On Dynamics and Control of Elastic Robots

Copyright © IFAC Robot Control Karlsruhe, FRG, 1988

ON DYNAMICS AND CONTROL OF ELASTIC ROBOTS F. Pfeiffer, B. Gebler and U. Kleemann Lehrstuhl B fur ,Hechal/ik, Techl/isch e L'I/il'ersitiit MUI/cil el/, MUI/ chel/, FRG

Abstract. Robots with flexible joints and links are considered. On the bas1s of a complete model of the robot dynamics, regarding the elastic deformations of the arms by a Ritz approach, a multistage control scheme is developed. An optimized reference trajectory for the rigid robot is realized in a nonlinear feedforward decoupling concept. Then, in a first step, the elastic deviations from that path are counterbalanced by an additional feedforward-Ioop and, in a second step, the remaining elastic vibrations are actively damped by strain gauge measurement feedback for each arm and joint. Theoretical and experimental results are compared for a three degree of freedom laboratory robot. Ke1words. Elastic Robots, Dynamics and Control of Manipulators, Mu t1stage-Control, Active Vibration Damping, Multibody Systems

INTRODUCTION

(1986), Craig (1984) and Furuta, Yamakita (1986). The main ideas in this paper are based on Gebler (1985a, 1985b, 1987a, 1987b) .

For robots with high precision requirements or for lightweight and fast robots the problem of elastic deformations must be taken into consideration. It involves two aspects. Firstly, the aspect of modelling a highly nonlinear system superimposed by small elastic vibrations of links and joints, and secondly, the problem of an appropriate control design. In the following the first problem is solved by applying hybrid multibody theory, which gives an exact representation of the robot's gross motion and approximates the small elastic vibrations of the links by a Ritz approach. The second problem is solved by adding to the nonlinear feedforward decoupling scheme for the rigid robot an elastic correction and by feeding back strain gauge measurements at the links, thus actively damping the elastic vibrations. The combined concept worked exceptionally well.

MECHANICAL MODELS Path Planning All reference trajectories are optimized applying the procedure of Pfeiffer, Johanni (1986). The main idea of this method consists of looking at the rigid robot following ideally a trajectory as an onedegree-of-freedom system, where all equations of met ion may be transformed to this one path coordinate. The resulting set of equations can be solved analytically by a geometric approach, which considers all constraints of joint torques, of path velocities or of path accelerations. Theory and experiment have been compared successfully for a three link laboratory robot (Fig. 1).

In literature various contributions may be found to these problems, though elastic robots became of general interest not before the last few years. Modern concepts of path planning may be found in Dubovsky, Shiller (1985), Pfeiffer, Johanni (1986) and Shin, McKay (1985), some interesting procedures to the problem of modelling elastic robots in Book (1984), Cannon, Schmitz (1983), DeMaria, Siciliano (1987), Dubovsky, Matuuk (1975), Kuntze, Jacubasch (1984) and Tomei, Nicosia, Ficola (1986). Many of these authors consider only an one arm robot or only elastic joints. The derivation of the equations is often performed by using Lagrange's equations, which has proven to be a rather time consuming approach. Applying d'AlembertJourdain's principle seems to be more promising (Johanni, 1986). Some convincing contributions to the control problem of elastic robots are qiven by An, Atkeson

Robot with Elastic Joints and Links The equations of motion for an elastic robot are derived by d'Alembert-Jourdain's principle (Fig. 2)

i€n

B. 1

(1 )

where the sum includes all bodies Bi of the robot, which may be rigid or elastic. The vector z represents the generalized coordinates, the vector ri is the cartesian vector from an inertial frame to a

-11

42

F. Pfe iffer. B. Gebler and C . Kleemann

masselement of body Bi and Fi are the applied forces. All dotted values are absolute time derivatives. Elasticity enters through the vector ri' which comprises rigid and elastic parts, and through the generalized coordinates z, which contain the rigid degrees of freedom, for example joint angles, and the elastic degrees of freedom via Ritz approach.

The Ritz approach models each small elastic deformation by a linear combination of shape functions each premultiplied by a time function which then is the elastic coordinate. According to Fig. 2 each link exhibits twofold bending and torsion. Then, the appropriate Ritz approach for body Bi would be expressed in the form

max. po s sib l e v e loci t y

\\

ma x . d e c e ler ati on

'optima l velocity profil e

,

- --

\

(rpi)]

i.

qli (t) vli q2i (t)T • w2i (rOi) [ qT (t) . " i ( r'o i )

md- x . a cc e l erat ion

(2 )

s· is the deformation vector for body B·, tRe q-vectors are elastic coordinates, tRe (v, w, 6e )-vectors shape functions. They d~pend on the relative cartesian vector r01 of the undeformed body Bi. From equations (1) and (2) the equations for the gross motion of the robot may be derived in the form M(Z,t)

P at h Coorin a t e

Joint 1

.,•

0.0

OA

0,2

0,6

0,8

~Q,8

.

o

z+

f(z,z,t)

=

B u(t),

(3)

where M represents a symmetric mass matrix, the vector f contains all gravitational, centrifugal and Coriolis forces, B is the input matrix and the control vector u includes all motor torques. (They appear in a linear relationship on the right hand side). The fiquations for the deviations from the reference path are time-variant but linear. They are obtained by linearizing (3) around the reference and taking into account all elastic influences, where with respect to the problem of quadratic elastic terms the reader is referred to Gebler (1987b) and Johanni (1986). The resulting equations are

~ D,A

..,o

M(Zo)Y + P(zo,zo)y + Q(zo'zo'zo)y

a,o

T

(4 )

-D,A

[yeRf, Zo€R f , (M,P,Q)€.Rf,f, h€Rf,

-Q,8

0.0

0,2

OA

T ime

Fig.

0,6

0,8

[s]

1. Comparison of Theory and Measurements for the "rigid" Reference Trajectory, Theory from Pfeiffer, Johanni (1986). Extended to Dry Friction, ------ Theory, ----Measurements

B€Rf,r,

AT€Rr, r
The vector y = z - Zo represents all deviations from the reference state, M, P, Q are the mass, the velocity- and positiondependant matrices, respectively, h contains external forces, those forces produced by the robot's nominal gross motion and the nominal joint torques T . AT are additional control torques regar~ing elasticity and other deviations from the reference . CONTROL SCHEME FOR AN ELASTIC ROBOT Elasticity of a robot causes firstly a deviation of the robot end point from the reference trajectory and secondly vibrations of the joints and links. The first effect will be reduced by bringing back the robot tip to the nominal path by an appropriate correction, the second effect will be counterbalanced by measuring the curvatures of the links and using this information for a proportional joint control. Elastic Feedforward Correction

Fig. 2. Mechanical Model of a Robot with Elastic Links and Joints

The principle is simple. For a given reference path the elastic deviations from that path are calculated. By inverse kine-

On Dvnamics and Control of Elastic Robots

matics it is then possible to evaluate the correction angles for each joint taking back the tip to its nominal position (Fig.

43

enter the robot. For a rigid robot they produce the appropriate joint angles to follow the nominal path.

3) •

y

y

x Yo

+

ll yo =

Fig. 3. Principle of Elastic Correction

In a second step the quasi-static correction is added (Fig. 4b). The nominal joint values )riO' iriO' ~iO enter a correction model {MC) accord~ng to eq. (5) which together with some transformations gives an estimate ~ for the "elastic" end-point position. The difference (~ - ro) is then used (CP) to evaluate a ~oint angle correction (4~iO' 4 ir iO' l>~ iO) according to fig. 3. It ~s used t.o cor~!,!ct the "rigid" nominal values ~iO' ~iO' ~iO' Thus, this second step might be looked at as a feedforward correction due to elasticity. _~:-~_lT10

RBK

Equations (4) may be used to evaluate these correction angles in a simple way. The main influence of the reference trajectory is contained in the vector h(zo'zo'zo) representing all applied and all nominal dynamic forces along the reference trajectory. The forces of this gross motion are obviously much larger then the dynamical deviation forces (M y + p y), which are only proportional to the deviation vector y. Thus, a good approximation to the deviations and from this to the joint corrections must be a quasi-static one Q(zo'zo'zo)' y ~ h(z,zo'zo)'

(5)

which is used as a feedforward correction model in the nonlinear decoupling concept (Fig. 4). Elastic Vibration Control From experience it can be seen, that the above correction works very satisfactorily for a nearly perfect reduction of elastic deviations from a reference state. But, due to its quasi-static character, it is not able to influence very much the elastic vibrations of the system. Therefore, they must be controlled separately, which from the physics of the elastic robot system should be preferably performed on a joint contro l level. These considerations led to a control design, which consists of strain gauge measurements at the elastic links (Kleemann,1986). These measurements must be performed at locations guaranteeing best observability. They are used in a feedback control loop for the joints.

RBK: Rigid Body Kinematic MC: Model of Correction CP: Correction Procedure NEM: Nonlinear Equation of Motion JC: Joint Control EC: Elasticity Control EB~:

~lastic

Body Kinematic

\0'\0'\0 :

Combined Control of an Elastic Robot The combined control takes place in three steps (Fig. 4). In a first step a reference trajectory is generated by a feedforward nonlinear decoupling scheme (Fig. 4a). The nominal end point position r of the robot enters inverse robot kinema~ics (RBK) prod~cing .the nominal joint magnitudes 'tiO' 'fiO' 'lr,i.O' These values are used in an inverse robot model (NEM) to give the joint torques T iO ' An additional linear PD-joint control is fed by the difference of measured and given position and velocity data to stabilize the system and to counterbalance for rigid robot disturbances. All torques are summed up and

Nominal Joint Angles, Angular Velocities, Angular Accelerations K1 : Curvatures (measured by strain

gauges) r o : Nominal Position of Robot Endpoint r: Actual Position of Robot Endpoint ~: Estimated Position of Robot Endpoint lly10 ' ll)-1O' lly i O :

Correction Angles, Correc~ion Angular Velocities, Correction Angular Accelerations ~ o : Nominal Torques Fig. 4: Multistage Control Concept

44

F. Pfeiffer. B. Gebler and L. Kleemann

In a third step an additional joint control is established counterbalancing elastic vibrations (Fig. 4c). The curvatures of the links are measured by strain gauges. The signals are used to perform a joint control especially for the small elastic vibrations.

4 2

MEASUREMENTS A laboratory robot with three rotatory joints has been designed and realized with the goal to verify the above concept (Fig. 5). It is centrally controlled by a Perkin Elmer 3210 processing unit together with a 8-channel A/D-converter with 200 kHz maximum sampling rate. Details of signal flow schemes and additional electronic equipment may be found in (Gebler,1987b). Joint 2

1,8 t

-2 -4

. .. 2~ Cs]

lend of trajectory

-6 -8

-10

measurement with correction simulation with correction measurement without correction Fig. 6. Deviation of Robot Endpoint with/ without Elastic Correction, Art = Deviations in Tangential Path Direction

4

2

-2 -4 -6

Fig. 5. Laboratory Robot with Joint Design In 0 first series of measurements a simple path according to fig. 1 has been investigated. Fig. 1 shows a time optimization applying the methods of Pfeiffer, Johanni (1986) and a comparison of theory and experiment for the rigidly assumed robot. Considering the problem of jumps in the time-optimal case, the comparison looks very good with the exception of some local deviations. A second series of measurements tested the quasi-static correction in form of the feed forward loop (MC-CP) in figure 4b. Figure 6 shows a good comparison of theory and measurements and, additionally, the efficiency of the concept. Deviations are practically reduced to zero. The effect of the strain gauge measurement feedback is represented by figure 7. Elastic vibrations are nearly completely damped out after the end of the trajectory. Therefore, the combination of both control loops, feedforward correction and strain gauge measurement feedback, will result in an excellent dynamic behaviour in the presence of elastic robot components.

lend of trajectory

-8

measurement with elastic control measurement without elastic control

Fig. 7. Positional Deviation of Robot Endpoint with/without Elasticity Control, Art = Tangential Path Deviations CONCLUSIONS Dynamics and control of a robot with elastic joints and links are considered. The robot is modeled by applying elastic multibody-system techniques, where the elastic deformations are represented by a Ritz-approach anticipating small elastic deviations. The control is established in several steps. Firstly, a feedforward decoupling variant is applied to control the rigid robot along an optimized reference trajectory, which is generated by known methods. Deviations from this path due to structural

On Dynamics and Control of Elastic Robots elasticity are counterbalanced in a second step by a quasi-static correction of the path feedforward control, which compensates already for the biggest share of the elastic influence. The remaining tendencies of the system to perform small elastic vibrations are then met by an additional elasticity control using the signals of strain gauge measurements at the elastic links. REFERENCES An,

Ch., Atkeson, C.G., Hollerbach, J.M., (1986). Experimental Determination of the Effect of Feedforward Control on Trajectory Tracking Errors, Proc. IEEE Conf. on Robotics and Automat10n, San Franc1sco, USA, 55-60. Book, W.J., (1984). Recursive Lagrangian Dynamics of Flexible Manipulator Arms, Journal of Robotic Research, Vol. 3, No.3, B7-101. Cannon, R.H., Schmitz, E., (1983). Precise Control of Flexible Manipulators, Robotics Research, (Brady & Paul, Eds.j, MIT Press, 841-861. Craig, J.J., (1984). Adaptive Control of Manipulators through Repeated Trials, Proc. of the American Control Conference, San D1ego, CA, (June 6-9, TI"6"6"=1574.

DeMaria, G., Siciliano, B., (1987). A Multilayer Approach to Control of a Flexible Arm, Proc. IEEE Int. Conf. on Robotics and Automat10n, Rale1gh, USA, 774-77B. Dubowsky, S., Maatuk, J., (1975). The Dynamic Analysis of Elastic Spatial Mechanisms, Journal of Mechanical Engineering, 927-932. Dubowsky, S., Shiller, Z., (1985). Optimal Trajectories for Robotic Manipulators, Proc. of the 5. CISM-IFToMM Symp. on Theory and Pract1ce of Robots and Man1~ulators, London. Furuta, K., Yama 1ta, M., (1986). Iterative Generation of Optimal Input of a Manipulator, Proc. IEEE Conf. on Robotics and Automat10n, San Francisco, USA, 579-5B4. Gebler, B., (1985a). Mechanisches Ersatzmodell und Bewegungsgleichungen fur einen Industrieroboter mit elastischen Komponenten, ZAMM 65 4, T53-55. Gebler, B., (1985b). Model11ng and Control of a Lightweight Robot, Proc. of the 2nd Euro ean S ace Mechan1csm & Tr1 0 ogy Symp., Meers urg, FRG, ESA SP-231j. Gebler, B., (1987a). Feed-Forward Control Strategy for an Industrial Robot with Elastic Arms, Proc. IEEE Int. Conf. on Robotics and Automat10n, Rale1gh, USA, 923-92B. Gebler B., (1987b). Modellbildung, Steuerung und Regelung fur elastische Industrieroboter, Fortschritt-Berichte VDI, Reihe 11, Nr. 9B, VDI verlag Dusseldorf. Johanni, R., (1986). On the Automatic Generation of the Equations of Motion for Robots with Elastically Deformable Arms, Prec. of the IFAC/IFIP/lMACS International Sympos1um on Theory of Robots, V1enna, Austr1a. Kleemann, U., (1986). Dynamics and Control of a Robotic System with Elastic Arms, Proc. Int. Symposium on Robot Manipulators, Albuquerque, USA, 4B7492.

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Kuntze, H.-B., Jacubasch, A., Salaba, M., (1984). On the Dynamics and Control of a Flexible Industrial Robot, Proc. 9th World Congress of the IFAC, Budapest, B6-92. Pfeiffer, F., Johanni, R., (1986). A Concept for Manipulator Trajectory Planning, Proc. IEEE Conf. on Roboics and Automat10n, San Franc1sco, USA, 1399-1405. Shin, K.G., McKay, N.D., (1985). Minimum-time control fo robotic manipulators with geometric path constraints, IEEE Trans. Automat. Contr. 30,531541. Tomei, P., Nicosia, s., Ficola, A., (1986). An Approach to the Adaptive Control of Elastic at Joints Robots, Proc. IEEE Conf. on Rovotics and Automat10n,San Franc1sco, USA, 552-558.