From: AAAI Technical Report SS-99-05. Compilation copyright © 1999, AAAI (www.aaai.org). All rights reserved.
On the use of Hybrid Control for LeggedLocomotion
S.V. Shastri, ne S.T. Venkataraman
SRI International
333 Ravenswood
Avenue, Menlo Park, CA94025
e-mail: venkat@erg.sri.com
robot called HEX(SectionII). Thesecondpart utilizes the experimentalresults anddevelopsa
hybrid control architecturefor leggedlocomotion
(SectionIV). Thehybrid control problemconsideredasa result is uniquein that it considers
switching betweenreference modelsin a model
referencecontrol architecture.Thethird part
describesongoinganalytical workin the use of
hybridcontrol for leggedlocomotion.
Abstract
In this paper, wedevelopa hybrid control approachfor
legged locomotion. Wemotivate the developmentof
the control architecture using the results of a series of
walking, running and obstacle climbing experiments
conductedusing a six legged robot called HEX.Our initial simulationresults indicate the potential stability of
the control approach,and our future analytical work
should provide the formal proof of these results.
I. Introduction
It is well known
that leggedlocomotion
involves
the use of prototypical movements
whereinthe
phase,frequencyandamplitudeof individual leg
motionsare related to oneanotherin specific
ways.In the literature, suchmovements
are
referredto asgaits. It is alsowell known
that the
neuralcircuitry in mostbiological systems
contain
programs
for multiple gaits. Typically, animals
appearto select andswitch between
gaits
dependingon a rangeof factors such as body
speed,unknown
terrain geometryandvariations
in terrain physics. Froma phenomenological
point
of view, this maybe viewedas an example
of a
control systemthat attemptsto maximizesystem
performance
and robustnessin the presenceof
large disturbances(and possibly modeluncertainties) by possiblyswitchingbetween
control laws.
As suchtherefore, leggedlocomotioncould well
serveas a problemareafor the studyof hybrid
control. This papermaybe consideredas a first
step in our formal studyof the useof hybridcontrol techniquesfor leggedlocomotion.It maybe
viewedas a logical next step after our gait modeling work[Shastri 1997].
Werestrict our attention to hexapedal
locomotionin this paper.Thepaperis dividedinto
threeparts. In the first, wedescribeopenloopgait
control experimentsperformedusing a hexaped
170
II.
ExperimentalResults
Figure 1: HEX, The LeggedRobot
Figure 2: Implementation Setup
Table 1: Walking, Runningand Climbing Experimental Results
Gait
Terrain
Speed
%Success
Both
Level
Range
100
Both
> 1" Single Step
Fixed
0
Both
< 1" Single Step
Fixed
100
Tripod
< 1" DoubleStep
Fixed
25
Wilson
< 1" DoubleStep
Fixed
40
Tripod
< 1" Triple Step
Fixed
5
Wilson
< 1" Triple Step
Fixed
20
Aspart of this ongoingeffort, a leggedrobot has
beenbuilt for experimental
work(Figure1). It has
six legs, eachwith four-degrees-of-freedom.
Threeof theseare at the hip, andoneat the knee.
Twoof the three degrees-of-freedom
at the hip is
actuatedusinga differential geararrangement.
All
the joints are drivenby Futabaservomotors,
which are commanded
with position commands.
Eachservohas an internal PIDposition control
loop in hardware
at a samplea rate of about2.5
KHz.There is a contact sensormountedon each
foot of the robot. Therobot is about50 cmslong,
30 cmswideand15 cmstall. Therobot is controlled by a data acquisition card mounted
on the
backplaneof a PC.All control programsare
developedin C Programming
Language,cross
compiledto the native assemblylanguageof the
processoron the data acquisition card andused
for commanding
the Futabaservos.
Results from the numerous
runs on level
groundas well as climbing over obstaclesmaybe
summarized
as shownin Table 1. Theseexperimentsconsist of walkingand runing experiments
usingboth the Wilsongait andthe tripod gait, and
climbing over unknown
obstacles. Theimplementation architectureutilized for the purpose
(describedin Figure2) generatesnominalleg
motionsusing the gait modelssuggestedin our
earlier work, andperformsPIDtype servocontrol
at the joints. Testson level groundwereperformed
usinga tripod gait over a rangeof bodyspeeds,
starting from about1 bodylength per minuteto
about30 bodylengthsper minute. Tests with the
Wilsongait wereperformedbetween1 and 8
171
bodylengths per minute. Obstacleswereconfiguredin the formof a single, doubleor triple step
anda fixed climbing speedof about4 body
lengthsper minutewasutilized. All the tests on
level groundwerecompletelysuccessful,implying
that the locomotion
wasstable, withoutsignificant
slippage betweenthe legs andthe ground
(despitethe absence
of anyfoot force sensors).
Foranyof the stepconfigurations,if the step
heightwas> 1 ", the legs couldnot clear pastthe
obstacle. This is indicatedby the 0%successin
Table1. For single step obstacleswhenthe step
height was< 1 ", the robot wasalwayssuccessful,
independentof the approachangle. For the two
andthree step configurations,the successrate
dependedvery muchon the approachangle.
When
the robot tried to climb the obstaclewith e
= 0, the motorssaturatedmoreoften (input saturation) than otherwise.Asa result, successrates
werequite low (as indicatedin Table1). When
the
approachangleswerevery large (e > 50°), input
saturation appeared
to lower the successrate as
well. Whilea uniformdegradationin performance
wasseenas the obstaclesgot worse,experimentswith the Wilsongait appearedto havebetter resultsthanwith tripod gait. Notethat in all of
the experimentsthe robot wasable to maintain
stability (e.g., did notfall or flip over),butthis may
be largely dueto sufficient number
of legs supporting the bodyat all times. Similar experiments
with biped or quadruped
robots cannotbe
expected
to yield suchstability properties.
III. Discussion
mentingthe twogaits. Asfar as tripod gait is conIn all of the climbingexperiments,
input saturation cerned,with a duty factor of 0.5, the frequencyof
oscillation is roughlythat of theforcingsignal.In
occursdueto, (i) an increasein the magnitude
the caseof Wilsongait, however,the needto
controlsignalsrequiredfor climbing,and(ii)
decrease
in the coefficient of friction between
legs maintaina dutyfactor of 1/6 results in higherharandthe grounddueto changein terrain in geom- monics.Thepowerspectraof a typical Wilson-like
tery. Theformeris relatedto the fact that climbing responsewhenforced with a 1Hzharmonicsignal
was shownto be spread out betweenOHzand
requires overcoming
larger potential energy,and
22Hz(with mostof powerresiding in frequencies
maybe whythere is a slight improvement
in perunderlOHz)in our earlier work.Asa result, gait
formancewhenemployingthe Wilsongait (greater
signals wouldhaveto be generatedat atleast
numberof legs on the groundin generalimplies
about 45x the leg movement
frequencydesired.
larger total bodyforcesfor a givenleg-ground
This is likely to posea tremendous
computational
force value). Thedecrease
in friction coefficient
overhead,
and
as
a
result
is
not
likely
to be the
results from prematurecontact betweenground
preferred
gait
at
higher
speeds.
As
a
result,
andthe heel of a foot. For example,as shownin
depending
on the operatingconditions, the preFigure3, a stepobstacleresults in reductionof
ferred
gait
mightvary between
tripod andthe Wilthe friction areafromthat of thefootprint, to a line
sonas a result of athe tradeoff by, anda
that is as long as the widthof the foot. Suchconlocomotion
architecturethat hasthe ability to
tact effects occurdueto the absence
of detailed
switch
between
gait modelsand tune their paramknowledge
aboutterrain geometry(and the resulteters on line wouldbe vital to achievinguniform
ing inability to shape
gait trajectoriesto fit thetersystemperformanceover a range of speedsand
rain), combined
with our assumption
about
terrain conditions.Ourworkin hybridcontrol is
treating changes
in terrain geometry
as an extermotivatedprecisely by this observation.Wehave
nal disturbance.Onewayto overcome
this probalready developed
a hybrid control architecture
lemis to placeforce sensorson the feet of the
for
locomotion,
and
havestarted to test the potenrobot, andtreat gait trajectories as compliant
tial of hybridcontrol techniques
for leggedrobots.
motiontrajectories. Analternate waymightbe to
In
the
following
sections,
we
report
the state of
integrate inclinometersandrate gyroson to the
this
work
in
progress.
bodyof the robot, andusethis informationfor
modifyinggait trajectories. A third, andperhaps
the mostcomputationallyatractive method
is to
drive the legs usingcompliantactuators(e.g., artificial muscles),andusetheir passivecompliance
to re-orient the legs andimprovecontactfriction.
Thelast approach
is evidencedin manybiological
systems,andis arguedas the basis for passive
stabilization of the bodyas well. Weare currently
investigatingthe useof Electrostrictive Polymer
Artificial Muscles
(EPAMs)
for this purpose.
It is alsoclear fromdiscussions
in the previoussectionthat the Wilsongait tendsto outperformthe tripod gait, andas a result shouldbe the
preferredchoicein variousinsects. Evidencefrom
biology, on the other hand,appears
to indicate the
Figure 3: Premature Leg-GroundContact
useof Wilsongait only in stick insects, andcertain arthropodsduringvery slowwalking.In this
paper,wearguethat the reasonfor this mightlie
in the computational
complexityinvolvedin imple-
172
IV, A HybridControl Architecture for
Legged Locomotion
Discussionsin the preceedingsection demonstrate that leggedlocomotioninvolvesthe use of
multiple gaits. It is also reasonable
to assume
that
thesegaits wouldneedto tunedon line to match
the geometricandphysicalcharacteristicsof terrain. In the experiments
conducted
thus far, we
havegeneratedthe desiredmotionof various legs
usinggait models,discretizedtheminto position
setpoints andperformedPIDcontrol of actuators
for realizing the desiredmotionon variouslegs.
Despitethe simplicity of suchan approach,its
applicability to situations wheregait modelsmay
be switchedandtunedon line maybe quite limited. In this section, wediscussthis issuein some
detail, andpresenta control architecturefor
leggedlocomotionthat has the ability to accomodategait switchingandgait tuning.
Figure 4: A Control Architecture for
Legged Locomotion
In systemstheory, control objectivesare
statedtypically usingcertain typesof error functions. Some
of theserelate to howclosely various
states (outputs) of a dynamicalsystemmay
maintainedaroundprescribedquantities. Specificationof the latter couldoccurin oneof threedistinct ways:referencesetpoints, reference
trajectories andreferencemodels.Theinitial
experimentshas consideredsetpoint control due
173
to its inherentsimplicity in design.Onthe other
hand,it is well knownthat this approach
maybe
too limiting a framework
for specifying complex
robot behaviors.Reference
trajectory following
schemes
allow specification of complicated
maneuvers
and is in fact mostcommonly
usedin
robotics.Modificationof gait trajectorieson line
mightresult in a change
in the orderof reference
trajectories andmustbe dealt with in the information specifiedof a controller. Suchan eventuality
maybe accountedfor by consideringthe highest
orderof the trajectories generated
by all gait models (with sometuning schemes).A reference
modelapproach
wouldalso havethe ability to
capturecomplex
robot behaviors.In addition, it
wouldpermitthe investigationof control designfor
classesof tuning laws, andclassesof gait model
switchinglaws. In viewof this, weproposethe use
of a modelreferencecontrol approachfor locomotion andsuggestthe architecturein Figure4 for
the control of locomotion.
In sucha setting, gait
control wouldinvolve the designof a modelreferencecontroller. In the eventthat the plant has
modeluncertainties,gait control wouldinvolve the
designof modelreferenceadaptivecontrol with
appropriateadaptationlaws. Gait tuning wouldbe
implementedthrough changesmadeto the
parameters
of gait models,andgait switching
wouldoccuras a result of switchingbetween
referencemodelson line. Further, realization of a
gait wouldinvolve the demonstration
of the existenceof a modelreferencecontroller. Stablegait
tuninglawsfor whichcontroller exist wouldbe permissibleandrules that guarantee
stability of the
overall systemduring transitions wouldbe consideredthe allowableswitchingbetween
gaits.
IV.1. Initial SimulationResults
Assuming
that the use of compliantactuators
passivelystabilizes the contactbetweenlegs and
the ground,the problemof controlling leg motions
maybe representedin a mannersimilar to
multifingered handcontrol problemdescribedin
[Li andSastry 1989]. As far as leg movements
are concerned,
the plant modelfor eachleg is
givenby:
= f(x)
+ g(x)u + ,I(x)F
then extendresults to the class of nonlinearsystemswith analytic vectorfields. Wewill present
theseresults to date, andmotivatethe conditions
positionsandvelocities), u ~ R,, is the joint
underwhichthe overall dynamicalsystem
remainsstable duringswitching.In addition, the
torquevector, F ~ RI’ is the foot to ground
applicability of switchingrules suchas cyclic
forces, andfand g are smoothfunctions. For
switching andrandomswitching will be explored
HEX,n = 8, m= 4, andp = 3. It is quite well known for locomotionover ruggedterrain.
that the dynamicmodeldescribedabovemaybe
re-qritten in the form
VI. Acknowledgements
Theworkdescribedin this paperis supportedin
2 = P lf(x) + P2g(x) u + J(x)F
part through the NSFGrant IRI-9530915and
through an ONRGrant (SRI project 1686). The
wherePl andP2are vectorsof fixed value
authorwouldlike to thankJohnKoofor his help in
parameters
that are related to link masses
and
the controls analysiswork.
link lengths. To studythe performance
of the
multiple modelreferencecontrol problem,we
VII. References
simplify the modeldescribedabovewith the
[Kanellakopoulos
1991]Kanellakopoulos,
I.,
assumption
that the legs are in free motion(i.e.,
Kokotovic,
P.V.,
Morse,
A.S.,
"’Systematic
Design
of
= 0). Considerthe simplified plant modelwith
AdaptiveControllersfor Feedback
Linearizable
scalar parameters
a andb. Let the true valuesof
Systems",
IEEE
Transaction
on
Automatic
Control,vol.
=
these parameters(which are unknown)be Pl
36, 1991,pp. 1241-1253.
0.25andP2= 1.0. Let the control objectivesbe to
track the referencetrajectory xd generated
by
[Li andSastry1989]Li, X., Sastry,S.S.,"’A Unified
cyclically switchingbetween
two reference
Approach
for the Controlof Multifingered
Robot
modelsR1 and R
2.
Hands,"Contemporary
Mathematics,
97:217-239,
In the actual simulation,both reference
1989.
modelsare forced vander Pol oscillators with
different parameters
(andhencedifferent limit
[Narendra1989][Narendra1989]Narendra,K.S.,
cycles). Control designandcontrol parameter
Annaswamy,
A.M., StableAdaptiveSystems,
Prentice
updatelaws werederivedfor the casewhenall of
Hall, 1989.
the states werefully accessible.Controldesigns
wereperformedusing the designprinciples
[Seto 1994]Seto,D., Annaswamy,
A., Baillieul, J.,
providedin [Narendra1989]and[Seto 1994],
"Adaptivecontrol of nonlinearsystemswith trian[Kanellakopoulos
1991]. FromFigure5, it is
gular structure," IEEETransactionsof Automatic
evidentthat the controller is ableto quicklyadapt Control, 1994.
as the referencemodelsswitch. Thereis,
however,sometransient error every time a switch [Shastri 1997]Shastri, S.V., "A biologically consisoccurs. Figure4(a) showsthe modelfollowing
tent modelof leggedlocomotiongaits," Biological
performance,while Figure 4(b) showsthe
Cybernetics,1997.
tracking errors. Figures4(c) and4(d) show
parameterlearning performance.
wherex ~ Rn is the state of the system(joint
V. Ongoing Systems Theory Research
Weare currently derivingformally results related
to proof of stability for systems
that are placed
undermultiple modelreferencecontrol. Ouranalytical workconsiderslinear systems
first, and
174
I
"~’/
I-
0
[
~"2
L
200
600
800
1000
1200
I
I
J
I
I
I
I~
’,
400
600
800
1000
I
I
I
I
I ’’-~"
%
400
200
I
200
I
400
600
I
800
1000
1400
I
I
I
1600
1400
I
I
1400
1600
Figure 5: AdaptiveMultiple ModelReferenceControl Simulations
175
1800
I
1200
1200
1600
/J
J
1800