2008 8th IEEE-RAS International Conference on Humanoid RobotsDecember 1 -- 3, 2008 / Daejeon, Korea TP1-10

Kinematic and dynamic analogies between planar biped robotsand the reaction mass pendulum (RMP) model

Ambarish Goswami

Honda Research Institute, Mountain Viel-v, CA 94041, U.S.A.agoswami@honda-ri.com

Abstract-In order to simplify dynamic analysis, humanoidrobots are often abstracted with various versions of the invertedpendulum model. However, most of these models do not explicitlycharacterize the robot's rotational inertia, a critical componentof its dynamics, and especially of its balance. To remedy this, wehave earlier introduced the Reaction Mass Pendulum (RMP), anextension of the inverted pendulum, which models the rotationalinertia and angular momentum of a robot through its centroidalcomposite rigid body (CCRB) inertia. However, we presented onlythe kinematic mapping between a robot and it~ correspondingRMP.

Focussing in-depth on planar mechanisms, here we derivethe dynamic equations of the RMP and explicitly compute theparameters that it must possess in order to establish equivalencewith planar compass gait robot. In particular, we show that, a)an angular momentum equality between the robot and RMP doesnot necessarily guarantee kinetic energy equality, and b) a cyclicrobot gait may not result in a cyclic RMP movement.

The work raises the broader question of how quantitativelysimilar the simpler models of humanoid robot must be in orderfor them to be of practical use.

I. BACKGROUND AND MOTIVATION

Humanoid gait is often modeled with various versionsof the inverted pendulum, such as the 20 and 3D linearinverted pendulums (LIP)[14], [13], the cart-table model[l2],the variable impedance LIP[26], the spring-loaded invertedpendulum[2], and the angular momentum pendulum model(AMPM)[I5], [16]. These reduced biped models have beenvery beneficial in the analysis and control of human andhumanoid gait. The inverted pendulum models allow us toignore the movements of the individual limbs of the humanoidand instead focus on two points of fundamental importance ­the center of pressure (CoP) and the center of mass (CoM) ­and the line joining them.

A limitation of the above models is that they representthe entire humanoid body only as a point mass and donot characterize the significant rotational inertia (except [15],[16]). The rotational inertia and the associated angular mo­mentum are important components of humanoid movementand especially of balance, as have been reported [1], [9]).Direct manipulation of angular and linear momentum has beensuggested as a reasonable, and sometimes preferable, way tocontrol a robot [10], [11], [27].

Reaction Mass Pendulum: In order to explicitly model therotational inertia and angular momentum of a humanoid wehave earlier proposed the Reaction Mass Pendulum (RMP)model[18]. An RMP is essentially an inverted pendulum with

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3D Planar

Fig. 1. Conceptual mechanical realization of the 3D and planar RMP. TheRMP consists of a massless leg connecting the CoP and CoM. and a rotatingbody. In 3D. the rotating ellipsoidal body is mechanically equivalent to sixequal masses on three mutually perpendicular rotating tracks. The body of aplanar RMP reduces to a wheel and is equivalent to a pair of equal masseson a linear track. The CoM of the RMP is fixed at the common mid point ofthe track(s). This is the point about which the tracks themselves can rotate.

its point mass replaced by an actuated extended-body mass thatcan rotate about the robot CoM. Fig. 1 shows the mechanicalmodels of a 3D and a planar RMP. The RMP body masscaptures the centroidal composite rigid body (CCRB) inertiaof the robot, which is the instantaneous generalized inertiaof the entire robot projected at its CoM. Additionally, fora particular rotational velocity of this extended mass, thecentroidal angular momentum of the RMP equals to that ofthe robot.

The movement of a humanoid robot results in a correspond­ing movement of the RMP. The change in robot's CCRBinertia is reflected in 3D by the changing shape, size andorientation of the ellipsoidal reaction mass, see Fig. 2. In plane,the radius of the rotational velocity of the inertia wheel willcontinuously change.

Fig. 2. As the humanoid moves, so does its RMP. Depending on the particulargait, the length of the RMP leg as well as the shape, size and orientation ofthe 3D reaction mass ellipsoid changes.

978-1-4244-2822-9/08/$25.00 ©2008 IEEE

H(q)q+C(q,q)q+Tg(q)==T, (5)

Fig. 3. Compass gait biped robot model. See Section II for description. Thethree masses are located at the vertices of an isosceles triangle (right top).

(1)

(2)

A

m B

x

h

y

\. ~ B~/

.",,-,,~"" ..,'

~/' .. "

~//

,//:,/ ~-.;:-'

.,(

B. Dynalnics

The CoM coordinates (xc R, Yc R) of the robot are given by,

where p == (/1'{3' - 1) such that (mHI + ma + ml) == mbp.The governing equations of the robot consist of nonlinear

differential equations for the swing stage and algebraic equa­tions for the instantaneous ground impact. The swing stageequations can be derived using Lagrange's method startingfrom the potential and kinetic energies of the robot, PER andK E R , respectively,

PER = bmg (p cos (}1 - cos (}2) (3)

K E R == O.5m b2[( O~ - 2{3' cos 20 O} O2 + O~] (4)

where, ( == (/1' (3,2 - 2{3 - 1). The swing stage equations,which are similar to those of a frictionless double pendulum,can be written as[8]:

{3'b. We use the nominal values of /1 == 2 and {3 == 1. Note thatthe robot's geometry is completely determined by its interlegangle 20, where 2a == (0 1 - ( 2 ). In this paper we will makefrequent use of the parameters /1' and {3'.

II. COMPASS GAIT BIPED

The biped robot model used in this paper is the so-calledcompass-gait model[8], see Fig. 3. The model has a doublependulum geometry and consists of two kneeless legs eachhaving a point mass, and a third point mass is coincident withthe hip joint. Despite its simplicity, this robot may exhibitautonomous "dynamic gait" [19] on a descending slope, evenwhen the joints are unpowered. Researchers continue to ac­tively study this model and suggest control laws of increasingcomplexity, depth and elegance[4], [3], [24].

The robot gait consists of successive stages of swing andground impact. The robot can be powered through one motor atthe hip and another at the ground contact point of the stanceleg ("stance ankle"). After briefly reviewing the kinematicsand dynamics of compass gait robot, which are known, wewill introduce the expressions for the aggregate moment ofinertia, the angular momentum and the momentum matrix ofthis model, all defined at the CoM.

Focus of this paper: Tn our previous work we only presentedthe technique of kinematic mapping necessary to generate theRMP model starting from a robot at a given configuration.In this paper, we additionally derive the dynamic and energyparameters of a planar RMP and compare them with a planarrobot. The decision to focus on planar robot models, andin particular the compass gait model, is to take advantagesof analytical exploration. Many of the present findings are,nevertheless, valid for 3D models.

The compari son between humanoid robots and their RMPcounterparts brings us to the broader topic of the extent ofvalidity of reduced models in the dynamic analysis and controlof humanoids. A humanoid controller based on a reducedmodel assumes that the humanoid has the same dynamics asthat of the reduced model. The differences between the robotdynamics, which is substantially more complex than that ofthe reduced model, is treated as an error and compensated bythe controller. If the robot behavior is reasonably captured bythe reduced model, the discrepancy between their dynamics isnot dramatic, and the compensation is generally successful.

What makes a dynamic model an acceptable reduced modelof a more complex system? Are there criteria that the reducedmodel must satisfy for it to be practically admissible? To ourknowledge, a comprehensive answer to this question has notemerged and each case is treated individually. While we donot seek to address this question in this paper, the process ofa thorough comparison of two dynamic systems puts us in theright reference frame to pose such a question.

A. Kinematics

The two legs of the robot are identical. The robot has threepoint masses, one at hip (mH) and one on each leg (Tn). Theleg masses are located at distances a and b, respectively, fromthe leg tip and the hip. The total mass of the robot is M ==2m+mh and the mass ratio is defined as J-L == r~:: .Therefore,M == (2 + J-L)m == J-L'm. The leg length is I == a + b and thelength ratio is defined as {3 == %' resulting in I == ({3 + l)b ==

where q == [01 02]T, H(q) is the 2 x 2 inertia matrix, C(q, q)is a 2 x 2 matrix with the centrifugal coefficients, and T 9 (q)is a 2 x 1 vector of gravitational torques at the joints. T is a2 x 1 vector of joint torques J•

IThe form of Eq. 5 is valid for both the robot and the RMP. We will usesubscript R to denote the robot and p to denote the planar RMP. H RandH p thus means H (q) for the robot and the RMP, respectively.

183

C. Centroidal composite rigid body (CCRB) inertia

Instead of using general equations for deriving the CCRBinertia2 of the compass gait robot[18], [21], we exploit thesimplicity of the planar robot and adopt a more direct method.At any given configuration, the spatial arrangement of thethree masses of the compass gait robot can be representedon an isosceles triangle ABC, as seen in Fig. 3. The globalorientation of the triangle is given by 01, the support leg angleof the robot.

The CCRB inertia of the three point masses is I C R

m(BGR)2 + m(CGR)2 + mH(AGR)2. We can show,

Ic == 2mb2 [1 _ 2 cos20] (6)

R ~'

where we have used the fact, AGR == AD - GRD == ~ cos aJ..t

and CGR = BGR = bJ1 - 4(~;;1-') cos2 0:.

For a planar robot the centroidal moment of inertia is ascalar. There is no inherent way to visualize it, unlike in case of3D robots, which has three perpendicular axes naturally givenby the eigenvectors of the centroidal inertia ellipsoid[18]. Themaximum value of ICR is 2mb2 which occurs when ° == 900

•

III. PLANAR REACTION MASS PENDULUM (RMP) MODEL

The planar RMP is dynamically analogous to the reactionwheel pendulum [5], [20], [25], except that the RMP has avariable CCRB whereas the wheel's inertia is constant3. Atany given configuration of the RMM, its CCRB inertia can berepresented by the pair of point masses mp • The generalizedcoordinates and generalized forces of the planar RMP areqp == (0,0, rz, rp) and T p == (T(), Iz, Ta , Ir), respectively(refer to Fig. I). When rp == 0, the planar RMP reduces toa traditional inverted pendulum. If, in addition, we constrainthe CoM to stay at a constant horizontal level, we obtain theplanar LIP.

A. Dynamic Equations

The motion equations of the planar RMP can be obtainedstarting from its potential and kinetic energies, PEp andKEp .4 These are given by

PEp == 2mpghp == 2mpglp sinOp (II)

KEp == mp(i~ + r~ + l~i;~ + r~a~) (12)

We can derive the dynamic equations as follows:

(10)

3Reaction wheel pendulum consists of an actuated spinning wheel pin­jointed to a rigid rod. A spinning reaction wheel or an inertia wheel has beensuggested as a means of stabilizing the lateral biped dynamics[l7], [23].

4Note that KEp does not contain any coupled terms of the form Opopand ipop . Neither does it depend on Op and ap, rendering them cyclicvariables([29], pp. 426).

5Angular momentum rate change referred to at the CoP ko = OGp x M 9from which follows ko = kcp + OGp x MOGp.

(14)

whereas its linear momentum is expressed as,

XC p == ip cos Op,

B. Linear and angular momenta

The CoM location of the RMP is given by,

lp == 2mp[(ipcosOp -lpiJpsinOp)i

+ (ip sin Op + IpiJ p cos Op )]] (15)

The CCRB inertia of the RMP has the compact form,

Icp == 2 mp r~ (16)

from which the centroidal angular momentum and its deriva­tiveS can be computed as,

kCp==Icpap==2mpr~ap (17)

kcp == 2mprp(rpiip + 2rpap). (18)

D. Centroidal momentun and centroidal momentum matrix

The linear momentum I R of the robot is computed bymultiplying the CoM velocity with the aggregate mass,

IR == jim(xcR i + iJCR]) (7)

The centroidal angular momentum kCR - the robot's angu­lar momentum referenced to its CoM - is the sum of angularmomenta of its individual mass elements, after projecting eachto the robot's CoM[18].

2 [ 2 cos2 0]. .kCR == mb 1 - -~-,- (01 + (2 ), (8)

A a R ( q ), the centroidal momentum matrix (CMM) ofthe robot maps its joint velocities to its centroidal angularmomentum[21] according to

hcR==AcR(q)q, (9)

where hCR == [kCR lRx lRy]T is the centroidal spatialmomentum vector. We can write,

[

b(1 - 2C~~2 a) b(1 _ 2Ca;2 a)]A c R == mb pcos ()1 - cos O2 •

-psinOl sin O2

The CMM matrix is recognized as an important matrix inangular momentum control of humanoids[II], [7] and itsstructure and properties are being studied[2 I].

Next we will study the dynamic equations and other phys­ical quantities of importance of the planar RMP.

2The composite rigid body (CRB) inertia was introduced in robotics toobtain efficient computational algorithms for robot dynamics[28]. In physicsand geometric dynamics the concept is called the locked inertia.

184

Similar to Eq. 10 the CMM of the planar RMP defining themapping relationship hG p == A G p (qp ) qp is given by,

Note that kGp is also expressed as, kGp == GpO x R, inwhich the ground reaction force, R, is computed from,

(20)

(19)

2r~ 0]~ ~.

ocosOpsinOp

R == M(OGp - g).

Fig. 4. Schematic diagram showing that planar robots of completely generalkinematics, both humanoid and fixed-based manipulators, can be representedby the same planar RMP model.

5) Centroidal composite rigid body inertia: CCRB inertiacontinuously changes for both the robot and the RMP. For theRMP, the CCRB inertia depends only on rp. In fact, rp canbe computed using the requirement I G R == I G pas,

bv12rp == -,- JJL' - 2 cos2

Q (24)JL

As an illustration, Fig. 5 shows the equivalence between acompass gait robot and the planar RMP.

6) Potential and kinetic energies: Potential energy dependsonly on the total mass and the CoM height. Since the CoMlocation is identical for both the robot and the RMP, itautomatically follows that PER == PEp. For kinetic energy,unfortunately, there is no such guarantee. K E R and K Ep maybe unequal.

Out of the four generalized coordinates of the planar RMP,we have so far determined only three. The fourth, Qp, is

1) Generalized coordinates and forces: The robot has twogeneralized variables (coordinates and forces) whereas theRMP has four. This might appear surprising and counter­productive to the underlying theme of model reduction. How­ever, note that the planar RMP model can represent not justthe compass gait robot considered here, but the entire rangeof planar humanoid robots, as well as planar manipulators nomatter how complex, as schematically shown in Fig. 4.

2) Geometry: While the robot has fixed geometry, thegeometry of RMP is continuously changing. In fact, the twolength parameters, l p and r p of the RMP, are generalizedcoordinates.

3) CoM location: The CoM of the robot and the RMP arerequired to coincide, which specifies the values of lp and Opas

lp = VX~R + Y~R = !!,Jp2 + 1 - 2pcos 2aR (21)JL

Op == arctan 2 (YGR) == arctan 2 [p c~s 01 - c~s (h ] (22)xG R pSln(}l - sln(}2

4) Mass and inertia: The total mass of the robot and theRMP are to be equal, from which we compute mp as,

IV. ESTABLISHING EQUIVALENCE BETWEEN COMPASS

GAIT ROBOT AND PLANAR RMP

The similarities and differences between a compass gaitrobot and a planar RMP are presented in Table. I. In asense, the robot and the RMP are two vastly different systemsgiven by their different sets of generalized coordinates andforces. Yet, by construction, the total mass, CoM position,centroidal moment of inertia and potential energy of the RMPare identical to those of the robot.

TABLE ICOMPARISON BETWEEN COMPASS GAIT BIPED AND PLANAR RMP

Variables and quan- Compass Gait Robot Planar RMPtities

Generalized qR = [e1, e2]T qp =

coordinates [ep, lp, Qp, rp]T

Generalized forces TR = [U1, u2]1' Tp =[TO, II, Ta, Ir]T

Geometry a, b, (b = (3a) None fixed

a+b=l

CoM position Function of qR' x = lp cosep,

see Eqs. 1 and 2 y = lp sinep

Mass m,mH, mp=¥,

(mH=j.lm) (M=2m+mH)

Centroidal moment ICR = Ic p = 2rnpr'~

of inertia (CCRBI) 2mb2[1- 2~Jl cos2 Q] JIGso rp = --E.., 2mp

Potential Energy PER = Mgh, PEp = 2mpghp ,

(hp = h)

Kinetic Energy KER : Eq. 4 KEp : Eq. 12

Linear momentum lR : Eq. 7 lp : Eq. 15

Angular momentum kCR : Eq. 8 kcp : Eq. 17

A. Computation of 0p, lp and r p

In this section we determine the quantities Op, lp and rp,these quantities depend on the joint angles, and not velocitiesof the compass gait robot.

mp == JL' /2 (23)

185

(26)

T T T T TFig. 5. Kinematic equivalence between compass gait robot and planar RM:P.Compass gait robot is shown with light blue mass and grey links in thebackground, its inter-leg angle increasing from 0° to 1800

• Superimposedon each robot configuration is the corresponding RMP. The leg of the RMPextends from the robot's CoP to CoM.

indirectly obtained from angular momentum equality betweenthe robot and the RMP, as we see next.

B. Angular momentum equivalence and computation of ap

To compute the RMP body orientation ap, one has to firstcompute ixp through angular momentum equivalence betweenrobot and RMP, and then integrate it. For compass gait robot,ixp can be obtained from Eq.l7, while setting kGp == kGR,

1ixp == IG~ kGR == -2--2 kGR (25)

mpTp

Next, the expressions for Tp and kG R can be plugged in usingEq. 24 and Eq. 8, respectively. From Eq. 25 we obtain,

. 81 + 82ap == --2-

which can be integrated to obtain a p (h ~(h. To beginintegration, we can arbitrarily set ap(O) == O.

In analogy to the RMP, what does ixp signify for thecompass gait robot? ap together with kGR is nothing butthe rotational counterpart of the linear momentum lRandthe CoM translational velocity. The same way as the CoMvelocity depicts an average translational velocity of a robot,ixp is seen as its fictitious centroidal angular velocity or thewhole-body angular velocity[22]. Furthermore, ap is seen asthe whole-body angular excursion[22] although it cannot bea-priori linked to the geometry or dynamics of the robot.An alternative description of average angular velocity, derivedfrom kinetic energy equivalence, is presented in [6].

The physical interpretation of the centroidal angular velocityof a robot is therefore the following: At any instant, if therobot is replaced by its equivalent CCRB inertia, the angularvelocity with which this inertia must rotate in order to possessthe prescribed angular momentum is the centroidal angularvelocity.

c. Torque mapping between planar RMP and planar biped

We have derived a mapping of generalized coordinatesbetween the compass gait robot and the planar RMP. Acomplete mapping of state variables can be established byobtaining the time derivatives (analytic, in the current case)of the generalized coordinates.

186

The mapping of generalized forces is more involved. Onemust first determine the purpose of such a mapping. As anexample, we explore the case of torque mapping where theobjective is to maintain the equivalence of rate of change ofcentroidal angular momentum between the RMP and the robot.

Taking time derivative of Eq. 9 we obtain, for both robotand RMP,

hG == A G ij + A Gq (27)

from which we extract ij

ij == A~ (hG - AG q). (28)

Plugging ij in the dynamic equations Eq. 5 we obtain,

hG == A GH- 1 (T - Gq - Tg ) + AGq (29)

which is of the form

hG == WT + X (30)

where W == AGH- 1 and X == (AG - WG)q - WT g •

The position and velocity dependent terms are contained inX. Eq. 30 is an affine relationship between joint torques andthe rate of change of centoridal spatial momentum. Settingh GR == hGp, we readily obtain,

WRTR+XR==WpTp+X p (31)

which yields an expression for T p ---* T R mapping

TR == W~ [WPTp + (X p - X R)] (32)

Note that W R is a rectangular matrix and the issues regardingits appropriate inverse need to be resolved.

For the planar RMP, Eq. 29 can be shown to result kGp ==To:. It is precisely because the centroidal angular momentumrate change is explicitly captured by a single joint torque, thatwe have chosen the RMP model.

v. SIMULATION AND DISCUSSION

To support our analytical findings we have used simulationdata of a compass gait biped6• The data contain the kinematicvariables during a few stable walk cycles of a compassgait biped robot on an inclined slope. Using the techniquesdescribed in this paper we have mapped the biped robotdata to the equivalent planar RMP variables. Fig. 6, Fig. 7,and Fig. 8 show the time evolution plots of the generalizedcoordinates, generalized forces and the CCRB inertia of theRMP, respectively. The generalized forces were computedusing Eq.13 through intermediate steps computation steps ofgeneralized accelerations.

Fig. 9 shows the plots of kinetic energies of the robot and theRMP. Although there were no KE-specific restriction placedon the RMP model variables, the kinetic energies of the modelstend to be in the neighborhood of each other.

Fig. 10 presents the plots of both ap and /J;p. The remark­able finding in this plot is that although ap is a cyclic quantity,

6The data was given by Dr. Fumihiko Asano. The bipedal gait was gener­ated in accordance with the method of virtual passive dynamic walking[41.

-0.015

0.1

004, 0h -0.1

r~61C't -0.2

01- -0.3

'..... -0.4 0.02

0.725

0.3

VI. CONCLUSIONS AND FUTURE WORK

We have explored the kinematic, dynamic and energyequivalence between planar biped robots and planar reactionmass pendulum (RMP) model. The RPM model is similar tothe existing inverted pendulum humanoid models, but has anadditional rotational mass to explicitly identify the centroidalrigid body inertia of the robot and, by extension, its centroidalangular momentum.

The RMP is an instantaneous capture of the aggregatekinematics and dynamics of a humanoid robot. We havederived the analytical dynamic equations of the planar RMP.By imposing equivalence between the robot and the RMP, wehave expressed the mass and geometry parameters and the gen­eralized coordinates of the RMP as functions of those of the

0.73

Fig. 11. Centroidal angular momentum of robot and RMP superposed onCoM trajectory. The magnitude of angular momentum is given by the lengthof the line projected outwards from the CoM trajectory.

0.755

0.76

]' 1.5___ 1

~~ 0.5

~ 0~(t -0.5

be~ -1

~~ -1.5

Time (sec)

-20~-----:-7-----.L-----c'-::-----7------::-'

Fig. 10. Rotational angle QP (in red) and angular velocity up (in blue) ofRMP.

N 0.75

i-~ 0.745Q)

~ 0.74

~ 0.735()

a p is not. Applied to the RMP, this implies that the pendulumbody does not repeat its trajectory at each gait cycle. In fact,as the plot shows, the body rotation fluctuates but steadilyrotates backwards as time goes on. The significance of thisfinding vis-a-vis the centroidal angular velocity of the robotis not clear.

Finally, Fig. 11 shows the evolution of the centroidal angularmomentum of the robot superposed on its 3D CoM trajectory.The positive to negative reversals of the angular momentumis a curious factor that deserves further study.~:~~~-_.._ _~--j

~~~-~_.....•••••••••.._..••..•..••.•~j~]~-~-- -~ _l~:: l

o 01 0.2 03 0.4 0.5 0.6 0.7 08

Time (sec)

Fig. 7. The generalized forces of the RMP from one gait cycle.

1.64L...-----'----'------'--------'----'-----'---.....L-------'o

~:::t...... :jc.. 80········~

60 ------------------

0'8~ j~ :;: _:.:------------- .0.74---------------------

lI_..· .._~_~._..... _.=j~O::c==H: :•..~ H.:.HH.jt O~~~~HHH~~H~~HHHHHHH_

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time (sec)

Fig. 8. CCRBI plot of both robot and planar RMP.

5: 10

~ 9

a::: 8

-gl'I:5 7

15.c 6e'5 5

~ 4

Fig. 6. The generalized coordinates of the RMP from one gait cycle.

30~-~--~---=":~--='"0.4:----~----:-7------:-'::--------='

Time (sec)

Fig. 9. Kinetic energy plot of robot (in red) and planar RMP (in blue).

187

robot. Interestingly, one of the generalized coordinates can beascertained through integration, only after the correspondinggeneralized velocity is determined. Through forward dynamicswe can also compute the joint torques necessary for the RMPto mimic the corresponding motion of the planar robot.

We have shown that, a) an angular momentum equalitybetween the robot and RMP does not necessarily guaranteekinetic energy equality, and b) a cyclic robot gait may notresult in a cyclic RMP movement. A non-cyclic gait hasramifications for the humanoid gait planning and this issueneeds to be further explored.

Regardless of the complexity of a planar robot, the RMP hasa fixed 4-dimensional equivalent RMP. Having this standardmodel makes it easier to uniformly compare the dynamics ofdifferent humanoids. This is a task for the future. Also it willbe interesting to consider the RMP equivalent of a fixed-basemanipulator robot.

We are also exploring appropriate torque mappings betweenthe compass gait robot and the RMP, as well the kinetic energyequivalence between the two models. Final objective of ourwork remains the formulation of control law for humanoidrobots to which the current paper is another step.

ACKNOWLEDGMENT

I am grateful to Dr. Fumihiko Asano of RIKEN, Nagoya,Japan, for discussions on compass gait dynamics, and espe­cially for providing me with his simulation data, which I usedin Section. V. Fruitful discussions with David Orin and JerryPratt are acknowledged.

REFERENCES

[1] M. Abdallah and A. Goswami. A biomechanically motivated two-phasestrategy for biped robot upright balance control. In IEEE InternationalConference on Robotics and Automation (ICRA), pages 3707-3713,April 2005, Barcelona, Spain.

[2] R. Altendorfer, Uluc Saranli, H. Komsuoglu, D.E. Koditschek, H. Ben­jamin Brown, Martin Buehler, N. Moore, D. McMordie, and R. Full.Evidence for spring loaded inverted pendulum running in a hexapodrobot. In D. Rus and S. Singh, editors, Experimental Robotics VII,pages 291 - 302. Springer-Verlag, 2001.

[3] F. Asano and Z.-W. Luo. On energy-efficient and high-speed dynamicbiped locomotion with semicircular feet. In IEEE/RSJ InternationalConference on Intelligent Robots and Systems (IROS), pages 5901-5906,October, Beijing, China 2006.

[4] F. Asano, Z.-W. Luo, and M. Yamakita. Biped gait generation and con­trol based on a unified property of passive dynamic walking. 21(4):754­762,2005.

[5] K. Astrom, A. Block, and M. Spong. The Reaction Wheel Pendulum.Downloadable from: http://decision.csl.uiuc.edu/ spong/main.htm, 2001.

[6] H. Essen. Average angular velocity. European Journal of Physics,14:201-205, 1993.

[7] A. C. Fang and N. Pollard. Efficient synthesis of physically valid humanmotion. 22(3):417-426, 2003.

[8] A. Goswami, B. Thuilot, and B. Espiau. Compass-like biped robot parti: Stability and bifurcation of passive gaits. Technical report, INRIA,No. 2996, Oct. 1996.

[9] H. Herr and M. Popovic. Angular momentum in human walking. TheJournal of Experimental Biology, 211:467-481, 2008.

[10] A. Hofmann. Robust execution ofbiped walking tasks from biomechan­ical principles. PhD thesis, MIT, AI Lab, 2006.

188

[11] S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada, K. Yokoi, andH. Hirukawa. Resolved momentum control: humanoid motion planningbased on the linear and angular momentum. In IEEE/RSJ InternationalConference on Intelligent Robots and Systems (IROS), volume 2, pages1644-1650, 2003, Las Vegas, NV, USA.

[12] S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada, K. Yokoi, andH. Hirukawa. Biped walking pattern generation by using preview controlof zero-moment point. In IEEE International Conference on Roboticsand Automation (ICRA), pages 1620-1626, 2003, Taipei, Taiwan.

[13] S. Kajita, F. Kanehiro, K. Kaneko, K. Yokoi, and H. Hirukawa. The 3dlinear inverted pendulum mode: A simple modeling for a biped walkingpattern generator. In IEEE/RSJ International Conference on IntelligentRobots and Systems (IROS), pages 239-246, 2001, Maui, Hawaii.

[14] S. Kajita and K. Tani. Study of dynamic biped locomotion on ruggedterrain. In IEEE International Conference on Robotics and Automation(ICRA), pages 1405-1411, 1991.

[15] T. Komura, H. Leung, S. Kudoh, and J. Kuffner. A feedback controllerfor biped humanoids that can counteract large perturbations during gait.In IEEE International Conference on Robotics and Automation (ICRA),pages 2001-2007, 2005, Barcelona, Spain.

[16] T. Komura, A. Nagano, H. Leung, and Y. Shinagawa. Simulatingpathological gait using the enhanced linear inverted pendulum model.IEEE Transactions on Biomedical Engineering, 52(9):1502-1513, 2005,September.

[17] A. Kuo. Stabilization of lateral motionin passive dynamic walking. Inter­national Journal ofRobotics Research, 18(9):917-930, 1999, September.

[18] S-H. Lee and A. Goswami. Reaction mass pendulum (RMP): An explicitmodel for centroidal angular momentum of humanoid robots. In IEEEInternational Conference on Robotics and Automation (ICRA), pages4667-4672, April 2007.

[19] T. McGeer. Passive dynamic walking. International Journal ofRoboticsResearch, 9(2):62-82, 1990.

[20] R. Olfati-Saber. Stabilization of a flat underactuated system: the inertiawheel pendulum. In 40th Conference on Decision and Control, pages306-308, December 4-7 2001, Orlando, FL.

[21] D. Orin and A. Goswami. Centroidal momentum matrix of a humanoidrobot: Structure and properties. In IEEE/RSJ International Conferenceon Intelligent Robots and Systems (IROS), 2008, Nice, France.

[22] M. Popovic, A. Goswami, and H. Herr. Ground reference pointsin legged locomotion: Definitions, biological trajectories and controlimplications. International Journal of Robotics Research, 24(12):1013­1032,2005.

[23] J Pratt. Exploiting inherent robustness and and natural dynamics in thecontrol of bipedal walking robots. PhD thesis, MIT, 2000.

[24] M. W. Spong and F. Bullo. Controlled symmetries and passive walking.IEEE Transactions on Automatic Control, 50(7):1025-1031, July 2005.

[25] M. W. Spong, P. Corke, and R. Lozano. Nonlinear control of inertiawheel pendulum. Automatica, 37:1845-1851, February 2001.

[26] T. Sugihara and Y. Nakamura. Variable impedant inverted pendulummodel control for a seamless contact phase transition on humanoidrobot. In IEEE International Conference on Humanoid Robots (Hu­manoids2003), pages 0-0, Oct 2003, Germany.

[27] J. Vermeulen, B. Verrelst, B. Vanderborght, D. Lefeber, and P. Guil­laume. Trajectory planning for the walking biped.

[28] M. W. Walker and D. Orin. Efficient dynamic computer simulation ofrobotic mechanisms. ASME Journal ofDynamic Systems, Measurement,and Control, 104:205-211, Sept. 1982.

[29] E. Westervelt, 1. W. Grizzle, C. Chevallereau, J. O. Choi, and B. Morris.Feedback Control of Dynamic Bipedal Robot Locomotion. CRC Press,Boca Raton, FL, 2007.