IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 5, NO. 4, DECEMBER 2000 437

Short Papers_______________________________________________________________________________

Enhanced Contour Control of SCARA Robot UnderTorque Saturation Constraint

Masatoshi Nakamura, S. Rohan Munasinghe, Satoru Goto, andNobuhiro Kyura

AbstractContour control of a robot arm is an act of the end-effectortip being moved along a reference Cartesian path, with an assigned velocity.When torque saturation occurs, and lasts for some time, contour deterio-rations result. In addition, system delay dynamics also causes contour de-teriorations. In this paper, an offline trajectory generation algorithm is de-scribed, which achieves both optimum avoidance of torque saturation anddelay dynamics compensation.

Index TermsContour control, feedforward compensator, industrialrobot arm, offline trajectory generation, torque saturation.

I. INTRODUCTION

The main objective of this paper is to enhance contour controlperformance of a selective compliance assembly robot arm (SCARA)under torque constraint. Its implementation on a SCARA AR-M270(Hirata Company Ltd.) robot arm is also presented. Contour controlis a twofold task. It involves offline trajectory generation and onlinetrajectory tracking. Trajectory generation starts with trajectorysegmentation, in which a set of way-points is declared. Trajectorysegments are approximated with either straight lines and/or somefunctions as proposed by Linet al. [1]. Finally, the segments arelinked in a sequence, which forms the generated trajectory. Trajectorytracking is implemented by the servo drive.

Research on contour control of serial link robot arms dates back tothe 1950s. Whitney [2] proposed forward and reverse Jacobian trans-formation for velocity programming in straight-line path tracking. Paul[3] assigned knot points in real time. However, these could not apply inindustry due to the inherent vulnerability to disturbances and the exces-sive computation burden involved in real time. Taylor [4] proposed anoffline path planning with enough knot points so that the kinematic anddynamic constraints are satisfied. Luhet al. [5] derived time schedulesfor velocity and acceleration to minimize tracking time.

There are two major causes of contour deteriorations, namely, torquesaturation and system delay dynamics. When torque saturation takesplace, desired and actual joint dynamics become significantly different.These differences transform into Cartesian space as contouring dete-riorations. System delay dynamics cause contour deteriorations, usu-ally in the form of overshoots at trajectory corners. There is no widelyaccepted discipline in the industry for contouring operation of robotarms. Instead, contour deteriorations are partially compensated by ex-pert practitioners, by incorporation of variousad hocmethods.

In this paper, an application-oriented trajectory generation method-ology is presented for contour control of industrial robot arms. BothCartesian and joint constraints are addressed, referring to actual indus-

Manuscript received February 12, 1999; revised September 20, 1999. Rec-ommended by Technical Editor K. Ohnishi.

M. Nakamura, S. R. Munasinghe and S. Goto are with the Department of Ad-vanced Systems Control Engineering, Saga University, Saga 840-8502, Japan.

N. Kyura is with the Department of Electrical Engineering, Kinki University(in Kyushu), Iizuka 820-8555, Japan.

Publisher Item Identifier S 1083-4435(00)11067-1.

(a) (b)

Fig. 1. Schematic of SCARA system. (a) Apparatus. (b) Coordinates.

trial applications. It has been proven from experimental results that thismethod would significantly improve contour control performance.

II. SYSTEM ARCHITECTURE OF ANINDUSTRIAL ROBOT ARM

A. Overview of the Robot Arm System

A SCARA system is shown in Fig. 1(a). A reference input gener-ator generates the sampled Cartesian trajectory and transforms it intosampled joint trajectories. Sampled joint trajectories are written ontothe servo controller in real-time operation. The servo controller readsjoint trajectory data and actuates joint motors accordingly. Accordingto Fig. 1(a), in the two-degree-of-freedom configuration, robot arm pa-rameters arel1 = 0:27 m, l2 = 0:24 m, m1 = 20 kg, andm2 = 15kg, wheremi andli denote the mass and length of theith link, respec-tively. Referring to Fig. 1(b), the robot arm kinematics that transformCartesian coordinates to joint coordinates are given by

x = l1 cos 1 + l2 cos(1 + 2) (1)

y = l1 sin 1 + l2 sin(1 + 2): (2)

Corresponding inverse kinematics are given by

1 = sin1 y

x2 + y2 sin1

l2 sin 2

x2 + y2(3)

2 = cos1 x

2 + y2 l21 l2

2

2l1l2: (4)

B. Robot Arm Dynamic Model

It is assumed that each joint operates independently and coupling be-tween links is neglected. Also neglected are the Coriolis and centripetaltorques, which cause significant contributions to contouring perfor-mance only at high speeds [6]. In this paper, we do not address the con-touring problem at high speeds, but contouring at normal speeds. Actu-ally, under low-speed operation, which is below about 1/20 of the ratedmotor speed, torque saturation hardly occurs, even at sharp trajectorycorners. On the other hand, under high-speed operation, above about1/5 of the rated motor speed, more precise joint dynamics described byeither Lagrange/Euler or Newton/Euler formulations should be used.However, industrial robot arms are not usually operated at such highspeeds for contouring applications. Under a normal speed conditionwhich is below about 1/5 of the rated motor speed, joint dynamics couldbe described by the second-order linear model, using only kinematics,as shown in Fig. 2. Under the same condition, however, torque satura-tion at sharp trajectory corners remains a significant problem. Under

10834435/00$10.00 2000 IEEE

438 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 5, NO. 4, DECEMBER 2000

Fig. 2. Second-order joint dynamic model.

normal speed, Coriolis and centripetal torque disturbances can be ne-glected, whereas gravity torques do not exist as the SCARA has onlyvertical axis joints. It is also assumed that the joint torque and accel-eration are proportionally related as given by(1; 2)T / (1; 2)T ,wherei, i = 1; 2 represents the torque of theith joint motor. Thisrelationship has been used extensively in modeling of contouring robotarms [6] and, consequently, torque saturation phenomena can be rea-sonably modeled by proportional acceleration saturation. The corre-sponding joint dynamic model of industrial robot arms is shown inFig. 2. The mathematical description of joint dynamics is given by

i(t) = sat[KvifKpi(Ui(t) i(t)) _i(t)g] (5)

sat(z) =

i;max ; z > i;maxz; i;max z i;maxi;max; z < i;max

(6)

wherei;max represents the angular acceleration limit of theith joint.In Fig. 2,(Ux; Uy) represents the position command in Cartesian coor-dinates and(U1; U2) represents the corresponding position commandin joint coordinates, which is the input to the servo controller.Kpitransforms position error into the velocity command andKvi trans-forms velocity error into the acceleration command, wherei = 1; 2represents the joint. Servo controller parametersKp1 = 17 1/s,Kp2 =15 1/s,Kv1 = 102 1/s, andKv2 = 90 1/s represent the SCARAAR-M270 system. When the operation is within the bounds of torquesaturation, the dynamics of each independent joint are given by

i(s)

Ui(s)= G(s) =

KpiKvis2 +Kvis+KpiKvi

: (7)

III. ENHANCED CONTOURCONTROL UNDER TORQUECONSTRAINT

A. Methodology of Trajectory Generation

1) Sharpness Elimination at Corners:At the corners of an objec-tive trajectory, velocity and acceleration demands become excessive.Consequently, torque saturation is frequently observed. To eliminatethis occurrence, sharp corners are rounded up by substitution of cir-cular arcs, as shown in Fig. 3(a). In Fig. 3(a),6 RST = (1 2)=2.Referring toRST , the trigonometric relationship between the radiusof arc r and introduced error is given byRS=ST = r=(r + ) =cosf(12)=2g. Then,r can be expressed in terms of as given by

r = cosf(1 2)=2g

1 cosf(1 2)=2g: (8)

2) Determination of Tracking Velocity at Corners:The tracking ve-locity profile is modified as shown in Fig. 3(b), where the circularcorner is traversed with a low velocityvc. The entire trajectory is rep-resented by segments as marked byti wherei = 0; 1; ; 7. Ac-

(a) (b)

Fig. 3. Trajectory generation at a corner. (a) Elimination of sharp corners.(b) Tracking velocity.

cording to the trajectory geometry and tracking velocity profile shownin Fig. 3(a) and (b), the generated trajectory is given by

x(t) =

Amaxt2 cos 1=2;

t0 t < t1x(t1) + v(t t1) cos 1;

t1 t < t2x(t2) + [v(t t2)Amax(t t2)

2=2] cos 1;

t2 t < t3x(t3) + r[sin(1 + vc(t t3)=r) sin 1];

t3 t < t4x(t4) + [vc(t t4) + Amax(t t4)

2=2] cos 2;

t4 t < t5x(t5) + vt cos 2;

t5 t < t6x(t6) + v(t t6) Amax cos 1(t t6)

2=2;

t6 t t7(9)

y(t) =

Amaxt2 sin 1=2;

t0 t < t1y(t1) + v(t t1) sin 1;

t1 t < t2y(t2) + [v(t t2)Amax(t t2)

2=2] sin 1;

t2 t < t3y(t3) r[cos 1 cos(1 + vc(t t3)=r)];

t3 t < t4y(t4) + [vc(t t4) + Amax(t t4)

2=2] sin 2;

t4 t < t5y(t5) + vt sin 2;

t5 t < t6y(t6) + v(t t6)Amax sin 2(t t6)

2=2;

t6 t t7(10)

whereAmax is the specification for maximum Cartesian acceleration.3) Construction of Trajectory Corners for Different Condi-

tions: Construction of corners is application oriented. There are twomajor categories of applications found in most industries. In the firstcategory, the maximum tracking errormax is prespecified. In thesecond category, the minimum tracking velocity denoted byvc;min isspecified.

Category 1 (givenmax): The tracking error at the cornershould bound to its maximum valuemax as given by max.Then, the radius of the circular arc is constrained according tor (max cosf(12)=2g)=(1 cosf(12)=2g). Centripetalacceleration along the circular arc isv2c=r. The maximum CartesianaccelerationAmax is already known. Therefore, the maximum

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 5, NO. 4, DECEMBER 2000 439

Fig. 4. Compensation of delay dynamics with feedforward compensator.

Fig. 5. Proposed contour control system.

centripetal acceleration is given byv2c=r = Amax. Tangential velocityvc is then determined byvc =

pAmaxr.

Category 2 (givenvc;min): The constraint for tracking velocityis given byvc vc;min. Then, the radius of the circular arc is con-strained according tor v2c;min=Amax. Finally, is given by (v2c;min(1cosf(12)=2g))(Amax cosf(12)=2g). The pro-posed trajectory in (9) and (10) can be numerically realized with thevalues determined forAmax, r, andvc, as described above.

B. Compensation for System Delay Dynamics

Joint coordinate trajectories are compensated for system delay dy-namics, by the modified taught data (MTD) algorithm [7], which is il-lustrated in Fig. 4. The MTD algorithm is a feedforward compensator,which improves transient performance of joint dynamics. The com-pensator dynamics are given byFi(s) = (s +Kpi)=Kpi(s ),where is the pole of the compensator. The pole should be selectedappropriately, so that a fast stable system is realized. As a constraintinequality for stability, Kp. The exact value of has to be de-termined from a computer simulation which evaluates transient perfor-mance of joint dynamics bound to rated motor speed. Since samplingtime s is fixed, the time constant of the modified system1= shouldbe bound to1= > 10s to guarantee that waveform deterioration iseliminated. A detailed explanation of the MTD algorithm can be foundin [7].

IV. RESULTS AND DISCUSSION

A. Implementation of the Proposed Contour Control Method

Referring to Figs. 1 and 4, the schematic of the contour control ex-perimental setup is illustrated in Fig. 5.

The limits for joint acceleration for the two joints were1;max =1:432 rad/s2 and 2;max = 1:732 rad/s2, respectively. The right an-gular corner with vertices (0.2 m, 0.2 m), (0.25 m, 0, 2 m), and (0.25m, 0.25 m) was selected as the objective trajectory. Maximum errormax was set to 0.5 mm. According to category 1,r = 1:2 mm andvc = 0:05 m/s withAmax = 0:05 m/s2. The corresponding trajectorysegmentation was implemented with time stamped att0 = 0:000 s,t1 = 0:100 s, t2 = 0:988 s, t3 = 1:038 s, t4 = 1:110 s, t5 = 1:160s, t6 = 2:048 s, andt7 = 2:148 s.

(a) (b)

Fig. 6. Contouring performance. (a) Conventional. (b) Proposed.

(a) (b)

(a) (b)

Fig. 7. Simulation results of joint acceleration for both methods. (a), (b)Conventional. (c), (d) Proposed. [(a), (c) joint 1 and (b), (d) joint 2.]

(a) (b)

Fig. 8. Results of tracking velocity. (a) Conventional. (b) Proposed.

B. Contouring Results

Fig. 6(a) and (b) illustrates contouring results of the conventional andproposed methods. In the conventional method, the reference input gen-erator carries out only sampling of the objective trajectory at a uniformtracking velocity along the entire path. Simulation and experimentalresults are comparable for both methods. Under conventional control,a significant overshoot occurs, as indicated by!A in Fig. 6(a). Thisovershoot can be approximated to 2 mm. In addition, significant con-touring fluctuations are observed near the corner. In contrast, underthe proposed control, overshooting is fully eliminated and the existingcontour deteriorations are minute. Simulation results of joint accelera-tions are illustrated in Fig. 7(a)(d). Under conventional control, joint1 shows torque saturation, indicated by B in Fig. 7(a), which lastsapproximately 0.2 s, whereas joint 2 saturates for 0.1 s as indicated by C in Fig. 7(b). In contrast, the proposed control causes no torque sat-uration, except a momentary demand of maximum torque at the trajec-tory corner, as indicated by!D in Fig. 7(c). Fig. 8(a) and (b) illustratestracking velocity results of the conventional and proposed methods.Under conventional control, significant velocity overshoot is observedat the trajectory corner, which is indicated by E in Fig. 8(a). In con-trast, the proposed method does not cause such behavior.

440 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 5, NO. 4, DECEMBER 2000

(a) (b)

Fig. 9. Behavior of contouring performance asK changes. (a) Trajectory. (b)Contouring error.

C. Analysis of System Robustness

The behavior of contouring errorEc was obtained by a simulationstudy of contour performance, in response to a change of servo pa-rameter settings. The delay compensator was set to nominal servo pa-rameter settings, whereas the servo controller settings of the two jointswere changed in 5% steps from70% to 130% of the nominal settings.The corresponding contouring performance was obtained as shown inFig. 9(a). Each solid line illustrates the following trajectory obtained fora particular offset of servo parametersKpi, in steps of 5%. The offset ofservo parameters is denoted byKp and is defined byKp = (KpiKpi)=Kpi, whereKpi andK

pi refer to the position loop gains ofthe modification term and servo controller, respectively. WhenKpi Kpi,the following trajectory shifts outward, resulting in an overshoot. Underthis condition, contouring performance is evaluated by the amount ofovershoot.

Fig. 9(b) illustrates the corresponding contouring error variationwhich was constructed from Fig. 9(a). It shows that contouring errorincreases with the decrease of servo parameters, whereas overshootincreases with the increase of servo parameters.

D. Discussion

Torque saturation indicated by B in Fig. 7(a) causes velocity over-shoot indicated by E in Fig. 8(a). Such velocity overshoot conse-quently results in significant contour deteriorations in the followingtrajectory, as indicated by!A in Fig. 6(a).

Robustness of the proposed contour control system has been guar-anteed. A maximum overshoot of 0.17 mm is introduced for a+30%change in position loop gain. Nevertheless, 0.17-mm overshoot for a30% change in system parameters could be accepted for all or most in-dustrial applications. Contouring error for a30% change in systemparameters does not exceed 0.6 mm. For most frequent small changesof system parameters, the corresponding contouring error or overshootcould be simply neglected.

In [8], contour control under torque constraint was realized for ver-tical-type articulated robot arms in which the position loop of the servocontroller was open. In this paper, the method has been extended toSCARA robot where position loop of servo controller is closed.

In this paper, the proposed method has been realized for two-dimen-sional applications. Its extension to more realistic three-dimensionalapplications is possible on the basis of the following three guidelines.

1) The third joint of the SCARA is a prismatic joint with a ver-tical axis. For three-dimensional applications, this joint can be

manipulated. Since the first two joints are rotary joints with ver-tical axes, operation of the prismatic joint has no coupling torquedisturbances. Therefore, the third joint can be treated as anotherindependent joint.

2) Almost all practical trajectories can be decomposed into a con-nected sequence of straight lines and corners. Therefore, trajec-tory generation can be carried out by parts, in Cartesian space. Athree-dimensional corner, for example, can be again decomposedinto its horizontal projection and vertical stretch. The prismaticjoint of the SCARA can realize the vertical stretch independentlyfrom the horizontal projection, which could be tracked by the tworotary joints the same way as described above.

3) Even with articulated-type robot arms, on the assumption thatjoints have independent dynamics, three joints could be servedin the same way as explained for the two links. The trajectoryaddressed in this paper is two-dimensional, however, a three-dimensional trajectory can be decomposed into a sequence ofstraight lines and two-dimensional corners that lay on differentplanes. Then, the solution could be applied on each segment inthe sequence in order to contour a three-dimensional trajectory.

V. CONCLUSIONS

Contour control performance of industrial robot arms under torqueconstraint has been enhanced by the proposed method. Optimum avoid-ance of torque saturation and delay dynamics compensation have beensuccessfully implemented and reduced into a single offline trajectorygeneration algorithm. Therefore, the proposed method can be conve-niently implemented in industrial robot arms. Extension guidelines forthe proposed method to address more realistic three-dimensional ap-plications have also been discussed briefly.

Being offline, the necessary hardware changes for the implementa-tion of the proposed method are minute and easily carried out. More-over, the proposed trajectory generation method is highly flexible andcould be easily customized for a particular application, in that it is al-ways possible to figure out the optimum trajectory for a given con-touring operation. Since torque saturation is avoided, the utility of in-dustrial robot arms could be significantly improved by the proposedmethod.

REFERENCES

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[2] D. E. Whitney, Resolved motion rate control of manipulators andhuman prosthesis,IEEE Trans. Man Mach. Syst., vol. 10, pp. 4753,June 1969.

[3] R. P. Paul, Manipulator Cartesian path control,IEEE Trans. Syst.,Man, Cybern., vol. 9, pp. 702711, Nov. 1979.

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[5] J. Y. S. Luh and C. S. Lin, Optimum path planning for mechanicalmanipulators,ASME Trans. Dynam. Syst., Meas., Contr., vol. 14, no.3, pp. 444451, June 1984.

[6] M. Nakamura, S. Goto, and N. Kyura,Mechatronic Servo System Con-trol (in Japanese). Tokyo, Japan: Morikita Shuppan, 1998, ch. 1 and 4.

[7] S. Goto, M. Nakamura, and N. Kyura, Modified taught data method forindustrial mechatronic servo systems to achieve accurate contour con-trol performance, inProc. IEEE/ASME Int. Conf. Advanced IntelligentMechatronics (AIM), June 1997, CD-ROM.

[8] M. Nakamura, S. Aoki, S. Goto, and N. Kyura, Upper limit of perfor-mance for contour control of industrial articulated robot arm with torqueconstraints and its realization (in Japanese),Trans. Soc. Instrum. Contr.Eng., vol. 35, no. 1, pp. 122129, Jan. 1999.