Table Of ContentBiologically Inspired Kinematic Synergies Provide
a New Paradigm for Balance Control of Humanoid
Robots
Helmut Hauser#1, Gerhard Neumann #2, Auke J. Ijspeert ∗3, Wolfgang Maass#4
#Institute for Theoretical Computer Science
Graz University of Technology
Inffeldgasse 16b/I, A-8010 Graz, Austria
1,2,4{hauser,neumann,maass}@igi.tugraz.at
∗Ecole Polytechnique Fe´de´rale de Lausanne (EPFL)
School of Computer and Communication Sciences
Station 14, CH-1015 Lausanne, Switzerland
[email protected]
Abstract—Nature has developed methods for controlling the addition during development, but also through injuries. But
movements of organisms with many degrees of freedom which the available experimental data suggest that biological organ-
differ strongly from existing approaches for balance control
isms employ a radically different strategy for controlling a
in humanoid robots: Biological organisms employ kinematic
movementapparatuswith manyDoF, inparticularforbalance
synergies that simultaneously engage many joints, and which
areapparentlydesignedinsuchawaythattheirsuperpositionis control. Numerous studies from the Lab of Bizzi at MIT
approximately linear. We show in this article that this control [4], [5], [6] have shown that the central nervous systems
strategy can in principle also be applied to balance control of a variety of organisms employ a modular architecture
of humanoid robots. In contrast to existing approaches, this
for motor control, whereby many different movements (arm
controlstrategyreducestheneedtocarryoutcomplexcomputa-
movements,walking,jumping,swimming)can beconstructed
tions in real time (replacing the iterated solution of quadratic
optimization problems by a simple linear controller), and it as largely linear (but non-negative) combinations of a rather
does not require knowledge of a dynamic model of the robot. small repertoire of movement primitives (also referred to as
Therefore it can handle unforeseen changes in the dynamics of muscle synergies, or kinematic synergies; we use the latter
therobotthatmayariseforexamplefromwindorotherexternal
term in this article). Figure 4 in [6] shows that kinematic
forces. We demonstrate the feasibility of this novel approach to
synergies in frogs engage largely disjoint sets of muscles,
humanoid balance control through simulations of the humanoid
robot HOAP-2 for tasks that require balance control under which may facilitate an approximate linearization of move-
disturbances by unknown external forces. ment control through non-negative linear combinations of
these kinematic synergies. Also recent work on whole-body
I. INTRODUCTION movements of humans [7], [8] shows that balance control
Balance control for humanoid robots is known to be a and other human body movements during standing can be
difficult problem because of the large number of degrees understoodascombinationsofaverysmallsetofstereotypical
of freedom (DoF) that are involved. It has been simplified kinematic synergies, that each affect several joints.
through the discovery that it suffices for balance control to We explore in this article the question whether an analo-
keep the center of pressure (CoP) (or related points on the gousmodularizationandsimplificationstrategymightprovide
ground[1]) withinthe 2-dimensionalsupportpolygondefined a viable alternative to existing balance control methods in
bytheconvexhullofthosepointswherethefeetaresupported humanoid robots. The first problem that one faces here is the
by the ground. Powerful computational methods have been designofsuitablekinematicsynergies.Inbiologicalorganisms
developed for achieving this (see e.g. [2], [3]) by solving they are assumed to result from a combination of genetic
suitable optimizationproblems.While [2] managesto achieve encoding and developmental learning. For humanoid robots
in this way even online balance compensation (by solving one could apply PCA to recordings of the movements of
a quadratic programming problem), these approaches require multiple joints of humans(this is the way in which kinematic
knowledge of the precise dynamic model of the robot, and synergies were extracted in [7], [8]) or one can use an even
therefore cannot be applied in situations where the dynamic simpler strategy that we describe in section (III), and apply
model of the robot may change online, due to wind, picking to the humanoid robot HOAP-2. We show in section (IV)
up loads, contact with other robots, or other external forces. how the resulting two kinematic synergies for moving the
Biological organisms face similar problems, in fact, the CoP left-right and forward-backwards can be used to design
dynamic model of their movement apparatus changes in a control loop with a linear controller for balance control of
qKS ∈Rm q∈Rn
the HOAP-2. We show in section (V) that a simple linear
controller is able to compensate for random movements of
Φ(s) ϕ(qKS) f(q)
a surfboard on which the robot stands, even in the presence
of unforeseeable online changes in the dynamic model of the
robot and its environment. Additionally we demonstrate that
the proposed framework can be used to follow any desired s∈R y ∈R
trajectory for the CoP.
linearity
II. FORMAL DEFINITION OFKINEMATICSYNERGIES
We regard a kinematic synergy as a fixed number m of Fig. 1. Scheme forthecomposition ofthe functions ϕand f according to
(2)and(3)withthekinematic synergies Φ.
degrees of freedom (joint angles) which cooperate under
the regime of a one dimensional input parameter s. It can
be represented as a nonlinear function Φ depending on one
Definition 2: Alinearizingkinematicsynergyisakinematic
controlling parameter s∈R.
synergyaccordingto(1)suchthatthereexistsalinearrelation-
shipbetweenitscontrollingparametersandthecorresponding
Definition 1: A kinematic synergy (KS) is a function Φ:=
(to be controlled) output y.
Φ(s) which maps a parameter s ∈ R onto a m dimensional
vector of joint angles qKS =Φ(s): y =(f ◦ϕ◦Φ)(s) = k·s (4)
forsomek ∈ R .
Φ: R→Rm . (1)
We restrict our attention in this article to such linearizing
The superscript KS denotes the subset of m joints of the KS, to which we simply refer as KS.
robot, which are controlled by the KS. The total number of In practice we deal with dynamics, and we want to control
joints in the robot is denoted by n. Further we define the the robot to accomplish dynamical tasks. Let us take a closer
function ϕ lookatwhyitisstillanadvantagetousekinematicsynergies:
ϕ: Rm →Rn (2) Let q ∈ Rn be the positions (angles), q˙ ∈ Rn be the
velocities and q¨ ∈ Rn be the accelerations of all n joints
which embeds the m-dimensional subspace spanned by Φ
of the robot. Then the state of the whole robot is defined
into the n-dimensional space of all joints of the robot. This
as p = [qT,q˙T,q¨T]T and therefore p ∈ R3n. Note that in
embedding copies the positions for all joints affected by Φ
generalanydynamicaltaskwillincludethewholestatespace.
and leaves the remaining joints constant.
TheKSjustcontrolsm ofnjointsandonlymapsonthe joint
A KS is typically applied in order to control a low-
positions qKS ∈ Rm. By definitions (1) and (2) we restrict
dimensional, or even one-dimensional, variable y. In this
the joint space of m joints. Loosely speaking we just allow
article we use two KS’s that each control one dimension of
defined postures for the robot. This makes sense since most
the center of mass projected to the ground (PCoM). Since
of the possible postures are not valid due to stability issues,
the PCoM depends on the posture of the whole robot, the
crossing of limbs, etc. . If one assumes that the i = n−m
outputy dependsonallnjointpositionsq∈Rn oftherobot.
joints which are not dominated by Φ do not move at all,
Generallyspeakingwe have a nonlinearfunction f depending
then we can neglect the embedding ϕ for the subsequent
on n degrees of freedom
derivations. Any change ds := s˙ of the control parameter
dt
f : Rn →R (3) causes a corresponding change of joint positions q˙KS
∂Φ ∂Φ ∂s ∂Φ
and we want the KS to control its corresponding output q˙KS = = · = ·s˙ .
y =f(q). ∂t ∂s ∂t ∂s
Figure1showsthemappingofΦfromtheonedimensional The same holds for the acceleration d2s :=s¨
s-spacetothem-dimensionaljointspace,theembeddingwith dt2
the function ϕ and the mapping of function f from the n- ∂2Φ ∂2Φ ∂Φ
q¨KS = = ·(s˙)2+ ·s¨.
dimensional space of all joints to the one dimensional output ∂t2 ∂s2 ∂s
y. Therefore the velocities q˙ and the accelerations q¨ of the
Since such KS affects m degrees of freedom that depend jointsarealsorestrictedtoasubspace.Obviouslythisstrategy
just ona one dimensionalparameter s, we can imposefurther facilitates the control of the system.
constraints on the function Φ. A reasonable choice for such
a constraint is a linear relationship between the controlling III. APPLICATION TOTHEHUMANOID ROBOT HOAP-2
parameter s and its corresponding output y. This will reduce We now test whether kinematic synergies provide an al-
the nonlinearities inherent to any kinematic chain and hereby ternative to existing methods (see e.g. [2], [3]) for balance
facilitate controlling and learning. Hence we are particularly control of the humanoid robot HOAP-2 [9]. This robot has
interested in the following type of KS: n = 25 degrees of freedom (rotational joints). It’s structure
andthepCoParelikelytomovewithin(andpossiblybeyond)
the support polygon when the platform tilts. We will assume
thatthefeetoftherobotdonotslip,andhencewecanusethe
samestabilitycriterionasforthecaseofahorizontalplatform.
L hip 1 The robot does not fall over as long as its pCoP stays within
L hip 3 R hip 1 thesupportpolygon(evenifthesupportpolygonisnowtilted).
L hip 2 R hip 2
Indefinition(2)wedefinedtheoutputtobeonedimensional
L knee R hip 3 (y ∈ R). Since dynamic stability points correspond to points
onthecontactsurface,andarethereforetwodimensional,itis
R knee
L ankle 1 naturaltodefinetwoKS’sΦ andΦ forcorrespondingoutput
L ankle 2 x z
R ankle 2 dimensionsxandz.Theoutputfunctionsf andf correspond
R ankle 1 x z
to the two dimensions x and z of the PCoM. The kinematic
(a) (b) synergy Φx controls the movement forward/backward and
the kinematic synergy Φ left/right relative to the robot (see
z
forward Figure 2(c)).
x The number of degrees of freedom controlled by the kine-
support polygon
border matic synergies is m = 12: three hip joints, one knee joint
and two ankle joints for both legs (see Figure 2(b)).
Sincen>m,notalljointsoftherobotarecontrolledbythe
z
left right kinematicsynergies.Theadditionali=n−m jointscouldbe
left foot foot right used for othertasks (grasping,waving, tracking objects, etc.).
Their movement will change the outputs y and y to some
x z
extent,butwillbeinterpretedbythekinematicsynergiesasan
touch sensors external perturbation. We will demonstrate in one experiment
(see section V) that our approach is capable to deal with this
backward
type of disturbances too.
(c)
A. Obtaining Motion Primitives with Inverse Kinematics
Fig. 2. (a) The real HOAP-2 robot and (b) its schematic structure. Red The desiredkinematicsynergiesare constructedvia inverse
markedandlabeledjointrotationaxesareusedforΦ.(c)Supportpolygonon
kinematics. They are constructed just once for the robot and
thesupportsurfacefortherobot,withtouchsensorsthatareusedtoestimate
thepseudocenterofpressure(pCoP).Blackarrowsindicatethexdimension are fixed during control action. With the desired linearization
(forward/backward: range 9.5 cm) and z dimension (left/right: range: 14.3 property (4) one still has an infinite number of possible
cm)formovements ofthecenterofpressure.
kinematic synergies. Therefore their construction can satisfy
extra constraints which depend on the set of tasks the KS
should accomplish. For our balancing task we want to assure
is shown in Figure 2(b). The task is to balance dynamically upright posture for the upper body and double support (both
theHOAP-2againstexternalperturbations.Thereforeanatural feet have contact with the ground).
choicefortheoutputfunctionf couldbeanydynamicstability Based on the given kinematic model and the mass distri-
pointlikethecenterofpressure(CoP),thezeromomentpoint bution of the HOAP-2 the kinematic synergies Φ and Φ
x z
(ZMP)orthefootrotationindicator(FRI)[1],[10].Wewantto have been calculated as follows: An initial posture q has
init
controlthe humanoidrobot standing in double supportphase. beendefinedwhichresults(forthecaseofahorizontalsupport
In practice these stability points are estimated by pressure surface) in a PCoM at the center of the support polygon S.
sensors. HOAP-2 has four such sensors per foot, located at By definition we set the origin of the coordinate system to
the corners (see Figure 2(c)). We call the dynamic stability the center of the support polygon and therefore the resulting
pointthat one computesfrom the outputsof these sensors the outputsintheinitialpostureare f (q )=f (q )=0.To
x init z init
pseudo center of pressure (pCoP). The goal is to keep the reflectthenaturallimitofstaticstabilitytheKSparametersis
pCoP within the supportpolygonshownin Figure2(c). Since normalizedsuch,that−1and+1correspondtothebordersof
a KS presents a static mapping we are going to use the static the supportpolygon S. Thereforethe regionof acting without
model (i.e. kinematic model and mass distribution) and the fallingwillbe(forthecaseofahorizontalsupportsurface)the
PCoM to obtain the KS. Note at zero joint velocities q˙ and range s ∈ [−1,+1] for both dimensions x and z, see Figure
zerojointaccelerationsq¨ thePCoMcoincideswiththepCoP. 2(c).
In one of oursubsequentlydiscussed experimentsthe robot The construction of each KS consist of two optimization
standsonaplatform.Ifthisplatformtilts,thesupportpolygon steps,whichwedescribeonlyforΦ .Similarstepsleadtothe
x
moves with the platform, but the projection of the CoM and secondkinematicsynergyΦ .Wedividedarbitrarilytherange
z
the pCoP is no longerorthogonalto the support polygon(but of the s -parameter over the support polygon into 80 points.
x
orthogonal to some ideal horizontal plane). Hence the PCoM Thereforeeach step was 9.5cm ≈0.12 cm and correspondto
80
a step of ∆sx = 0.025. From the starting posture qinit the Fx(forward / backward) [rad] FZ(left / right)
nextposture q′ is calculatedto geta desiredoutputyx′, which Lhip 1
is one step (0.12 cm) away from the origin of the coordinate +0.3
Lhip 2
system in the x-direction.
Lhip 3
The first optimization step, which used all 12 joints of the +0.2
Lknee
legs, was used to move the PCoM of the robot in the x-
direction and simultaneously tried to keep the upper part of Lankle 1 +0.1
the body in an uprightposition. The coordinatesystem of the Lankle 2
0.0
left leg was always used as origin, all positions and rotations R hip 1
of the remaining body parts were calculated relatively to the
R hip 2
-0.1
left leg’s coordinateframe. In orderto calculate the new joint
R hip 3
positions, a Jacobian Pseudo-Inverseapproachwas used [11].
The applied Jacobian matrix consisted of two 3 × n sub- R knee -0.2
matrices, the Jacobian for the CoM-Position and the Jacobian R ankle 1
-0.3
for the rotation of the torso. Therefore, both constraints, R ankle 2
movingtheCoMinthedesireddirectionandkeepingtheupper -1 -0.5 0.0 +0.5 +1 -1 -0.5 0.0 +0.5 +1
s s
part of the body upright were fullfilled by the optimization. x z
The coordinate frame of the left foot was kept constant, but
the position of the right foot tended to change, so that it was Fig. 3. Kinematic synergies Φx and Φz for the HOAP-2 that move the
PCoM forward/backward and left/right over the range [-1,+1] for the KS-
likely to loose ground contact. parameters sx andsz.NotethatthesetwoKS’sprimarilyaffectdisjointsets
Therefore the second optimization step was used to move the ofjoints.
rightfootback into its originalposition. Forthis optimization
onlythe6jointsrelevantfortherightfootwereusedandagain
8 8
a Jacobian Pseudo-Inverse approach was applied. PCoMZ PCoMZ
These two steps were iterated until the output value y′ was 4 PCoMX 4 PCoMX
x
nreeawchsetda.rtTinhge preosiunltt.inTghneewnexptosptousreturqe′ qw′a′swthaesnfoseutntdobbye tthhee [cm] 0 [cm] 0
same procedure. This approach was applied iteratively until −4 −4
the output y reached the borderof the support polygon. The
x
same procedurewas appliedforthe opposite direction(i.e. s −−81 −0.5 0 0.5 1 −−81 −0.5 0 0.5 1
x s s
from 0 to −1). The process leads to a look up table for the x z
range s ∈ [−1,+1]. Joint positions in between the steps, if (a) (b)
x
needed, are calculated by linear interpolation.
Figure 3 shows the kinematic synergies Φ and Φ which Fig.4. (a)TheKSΦxmapsthecontrolvariablesxlinearlyontothex-and
x z
z-coordinates of the PCoM. It strongly affects the x-coordinates, but keeps
result from this procedure. One can clearly see that (anal-
the x-coordinates ofthe PCoM approximately constant. (b)Thesameholds
ogously as their biological prototypes, see Figure 4 in [6]) forthesecond KSΦz.
these two KS’s primarily affect disjoint set of joints, thereby
supporting linearity of superpositions. Figure 4(a) shows the
mapping from the parameter s that controls the KS Φ to are the cases when the PCoM is moved into a corner of the
x x
the coordinates y = PCoM and y = PCoM of the support polygon) the deviation from linearity is quite small.
x x z z
PCoM. Note the linear relationship, and that the other output
IV. BALANCE CONTROL WITHKINEMATICSYNERGIES
dimension y = PCoM is unaffected by s . Figure 4(b)
z z x
shows the same for the KS Φz. The kinematic synergies Φx and Φz were constructed
Since we have two kinematic synergies which are mapping to control the output function PCoM. However the robot
to the same 12 joints, we are not able to use joint positions can estimate from its pressure sensors in the feet only the
directly for the encoding of each KS. Therefore only joint pCoP, which is also affected by the dynamics of the robot.
offsets ∆qKS from the initial position q are stored in the Similarly, values from simulated pressure sensors are used in
init
lookuptable.Whenbothkinematicsynergiesareapplied,the our simulations of the HOAP-2 to estimate the PCoM.
resulting target joint positions are calculated by summing up Assumption: The robot moves sufficiently slowly so that
the initial position and the two offsets at every single joint. one can assume that approximately
This is possible since the joints mainly responsible for the pCoP ≈PCoM . (5)
movement in x-direction are orthogonal to the joints mainly
responsible for the z-direction(see Figure 3). Figure 5 shows We will show in section V that external forces that change
the results of an empirical evaluation of the approximate on a time scale of seconds, such as wind, can be handled
linearityof the KS’sΦ andΦ . Exceptforcaseswhere both by movements of the HOAP-2 which satisfy this assumption.
x z
of the variables s and s reach their extremal values (these With this assumption we are able to control the humanoid
x z
Error Function P(z)= K d
z−1
[cm]
y˜x=0 ex s˙x sx pCoPx
1.4 _ PID R Φx Robot
y
1.0 x
left
0.6 Fig. 6. Closed control loop for one kinematic synergy Φx. The reference
pointissettoy˜x=0sincewewanttohavethepCoPx atthecenter ofthe
supportpolygon.ExternalperturbationdisachangeofpCoPxnotintroduced
0.2 bythekinematic synergycontrol loop.
rear
following equation for the controller output u :
Fig.5. Empiricalevaluationoftheapproximatelinearityofthesuperposition x
oftheKS’sΦx andΦz.Foreverypoint<yx,yz >inthesupportpolygon de
wpoestauprepliwedhertehethKeSc’soowrditihnatveasluoefstshxe,PsCzoMthaatrewe<reysxu,pypzos>ed(tboasperdodouncethae ux =KP ·ex+KIZ exdt+KD dtx =s˙x (8)
linearity assumption 2 for each KS, and the assumption of linearity of the
superposition).TheEuclideandistancebetween<yx,yz >andtheresulting Figure 6 shows the closed control loop for one kinematic
actual position of the PCoM is shown in color code. One clearly sees the synergy (Φ ). Note that the structure of the control loop is
x
the error caused bynonlinearity is inthe mm-range formostofthe support
a standard feedback control loop which has the property to
polygon,andonlybecomeslargerinthecorners.Thewhitedottedlinesdepict
thecontours ofthefeet. suppress perturbations d.
The formulation of the control loop is very general. The
previously described control scheme is designed to control
robot with simple linear controllers in conjunction with the aroundansetpoint(y˜ =y˜ =0).Ofcourseonecouldusethe
x z
KS’s.Notethatevenifthisassumptiondoesnothold,onecan samecontrollooptomovethepCoPonanydesiredtrajectory
use KS’s for balance control. But then, instead of a simple (timevaryingy˜ (t))asweshowlaterinanexperiment.This
x,z
linear controller, a more sophisticated controller has to be isusefulforexampletoinitiateawalkingcycle.Therobothas
applied. to move the pCoP under the future supporting foot in order
WenowshowhowthekinematicsynergyΦ canbeusedto to be able to raise the other leg without falling.
x
constructalinearcontrolloopforbalancingtherobotinthex-
dimension.The Φ is analogous.If the assumptionholds, the A. Controller Design
z
function from the time derivative s˙x of the KS-parameter sx To find appropriate values for the PID-controller (8) we
to the valuepCoPx ofthe estimatedpCoP inthe x-dimension can use the rich set of tools which are offered by linear
can be sufficiently representedby a lineartransferfunctionof control theory. The values K = 15, K = 0 and K =
P I D
the form: 0.0001 we used, were empirically found to have a reasonable
P(z)= K with K ∈R+ , (6) performance. There exist of course a number of possible
(z−1) improvements to get better controllers: adaptive control or
robust control approaches, optimal control or different trial
with z being the time shift operatorfor discrete systems [12].
and error approaches to find good control parameters. Even
Thedenominatorpolynomialrepresentsanintegrator(onepole
higher order controllers or different control structures than
at z = +1) which integrates the velocity s˙ of the KS-
x in Figure 6 are possible. Here we just present a very simple
parameter to obtain s .
x and straightforward implementation with a PID controller to
By the assumption this approach has a natural limit, but as illustrate the new strategiesforbalancecontrolwhich become
long as the dynamical effects are small enough they can be feasible through the use of kinematic synergies.
seen as uncertainties in the linear model. Already a simple
linear controller can deal with these small uncertainties. To B. Balance Control Task
get a closed control loop with feedback we define a feedback
We consider three tasks. First we investigate the capability
error
oftheapproachunderstaticconditions.Secondwebalancethe
e :=y˜ −y (7) humanoidrobotHOAP-2onarandomlymovingplatform(i.e.
x x x
surfboard) with additional external forces (winds). Third we
with y˜x being the desired output value and yx being the demonstrate the capability to follow a general CoP trajectory
correspondingvaluepCoPxthatiscomputedfromthepressure and to deal with movementsof joints not underthe regime of
sensors. The goal is to prevent the robot from falling over. the KS’s.
Therefore the pCoPx should stay close to the center of the A detailed dynamical model of the HOAP-2 robot based on
support polygon S. Since we have defined the center of S at data from the vendor Fujitsu has been implemented in the
the origin, see Figure 2(c), the desired value is y˜x = 0 (and robot simulation software Webots [13] and has been used for
y˜z =0 for the z-dimension). our simulations. The basic simulation time step has been set
Now we can formulate a standard PID-controller with the to 1 ms. The time step for the control has been set to 8 ms.
x-direction z-direction
forward backward left right 10 10
withoutcontrol +10.6◦ −8.6◦ −14.3◦ +14.3◦
W1 W2 W1 W2
withcontrol +20.1◦ −22.3◦ −26.4◦ +26.4◦ 5 5
improvement 89.6% 159.3% 84.6% 84.6% g] g]
e e
d d
TABLEI e [ 0 e [ 0
gl gl
THEFIRSTTWOROWSSHOWTHETILTANGLE(OFTHEPLATFORMON an an
WHICHTHEHOAP-2WASSTANDING)ATWHICHTHEROBOTLOOSESITS −5 −5
BALANCE,BOTHWITHOUTACONTROLLERWHICHCHANGESTHE
POSTUREOFTHEROBOTINORDERTOCOMPENSATEFORTHETILTED −10 −10
0 5 10 15 20 0 5 10 15 20
PLATFORM,ANDWITHTHELINEARCONTROLLERBASEDONKINEMATIC time [sec] time [sec]
ESNYANBELREGSIETSHTEHRAOTBIOSTDTEOSCTROILBEERDAITNESAEBCOTUIOTNTWIVIC.TEHTHEECOTINLTTRAONLGLLEER. (a) tiltangleΘx (b) tiltangleΘz
10 10
UNcontrolled W1
controlled *
5 5
The surfboard can rotate about the x-axis with angle Θ
x
ΘandaanlsdoΘaboaurtethgeenze-raaxteisdwinidtheptehnedaenngtllyefΘorz.eTacrahjedcitmoreinessiofonr. [cm] 0 [cm] 0
x z
These were found by following procedure: We just generated −5 −5
a random trajectory of jumps with random amplitude and W1 W2 UcoNnctoronltlreodlled W2
random durations. In order to get a smooth trajectory we −100 5 10 15 20 −100 5 10 15 20
time [sec] time [sec]
appliedadditionallya discrete smoothingFIR-filterwith three
(c) x-coordinate ofthepCoP (d) z-coordinate ofthe pCoP
poles at 0.997. Typical resulting trajectories are presented in
Figures 7(a) and 7(b).
Since we have two kinematic synergies (Φ and Φ ) 10 10
x z W1 W2 W1 W2
we have two control loops. They react independently from
5 5
each other on their corresponding output dimension x and z
respectively. Both linear controllers calculate, depending on m] 0 m] 0
[c [c
their errors e and e , the velocities s˙ and s˙ of their KS-
x z x z
−5 −5
parameters.Theseareintegratedto s ands ,whichresultvia
x z UNcontrolled * UNcontrolled
the look up table into joint angle offsets. They are summed controlled controlled
−100 5 10 15 20 −100 5 10 15 20
up as described in section (III-A) to get the actual joint target time [sec] time [sec]
positions. Local PD-controllers at the servos transform then (e) errorsignal ex (f) errorsignalfor ez
the desired angles into torques.
1 1
V. RESULTS
0.5 W1 W2 0.5 W1 W2
In a first test of the capability of the linearcontrollerbased
0 0
on KS’s (that was described in the preceding section) we
[] []
tilted the platform on which the HOAP-2 was standing, and −0.5 −0.5
determinedat which tilt anglethe robotfalls over.This tilting
−1 −1
wascarriedoutveryslowlyinthisfirsttest,andseparately for
−1.5 −1.5
the two tilt axes that move the CoP in the x and z direction. 0 5 10 15 20 0 5 10 15 20
time [sec] time [sec]
Table I shows that our control strategy allows the robot to (g) activation sx ofKSΦx (h) activation sz ofKSΦz
tolerate about twice the tilt angle without loosing its balance
– comparedwith a robotwhichdoesnotchangeits posturein Fig.7. Resultfromthetestwithamovingsupportplatform(”surfboard”)and
unexpectedexternalforces(”wind”)W1andW2.ThebalanceoftheHOAP-
an adaptive manner.
2 is controlled by two linear controllers, based on two kinematic synergies.
We next considered the case where the platform on which Without balance control (red dashed line in (c) and (d) shows trajectory
the robot stands is tilted dynamically in random directions of pCoP without active balance control) the pCoP would leave the support
polygonattimepoint16s(inresponsetothewindW2),andtherobotwould
accordingto the stochastic process describedin the preceding
fallover.Withbalancecontrolthestabilityoftherobotismaintainedinspite
section–asonemighthaveforexampleonasurfboard–(see ofunexpected external forces.
Figure 7(a), 7(b)), and the previously described controllers
based on kinematic synergies for the x- and z-direction are
simultaneously active. In addition random external forces thedynamicmodeloftherobotinapplicable.Figure(7)shows
(which might for example arise from wind or contact with the results when an external force W1 of 0, 0, −5 N
other objects) were applied to the CoM (at the torso) of the (a force from the right side) is applied a(cid:2)t the torso of(cid:3)the
robot at various points in time. These external forces change robot during the interval [5s,10s], and another external force
the dynamic model of the robot in an online manner, and W2 of 5, 0, −5 N (a force from the right and the
therefore make control strategies that require knowledge of back) is(cid:2)applied during(cid:3)the interval [15s,20s](these two time
to counteract it. Figures 9(b) (green line) and 9(c) (blue line)
show that the robot is able to follow the desired trajectories
despite the arm movement. The control loops react properly
with sinusoidal movements of the KS-parameter to follow
the desired trajectory and with an offsets to counteract the
disturbance (see 9(d)). The second row of Figure 10 shows a
(a) at3second (b) at7second seriesofscreenshots.Noteatthefinalposition(lastscreenshot
on the right) the robot leans to the right in order to get the
pCoP in the center of the support polygon.
5
5
on [cm] 2.5 m] 2.5
(c) at12second (d) at16second dimensi−2−.50 CoP [cx 0
x− left foot right foot −2.5 dcoensitrreodlled
−5 UNcontrolled
Fig.8. Snapshotsofthepostureofthe(simulated)HOAP-2at4timepoints −5 z−dimen0sion [cm] 5 −50 5 10 15 20 25
during the balancing experiment of Fig. 8. In (b) the wind W1 is blowing time [s]
from the right side of the robot, and the robot is leaning against the wind (a) Desiredtrajectory in2D (b) CoPx trajectory realvs.desired
in order to move its pCoP back into the middle of the support polygon. In
td(hdye)naabmnaocitckh,emraonwddeinlt.dheWro2boist aaslssoumreesdpotondcsompreodpiearglyontaollythifsroomnltihneercighhantgseideinanitds 2205 dcUoeNnscitrroeondllterodlled 0.15 ssxz
time [s]1105 s [] 0
intervalswe shadedin gray).Note the onsetsofthe windsare 5 −0.5
asybsrtuepmt.aTndheretprarjeescetnotryaohfigthhelypdCyonPamisicsahlopwenrtuinrbaFtiigounreto7(tch)e, 0 −5 CoPz0 [cm] 5 −10 5 1t0ime [s1]5 20 25
7(d) for the case of random movements of the platform in (c) CoPz trajectory realvs.desired (d) KS-parameter sx andsz
combination with external forces, both for the case with the
Fig.9. Resultsofthetrajectoryfollowingtask:(a)presentsthedesiredfigure
describedcontroller,andwithout.Withoutcontrollerthe robot eighttrajectoryfrombirdview.Thegrayshadedareasdepictthecontactareas
lost balance after 16s (see black star in Figures 7(c), 7(e)), ofthefeetwiththeground.(b)and(c)presentthe pCoPtrajectories splitted
upintoitstwocorrespondingdimensionsxandzrespectively.Ourapproach
whereas with controller the balance is maintained. This was
is able to follow the desired trajectory (gray;dash-dotted). The red dashed
typicalfortherandommovementsoftheplatformandexternal linesdepictthetrajectoryofthepCoPifnocontrolactionisapplied.(d)The
forces that we have specified in the preceding text. Note that system reacts properly with an offset inthe KS-parameters to cope with the
disturbances introduced bymanipulating theweight.
these external movements and forces were interpreted by the
controlleras externalperturbations d (compareFigure 6), and
wereautomaticallycounteracted.Resultingerrorsignalswhich
VI. DISCUSSION ANDGENERALREMARKS
were fed back in the control loop for both dimensions x and
z can be seen in Figures 7(e) and Figure 7(f) respectively. Wedemonstratedthatbiologicallyinspiredkinematicsyner-
In a third task we demonstrate that the setup as described giesgiverisetointerestingnewcontrolmethodsforhumanoid
can be used to follow any desired trajectory(i.e. time varying robots. We showed in experiments that our proposed setup
y˜ (t)) in contrast to the preceding experiments where the is able to balance the humanoid robot HOAP-2 even under
x,z
robot has been controlled around a set point (y = y = 0). unknown external disturbances. The general strategy is to
x z
The robots stands now on flat ground. It should follow a modularize and approximately linearize the control task for a
desired trajectory of the shape of a figure ”eight” (Figure robotwithmanydegreesoffreedom,sothatsimplecontrollers
9(a)). The desired trajectory (gray;dash-dotted) splitted up in can be applied to the resulting virtual low-dimensional and
its corresponding dimensions x and z can be seen in Figures approximately linear control problem. Linear control theory
9(b) and 9(c) respectively.Additionally the robotmanipulates provides a large set of tools for improving the performance.
a heavy weight(20% of the robot’sweight) with his left arm. One can also target the remaining nonlinearity and apply
The trajectory of the extended left arm over the simulation nonlinear control techniques [14].
time of is shown in the first row of Figure 10. Note the arm The use of kinematic synergies of the type that we have
jointsarenotundertheregimeoftheKS’sbuttheirmovement discussed in this article may have an additional benefit for
change the pCoP. This can be clearly seen in an offset of the the control of humanoid robots, that needs to be explored
red dashed line in Figures 9(b) and 9(c) (when no control in subsequent research: They make it feasible to apply pow-
action is applied). For the control loop this is a disturbance erful methods from machine learning such as reinforcement
d as any other external force and our proposed setup is able learning [15] to robot control, which typically require a low-
Fig. 10. Screenshots of the simulation for the trajectory following task: The corresponding times are from left to right 0 (start), 12.5 ,15, 17.5, and 25
sec(end). In the first row no control action is applied. The plain arm trajectory of manipulating a heavy weight (20% ofthe robot’s mass)is visible. The
secondrowpresentsaseriesofscreenshotswhencontrolisactive.Noteatthefinalpositiontherobotleanstotherighttocopewiththeheavyweightinhis
extended leftarm.Thisresults inapCoPatthecenterofthesupportpolygon.
dimensional action space in order to escape the curse of [9] HOAP2 Design Specification. [Online]. Available:
dimensionality. http://jp.fujitsu.com/group/automation/downloads/en/services/humanoid-
robot/hoap2/spec.pdf
Another interesting new research direction suggested by
[10] A. Goswami, “Postural stability of biped robots and the foot rotation
the results of this article is the design of suitable kinematic indicator(FRI)point,”TheInternationalJournalofRoboticsResearch,
synergiesforavarietyofmovementtasksofhumanoidrobots. vol.18,no.6,pp.523–533, 1999.
[11] L.Sciavicco andB.Siciliano, Modelling andControlofRobotManip-
Besidesthesimpleandstraightforwarddesignmethodthatwe
ulators,2nded. Springer, 1999.
have applied in this article, one can apply PCA to motion [12] A.V.OppenheimandA.S.Willsky,SignalandSystems. Prentice-Hall
capturedataofhumans.Inadditionaninterestingnewresearch Inc.,Englewood Cliffs,1992.
[13] O.Michel, “Webots:Professional mobile robotsimulation,” Journal of
direction is the design of mathematical optimization methods
AdvancedRobotics Systems,vol.1,pp.39–42,2004.
for the construction of kinematic synergies with particular [14] J.-J.E.SlotineandW.Li,AppliedNonlinearControl,1sted. Prentice
properties. Hall,1991.
[15] R.S.SuttonandA.G.Barto,ReinforcementLearning-AnIntroduction,
1sted. TheMITPress,1998.
ACKNOWLEDGMENT
Written underpartialsupportby the Austrian ScienceFund
FWF project # P17229.
REFERENCES
[1] M.B.Popovic´,A.Goswami,andH.Herr,“GroundReferencePointsin
Legged Locomotion: Definitions, Biological Trajectories and Control
Implications,” International Journal of Robotics Research, vol. 24,
no.12,pp.1013–1032, December2005.
[2] S. Kagami, F. Kanehiro, Y. Tamiya, M. Inaba, and H. Inoue, “Au-
toBalancer: An Online Dynamic Balance Compensation Scheme for
Humanoid Robots,” in Algorithmic and Computational Robotics: New
Directions, B.Donald, K.Lynch, andD.Rus,Eds. A.K.Peters Ltd.,
2001,pp.329–340.
[3] J. J. Kuffner, S. Kagami, K. Nishiwaki, M. Inaba, and H. Inoue,
“Dynamically-stable Motion Planning for Humanoid Robots,” Au-
tonomous Robots (special issue on Humanoid Robotics), vol. 12, pp.
105–118,January2002.
[4] F. A. Mussa-Ivaldi, “Modular features of motor control and learning,”
CurrentOpinioninNeurobiology, vol.9,pp.713–717, 1999.
[5] A.d’Avella,P.Saltiel,andE.Bizzi,“Combinationsofmusclesynergies
in the construction of anatural motor behavior,” Nature, vol. 6,no. 3,
pp.300–308, March2003.
[6] A. d’Avella and E. Bizzi, “Shared and specific muscle synergies in
naturalmotorbehaviors,”ProcNatlAcadSci,vol.102,no.3,pp.3076–
3081,2005.
[7] S. M. S. F. Freitas, M. Duarte, and M. L. Latash, “Two kinematic
synergies in voluntary whole-body movements during standing,” J
Neurophysiol, vol.95,no.2,pp.636–645, Feb2006.
[8] V. Tricon, A. L. Pellec-Muller, N. Martin, S. Mesure, J.-P. Azulay,
and S. Vernazza-Martin, “Balance control and adaptation of kinematic
synergy in aging adults during forward trunk bending.” Neuroscience
Letter, vol.415,no.1,pp.81–86,Mar2007.
