Introduction to Dynamic Models for Robot Force Control

Introduction to Dynamic Models for Robot Force Control Steven D. Eppinger and Warren P. Seering control is often used to refer to any force ABSTRACT:Endpointcompliancestrategies for precise robot control utilize feedbackcontrol algorithm, sincethey use sensed force to alter controller commands. fromaforcesensorlocatednearthetool/ Endpoint compliant control strategies deworkpieceinterface. The closed-loopperformance of such endpoint force control sys- pend on force signals measured by a wrist sensor. The sensor output is fed back to the tems has been observed in the laboratory to controller to alter the system’s performance. be limited and unsatisfactory for industrial Manysuchclosed-loopsystemshavebeen applications. This article discusses the parbuilt using various force control algorithms, ticular dynamic properties of robot systems and many stability problems have been obthat can lead to instability and limit performance. A series of lumped-parameter models served. A theoretical treatment of environis developed in an effort to predict the closed- mentally imposed constraints is provided by Mason [3], whoalsosuggestsacontrol loop dynamics of a force-controlled singlemethodologytoaugmentthese‘.natural” axis arm. The models include some effects of robot stmctural dynamics, sensor compli- constraints with an appropriate set of “artificial” constraints. Raibert and Craig [4] deance, and workpiece dynamics. The qualitative analysis shows that the robot dynamics veloped a hybrid control architecture capable of implementing Mason’s theory. Saliscontribute to force-controlled instability. bury’s stiffness control [5] regulates the end Recommendations are made for models to effector’sstiffness in Cartesiancoordinates be used in control system design. using an appropriately formed joint stiffness matrix.Othercompliantcontrolstrategies Introduction include Whitney’s damping control [6] and Certain robot tasks demand precise interHogan’simpedancecontrol [7]. Thesimaction between the manipulator and its enplest force control strategy is explicit force vironment.Amongthese are many ofthe control [8], which makes no use of position operations required in the mechanical assem- feedback information in the controller, and bly process. Strategies for the execution of the servo loop is based on force errors. such tasks involve controlling the relationActive force control systems implemented ship between endpoint forces and displaceto test these strategies have demonstrated dymentsunder the environmentallyimposed namic stability problems. In practice, forceconstraints. This endpoint compliance can be controlled robotsdo not have bandwidth sufimplemented in many different schemes, and ficient for most industrial applications. HisWhitney [2] provides an overview of these. torically, some instabilities have been caused Strictly speaking, compliance is the inverse by digitalsampling,andWhitney [2] disof stiffness; however,theterm compliant cussestheseconditions.Researchershave also observed the effects of unmodeled (uncompensated) nonlinearities, such as friction StevenD.Eppingerand Warren P. Seering are or backlash [9]. RaibertandCraig [4] imwith the Artificial Intelligence Laboratory of the plemented their hybrid controller and found Massachusetts Institute of Technology,545 Techoscillations present in the controlled system. nology Square, Cambridge, MA 02139. This arInstabilities have been observed in the opticle describes research done at the Altificial Intelligence Laboratory of the Massachusetts eration of both of the force-controlled robots Institute of Technology. Support for thelaboracurrently in use at the MIT Artificial Inteltory’s artificial intelligence research is provided in ligence ( A I ) Laboratory.Theserobotsinpart by the System Development Foundation and clude a PUMA arm and the new MIT Prein part bythe Advanced Research Projects Agency cision Assembly Robot. Both arms display of the Department of Defense under Office of Naperformance differences whenthe workpiece Sup valResearchContract N00014-85-K-0124. (environment) characteristics are changed. port for this research project is also provided in Roberts et al. [lo] investigated the effect part by the TRW Foundation. The original version of wrist sensor stiffness on the closed-loop of this paper was presented at the IEEE Internasystemdynamics:theyalsoincludeddrive tional Conference on Robotics and Automation in San Francisco, California, April 7-10. 1986 [l]. stiffness (transmissioncompliance)intheir 0272-1708;87:0400-0048

48

301 OC

D

dynamicmodel.Transmissioncompliance causes the joint actuators and wrist sensors to be noncolocated, a condition discussed in detail by Gevarter [ 111. Noncolocation is the stability problem that occurs when a control loop is closed using a sensor and an actuator placed at different points on a dynamic stmcture. Cannon [12] has investigated the similar problem of the position control of a flexiblearmwithendpointsensing.Hehas shown that a high-order compensatoris able to stabilizethesystem, but withlimited bandwidth and high sensitivity to parameter changes. Sweet and Good [13] obtained realistic robotltransmissionsystemdynamics fromexperimentaltests and thenincluded the influence of these effects in the controller design. Researchers have named many suspected causes of the instabilities observed in forcecontrolled robot systems. Among these are: lowdigitalsamplingrate, filtering, workpiecedynamics,environment stiffness, actuatorbandwidth,sensordynamics, arm flexibility, impactforcesupontool/workpiececontact,anddrivetrainbacklash or friction. This article addresses the effects of arm, sensor, and workpiece dynamics. Using conventional modeling and analysis techniques, it is demonstrated that when the arm flexibility gives rise to a vibratory mode withinthedesiredclosed-loopbandwidth, instability can occur. In particular, a simple, one-axis force control algorithm exhibits stable behavior when the higher-order dynamics of the arm can be neglected, and it can be unstable if those effects are significant. This is believed to be the cause of the instabilities observed in the robots at the MIT AI Laboratory. Unstable behavior often takes the form of a limit cycle where the robot is making and breaking contact with the workpiece. In this article, the terms workpiece and environment are used interchangeably. The workpiece is the component of the environment contacted by theendeffector of the force-controlled robotsystem. The discontinuousnature of this response makes the system difficult to model using linear elements. However, for thepurposesofcontrollerdesign,wewill neglectthediscontinuityandstudylinear systemmodels.Nonlinearsimulations(not

1987 IEEE

I€€€ Corirroi Systems Mogozme

presented in this paper) suggest that if the linearizedsystemhassufficientdamping, then neglecting the discontinuityis justified. However, if the linearized system has unstable or highly oscillatory closed-loop poles, then the discontinuity should be included in the model so that limit cycles can be predicted in simulation. For control system design,thisanalysisassumesthattherobot endpoint remains in contact with the workpiece. Note that choosing control gains that give desirable response assures that we will not have unstable or highly oscillatory systems, and so the arm is likely to remain in contact.

Rigid-Body Robot Model To begin with a simple case, let us consider the robot to be a rigid body, with no vibrational modes. Let us also consider the workpiece to be rigid, having no dynamics. The sensorconnects thetwo with some compliance, as shown in Fig. 1. Throughout the model development, the term robot refers to the arm itself. The term robot sysrem refers to the total system, comprised of the robot, sensor, workpiece, and controller. We modeltherobot as a masswith a damper to ground. The mass m, represents the effective moving mass of the arm. The viscous damper b, is chosen to give the appropriate rigid-body mode to the unattached robot. While structural dampingis very low, b, includes the linearized effects of all of the other damping in the robot. The sensor has stiffness k, and damping 6,. The workpiece is shown as a “groundstate.”Therobot actuator is represented by the input force F, and the state variable x, measures the position of the robot mass. The open-loopdynamicsofthissimple system are described by the following transfer function: X,(s)lF(s) = l/[m,s2

+ (b, + bs)s + k,]

Since this robot system is to be controlled to maintain a desiredcontactforce,wemust recognize that the closed-loop system output variable is the force across the sensor, the contact force F,.

F,

=

k,x,

Include Workpiece Dynamics The simple robot system of Fig. 1 has been shown tobe unconditionally stable(for kf B 0). Force-controlled systems, however, have Fig. 2. Block diagram for the controller of been observed to exhibitvariations in dyFig.1. namic behavior, depending upon the characteristics of the workpiece with which the r o b o t is in contact. It is with this phenomeWe will now implement the simple propornon in mind that the robot system model is tional force control law: augmented to include workpiece dynamics, F = kf(Fd - F,), 4 5: O as shown in Fig. 4. This two-massmodelincludes the same which states that the actuator force should robotandsensormodelsusedpreviously, be some nonnegative force feedback gain kf withtheworkpiecenowrepresented bya times the difference between some desired mass Q,. The workpiece is supported by a contact forceFdand the actual contact force. spring and damper to ground with paramThis control law is embodied in the block eters k, and bw, respectively. The new state diagram of Fig. 2. The closed-loop transfer variable x, measuresthepositionofthe function then becomes workpiece mass. F,(s)/Fd(s) = $k,l[m,s2 + (b, bs)s Theopen-looptransferfunctionsof this two-degree-of-freedom system are: + k S ( 1 + $11

+

The control loop modifies the characterXr(s)/fTs) = Nw(s)/D,(s) istic equation only in the stiikess term. The X,(s)lF(s) = [b,s ks]/D4(s) force control for this simple case works like a positionservosystem.Thiscouldhave where been predicted from the model in Fig. 1 by N , ( s ) = [rn,s2 + (b, + b,)s + (k, + k,,,)] noting that the contact force depends solely on robot position x,. ~ ~ ( = s )[m,s2 + (b, + b,)s k,] For completeness, let us look at the root locus plot for this system. Figure 3 shows * N,(s) - [ b , ~ + k,]’ the positions in the s plane of the roots of the closed-loop characteristicequation as the The output variableis again the contact force F,, which is theforce across the sensor, given force feedback gain kf varies. For this qualby itative analysis, the model parameter values havebeenchosenonly to plotmot locus F, = k,(x, - x 3 shapes representative of robot systems. They Ifwenowimplementthesamesimple do notcorrespond to data taken from any force controller, the control law remains unspecific robot. For this reason, the plots do changed. not contain numerical markingson the axes. For kf = 0, the roots are at the open-loop F = $(Fd - Fc), kf 2 0 poles. The loci show that as the gain is incteased, the natural frequency increases, and The block diagram for this control systemis shown in Fig. 5. Note that the feedforward the damping ratio decreases, but the system path includes the difference betweenthe two remains stable. In fact, kf can be chosen to give the controlled system desirable response open-loop transfer functions. The root locusfor this system is plottedin characteristics. Fig. 6 as the forcefeedback gain kris varied. There are four open-loop poles and two openloopzeros.Theplotthen still has two asymptotes, at +90 deg The shape of this s-p’ane rootlocusplot tells us that even for high

+

+

I

fm I

---

-ROBOT

SENSOR

Fig. 1. Rigid-body robot model with compliant sensor and rigid workpiece.

April 1987

.

ROBOT

Fig. 3. Root locus plot shape for the controller of Fig. 1.

SENSOR WORKPIECE

Fig. 4. Rigid-body robot model including workpiece dynamics.

49

x,, x2, and x , measure ~ the positions of the

masses mi,m2. and in,,. This three-mass model has the following open-loop transfer functions:

xl(s)/F(s)= N.I(s)/D,(s) I

x2(s)/F(s)= N3(s)/D&)

Fig. 5. Block diagram for the controller of Fig. 4.

I

I

X,,(s)lF(s) = N2(s)/D&)

?

fm

spii

&(s)

=

s-p’ane Re

,

[rn2s2+ (b2 + b,)s

. N,,,(s)N&)

=

[b,s

+ (k2 + kJl

+ k,]’

N,,.(s)[b,s+ k2l

N2(s) = [b2s +

k2l

*

[b,s + k,l

+ b2)s + k21 [m2s2+ (b2 + b,)s + (k2 + k,)I

~ ~ ( = s )[mls2+

(bl

. N,.W

Fig. 6. Root locus plot shape for the controller of Fig. 4.

- Nn(s)[bzs +

Include Robot Dynamics Since the addition of the workpiece dynamics to the simple robot system model did not result in the observed instability, we will augment our systemwithamorecomplex robot model. If we wish to include both the rigid-body andfirst vibratory modes of the arm, thentherobotalonemust be represented by two masses. Figure 7 showsthenewsystemmodel. The totai robot mass is now split between m , and m2. The spring and d a m p e r with values k2 and b2 set the frequency and damping of the robot’s first mode, while the damper to ground, b l , primarily governs the rigidbody mode. The stiffness between the robot masses could be the drive train or transmission stiffness, or it could bethestructural m i and m2 stiffness of a link. The masses would then be chosen accordingly. The sensor and workpiece are modeled in the same manner as inFig. 4. The three state variables

ROBOT

SENSOR WORKPIECE

Fig. 7. Robot system model including Ebot first-mode and workpiece dynamics.

k21

+ k,]’ N,.(s) = [m,,s2 + (b, + b,,.)s + (k, + k , ~ ] *

Fig. 9. Root locus plot shape for the controller of Fig. 7 .

[b$

The contact force is again the force across

k,

F, = k ( x 2 - x,J and the simple force control law is

F

“-h

k2I2

- [mls2 + (b, + b2)s +

values of gain, the system has stable roots. Therefore,whilethecharacteristics of the workpiece affect the dynamics of the robot system, they donot cause unstable behavior.

I

Fig. 8. Block diagram for the controller of Fig. 7.

where

50

’ -

I

=

kf(F, - Fc), kf 2 0

The block diagram for this controller, Fig. 8, showsagainthatthefeedfonvardpath takes the difference between two open-loop transfer functions. This time, however, both ofthesetransferfunctionsrepresentpositions remote from the actuator force. Note that the auxiliary output is x , . which is not a measurable position. The root locus plot, Fig. 9, shows a very interesting effect. Thesystem is onlyconkr, the ditionally stable. For low values of system is stable; for high values of kf, the system is unstable;andforsome critical value of the force feedback gain. the system is only marginally stable. The +60-deg asymptotes result from the system’s having six open-looppoles, but only three openloopzeros.Inspection of theopen-loop transfer functions confirms this: The numerator of the transfer function relatingX&) to F(s) is a third-order polynomial in s. To provide some physical interpretation to this effect, note again that the input force F is applied to m,,which moves with xi. The sensor is attached to the robot at m2. which moves with x?. Here the controller attempts to regulate the contact force through the mz-

b2-k2dynamic system. In the preceding two examples, stability was achieved while the controller regulated the contact force on the single-robot mass. In practice, gains are chosen to give stable response for nominal values of the workpiece parameters. Unstable behavior can then be observed with variations in those parameters. This instability can bepredicted by drawinganewrootlocus plot for the varied system parameters.

Exclude Workpiece Dynamics To determine the influence of the workpiece dynamic characteristicson this system, theireffectsare now removedfromthe model. Figure 10 shows the workpiece modeled rigidly as a “wall.” The robot model still includesboththerigid-bodyand first vibrationmodes.Thesensorconsists of a spring and damper betweenthe robot andthe workpiece. This simpler two-mass modelhas only two state variables, x , and x2, which measurethe displacements of the two robot masses. The two open-loop transfer functions are

Fig. 10. Robot system model including robot first-mode and rigid workpiece.

!€€E Conrroi

Sysiems Magazlne

=

+

[mzs2 (b, + b J s

+ (k, + k2)]/Dz

fm

;;plane

X,(s)lF(s)= [bzs + kJD,*

- [b*s

+ kJ2

The contact force is given by

Fig. 12. Root locus plot shape for the controller of Fig. 10.

F, = kSx2 effects of robot and workpiece dynamics on the stability of simple force-controlled sysF = kJF, - Fc), 4 L 0 tems. An instability has been shown to exist for robot models that include representation The block diagram for this controller, Fig. of a first resonant mode for the arm. The 11, shows that no differences in open-loop modemodeledcanbeattributed toeither transfer functions are being taken. drive trainor structural compliance (or both). The root locus plot is shown in Fig. 12. Thepotential for instabilityispresentbeAgain, the system is conditionally stable, as this time there one is open-loop zero andfour cause the sensor is then located at a point remote from the actuator. The controller atpoles. The instability is therefore shown to tempts to regulatecontactforcethrougha be present regardless of the workpiece dythe systemsof namics (which may have been suspect in the dynamicsystem.Compare Figs. 7 and 10 with those of Figs. 1 and 4. preceding case of the model in Fig. 7 ) . It must be noted, however, that there are Comparison of thehvo-mass model ofFig. many causesof force-controlled instabilities. 4 with that of Fig. 10 showsthat the models are basically the same (note the different sub- The effect presented in this article, that of robot s t ~ ~ c t u r dynamics, al is animportant scripts), and the equations are therefore of If thedesired probleminsomesystems. the same form. One controlled system is staclosed-loop bandwidth is low compared to ble (Fig. 6), however, while the other isnot (Fig. 12). The difference is only in the place- the first mode frequency of thearm, then the target performance may be achievable. ment of the sensor. In the former, the feedHowever, if we require a machine capable backcomesfromthespring between the of higher performance, we must also invesmasses. In thelatter,thefeedbacksignal comes from the spring at the second mass to tigateotherissuescarefully.Inparticular, theworkpiecedynamics,higherstructural ground. A typical industrial robot with first mode near20 Hz and astiff force sensor may modes,actuatorlimitations,andcontroller implementation must be considered. have the following parameters: m, = 40 kg, The effect of the workpiece dynamics is m2 = 40 kg, k2 = 400,000 Nlm, k, = asyetunclear.Observationofforce-con315,000 N/m, b, = 500 N-sec/m, b2 = 500 trolledroboticsystemssuggeststhatthe N-sec/m, 6, = 400 N-sec/m. For a verystiff workpiece, when coupled through the force workpiece, the upper limit on gain for stasensor, can significantly change the dynambility is 4 = 1.16. ics. Certainly, if theworkpiecewerevery compliant and extremely light, there could Conclusion be no force across the sensor, degenerating A series of lumped-parameter models has the closed-loop systemto theopen-loop case, been developed in order to understand the which of course is stable. In this article, we have demonstrated the opposite extreme, that when the workpiece is modeled as a rigid wall, the systemcan be unstable. The sensor and workpiece (environment) dynamics are therefore important and should be modeled. Limitedactuatorbandwidth, filtering, and digitalcontrollerimplementationcanalso causeinstability.TheseperformancelimiFig. 11. Block diagram for the controller tations must also be included in the system of Fig. 10. model used for controller design. and the control law again will be

April 1987

In this article, we have not addressed the nonlinear effects of the discontinuity at the workpiececontact,theassociatedimpact forces that occur, axis friction, joint backlash, or actuator saturation. Nonlinear simulations suggest, however, that these effects can, undersome conditions,lead tolimit cycles in the otherwise stable linear systems, and under other conditions, they can even stabilize unstable linear systems. The stability bounds derived using the linear models should be used to set upper limits on .the controller gains, which should then be decreased if limitcycles are observedunder operating conditions. The modelingand analysis techniques presented are tools to aid in control system design. For their accurate use, however, the models must sufficiently describe the actual hardware. A topic of ongoing researchis the comparison of these model predictions with experimental result. . In particular, it is not clear how all the model parameters should be chosen in order to assure agreement between the analytical model and the experimental hardware.

References S . D.EppingerandW.P.

SeeMg, “On

Dynamic Models of Robot Force Control,” Proc. Int. Con& Robot. hitom., Apr. 1986. D. E. Whimey, “Historical Perspective and State of the Art in Robot Force Control,” Proc. Int. Con$ Robot. Autom., Mar. 1985. M. T. Mason,“ComplianceandForce Control for Computer Controlled Manipulators,” IEEE Trans. Syst., Man, Cyber., vol. SMC-11, pp. 418432, June 1981. [41 M. H. Raibertand J. J. Craig,‘:Hybrid PositiodForceControlofManipulators,” J. Dyn. Syst., Measurement and Control,

vol. 103, June 1981. 151 J. K. Salisbury,“ActiveStiffnessControl of Manipulator a inCartesianCoordinates,” Proc. 19th Con$ Decision and Control, vol. 1, Dec. 1980. D. E. Whitney,“ForceFeedbackControl of Manipulator Fine Motions,” J. Dyn. Syst., Measurementand Control, vol. 99, pp. 91-97, June 1977. N.Hogan,“ImpedanceControlofIndustrial Robots,” Robotics and Computer-Integrated Manufamring, vol. 1, no. 1, pp. 97-113,1984. J. L. Nevinsand D. E. Whitney,“The Force Vector Assembler Concept,” Proc., First CSIM-IITOMM Symp. lheory and Pracn’ce of Robots and Manipulators, ASME, Udine, Italy, Sept. 1973. [91 Y. H. S. Luh, W . D. Fisher, and R. P. C. Paul, “Joint TorqueControlbyaDirect IEEE FeedbackforIndustrialRobots,” AC-28, Feb. Trans. Auto. Contr., vol. 1983.

51

[lo] R. K. HillM. Roberts, Paul, B. and P.R. Steven D. Eppinger reForce Sensor Effect ceived of the S.B. degree in berry, “TheWrist Stiffness onControl Robot Maniputhe of S.M. degree the 1983 and Massathe from 1984in lators,” Proc. Int. Con$ Robot. Autorn., . Mar. “Basic Relations nology. ConDepartment for of [l 11 W. B. Gevarter, trol Flexible Vehicles,” of AZAA J., vol. 8, Mechanical Engineering. no. 4,app. candidate now 666-672. is He 1970. Apr. for [12] R . H. Cannon and E. Schmitz. “Initial Ph.D. degree the Exisand conducting End-Point periments the aresearch Control on at of Int. J . Robot. Artificial IntelliMIT One-Link Flexible Robot,” Res., vol. 3, no.Laboratory. gence 1984. 3, 62-75, Fall pp. Mr. [I31M.L. M. Sweet and C. Good, “RedefiniEppinger’s research intertion of theRobotMotionControlProbests lieintheareaofautomatedmanufacturing. lem.” ZEEE Cont. Sysr. Mug., pp.18-25,specificallyimprovingmachineperformancewith nory 1985. Aug.

Warren P. Seering received the B.S. dempein

1971 and the M.S. degree in 1972,bothfromthe University of Missouri at Columbia.Hecontinued his graduate work atStanford University. receiving the Ph.D. degree in 1978. In Fall 1978, Professor Seering was appointed to the faculty of the DepartmentofMechanicalEngineering at the Massachusetts Institute of Technology. His work has focused on machine design in machineperforandtheroleofcomputation mance. Professor Seering helped to establish the MIT Machine Dynamics Laboratory, of which he is currently Co-Director. Professor Seering is now conducting research at the MIT Artificial Intelligence Laboratory. His research interests are in the areas of dynamics, vibration, system design, and artificial intelligence. His work centers around the use of computation to extend the performance of machines, particularly robots.

1987 CDC TheIEEE ConferenceonDecisionand Control (CDC) is the annual meeting of the IEEE ControlSystemsSocietyand is conductedincooperationwiththeSocietyfor Industrial and Applied Mathematics (SIAM) andtheOpeiationsResearchSocietyof America(ORSA).Thetwenty-sixth CDC will be held on December 9- 11. 1987,at the Westin Century Plaza Hotelifi Los Angeles. California.TheGeneralChairmanofthe conference is ProfessorWilliam S. Levine of the University of Maryland. The Program Chairman is Professor John Baillieulof Boston University. The conference will include both contributed and invited sessions in all aspects of the theory and application of systems involving decision. control, optimization, and adaptation. We urge you to attend this premier control conference held in the dynamic and historic setting of Los Angeles. The pictureshere illustrate a few of the many attractions of the Los Angeles area. For further information, contact the General Chairman: Prof. William Levine Dept. of Electrical Engrg. University of Maryland College Park, MD 20742 Phone: (301) 454-6841

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The beautiful Malibu Beach community, embraced by the Pacific Ocean.

Hooray for Hollywood! This famous intersection of Hollywood and Vine is a focal point for the mystique of the Entertainment Capital. The Capitol Record Company in the background offers tours and shows how records are made.

F E E Control Sysrems Mogozine