4 will differ only slightly from those for the subsystem; they could be found by sub-system coupling methods if required. In the model the "arm" is shown as one rigid mass, consistent with the experimental observation that the forearm was effectively rigid for the frequencies considered here. Included in the total mass, M, and moment of inertia, l G = Mr will be contributions from the upper arms. The flexibility in the joints and linkages is represented by a dissipative spring with lateral and torsional stiffness, k x and k z, and damping coefficients, c x and c z. The elastic center of the upper-armlinkage, 0, is taken to lie on the rotation axis. The distances from this axis to the centre of mass of the arm, G, and to the wrist are denoted by c and a, respectively. Expressed in nondimensional form, the transfer functions relating the yaw and sway responses to base rotation inputs are: where (Bp + l) -cp 2 (Bp+\) (p 2 +ABp+K) -9, (6) D = r 2 p 4 + B[ + A(c 2 +r 2 )]p 3 + [ +B 2 A +K(c 2 +f 2 )]p 2 and p 2 = s l /[k z /Ma 2 } c = c/a,r = r/a,x=x/a B = c z /[k z Ma 2 ] v ' K = a 2 k x /k z A = a 2 c x /c z + B(A+K)p+K The lateral displacement at station /', a distance y, = ay, from 0, is given by the transfer function: X,/Q b =MBp + )[( - c/y,)p 2 +ABp+K]/D (7) Using approximate values for parameters in the above model results in dimensionless natural frequencies of 0.9 and 2.7, in reasonable agreement with the experimental values of approximately 3 Hz and 37 Hz, if the reference frequency [k z /Ma 2 ] v ' is taken to be 88 rad/s (4 Hz). The modal damping ratios implied by the above parameters are for the 3 Hz mode and 0.4 for the 37 Hz mode. Model frequency response plots for stations and 7 (at the wrist and rear elbow joints, respectively) are compared with the experimental Bode plots in Fig. 6. The model plots are dimensionless and normalized to a DC gain of. The general form of the experimental plots is reproduced quite well by the model, with the exception of the high-frequency phase behavior. The reason for the phase lead at high frequencies in the experimental frequency response functions is not understood; the nonlinear effects of clearance in the joints, together with the very small displacements at these frequencies, may be involved. In the four-dof system, the natural frequencies are shifted only slightly, to 4 Hz and 40 Hz, respectively (see Fig. 3), substantiating use of the two-dof model. Figure 7 shows the yaw and sway mode shapes predicted by the 2-DOF model together with those measured for the complete four-dof system. It is apparent that for this robot the beam rigid body modes are dominant relative to any beam bending modes. Nonlinear Models for Robot and Drive Systems Nonlinearities associated with robot arm geometry and dynamic interactions among the arm links have been longrecognized as important to the design of motion controls. In addition to robot arm nonlinearities, nonlinear characteristics of elements in the drive system play an important role in robot control system design. In this section nonlinear models for drive system dynamics are presented, as extensions of the linearized models from previous sections. The structure of the nonlinear model is shown in Fig. 8. Principal nonlinearities include: (A) Stiffening spring characteristic of the harmonic drive. (B) Viscous plus Coulomb friction damping for both the motor and load inertias. (C) Current limiters in the motor control loops. Backlash effects were found to be negligible in the drive systems studied. Parametric values needed to characterize these nonlinearities were determined through direct experimental measurement or through parameter identification techniques, the latter being described in [6]. The validity of the nonlinear model is tested by comparison of the magnitude and phase data measured under constant sinusoidal input testing with model predictions (Fig. 9). As signal level increases, there is a decrease in damping of the closed-loop poles of the equivalent linear system, a slight increase in the frequency of these poles, and a decrease in loop gain at higher frequencies. From a control design point of view, the current (torque) limiter is the most significant nonlinearity. By replacing the stiffening spring and Coulomb friction nonlinearities with equivalent linear terms (using an SIDF approach), a simplified nonlinear model for the drive system results that closely corresponds with the measured nonlinear behavior. However, for time domain simulations it is necessary to include all the above nonlinearities to obtain good agreement with measured robot motions for both small and large amplitude motions. Robot and Drive System Model Integration The subsystem models for the robot arm dynamics, drive Journal of Dynamic Systems, Measurement, and Control MARCH 985, Vol. 07/57

6 systems, and microelectronic controls have been integrated into a complete nonlinear robot simulation model, as shown schematically in Fig. 0. The principal elements in the integrated model include: (A) Microelectronic controls. Contains algorithms for performing path generation, interpolation, inverse coordinate transformation, and motion control functions. Used to evaluate alternative motion control strategies. (B) Velocity servo. Contains models for drive system power electronics and velocity feedback controls. (C) Actuator dynamics. Contains detailed nonlinear models for electromechanical drive systems, including motors, harmonic drives, and mechanical linkages. (D) Arm dynamics. Contains Newton-Euler model for multiple link arms, plus forward coordinate transforms to predict end effector motions in world coordinates. The application of this integrated dynamic simulation model to off-line robot programming is described in [7], while its use in designing robot motion controls with end effector sensor feedback is presented in [8]. Summary and Influence on Design of Robot Motion Controls The control models for integrated robot arm and drive systems exhibit the following properties: 3 <0% LIMIT TORQUE ~^>* EXPERIMENT O0 tooo FREQUENCY, RAD/SEC FREQUENCY. RAD/SEC Fig. 9 Comparison of nonlinear drive system model and measured sinusoidal input describing functions G(jw;a), for constant input amplitudes. Input amplitude is expressed as percent of torque limits on drive motors. (A) Flexibility in the harmonic drives is responsible for lightly-damped vibrational modes in the rotation, upper-arm, and forearm axis drives of the robots. The dynamic characteristics of these three axes are quite similar. (B) The natural frequency and effective damping of the vibrational modes depend on the robot configuration and payload, and the amplitude of the oscillatory motions. In the absence of servo control, natural frequencies in the range Hz were observed, with damping ratios less than 0.3. (C) The first two "structural" modes associated with the rotation axis drive were found to involve lateral "sway" and "yaw" motions of the forearm, at about 4 Hz and 40 Hz respectively. In these modes the forearm moves as a rigid body. These modes could be excited by tool/workpiece interaction forces or possibly by cross-axis inertia forces. (D) The dynamics of the wrist bend and twist axes are more heavily damped and are of high frequency compared with the other axes. The dynamics of these drives can be represented by a first-order lag. (E) Sources of nonlinearity in the drives include the "hardening" stiffness characteristic of the harmonic drives, Coulomb friction, backlash, and current limiting in the velocity servos. (F) Simple mathematical models have been developed which are successful in accounting for most of the foregoing phenomena and which allow control design to proceed using realistic representations of the arm dynamics. The dynamic characteristics of the robot arm and drive integrated system summarized above strongly influence the design of robot motion controls: (A) If non^adaptive controls are used, motion controls must utilize relatively low gains, to avoid excitation of drive system resonances. The net result is to constrain the effective bandwidth of the motion controls to be below the resonance. (B) If adaptive controls are used, they must be designed to suppress the interactions between the drive systems and the robot arm. Adaptive controls designed based on the assumption of idealized torque source drive systems may fail due to unmodeled dynamics. (C) For both non-adaptive and adaptive controls, nonlinearities in the drive system must be accommodated by the design, in addition to those present in the robot arm. The most significant drive system nonlinearity is the current limiter. (D) Due to drive system compliance, in most robots there are no direct measurements of the actual robot joint angles. Feedback signals that are usually available include drive motor velocities and angular displacements. Because of the large number and uncertain values of parameters describing drive system nonlinearities (such as Coulomb friction, stiffness variation), it is not practical to use motor current as an input to an observer for estimating real joint angles. Consequently, additional sensors mounted on the end effector or arm links are needed to improve the performance of robot motion controls. Fig. 0 Structure of integrated robot and drive system dynamic simulation model 58/Vol. 07, MARCH 985 Transactions of the ASME

7 References Hollerbach, J. M., "A Recursive Lagrangian Formulation of Manipulator Dynamics and a Comparative Study of Dynamics Formulation Complexity," IEEE Trans. Syst. Man, Cybern, Vol. SMC-0, No., pp , Vukobratovic, M., and Potkanjak, V., Dynamics of Manipulation Robots: Theory and Application, Heidelberg: Springer-Verlag, Silver, W. M., "On the Equivalence of Lagrangian and Newton-Euler Dynamics for Manipulators," Robotics Research, MIT Press, Vol., No. 2, pp , Summer Paul, R. P., Robot Manipulators: Mathematics, Programming, and Control, MIT Press, Cambridge MA, Thomas, M., and Tesar, D., "Dynamic Modeling of Serial Manipulator Arms," ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 04, No. 3, Sept. 982, pp Sunada, W. H., and Dubowsky, S., "On the Dynamic Analysis and Behavior of Industrial Robotic Manipulators with Elastic Members," ASME Journal of Mech. Transm. Autom, Design, Vol. 05, No., Mar. 983, pp Liegeois, A., Dombre, E., and Borrel, P., "Learning and Control for a Compliant Computer-Controlled Manipulator," IEEE Trans. Aut. Coat., Vol. AC-25, No. 6, Dec. 980, pp Freund, E., "Fast Nonlinear Control with Arbitrary Pole Placement for Industrial Robots and Manipulators," Robotics Research, MIT Press, Vol., No., Spring, 982, pp Horowitz, R., and Tomizuka, M., "An Adaptive Control Scheme for Mechanical Manipulators Compensation of Nonlinearity and Decoupling Control," ASME Paper 80-WA/DSC-6, ASME Winter Annual Meeting, Chicago IL, Nov Lee, C. S. G., and Chung, M. J., "An Adaptive Control Strategy for Computer-Based Manipulators," Proc. IEEE Conf. on Dec. and Coat., 982, pp Koivo, A. J., and Guo, T.-H., "Adaptive Linear Controller for Robotic Manipulators," IEEE Trans. Aut. Coat., Vol. AC-28, No. 2, Feb. 982, pp Dubowsky, S., and DesForges, D. T., "The Application of Model- Reference Adaptive Control to Robotic Manipulators," ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 0, No. 3, Sept. 979, pp Rohrs, C. L., Valavani, L. S., Athans, M., and Stein, G., "Stability Problems of Adaptive Control Algorithms in the Presence of Unmodelled Dynamics," Proc. 2th IEEE CDC, Orlando, Vol., 982, p Asada, H., "A Geometrical Representation of Manipulator Dynamics and Its Application to Arm Design," ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 05, No. 3, Sept. 983, pp Book, W. J., and Majette, M., "Controller Design for Flexible, Distributed Parameter Mechanical Arms Via State Space and Frequency Domain Techniques." ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 05, No. 4, Sept. 983, pp Taylor, J. H., and Strobel, K. L., "Applications of a Nonlinear Controller Design Approach based on Quasilinear System Models," American Control Conference, San Diego, Calif., June Imam, I., Sweet, L. M., Davis, J. E., Good, M., and Strobel, K., "Simulation and Display of Dynamic Path Errors for Robot Motion Off-Line Programming," Proc. of Robots & Conference, Detroit MI, June Good, M. C, and Sweet, L. M., "Structures for Sensor-Based Robot Motion Control, Proc. American Control Conf, San Diego CA, June 984. Journal of Dynamic Systems, Measurement, and Control MARCH 985, Vol. 07/59