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,”
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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
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inCartesianCoordinates,” Proc. 19th Con$ Decision and
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D. E. Whitney,“ForceFeedbackControl
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N.Hogan,“ImpedanceControlofIndustrial Robots,” Robotics and Computer-Integrated Manufamring, vol. 1, no. 1, pp.
97-113,1984.
J. L. Nevinsand D. E. Whitney,“The
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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
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.
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“Basic
Relations
nology.
ConDepartment
for of
[l 11 W. B. Gevarter,
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Flexible
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of
AZAA J., vol. 8,
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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
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Artificial
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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
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CDC
will be held on December 9- 11. 1987,at the
Westin Century Plaza Hotelifi Los Angeles.
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conference is ProfessorWilliam S. Levine
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both contributed and invited sessions
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We urge you to attend this premier control
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Phone: (301) 454-6841
52
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