Modelling and Control of an Elastic Robot Arm with Rotary and Prismatic Joints

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MODELLING AND CONTROL OF AN ELASTIC ROBOT ARM WITH ROTARY AND PRISMATIC JOINTS P. C. Muller,

J.

Ackennann and M. Gurgoze

Safety Control Engineering. University of Wuppertal, GaujJslr. 20, D-5600 Wuppertal 1. FRG Abstract. ~e. application of industrial robots. to. a~vanced manufacturing tasks requires highly accurate positIOn and/or force control. Actual Inrutatlons to these requirements are mainly caused by el.asticity, Coulomb friction and backlas~ in the system. Conventional control algorithms do not conSider these effects and can hardly prOVide sufficient control accuracy or yield limit cycles. In thiS paper the common multibody approach of models of industrial robots is extended to derive more realistic models including elasticity of both joints and links, Coulomb friction and backlash. For the rota~ory motion ?f an elastic robot arm. with ~ revolute joint an algorithm for a highly accurate posItIon control IS developed based on (I) multI-layer control to damp quickly the elastic motions "separately" from the rigid-body motion of the robot , and (ii) compensation of Coulomb friction and backlash by the method of disturbance rejection control. In contrast to the very successful resul~s in case <;>f rotatory motion, the modelling and a suitable design of a control system for the translatIOnal motIOn of an elastic arm with a prismatic joint is a much more difficult I'roble~. A co~plex. contro~ system. is obtained. represented approximately by a set of ordinary I~ear tlme-varl~t differentIal equa~IOl~s of vanable order depending on the actual arm length driven out. Certam approaches of deslgrung a feedback control are discussed. Keywords . Industrial decentralized control.

robots;

friction;

elasticity;

INTRODUCTION The application of industrial robots to advanced manufacturing tasks requires highly accurate position and/or force control. Actual limitations to these requirements are mainly caused by elasticity, Coulomb friction and backlash in the system. Conventional control algorithms do not consider these effects and can hardly provide sufficient control accuracy or yield limit cycles. Therefore, research activities are observed world-wide to overcome these problems, cf. e. g. the related sessions of the IEEE International Conferences on Robotics and Automation of the last years (IEEE 1987, 1988, 1989). A very early but far-reaching result on a singleaxis control of an elastic robot was presented by Truckenbrodt (1980). More recently the multi-axes control of elastic robots were considered in detail analytically and experimentally by Gebler (1987) and Henrichfreise (1989). The compensation of friction effects of an elastic robot was dealt with in detail by Ackermarm (1989). In almost all the investigations so far, elastic robots with revolute joints have been considered only, i. e. the flexible members of the robot has been assumed to have fixed lengths. Just recently first results on translational moving flexible robot arms with prismatic joints were published. Wang and Wei (1987 a, b) considered the vibrations of a moving flexible robot arm with prismatic joint and its feedback control. Here, the driving motion of the prismatic joint was assumed to consist only of a translation along and a rotation about the vertical axis. In contrast, in (Giirgoze and Miiller, 1989) the problem of a general driving motion of the prismatic joint was considered showing a number of difficulties in modelling and control of elastic robot arms with prismatic joints. In the following a lot of results are presented which were found by the research group of safety control engineering at the University of Wuppertal. Therefore, in this paper the common multi body approach of models of industrial robots is extended to derive more realistic models including elasticity of both joints and links, Coulomb friction and backlash. In case of revolute joints an algorithm for a single

position control;

disturbance

rejection;

axis robot axis is developed based on the compensation of Coulomb friction by the method of disturbance rejection control. The control of multi-axes robots consists of a robust decentralized control of each robot arm. Results from simulations and from experiments show the high accuracy by the proposed algorithm. In case of prismatic joints a reasonable modelling of the motion of the elastic link leads to a time-variant nonlinear dynamical system with time-variant order. Because of this complexity a new control concept has to be developed having regard to reducing residual vibrations in systems with time-varying resonances as suggested by Meckl and Seering (1987). ELASTIC ROBOT ARM WITH REVOLUTE JOINT Subject of this chapter is the decentralized control of each axis of an industrial robot with three rotational degrees of freedom for the main motion, see Fig. 1. Firstly the singleaxis control problem is considered which represents the lower level of the multi-axes control of the complete robot. Secondly the decentralized control of the multi-axes robot is shown. Single Axis Control Modelling. In the contrary to the common rigid body approach, the torsional behaviour of the flexible drive, the Coulomb friction of the d. c.-motor and of the harmonic drive gear in each joint as well as the elasticity of each robot arm are taken into account. The physical model of the single robot axis is shown in Fig. 2. According to Ackermarm, Miiller (1986) a representative model of the dynamics of this axis is given by

x= Ax + bu + Gr,

(1)

y=Cx,

(2)

z = fTx

(3)

where the state vector consists of the motor angle \Pl, the relative gear angle t1~, the relative shoulder angle t1!ps, the amplitude function v of the first elastic mode of the arm, of their time derivatives, respectively, and of the motor current i,M. The measurement vector y contains measurements of !P2, \Pl, and v, v (by strain gauges), and additionally of the torque at the output of the gear to decouple the. joint from the connecting link. The scalar variable z descnbes the position of the end of the robot arm which is the interesting variable to be regulated. The control input u is the voltage of the electric motor. At last, the vector r represents the nonlinearities (Coulomb friction, backlash) effecting the robot dynamics; G is the related input matrix. The matrices A, b, G, C and f are described in detail in (Mi.iller, Ackermann, 1988). Due to the Coulomb friction in the motor and in the gear stops may appear in the region of the end position of the robot (x=O). Dependent on the control design the stick friction leads to steady-state inaccuracy (PO-control) or to limit cycles about the equilibrium position (PI-control), cf. (Mi.iller, Ackermann, 1986). Additionally, t~e controllabity of the arm vibrations can be lost, because dunng the stops the friction torque is reacting the motor torque yielding a decoupling of the control loop. The Coulomb friction depends on velocity, actuating force and normal pressure. Its characteristic is described as usual, cf. (Mi.iller, Ackermann, 1986). As mentioned above, conventional control algorithms lead to steady-state inaccuracy or to limit cycle behaviour due to sticking. To avoid these shortcomings a new feedback control IS deslgneo to compensate the Coulomb friction torque. Compensation of friction. If in the dynamical system (l) the vector r is interpreted as a vector of extemal disturbances, then the theory of disturbance rejection control gives conditions and rules for the design of a suitable control to decouple the interesting variable (3) from the disturbances, i. e. in the closed loop system the disturbances do not effect the interesting variable any more. Particularly in the case if some information on the type of the disturbance signals is available, e. g. step, ramp or harmonic function behaviour, a set of linear time-invariant differential equations is used to characterize the disturbances. A detailed discussion of the disturbances rejection control shows how the actual disturbances will be estimated by a disturbance observer and how the estimates will be fed back to compensate the influence of disturbances on the interesting variables, cf. (Miiller, Liickel, 1977). Therefore, the friction operating in motor and gear of the drive are interpreted as "external disturbances" which are assumed to be piecewise constant almost everywhere: r = w, ~ = 0 piecewise.

(4)

By the disturbance rejection control an extended state feedback (5)

can be determined. The gain vector k, can be designed by standard methods such as pole placement or linear optimal regulator (Riccati), while the gain vector kw results in kw' =-

kW2

k" = - --- --KM K, C23

(6)

observer" is used only for the reconstruction \';.) of the actual frictions (and backlash). Making available the measurement Ys of the torque at the driven end of the gear to the observer, it is possible to decouple the joint from the connected arm. Thus a functional observer may be only designed with respect to the subsystem consisting of motor, gear and friction models. It dose not need any data of the elastic arm (and additional joints and arms). The main advantage of this design is the reconstruction ~ of the frictions in each joint of a multi-axes robot depart from the other ones. Unfortunately, the analysis of the observability of the mentioned subsystem shows that the components w, and W2 are not reconstruct able separately but only the sum w, + W2 can be estimared. Therefore, the disturbance rejection control (5) cannot be realized exactly. But taking into account

(8) because the torsional spring C23 is very stiff, the disturbance rejection is performed approximately by I kwTw=----(~,+~2) . KMKr

(9)

By this, the control (5) can be given by a static output feedback and by the estimated sum of frictions in motor and gear. Simulation results of the proposed position control of a single robot axis including the compensation of Coulomb friction are shown in (Ackermann, 1989). The reconstruction of a friction torque by a disturbance observer is illustrated in Fig. 3. The regulation of the position z of the end of the robot arm is demonstrated in Fig. 4. Both figures show the efficiency of the proposed control. Multi-Axes Control In principal the control system for a multi-axes robot will consist of two control levels . On a frrst level each axis is controlled by the feedback control developed above. On a second level the nonlinear couplings among the various axes are compensated by a coordination algorithm according to the theory of hierarchical decentralized control systems. The main advantage of this control concept consists of its realization by application of several parallel processors. The time critical operations can be realized by fast signal processors while the coordination of the coupled arm motions are controlled by a coordination processor. This concept leads to a fast and highly accurate control of the elastic industrial robot. The strong influence of variable moments of inertia depending on the actual position of the robot arms to the dynamics of the elastic links has to be considered carefully to guarantee a robot position control. For this, a control algorithm was developed by applying the technique of robust control design in multi-model problems. Design simultaneously the gains k, of a static output feedback with respect to four different typical end positions of the robot, a fixed tuning of the gains can be found which guarantees stable behaviour of the complete robot in the whole region of operation (Ackermann, 1989). Because the disturbance rejection control of each axis is not influenced by the nonlinear coupling among the various axes, the total control system of the multi-axes robot consists of a robust decentralized control of each robot arm includmg the decoupled compensation of Coulomb friction (and backlash) in each joint.

(7)

by the theory of disturbance rejection. Here, KM, K, are gains of the electric motor, C23 is the stiffness of the torsional spring between motor and gear, and k" is the first element of the state gain vector. In a second step the control (5) has to be realized by feeding back the measurements (2). Even if the first part of the control (5) is a static output feedback, for the second part of (5) an observer has to be included. But this "disturbance

Using an experimental laboratory robot with elastic elements the theoretical and simulation results could be confirmed experimentally (Ackermann, 1989). There the efficiency of the new robust decentralized position control of industrial robots is convincingly illustrated. The tracking of quite complicated robot trajectories is performed with very small overshooting and very small absolute errors. The static and dynamic performance of the robot is improved even if friction and torsional elasticity in the joints and elasticity in the links arise. The implementation of the new robust decentralized position control could be realized by presently

available signal processors . A typical result is illustrated in Fig. 5 showing the position error for two controllers applying the compensation of friction or not. ELASTIC ROBOT ARM WITH PRISMATIC JOINT In this chapter we consider the modelling and control of an elastic robot arm with prismatic joint, cf. Fig. 6. The prismatic joint connecting the elastic beam and - in general the preceding robot link is built such that the beam axis and the link axis are made to coincide at two or more points by bearings which allow only relative translational motion y.(t). The orientation and the motion of the preceding link and with that the orientation and motion of the joint may be arbitrarily given; they characterize orientation and motion of a reference coordinate system, cf. {xB' YB, ZB} and vB(t), roa(t) in Fig. 6. The elastic vibrations of the beam are composed of bending motions in the x- and z-directions perpendicular to the y-axis of the beam and of a torsional motion about the y-direction. Firstly we remember the problem of a rigid robot arm with prismatic joint. Here, only the equation of translational motion of the arm relatively to the prismatic joint has to be derived. This results in my. - m (~Bx + O)2By) y, = - m (VBy + VBx
(10)

Usually the t~k of the robot will defme certain time functions Ci(t), l3(t), :y(t) for the orientation and vB(t) and roa(t) for the motion of the joint as well as y ,( t) for the relative position of the robot arm. Then equation (10) defines the required control force F(t). Secondly the robot with a flexible arm is considered. By Giirgoze and Miiller (1989) two different approaches were discussed. On the one hand side Hamilton ' s principle was applied to model a distributed parameter system resulting in three coupled partial differential equations for the. two bendin~ displacements w,(y ,t) , w.(y ,t) and for the torsIOnal angle l5(y,t). Because of the prismatic joint there are timevariant intermittency conditions, e. g. with respect to Wx we have

Iy=-y,(t) w~(y,t) I y=-y,(t)

The second approach is based on the theory of multibody systems. For this, the beam will be physically discretized and it will be considered as a chain of small beam-like rigid subbodies coupled by fictitious Cardan joints and fictitious springs and dampers representing elasticity and material damping of the beam. Applying Newton 's and Euler's equation of motion to each subbody, eliminating the constraint forces , and introducing state space notation with respect to subbody displacements, to arm elongation, and their time derivatives, respectively, then a set of differential equation of first order is obtained: Xi = Ai (VB, roa , ~B,

cOs, F) Xi + bi F(t).

0

(1la)

= 0

(1lb)

Additionally, there still is a coupled ordinary differential equation for the translation motion which is a modification of (10):

(13)

This model applies as long as subbody no. i contacts the prismatic joint. The system matrix Ai depends on the kinematic control inputs VB and roa and their time derivatives as well as on the axial control force F. Again an untypical control problem has been encountered. A change of the description no. i to that of no. i-I or i+ I will appear if y, crosses the values L

y,=(2i - 2 - N)N

where m is the mass of robot arm; VB .. VBy, VBz, and roa .. roay,
wx(y,t)

we have a nonlinear control problem also including trrne derivatives of the control inputs.

or

L

y, = (2i - N) N

(14a)

(14b)

Both approaches end up with an unconventional and complicated control problem. Unfortunately, the robust decentralized feedback control of elastic robots with revolute joints cannot be applied to elastic robots with prismatic joints. There is a different structure of control inputs . Therefore, in the future the authors shall apply two different control design methods to the problem. On the one hand side the design of robust control with respect to multi-model problems will be considered. On the other hand side a suitable determination of the kinematic input functions will be investigated regarding the method of Meckl and Seering (1987) to avoid the excitation of the elastic motions. CONCLUSIONS In this contribution an attractive posilIon control of an industrial robot including elasticities and Coulomb friction was dealt with in the case of robots with rotational degrees of freedom only. This control is based on (i) a multi-layer control to damp quickly the elastic motions "separately" from the rigid body motion of the robot, and (ii) a compensation of Coulomb friction by the method of disturbance rejection control. The robust decentralized position control was successfully verified by experiments. In contrast to this very successful results, the modelling of the translational motion of an elastic arm with a prismatic joint leads to a very difficult and unconventional control problem, which is still under consideration. Robust control design based on the multi-model-approach and the determination of suitable inputs to avoid the excitation of elastic motions will be helpful tools to overcome the difficulties.

• y,(t) + S22Y,(t). - Y2 + Sg2 + WzOOax - WxOOaz

+ P~L IQy + F(t)]

(12)

Here, Y2 is an acceleration according to the body velocity VB , s 2 has regard to gravitational effects, S22 contains squares of c~mponents of roa and characterize.s centrifugal effects, p is mass density, A denotes cross-sectIOnal area, and L IS total length of the arm. The set of coupled partial and ordinary differential equations describes an unconventional and troublesome control problem. Considering the components of VB ~d roa as kinematical control inputs and F(t) as force control mput then

REFERENCES Ackermann, J. (1989). Positionsregelung reibungsbehafteter elastischer Industrieroboter, Fortschr .-Ber. VDI, Reihe 8, Nr. 180. VD! , Diisseldorf. Ackermann, J., and P. C. Miiller (1986). Dynamical behaviour of nonlinear multibody systems due to Coulomb friction and backlash. In Preprints IFAC/lFIP(IMACS Internat. Symp. Theory of Robots . Austrian Center for Productivity and Efficiency, Wien. pp. 289-294. Gebler, B. (1987). Modellbildung, Steuerung und Regelung

fur elastische Industrieroboter, Fortschr.-Ber. VDI, Reihe 11 , Nr. 98 . VDI, Diisseldorf. Giirgoze, M., and P. C. Miiller (1989 ). Modeling and control of elastic robot ann with prismatic joint. In O. Schweitzer and M. Mansour (eds.), Dynamics of Controlled Mechanical Systems. Springer, BerlinHeidelberg-New York.. pp. 235-245 . Henrichfreise, H. (1989). Aktive Schwingungsdampfung an einem elastischen Knickarmroboter. Vieweg, Braunschweig-Wiesbaden. IEEE (1987, 1988, 1989). Proceedings of International Conference on Robotics and Automation. IEEE Computer Society Press, Washington, D. C .. Meckl, P. H. , and W. P. Seering (1987). Reducing residual vibration in systems with time-varying resonances . In Proc. 1987 IEEE Int. Conf. Robotics and Automation. IEEE Computer Society Press, Washington, D. C .. pp. 1690-1695. Miiller, P. C., and 1. Ackermann (1986). Nichtlineare Regelung von elastischen Robotem. In VDI/VDEGesellschaft fur MeB- und Regelungstechnik (ed.), Steuerung und Regelung von Robotern, VDI-Berichte 598. VOI, Diisseldorf. pp. 321-333 . Miiller, P. C. , and J. Ackermann (1988). Robust decentralized position control of industrial robots with elasticities and Coulomb friction. In K. Warwick and A. Pughs (eds.), Robot Control - Theory and Applications. Peter Peregrinus, London. pp. 176-184. Miiller, P. C., and 1. Liickel (1977). Zur Theorie der StorgroBenaufschaltung in linearen MehrgroBensystemen. Regelungstechnik, 25, 54-59. Truckenbrodt, A. (1980). Bewegungsverhalten und Regelung hybrider Mehrkoq>ersysteme mit Anwendung auf Industrieroboter, Fortschr.-Ber. VOI, Reihe 8, Nr. 33, Diisseldorf. Wang, P. K. C., and I .-D. Wei (1987a). Vibrations in a moving flexible robot ann. 1. Sound Vibr., ill, 149160. Wang, P. K. C., and J.-D. Wei (l987b). Feedback control of vibrations in a moving flexible robot ann with rotary and prismatic joints. In Proc. 1987 IEEE Int. Conf. Robotics and Automation. IEEE Computer Society Press, Washington, D. C .. pp 1683-1689.

l oad arm

gear bo x Cou l omb f ri ct i on

back l ash moto r tacho mete r

Fig . 2 . Physical model of single robo t ax i s . Nm .5

o. - .5

.2

s

.8

.6

.4

Fi g . 3 . Fri c ti on and estimated sig nal.

m

a b

-. 005

- . 01 0

.2

.4

.6

.8

s

Fig . 4 . Position error of robot arm end : a . Conventional control ; b . control with feedback v , v without frict i on compensRtion ; c . control with feedback v , v with friction compensation .

I

load elastic fore an'!

, +

I elastic I deformations

Fig . 1 . Three - joint elastic robot .

226

.1

cm

o.

-.1

5.

o.

s

10.

Fig. 5. Experi ment: r1easur ed positio n error: a. Withou t frictio n compe nsation ; b. with frictio n compe nsation .

prismat ic joint

x

Q(t)

Fi9. 6. Elasti c robot arm with p rismat ic jOint.

227