Automatica 40 (2004) 1647–1664www.elsevier.com/locate/automatica

Modeling, stability and control of biped robots—a general framework�

Yildirim Hurmuzlua ;∗, Frank G*enotb, Bernard Brogliatoc

aMechanical Engineering Department, Southern Methodist University, Dallas, TX 75252, USAbINRIA Rocquencourt, Domaine de Voluceau—BP 105, 78153 Le Chesnay, Cedex, France

cINRIA Rhone-Alpes, ZIRST Montbonnot, 655 Avenue de l’Europe, 38334 Saint-Ismier, France

Received 15 October 1999; received in revised form 15 September 2003; accepted 7 January 2004

Abstract

The focus of this survey is the modeling and control of bipedal locomotion systems. More speci6cally, we seek to review the developmentsin the 6eld within the framework of stability and control of systems subject to unilateral constraints. We place particular emphasis onthree main issues that, in our view, form the underlying theory in the study of bipedal locomotion systems. Impact of the lower limbswith the walking surface and its e:ect on the walking dynamics was considered 6rst. The key issue of multiple impacts is reviewed indetail. Next, we consider the dynamic stability of bipedal gait. We review the use of discrete maps in studying the stability of the closedorbits that represent the dynamics of a biped, which can be characterized as a hybrid system. Last, we consider the control schemes thathave been used in regulating the motion of bipedal systems. We present an overview of the existing work and seek to identify the neededfuture developments. Due to the very large number of publications in the 6eld, we made the choice to mainly focus on journal papers.? 2004 Published by Elsevier Ltd.

Keywords: Biped robots; Non-smooth mechanics; Unilateral constraints; Complementarity conditions; Multiple impact laws; Hybrid system; Gaitstability; Control synthesis

1. Introduction

In general, a bipedal locomotion system consists of sev-eral members that are interconnected with actuated joints.In essence, a man-made walking robot is nothing more thana robotic manipulator with a detachable and moving base.Design of bipedal robots has been largely inDuenced by themost sophisticated and versatile biped known to man, theman himself. Therefore, most of the models/machines devel-oped bear a strong resemblance to the human body. Almostany model or machine can be characterized as having twolower limbs that are connected through a central member.Although the complexity of the system depends on the num-ber of degrees of freedom, the existence of feet structures,

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Manfred Morari Editor.In this survey, we review research e:orts in developing control algorithmsto regulate the dynamics of bipedal gait. We focus on issues that are relatedto modeling, stability, and control of two legged locomotion systems.

∗ Corresponding author.E-mail address: hurmuzlu@seas.smu.edu (Y. Hurmuzlu).

upper limbs, etc., it is widely known that even extremelysimple unactuated systems can generate ambulatory motion.A bipedal locomotion system can have a very simple struc-ture with three point masses connected with massless links(Garcia, Chatterjee, Ruina, & Coleman, 1997) or very com-plex structure that mimics the human body (Vukobratovic,Borovac, Surla, & Stokic, 1990). In both cases, the systemcan walk several steps. The robotics community has beeninvolved in the 6eld of modeling and control of bipeds formany years. The books (Vukobratovic, 1976; Vukobratovicet al., 1990; Raibert, 1986; Todd, 1985) are worth readingas an introduction to the 6eld. The interested reader mayalso refer to the following web pages:http://www.androidworld.com/prod28.htm,http://robby.caltech.edu/∼kajita/bipedsite.html,http://www.fzi.de/divisions/ipt/WMC/preface/

preface.html,http://www.kimura.is.uec.ac.jp/faculties/legged-robots.

html.Nevertheless, and despite the technological exploit

achieved by Honda’s engineers (Japan is certainly thecountry where bipedal locomotion has received the mostattention and has the longest history), some fundamental

0005-1098/$ - see front matter ? 2004 Published by Elsevier Ltd.doi:10.1016/j.automatica.2004.01.031

1648 Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664

modeling and control problems have still not been addressednor solved in the related literature. One may notice, in par-ticular, that the locomotion of Honda’s P3 prototype re-mains far from classical human walking patterns at the samespeeds. Although Honda (HONDA) did not publish manydetails either on the mechanical part or on the implementedcontrol heuristic, it is easy to see on the available videos thatP3’s foot strike does not look natural and leads to some tran-sient instability (http://www.honda-p3.com). The number offoot design patents taken out by Honda (up to an air-bag-likeplanter arch) reveals again that foot–ground impact remainsone of the main diLculties one has to face in the design ofrobust control laws for walking robots. This will become thekey issue with increasing horizontal velocity requirement.This problem, however, is more sensitive for two-leggedrobots than for multi-legged ones due to the almost straightleg con6guration and the bigger load at impact time for theformer, leading to stronger velocity jumps of the center ofmass. While Honda’s engineers seem to consider these ve-locity jumps as unwanted perturbations and thus appeal tomechanical astuteness to smooth the trajectory, we arguethat impact is an intrinsic feature of mechanical systems likebiped robots and should be taken as such in the controllerdesign. Other bipedal robots have been designed. Amongthe most advanced projects, we cite the Waseda UniversityHumanoid Robotics Institute biped, the MIT Leg Labora-tory robots, the LMS-INRIA BIP system (Sardain, Rostami,& Bessonnet, 1998; Sardain, Rostami, Thomas, & Besson-net, 1999), the CNRS-Rabbit project (Chevallereau et al.,2003), and the German Autonomous Walking programme(Gienger, LNoOer, & Pfei:er, 2003), which can be found athttp://www.humanoid.rise.waseda.ac.jp/booklet/

kato 4.html,http://www.ai.mit.edu/projects/leglab/robots/robots.html,http://www.inrialpes.fr/bip,http://www-lag.ensieg.inpg.fr/PRC-Bipedes/,http://www.fzi.de/ids/dfg schwerpunkt laufen/

start page.html.respectively. Among all these existing bipeds, the Honda

robots seem to be the most advanced at the time of writingof this paper according to the information made available bythe owners. However, the solution for control designed byHonda does not explain why a given trajectory works nordoes it give any insight as to how to select, chain together,and blend various behaviors to e:ect locomotion throughdiLcult terrain (Pratt, 2000). It is the feeling of the authorsthat the problem of feedback control of bipedal robots willnot be solved properly as long as the dynamics of such sys-tems is not thoroughly understood. In fact, the main moti-vation for the writing of this paper has been the followingobservation about walking: there is no analytical study ofa stable controller with a complete stability proof availablein the related literature. It is our belief that the main rea-son for this is the lack of a suitable model. We propose aframework that is not only simple enough to allow subse-quent stability and control studies but also realistic as some

experimental validations prove. In addition, the frameworkprovides a uni6ed modeling approach for mathematical, nu-merical, and control problems, which has been missing. It isfor instance signi6cant that the main e:orts of the MIT LegLab (Pratt, 2000) have been directed toward technological(actuators) improvement and testing of heuristic control al-gorithms similar to Honda’s works.We should emphasize that the main thrust of this sur-

vey does overlook several practical aspects that may ariseduring the design and development of walking machines.Admittedly, a walking machine can be built without payingattention to many of the main ideas of this survey. There arenumerous toys that walk in a certain fashion. There are quitea few bipedal robots that are designed to avoid impacts al-together during walking. The fact remains that the stability,agility, and versatility of any existing bipedal machine doesnot even come close to that of the human biped. The sur-veyed concepts will better enable the design and evaluationof such machines through more suitable control algorithmsthat take into account impact mechanics and stability. Thepractical issues that arise in the design and development ac-tual machines deserve another survey article. In the ensuingpart of this survey we, therefore, will mainly focus on a the-oretical framework.

2. General description of a bipedal walker

A biped can be represented by an inverted pendulumsystem that has a constrained motion due to the for-ward and backward impacts of the swing limb with theground (Cavagna, Heglund, & Taylor, 1977; Hurmuzlu &Moskowitz, 1986; Full & Koditschek, 1999). Although sim-ilar to the structure of vibration dampers in many aspects(Shaw & Shaw, 1989), which are relatively well studied,structure of bipedal systems have a fundamental di:erencearising from the unconstrained contact of the limbs with theground (see Fig. 1). While the vibration damper remains incontact with the reference frame at all times because of thehinge that is located between the inverted pendulum andthe vibrating mass, the limbs of the biped are always freeto detach from the walking surface. Detachments occur fre-quently and lead to various types of motion such as walking,running, jumping, etc. As a matter of fact, one can classifybipedal locomotion systems as complementarity systems(LNotstedt, 1984; Brogliato, 2003). Such a modeling frame-work does not at all preclude the introduction of Dexibilitiesat the contact points. Also it allows one to include other ef-fects like Coulomb friction in a single framework, which canbe quite useful for numerical simulations in order to validatethe controllers. Fig. 1 depicts other systems that fall into thesame category. In the latter part of the article, we will showthat such systems can be analyzed by the use of Poincar*emaps (Hurmuzlu & Moskowitz, 1987), and possess com-mon features in terms of motion control. We would like tostress that the focus of this article is on motions that include

Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664 1649

•

(a) (b)

(c)

(d) (e)

Fig. 1. Bipeds as complementarity systems: (a) biped, (b) pendulum,impact damper, (c) mass impact damper, (d) juggler, and (e) manipulatorin contact with a rigid wall.

contact and impacts. For example, a biped can rock backand forth while the swing limb remains above the walkingsurface for all times. Such motions will be outside the focusof the discussions presented here.A typical walking cycle may include two phases: the

single support phase, when one limb is pivoted to theground while the other is swinging in the forward direc-tion (open kinematic chain con6guration), and the doublesupport phase, when both limbs remain in contact with theground while the entire system is swinging in the forwarddirection (closed kinematic chain con6guration). Whenboth limbs are detached, the biped is in the “Dight” phaseand the resulting motion is running or some other type ofnon-walking motion (Kar, Kurien, & Jayarajan, 2003). Anye:ort that involves analytical study of the dynamics of gaitnecessitates a thorough knowledge of the internal structureof the locomotion system. When the system is human oranimal, this structure is extremely complicated and little is

(a) (b)

Fig. 2. (a,b) Types of oscillation of a three-element biped.

known about the control strategy that is used by human be-ings and animals to realize a particular motion and achievestable gait (Full & Koditschek, 1999; Vaughan, 2003). Ifthe system represents a man-made machine or a numericalmodel representing such a machine, one has to synthesizecontrol strategies and performance criteria that transformmulti-body systems to walking automata. Devising prac-tical control architectures for bipedal robots remains tobe a challenging problem. The problem is tightly coupledwith the control studies in the area of robotic manipulators.Unlike manipulators, however, bipedal machines can havemany types of motion. The control objectives should becarefully selected to conform with a speci6c type of motion.A control strategy that is selected for high-speed walkingmay cause the system to transfer to running, during whichan entirely di:erent control strategy should be used, similarto jugglers control (Brogliato & Zavala-Rio, 2000).Consider the biped depicted during the single support

phase in Fig. 1(a). This biped is equivalent to a pendulumattached to the foot contact point with a mass and a lengththat are con6guration dependent. Inverted pendulum mod-els of various complexities, therefore, have been extensivelyused in the modeling of gait of humans and bipedal walk-ing machines. The dynamics of bipedal locomotion is intu-itively similar to that of an inverted pendulum and has beenshown to be close to it according to energetical criteria (Karet al., 2003; Cavagna et al., 1977; Full & Koditschek, 1999;Blickman & Full, 1987).A simple two-element biped (Hurmuzlu & Moskowitz,

1987) may operate in two modes (see Fig. 2): (a) impactlessoscillations, (b) progression with ground contacts. The im-portance of the contact event can be better understood if themotion is depicted in the phase space of the state variables.We simplify the present discussion by describing the

events that lead to stable progression of a biped for asingle-degree-of-freedom system, however, this approachcan be generalized to higher-order models. The phaseplane portrait corresponding to the previously describeddynamic behavior is depicted in Fig. 3(a). The sampletrajectories corresponding to each mode of behavior arelabelled accordingly. The vertical dashed lines representthe values of the coordinate depicted in the phase planefor which the contact occurs. For the motions depicted inFig. 3(a) or (b), the only trajectory that leads to contact isC. The contact event for this simple model produces two

1650 Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664

(a)

12

Backward ContactLimit Forward Contact

Limit

C

AB AB

IMPACTSWITCHING

1

2

C

(b)

(c)

45o Accumulation point

(d)

pe

�

�

1

n+1

n

.

1 .

.

�.

.

1�1

�

�

Fig. 3. Impact on the phase plane portrait.

simultaneous events:

(1) impact, which is represented by a sudden change ingeneralized velocities,

(2) switching due to the transfer of pivot to the point ofcontact.

The combined e:ect of impact and switching on the phaseplane portrait is depicted in Fig. 3(b). As shown in the 6g-ure, the e:ect of the contact event will be a sudden transferin the phase from points 1 to 2, which is generally locatedon a di:erent dynamic trajectory than the original one. If thedestination of this transfer is on the original trajectory, thenthe resulting motion becomes periodic (Fig. 3(c)). This typeof periodicity has unique advantage when the inverted pen-dulum system represents a biped. Actually, this is the onlymode of behavior that this biped can achieve progression.The most striking aspect of this particular mode of behav-ior is that the biped achieves periodicity by utilizing only aportion of a dynamic trajectory. The impact and switchingmodes provide the connection between the cyclic motionsof the kinematic chain and the walking action.We can clearly observe from the preceding discussion

that the motion of a biped involves continuous phases sep-arated by abrupt changes resulting from impact of the feetwith the walking surface. During the continuous phase, wemay have none, one, or two feet in simultaneous contactwith the ground. In the case of one or more feet contacts,the biped is a dynamical system that is subject to unilateralconstraints. When a foot impacts the ground surface, weface the impact problem of a multi-link chain with unilat-eral constraints. In fact, the overall motion of the biped mayinclude a very complex sequence of continuous and discon-tinuous phases. This poses a very challenging control prob-

lem, with an added complication of continuously changingmotion constraints and large velocity perturbations resultingfrom ground impacts.

3. Mathematical description of a biped as a systemsubject to unilateral constraints

3.1. Dynamics of the complementarity model

Bipedal locomotion systems are unilaterally constraineddynamical systems. A way to model such systems is to in-troduce a set of unilateral constraints in the following form:

F(q)¿ 0; q∈Rp; F :Rp → Rm;

where q represents the complete vector of independent gen-eralized coordinates. In other words, p denotes the numberof degrees of freedom of the system without constraints, i.e.when F(q)¿ 0. The constraints mean that the bodies thatconstitute the system cannot interpenetrate (irrespective ofthe fact that they are rigid or Dexible). The dynamics of ap-degree-of-freedom mechanical system subject to m uni-lateral constraints may be written as the following system(S) of equations, named a complementarity dynamical sys-tem (Brogliato, 2003; Heemels & Brogliato, 2003):

M (q) Nq + N (q; q) = Tu + ∇F(q)�n + Pt(q; q); (1)

�TnF(q) = 0; �n¿ 0; F(q)¿ 0; (2)

Restitution law + shock dynamics; (3)

Dry Friction Amontons–Coulomb′s model; (4)

where M (q) is the inertia matrix, N (q; q) includes Coriolis,centrifugal, gravitational, and other terms, u is an externalinput, �n ∈Rm is the Lagrange multiplier corresponding tothe normal contact force. The orthogonality �TnF(q) = 0means that if F(q)¿ 0 then �n=0, whereas a non-zero con-tact force �n ¿ 0 is possible only if there is contact F(q)=0.Such a contact model therefore excludes gluing, magneticforces. Complementarity Lagrangian systems as in (1)–(4)have been introduced by Moreau (1963, 1966), and Moreauand Panagiotopoulos (1988). The rest of the terms and vari-ables are de6ned next. For the bipeds the Lagrange dynamicscan be rewritten in a speci6c way that corresponds to con-trol objectives and allows the designer to get a better under-standing of their dynamical features (Wieber, 2000; Grizzle,Abba, & Plestan, 2001; Werstervelt, Grizzle, & Koditschek,2003). In other words, the choice of the generalized coor-dinates q is crucial for control purpose, and certainly muchless obvious than it is for serial manipulators. For examplein Fig. 4, q can be split in two subsets q1 and q2. The vec-tor q1 = (x; y; �) describes the global position of the robotin space whereas the vector q2 = (�1; : : : ; �6) encapsulatesthe joint coordinates. The vector q1 could be attached to anypoint of the biped. Nevertheless, it is known from biomedi-cal studies of human gait that one of the primary objectives

Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664 1651

(x, y, �)

y

B2A2B1

A1 �6

�1, �2

�3

�5

�4

Fig. 4. A 9-degree-of-freedom planar biped.

during locomotion is the stabilization of the head wherethe exteroceptive sensors (inner ear, sight) are located. Set-ting q = (q1; q2)T, (1) splits in an upper part and a lowerpart corresponding to head motion and joint dynamics,respectively,(

M1(q)

M2(q)

)Nq +

(N1(q; q)

N2(q; q)

)

=

(0

T2

)u + ∇F(q)�n + Pt(q; q); (5)

F(q) = (yA1 ; yB1 ; yA2 ; yB2 )T ∈R4; (6)

where yA1 denotes the y-coordinate of point A1, and so on.In the sequel we will review the available modeling toolswhich will allow the designer to complete (5) with (3) and(4). In (5), T1=0 since the biped has only joint actuators. Asa matter of fact head motion can only be achieved thanks to acoordinated action of joint actuation and contact forces. Thisfact is probably more apparent for the Dight phases (running)during which there are no contact forces and the trajectoryof the center of mass (attaching this time q1 to the center ofmass) yields only to gravity. All complementarity systemsdepicted in Fig. 1 can be represented in the form given by(1)–(4). However, biped dynamics possesses some speci6c

features that make the control study di:er signi6cantly fromthe control of other complementarity systems. Bipeds sharethe following features:

(1) with systems (b), (d) and (e): the center of mass is notcontrolled when F(q)¿ 0, as the object of a juggler(Brogliato & Zavala-Rio, 2000),

(2) with system (e): their dynamics is that of a manipulatorwhen one foot sticks to the ground,

(3) with system (b): they may be underactuated (no ankleactuator) and act as an inverted pendulum when onefoot sticks to the ground.

The fact that bipeds merge all these characteristics makestheir control analysis complex. Eq. (1) represents the dy-namics when either the system evolves in free-motion or ina phase of permanent contact, i.e.{

Fi(q) ≡ 0 for some i ∈I(q) ⊆ {1; : : : ; m};Fj(q)¿ 0 for j ∈ I(q):

(7)

Then, in Eq. (1) (∇Fi(q)=(9FTi =9q)(q)∈Rp is the gradient

vector)

∇F(q)�n =i=m∑i=1

∇Fi(q)�n; i:

Notice that for i ∈I(q), one can express (2) as (whereF i(q; q) = d=dt[Fi(q(t))])

�n; iF i(q; q) = 0; �n; i¿ 0; F i(q; q)¿ 0 (8)

or, if F i(q; q) ≡ 0 for i ∈I(q), as (where NFi(q; q; Nq) =(d=dt)[F i(q(t); q(t))])

�n; i NFi(q; q; Nq) = 0; �n; i¿ 0; NFi(q; q; Nq)¿ 0: (9)

As we shall see, when the constraints are frictionless theconditions in (9) de6ne a linear complementarity problem(LCP) with unknowns �n; i. An LCP is a system of theform Ax + B¿ 0, x¿ 0, xT(Ax + B) = 0 (Cottle, Pang, &Stone, 1992). LCPs are ubiquitous in many engineering ap-plications (Ferris & Pang, 1997), and particularly in uni-lateral mechanics in which they have been introduced byMoreau (1963, 1966). Finally, the models in (3) and (4) areneeded to complete the dynamics. In particular, it is nec-essary to relate the post-impact velocities to the pre-impactdata to be able to compute solutions that are compatiblewith the constraints (integrate the system and render the do-main� , {q |F(q)¿ 0} invariant). The classical Coulombfriction model in (4) provides the form of Pt(q; q) in (1).It is apparent that the set of equations in (1)–(9) de6nesa complex hybrid dynamical system, in the sense that itmixes both continuous and discrete-event phenomena. Thestates of the discrete-event system (DES) are de6ned bythe 2m modes of the complementarity conditions in (2). Itis also important to point out that such systems fundamen-tally di:er in nature from those studied in Bainov and Sime-onov (1989), which consist of mainly ordinary di:erentialequations with impulsive disturbances. Some discrepancies

1652 Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664

between both models are recalled in Brogliato (1999, Sec-tions 1.4.2, 7.1) and Brogliato, ten Dam, Paoli, G*enot, andAbadie (2002). Especially, one should always keep in mindthat the complementarity conditions (2) play a major rolein the dynamics of complementarity systems (for instancethey are necessary to properly characterize the equilibriumstates). From a mechanical engineer point of view, two ques-tions have to be answered to, when one wants to integratesystem (1)–(4) (Moreau & Panagiotopoulos, 1988):Q1. Assume that I(q) in (7) is non-empty at a time

instant �, and that the velocity q(�−) points inwards � ortangentially to 9�: determine which contacts i ∈I(q), willpersist at �+. In other words, determine the subsequent mode(or DES state).Q2. At an impact time tk , one has q(t−k )T∇Fi(q(tk))¡ 0

for some i ∈ {1; : : : ; m} and Fi(q(tk))=0. Determine the rightvelocity q(t+k ). In other words, determine a re-initializationof the (continuous) state as soon as one (or several) of theunilateral conditions in (2) is going to be violated.In the sequel tk , k = 0; 1; 2 : : : will generically denote the

impact instants. The answer to those questions is far frombeing trivial and has been the object of many researches.The 6rst one is related to solving the LCP associated tothe system, i.e. being able to calculate at each time of acollision-free phase the interaction forces Pq , ∇F(q)�n+Pt(q; q); hence, the acceleration Nq and the subsequent mo-tion. It has been raised initially by Delassus (1917) (seealso Pfei:er & Glocker, (1996) for a very nice and sim-pler example). The second question is that of de6ningproper restitution rules, or collision laws. This goes back tothe 17th century and the celebrated Newton’s conjecture,see Brogliato (1999) and Kozlov and Treshchev (1991)for more details. In particular, if several hypersurfacesZi = {q |Fi(q) = 0} are attained simultaneously, a multipleimpact occurs. Such an event occurs typically during walk-ing at the end of a single support phase when the swingingleg foot hits the ground. For example, for the planar bipeddepicted in Fig. 4, a 3-impact takes place at foot strike, thatis yA2 (t)=yB2 (t)=0, yA1 (t)¿ 0, and yB1 (t)¿ 0 for t ¡ tk ,while yA2 (tk) = yB2 (tk) = 0, yA1 (tk) = 0, and y A1 (t

−k )¡ 0.

From our point of view, only too few attention was payedin the literature to the existence of unilateral constraints.In fact, most of the works on the control of biped robotsmodel the robot in the single support phase as a manipula-tor whose base corresponds to the supporting foot and addsome closed loop constraints for the double support phase.Even if this approach is very convenient to derive trajectorytracking laws via the computed torque technique, it doesnot account for possible slippage nor detachment at groundcontacts nor say anything about the inDuence of impacts onstability. Notice also that the human walking pattern andthe underlying control strategy di:ers notably depending onthe ground characteristics (ice arena, basketball playground,trampoline, etc.), which makes the walking pattern (or tra-jectory planning) an important topic of research (El Ha6 &Gorce, 1999; Lum, Zribi, & Soh, 1999; Yagi & Lumelsky,

2000; Rostami & Bessonnet, 2001; Shih, 1999; Huang et al.,2001; Chevallereau & Aoustin, 2001; Saidouni & Besson-net, 2003).

3.2. Frictionless contacts (continuous motion)

The succeeding sections are devoted to the study of theLCP in (9) (i.e. the calculation of the contact forces) andmultiple impacts respectively. In the frictionless case (S)reduces to (1), (2) and (3) with Pt(q; q)=0. Moreau (1963,1966) has been the 6rst to show that in the multiple con-straints case m ≥ 1, using that NF(q; q; Nq) = ∇FT(q) Nq +f(q; q), the complementarity relation (9) combined with(1) yields an LCP of the form (assuming here that I(q) ={1; : : : ; m} in (7))

NF(q; q; Nq) = A(q)�n + b(q; q)¿ 0;

�n¿ 0 and �Tn NF(q; q; Nq) = 0; (10)

where

A(q) = ∇FT(q)M−1(q)∇F(q)

and

b(q; q) = ∇FT(q)M−1(q)h(q; q) + f(q; q)

and setting h(q; q) = Tu − N (q; q). If the active constraints(i ∈I(q)) are independent, then A is positive symmetricde6nite (PSD) and it is known that the LCP in (10) pos-sesses a unique solution �n (Cottle et al., 1992). If some con-straints are dependent, then A is only semi-PSD and unique-ness only holds for the acceleration NF(q; q; Nq). Moreover,∇F(q)�n is also unique, see Moreau (1966) and LNotstedt(1982), and thus from dynamics (1) Nq is unique too. As aclassical example, one may think of a chair with four legson a rigid ground: even if the interaction forces cannot beuniquely determined, the acceleration of the mass center isunique (upwards). Using Kuhn–Tucker’s theory (Kuhn &Tucker, 1951) it is possible to show that any solution �n tothe LCP in (10) is also a solution to the quadratic problem

min�n¿0

12 �TnA(q)�n + �Tn b(q; q); (11)

which is equivalent to Gauss’ principle of least deviation(Moreau, 1966; LNotstedt, 1982). To summarize, it is clearthat if one is able to obtain at time � a value for �n, thenintroducing this value into (2) allows one to determine whichcontacts persist and which ones are going to break (becomeinactive) on (�; � + ), ¿ 0, small enough.

Remark 1. The answer to Q1 does not necessarily re-quire the explicit calculation of the contact forces �n. Letus consider the simple example (p = 1) of a ball restingon the ground (m = 1) at �0. Suppose that an externalforce f is applied to the ball at �0. Then the dynamics isNq(t) = max(0; f) (recall that max functions can be writtenwith complementarity). Let us discretize the unconstrained

Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664 1653

A

BC

R(t)R(t) R(t)

(a) (b) (c)

Fig. 5. Possible motions and shock outcomes in an abstract con6gurationspace.

motion (q¿ 0) as

qi+1 = qi + hqi;

qi+1 = qi + hf: (12)

Using (12), the dynamics is discretized as

qi+1 = prox[R+; qi + hf]; (13)

where prox denotes the proximal point to qi + hf in R+.This implicit scheme can be generalized to more complexsystems (p¿ 1, m¿ 1), as proposed in Moreau (1986).Moreover, it can be generalized to provide one solution tothe multiple impact problem as will be shown in Section 3.3.

3.3. Frictionless contacts (multiple impacts)

Section 3.2 was devoted to partially answer to Q1. Wenow focus on Q2. In general, walking robots use severalsupport points during locomotion (biped robots; Hurmuzlu& Moskowitz, 1986, 1987; Hurmuzlu, 1993, quadrupedrobots, Chevallereau, Formal’sky, & Perrin, 1997; Perrin,Chevallereau, & Formal’sky, 1997). In other words m¿ 2.This means that the boundary 9� of the admissible domain� is not di:erentiable everywhere. Its singularities corre-spond to surfaces with codimension¿ 2. Thus, the eventu-ality that the state collides in a neighborhood of a singularitycannot be excluded in mechanical systems as (S). In theframework of bipedal locomotion, such events intrinsicallybelong to the dynamics of walking. Let us view for simplic-ity the bipedal robot as a point R(t) in a two-dimensionalgeneralized con6guration space. The classical standardwalking assumption can be seen as a “bilateral slidingmotion” of the generalized point R(t) con6ned to the twoconstraint boundaries, see Fig. 5(a). Single support phasescorrespond to R(t)∈ [A; B) ∪ (B; C], the double supportphase to R(t) = B. However one cannot a priori exclude

• rebounding multiple shocks at B, see Fig. 5(b);• detachment during single support phases, see the forego-ing paragraph, and Fig. 5(c).

The most recent studies in the 6eld can be divided in twodi:erent approaches.

• The 6rst one (Hurmuzlu & Marghitu, 1994; Marghitu &Hurmuzlu, 1995; Han & Gilmore, 1993) consists of anenumeration procedure. They apply a restitution law forsimple impact at Aj and sequentially propagate its e:ectson the other contact points Ai, i ∈I(q). Nevertheless, theoverall process is assumed to be instantaneous, i.e. allshocks occur simultaneously at all the contact points: thisis therefore really a multiple impact. At each point theyinvestigate whether assuming zero or non-zero local per-cussions yield consistent outcomes. The main drawbackof these studies is that they do not rule on whether or notthe proposed algorithms always terminate with a uniquesolution. In the case of several admissible solutions, theydo not give a criterion to make the choice between thesesolutions. However, this should not be seen as a real draw-back (except for simulation tool design purpose) sinceany heuristic to make the “good” choice, excluding in thesame time several admissible solutions, drops the rigidbody assumption.

• The second approach corresponds to the de6nition of acollision mapping

Pc : 9� × {−V (q(tk))} → 9� × {V (q(tk))};

(q(tk); q(t−k )) �→ (q(tk); q(t+k )); (14)

whereV (q)={v∈Rp: ∀i ∈I(q), vT∇Fi(q)¿ 0} denotesthe tangent cone to 9� at q(t) (Moreau & Panagiotopou-los, 1988), i.e. the set of admissible post-impact velocities.It is clear that the choice of Pc should yield a mathemat-ically, mechanically and numerically coherent formula-tion of the studied phenomenon. The so-called “sweepingor Moreau’s process” (Brogliato, 1999; Moreau, 1985;Moreau & Panagiotopoulos, 1988) is a general formula-tion of the dynamics in (1)–(4) based on convex analysistools. It allows one to write the dynamics as a particu-larMeasure DiBerential Inclusion, a term coined by J.J.Moreau. It implicitly de6nes a collision mapping basedon the computation of the post-impact motion via a prox-imation procedure in the kinetic metric

q(t+k ) = proxM (q(tk ))[q(t−k ); V (q(tk))]: (15)

It is noteworthy that this can also be equivalently writ-ten as a quadratic programme under unilateral constraints,and consequently under a LCP formalism, see (10), (11).Let us note that the mapping in (15) applied to the shockof two rigid bodies correspond to taking a zero restitu-tion coeLcient. Hence it may be named a “generalizeddissipative impact rule”. However, it is possible to intro-duce some restitution %∈ [0; 1] by substituting q(t+k ) in(15) by 1

2 (1+%)q(t+k )+12 (1−%)q(t−k ) (Moreau & Pana-

giotopoulos, 1988; Mabrouk, 1998). The interested readermay have a look at Brogliato (1999, Section 5.3) for anon-mathematical introduction to this material. Similarresults have been obtained in LNotstedt (1982). In Pfei:erand Glocker (1996), an extension of Poisson’s impact lawis proposed that takes the form of two LCPs (see, e.g.,

1654 Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664

Brogliato, 1999, Section 6.5.6 for a simple example ofapplication of this impact mapping).

3.4. Contacts with Amontons–Coulomb’s friction

In the frictionless case, we saw that the LCP (10) waswell-posed since the only possible indeterminacies, result-ing from dependent active constraints, do not inDuence theglobal motion of the mechanical system (uniqueness of theacceleration still holds). The problem becomes more com-plicated through the introduction of dry friction, an essentialparameter of the legged locomotion. The next step relies onthe fact that the relay characteristic of Coulomb’s friction,can be represented analytically in a complementarity for-malism by introducing suitable slack variables (or Lagrangemultipliers). This allows one to construct a set of comple-mentarity conditions which monitor all the transitions fromsticking to slipping, and from contact to detachment. The dy-namics in (1)–(4) can consequently be rewritten as (Pfei:er& Glocker, 1996)

M (q) Nq = h(q; q) + [W (q) + Nslide(q)]�; (16)

y = A(q)� + b(q; q);

y¿ 0; �¿ 0; yT� = 0 (17)

for appropriate matrices A(·) and b(·), W and Nslide, wherethe components of � are suitable slack variables. The pow-erfulness of the complementarity formalism clearly appearsfrom (16), (17) in which the overall dynamics is writtenin a compact way: the continuous dynamics plus comple-mentarity conditions. This perfectly 6ts within the generalcomplementarity dynamical framework (Brogliato, 2003).

4. The stability framework

The most crucial problem concerning the dynamics ofbipedal robots is their stability, see e.g. http://www.ercim.org/publication/Ercim News/enw42/espiau.html. As hasbeen explained in Sections 2 and 3, a biped is far frombeing a simple set of (controlled) di:erential equations.Moreover, the objectives of walking are quite speci6c.One is therefore led to 6rst answer the question: whatis a stable biped? And, consequently, what mathematicalcharacterization of this stability can be constructed fromthe complementarity models? As we will see next, this isclosely related to the fact that bipeds can be considered ashybrid dynamical systems, the stability of which can beattacked from various angles. The goal of this section is topresent some tools which can serve for the stability analysisof models as in (2)–(5), and which are suitable for bipedsbecause they encapsulate their main features. Firstly, wespend some time on describing invariant sets for comple-mentarity systems. The point of view that is put forth is thatvarious existing, or to be investigated, stability frameworks

x

um

Admissible Region

Inadmissible Region

A

(a)

(b)0

FCIS CVIS UIS

Fig. 6. Controlled mass subject to a unilateral constraint.

are better understood when invariant sets are classi6ed. Sec-ondly, we review the so-called impact Poincar*e maps, whichhave been used extensively in the applied mathematics andmechanical engineering literature for vibro-impact systems(Masri & Caughey, 1966; Shaw & Holmes, 1983; Shaw &Shaw, 1989). This point of view seems natural if one con-siders bipeds as jugglers (Brogliato & Zavala-Rio, 2000;Zavala-Rio & Brogliato, 2001). However, it presents limi-tations which we point out.

4.1. Invariant sets of systems subject to unilateralconstraints

Application and use of mapping techniques is tightly cou-pled with the structure of the invariant set that representsthe steady-state motion. Systems subject to unilateral con-straints behave in a more complex manner than the onesthat are not (Budd & Dux, 1994). For example, let us con-sider the system given in Fig. 6(a). Suppose we would liketo develop a controller to place the mass at a time varyingposition xd(t)=A+B sin(!t) starting from an arbitrary ini-tial condition in the admissible region. One can use a simplecontroller that yields the following closed-loop dynamics:

m( Nx − Nxd(t)) + k1(x − xd(t)) + k2(x − xd(t)) = �n;

06 �n ⊥ x¿ 0; x(t+k ) = −ex(t−k ) with e ∈ [0; 1):(18)

Our objective here is not to explore all possible typesof invariant sets that may be attained by the systemgiven by (18). Instead, we would like to show that the

Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664 1655

invariant sets of the closed loop system can be classi6edunder three categories (see Fig. 6(b)):(i)Constraint violating invariant sets (CVIS): An invari-

ant set that includes at least one collision with the constraintsurface per cycle of motion (including orbits that stabilizein 6nite time on the constraint surface after an in6nite num-ber of collisions). Hence, an impact PoincarDe mapping P iswell de6ned that captures these orbits. This type of invari-ant set is a unique feature of systems subject to unilateralconstraints. All the systems shown in Fig. 1 can be made toexhibit this behavior with a set of properly selected param-eter values. Speci6cally, for locomotion systems, this willbe the only mode of motion that can describe running andwalking.(ii) Unconstrained invariant sets (UIS): An invariant set

that does not include collisions with the constraint surface.In the single mass case, this corresponds to the cyclic mo-tions of the mass that occur to the right of the constraintsurface (x¿ 0 ∀\t6 t6∞). This type of invariant set canbe observed for all systems of Fig. 1 except the juggler.For the biped, this corresponds to rocking when one or twolimbs are in contact with the ground.(iii) Fully constrained invariant sets (FCIS): An in-

variant set that never leaves the constraint surface. Thiscorresponds for the system in (18) to a static equilibriumwhere the system rests on the constraint surface (x(t) =0 ∀\t6 t6∞). All the systems in Fig. 1 can exhibit this be-havior with a speci6c choice of the control parameter valuesand initial conditions.It is easy to imagine that trajectories of a system may

be CVIS, FCIS and UIS simultaneously. In particular, no-tice that although the complementarity relations in (18),i.e. x¿ 0, �n¿ 0, x�n = 0, a priori de6ne two modesx¿ 0 and x = 0 (hence a bimodal system), for controland dynamic systems analysis purpose one is led to con-sider those phases that correspond to CVIS as independentones (Brogliato, Niculescu, & Monteiro-Marques, 2000;Bourgeot & Brogliato, 2003): they are described neither bythe free motion nor by the constrained motion dynamicsbut by the whole dynamics of the hybrid system.

4.2. Bipedal robots as hybrid dynamical systems

The above classi6cation of the invariant sets naturallyleads one to consider complementarity systems as in Fig. 1as hybrid dynamical systems whose DES states are de6nedfrom the described invariant sets. In the following, we shallgenerically denote the phases that correspond to CVIS as Ikand those that correspond to FCIS and/or UIS as +k . Withsome abuse of notation, we shall denote the DES states andthe corresponding time intervals in the same manner. As weshall see further subdivisions will be needed. As an examplelet us consider an impact damper as in Fig. 1(c) and with asinusoidal excitation applied to the basis mass: for a properchoice of the spring-dashpot, the excitation parameters and

of the initial conditions, the system possesses periodic tra-jectories with one impact per period (hence CVISs) (Masri& Caughey, 1966). Thus one has

R+ = I0: (19)

Consider now a manipulator as in Fig. 1(e) that performsa complete robotic task with a succession of free-motionand constrained motions phases, during which it isexplicitly required to track desired motion and/or con-tact force (Bourgeot & Brogliato, 2003; Brogliato, 1999;Brogliato, Niculsecu, & Orhant, 1997; Brogliato et al., 2000;Menini & Tornamb*e, 2001). During the force/position con-trol phases, the trajectories will in general be both UIS (inthe tangent direction to the constraint surface) and FCIS (inthe normal direction). During the free-motion phases, thetrajectories are UIS. It is therefore natural to split such taskinto three phases +2k , +2k+1 and Ik that correspond to UIS,FCIS and CVIS, respectively, i.e.

R+ = +0 ∪ I0 ∪ +1 ∪ +2 ∪ I1 ∪ · · · : (20)

Consider now the biped. It is clear that in order to describewalking, running and hopping motions, one needs more thanthe above three types of invariant sets (Kar et al., 2003).Moreover, one needs more than the three phases proposedfor the manipulator case. Indeed, as we already pointed outconcerning the choice of the Poincar*e section, describingfor instance a walking motion involves to take care of thenon-sliding conditions. Hence, one is led to di:erentiatecontact phases (FCIS and UIS) and impact phases (CVISand FCIS and/or UIS) that slide and those that stick. Noticemoreover that this may be done independently of the pres-ence of Amontons–Coulomb friction at the contact points:friction adds modes to the plant model, whereas our descrip-tion concerns the nature of the trajectories and is directlyrelated to stability and control objectives. But it is clear thatthe plant modeling will strongly inDuence the conditionsunder which those modes will be activated. Such a de6-nition yields generally a large number of DES states. Weshall de6ne only those that are needed to describe the threementioned types of motion:

• +fk : Dight phases (both feet detached);

• +slk : left foot sticks, right foot detached;

• +srk : right foot sticks, left foot detached;

• +dssk : double support phase, both feet stick;

• I rsk : impacts on the right foot, left foot sticks;• I ldk : impacts on the left foot, right foot detached.

Then we obtain the following:

Walking : R+ =+sl0 ∪ I rs0 ∪ +dss

0 ∪ +sr0 ∪ I rl0

∪+dss1 ∪ +sl

1 ∪ · · · ; (21a)

Hopping : R+ = +f0 ∪ I ld0 ∪ +f

1 ∪ I ld1 ∪ · · · ; (21b)

Running : R+ =+f0 ∪ I ld0 ∪ +sl

0 ∪ +f1 ∪ I rd1 ∪ +sr

1

∪+f2 ∪ I ld1 ∪ · · · : (21c)

1656 Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664

The conditions of activation of one mode have to be stud-ied. For instance, conditions such that sticking occurs at animpact or during a step have been studied in Rubanovichand Formal’sky (1981), Hurmuzlu (1993), Chang and Hur-muzlu (1994) and G*enot, Brogliato, and Hurmuzlu (1998).They evidently strongly depend on the process model likeAmontons–Coulomb friction and the multiple impact resti-tution law. The concatenation of phases in (21) correspondsto desired invariant sets of the DES. From a general viewpoint, any stability criterion should take into account boththe non-smooth and the hybrid natures of such complemen-tarity systems. As it is known, one may have several pointof views of hybrid dynamical systems: continuous-time,discrete-event, or mixed, see e.g. Automatica (1999).

4.3. Constraints that guarantee foot sticking, stabilitymargins

The control torques u(·) have to obey certain conditionsto ensure sticking of the contacting feet at all times. Theseconditions can be written as (G*enot, 1998; G*enot et al.,1998)

A(q; ,)u + B(q; q; ,)¿ 0 (22)

during smooth motion, and as

\A(q(tk); ,)q(t−k )¿ 0 (23)

at impact times (and for some choice of the double impactmodel), where , is Amontons–Coulomb’s friction coeL-cient at contact. The detailed calculations of inequalities (22)and (23) can be found in G*enot, Brogliato, and Hurmuzlu(2001). Both inequalities in (22) and (23) are necessary andsuFcient conditions to be satis6ed by the control input uso that during the whole motion (smooth and non-smooth)sticking is maintained and detachment is monitored. The in-equality in (22) can be rewritten as A(�)¿ 0 and can beused to compute stability margins (Wieber, 2002). The no-tion of stability margin has been introduced correctly in Seoand Yoon (1995), who formulated a set of constraints in thespirit of (22), (23). The “distance” from the trajectory to theconstraint boundary is de6ned as the maximal magnitudeof a disturbance that is applied on the biped. Stability mar-gins help understanding the di:erence between static gait(center of gravity located within its base of support), anddynamic gait (center of gravity may fall outside the sup-port base) (Vaughan, 2003). To the best of our knowledge,none of the control laws proposed in the literature until nowwas shown to be “stable” with respect to these fundamen-tal conditions, mainly due to the fact that the underlyingdynamics of the robot does not capture the unilateral fea-ture of the feet-ground contacts. To summarize, a controllerwhich guarantees both (22) and (23) implies that the systemevolves in the DES path in (21a).

4.4. PoincarDe maps and stability

Generally, the approach to the stability analysis takes intoaccount two facts about bipedal locomotion: the motion isdiscontinuous because of the impact of the limbs with thewalking surface (Hurmuzlu &Moskowitz, 1987; Hurmuzlu,1993; Katoh & Mori, 1994; Zheng, 1989; Grizzle et al.,2001), and the dynamics is highly nonlinear and non-smoothand linearization about vertical stance generally shouldbe avoided (Vukobratovic et al., 1990; Hurmuzlu, 1993;Grizzle, Abba & Plestan, 1999).A classical technique to analyze dynamical systems is

that of Poincar*e maps. In the three-link bipedal model ofSection 2, we have shown that periodic motions of a simplebiped can be represented as closed orbits in the phase space.Fig. 3(d) depicts a 6rst return map obtained from the pointsof the trajectory that coincide with the instant of heel strike.A Poincar*e map for a generalized coordinate -1 (which istypically a joint angle in bipedal systems) at the instant ofheel strike now can be obtained by plotting the values of -1

at ith versus the values at (i + 1)th heel strike.One can choose an event such as the mere occurrence

of heel strike, to de6ne the Poincar*e section (Hurmuzlu &Moskowitz, 1987; Kuo, 1999). We can construct severalmappings depending on the type of motion. In general, how-ever, the section can be written as follows:

Z+i = {(q; q)∈R2p | t = tck}; i = 1; : : : ; l; (24)

where the condition establishes the Poincar*e section.The discrete map obtained by following the procedure de-

scribed above can be written in the following general form:

�i = P(�i−1); (25)

where � is a reduced dimension state vector, and the sub-scripts denote the ith and (i − 1)th return values, respec-tively. Periodic motions of the biped correspond to the 6xedpoints of P where

�∗ = Pk(�∗); (26)

where Pk is the kth iterate. The stability of Pk reDects thestability of the corresponding Dow. The 6xed point �∗ is saidto be stable when the eigenvalues 0i, of the linearized map,

%�i =DPk(�∗)%�i−1 (27)

have moduli less than one. This technique employedin Hurmuzlu and Moskowitz (1987), Hurmuzlu (1993),Fran]cois and Samson (1998), Kuo (1999), Grizzle et al.(1999, 2001), Piiroinen and Dankowicz (2002), Dankowiczand Piiroinen (2002), Dankowicz, Adolfsson and Nordmark(2001), Piiroinen, Dankowicz, and Nordmark (2001, 2003)and Quint van der Linde (1999) has several advantages.Using this approach the stability of gait conforms with theformal stability de6nition accepted in nonlinear mechanics.The eigenvalues of the linearized map (Floquet multipliers)provide quantitative measures of the stability of bipedalgait. Finally, to apply the analysis to locomotion one only

Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664 1657

requires the kinematic data that represent all the relevantdegrees of freedom. No speci6c knowledge of the internalstructure of the system is needed. This feature also makes itpossible to extend the analysis to the study of human gait.Using this approach one can develop quantitative measuresfor clinical evaluation of the human gait (Hurmuzlu & Bas-dogan, 1994; Hurmuzlu, Basdogan, & Stoianovici, 1995).

5. Control of bipedal robots

The control problem of bipedal robots can be de6ned aschoosing a proper input u in (S) such that the system be-haves in a desired fashion. The key issue of controlling themotion of bipeds still hinges on the speci6cation of a de-sired motion. There are numerous ways that one can specifythe desired behavior of a biped, which in itself is an openquestion. The control problem can become very simple orextremely complex depending on the speci6ed desired be-havior and the structure of the system. Typical bipedal ma-chines are designed to perform tasks that are not con6ned tosimple walking actions. Such tasks may include maneuver-ing in tight spaces, walking or jumping over obstacles, andrunning. In this article, we place our main focus on tasksthat are primarily related to walking. Therefore, we will notbe concerned with actions such as performing manipulationtasks with the upper limbs.

5.1. Passive walking

Possibly the 6rst bipedal walking machine was built byFallis (1888). The idea of a biped walking without jointactuation during certain phases of locomotion was initiallyproposed in Mochon and McMahon (1980a). The authorstermed this type of walk as “ballistic walking”. The inspi-ration of this idea originated from evidence in human gaitstudies, which pointed out to relatively low levels of mus-cle activity in the swing limb during the swing period (seeBasmajian, 1976; Zarrugh, 1976). Two planar models wereused: (1) a three-element model with a single-link stance andtwo-link swing leg, (2) a four-element model with two-linkstance and swing limbs. In each case, the authors searchedfor initial conditions at the onset of the swing phase suchthat the subsequent motion satis6ed a set of kinetic and kine-matic constraints. Then, they isolated the initial conditionsthat satis6ed the imposed constraints. The simulation resultswere compared to experimentally measured knee angles andground reaction forces. They concluded that their outcomesand human data had the same general shape. In Mochon andMcMahon (1980b), they improved their model by adding thestance knee (two-element stance limb). With this more so-phisticated model, the authors analyzed the model responseby using three out of the 6ve gait determinants (Saunders,Inman, & Eberhart, 1953). The premise of walking with-out joint actuation, prompted McGeer (1990) to propose the“passive walking”. McGeer developed numerical as well as

experimental models of bipeds inspired by Fallis (1888) thathave completely free joints. He demonstrated that these sim-ple, unactuated bipeds can ambulate on downward planesonly with the action of gravity.Now, returning to (1)–(4), the typical passive control

scheme is concerned with the free dynamics of the system(S) given in Section 3.1 (i.e. the dynamics of the systemsubject to u=0). Then, as we have shown in Section 4.1, aPoincar*e section Z+ can be selected (see (24)) to obtain anonlinear mapping in the form of

�i = P(�i−1; ’); (28)

where the parameter vector ’ typically includes the slope ofthe walking surface, member lengths, and member weights.Then, the underlying question becomes the existence andstability of the 6xed points of this map (Kuo, 1999) andthe resemblance of the resulting motions to bipedal walk-ing. This task is generally very diLcult to realize with theexception of very simple systems. For example, in the caseof vibro-impact systems, analytical expressions to show ex-istence can be found (Shaw & Shaw, 1989). In the case ofslightly more complex systems, such explicit calculationsbecome impossible since the free-motion dynamics is nolonger integrable. Then one has to rely on numerical toolsto derive both the Poincar*e mapping and its local stabil-ity (Kuo, 1999; Piiroinen et al., 2001, 2003; Piiroinen &Dankowicz, 2002; Dankowicz & Piiroinen, 2002).The main energy loss in these bipeds is due to the repeti-

tive impacts of the feet with the ground surface. The gravi-tational potential energy provides the compensation for thisloss, thus resulting in steady and stable locomotion for cer-tain slopes and con6gurations. The ground impacts (Hurmu-zlu & Moskowitz, 1986) provide a unique mechanism thatleads to stable progression in very simple bipeds as has beenrecently demonstrated by several investigators. In Goswami,Thuilot, and Espiau (1996, 1998), the authors consider asimple model that includes two variable length memberswith lumped masses representing the upper body and thetwo limbs. A third lumped mass is attached to this point,which represents the upper body. They analyze the nonlineardynamics of (28) subject to prescribed variations in the ele-ments of ’. They primarily focus on the e:ect of the groundslope, mass distribution, and limb lengths. Numerical anal-ysis of the nonlinear map, results in the detection of stablelimit cycles as well as chaotic trajectories that are reachedthrough period doubling cascade. A simpler, two-link modelwas considered in Garcia et al. (1997). They also demon-strated that this simple biped can produce stable locomotionas well as very complex chaotic motions reached throughfrequency doubling cascade. Chatterjee and Garcia (1998)and Das and Chatterjee (2002) studied the existence ofperiodic gaits in the limit of zero slope. The addition ofpassive arms served to reduce side-to-side rocking in a 3Dpassive walker (Collins, Wisse, & Ruina, 2001). Quint vander Linde (1999) includes phasic muscle contraction as theenergy source, and vary the muscle model parameters to

1658 Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664

create new periodic gaits. The main challenge of the study ofpassive gait is to translate the understanding gained by study-ing passive systems to active systems. TheMIT Leg Lab pla-nar bipeds are controlled this way (Pratt, 2000) for periodicwalking. In other words, the desired trajectories qd(t) aredesigned from the study of passive walking. This is not thecase for the Honda robots where qd(t) are obtained from hu-man recordings. One of the main obstacles to build bipedalrobots remains to be the prohibitively high joint torques thatare often required to realize even routine walking tasks. Acomprehensive investigation that bridges the passive stud-ies to better design of active control schemes would be anatural extension of passive locomotion research. This isachieved in Piiroinen and Dankowicz (2002), Dankowiczand Piiroinen (2002), who propose a control method basedon discrete adjustments of the swing-foot orientation priorto contact, hence indirectly a:ecting the nature and timingof the subsequent impact. This results in the (local) stabi-lization of a motion that is naturally occurring in the system.

5.2. Walking with active control

5.2.1. Controller designThe control action must assure that the motion of a

multi-link kinematic chain, which can characterize a typicalbiped, is that of a walker. Although, the characteristics ofthe motion of a walker is still open to interpretation, wemay translate this requirement to a set of target/objectivefunctions given in the form

gi(q(t); q(t); qd(t); qd(t); �n; �t ; 3; u) = 0

i = 1; : : : ; k6p; (29)

where 3 is a vector of parameters that prescribes certain as-pects of the walking action such as progression speed, steplength, etc. Note that for the sake of simplicity of the nota-tions q will denote in the sequel the vector of generalizedcoordinates of the model considered by the referenced au-thors, i.e. either of the full order model, or of the reducedorder model when assuming that the ground contacts, whenactive, are bilateral contacts.The control problem can be described as specifying the

vector of joint actuator torques u in (1) such that the systembehaves in a certain way. The simplest way to proceed isto specify the time pro6les of the joint trajectories. Inves-tigators in the 6eld used kinematics of human gait as de-sired pro6les (see Hemami & Farnsworth, 1977; Khosravi,Yurkovich, & Hemami, 1987; Vukobratovic et al., 1990).One can also simply specify certain aspects of locomo-tion such as walking speed, step length, upright torso, etc.(Chudinov, 1980, 1984; Lavrovskii, 1979, 1980; Beletskii,1975; Beletskii & Kirsanova, 1976; Beletskii & Chudi-nov, 1977b, 1980; Beletskii, Berbyuk, & Samsonov, 1982;Grishin & Formal’sky, 1990; Novozhilov, 1984; Hurmuzlu,1993; Chang & Hurmuzlu, 1994; Yang, 1994). In Beletskiiand Chudinov (1980), the authors use the components of

the ground reaction forces in addition to kinematics inorder to completely specify the control torques. Once theobjective functions are speci6ed, one has to choose a con-trol scheme in order to specify the joint moments (controltorques) that drive the system toward the desired behavior.We encounter several approaches to this problem that canbe enumerated as follows:(1) Linear control: The equations of motion are lin-

earized about the vertical stance, assuming that the postureof the biped does not excessively deviate from this posi-tion. For example, in Jalics, Hemami, and Clymer (1997),Kajita and Tani (1996), Kajita, Yamaura, and Kobayashi(1992), Grishin, Formal’sky, Lensky and Zhitomirsky(1994), Zheng and Hemami (1984), Hemami, Zheng, andHines (1982), Golliday and Hemami (1977), Hemami andFarnsworth (1977), Gubina, Hemami, and McGhee (1974),Mitobe, Capi, and Nasu (2000), Seo and Yoon (1995) andGarcia, Estremera, and Gonzales de Santos (2002), a PDcontroller was used to track joint trajectories. The linearcontroller, however, cannot track time functions. Thus, theauthors discretized the desired joint pro6les and let thecontroller track the trajectory in a point-to-point fashion.van der Soest “Knoek”, Heanen, and Rozendaal (2003)study the inDuence of delays in the feedback loop on stancestability, including muscle model.(2) Computed torque control: This method was applied

(Hemami & Katbab, 1982; Lee & Liao, 1988; Hurmuzlu,1993; Yang, 1994; Jalics et al., 1997; Lum et al., 1999; Song,Low, & Guo, 1999; Park, 2001; Taga, 1995: Chevallereau,2003) to bipedal locomotion models with various levels ofcomplexities. In Chevallereau (2003), the computed torqueis combined with a time-scaling of the desired trajectoriesoptimally designed (Chevallereau & Aoustin, 2001), whichallows the 6nite-time convergence of the system’s state to-wards the desired motion. The 6nite-time convergence espe-cially allows one to avoid the tricky problems due to track-ing errors induced by impacts (Bourgeot & Brogliato, 2003;Brogliato et al., 1997, 2000).(3) Variable structure control: This method results in a

feedback law that ensures tracking despite uncertainties insystem parameters. In this approach, one chooses the controlvector as

ui = u i − ki sign(s); (30)

where u i is a trajectory tracking controller with 6xed esti-mated parameters. The second term, is the variable structurepart of the control input. The function s de6nes the slidingsurface that represents the desired motion. This is a highgain approach that is advantageous because it ensures con-vergence in 6nite time. In locomotion, the stability of theoverall motion relies on the e:ectiveness of the controllerin eliminating the errors induced by impact during the sub-sequent step. The reader can check Chang and Hurmuzlu(1994) and Lum et al. (1999) to see the application of sucha controller to a 6ve-element planar model.

Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664 1659

(4) Optimal control: Optimal control methods have beenused by researchers to regulate the smooth dynamic phaseof bipedal locomotion systems. Two approaches have beentaken to the optimization problem. The 6rst method is basedon computing the values of selected parameters in the ob-jective functions that minimize energy-based cost functions(Frank, 1970; Vukobratovic, 1976; Beletskii & Chudinov,1977a; Beletskii et al., 1982; Rutkovskii, 1985; Channon,Hopkins, & Pham, 1992; Saidouni & Bessonnet, 2003).We note that optimal trajectory planning including the dy-namics in (1)–(4) is equivalent to searching for an optimalopen-loop control u(t). The second approach is based onvariational methods to obtain controllers that minimize costfunctions (Beletskii & Bolotin, 1983; Bolotin, 1984; Fu-rusho & Sano, 1990; Channon, Hopkins, & Pham, 1996a, b).It is the direct application of classical optimal control meth-ods to bipedal locomotion, see Channon, Hopkins, & Pham(1996c) for the most advanced work in this topic. Herethe authors regulate the motion of the biped over a supportphase with a quadratic cost function. van der Kooij, Jacobs,Koopman, & van der Halm (2003) propose a model pre-dictive controller designed from a tangent linearization toregulate gait descriptors formulated as end-point conditions.The main obstacle towards real implementation is a too largecomputation time.(5) Adaptive control: The adaptive control approach has

received very little attention in biped control. Perhaps it isdoes not have real advantage in controlling bipedal locomo-tion. Nevertheless, Yang (1994) has applied adaptive con-trol approach to a three link, planar robot. Experiments havebeen led at the MIT Leg Lab (Pratt, 2000) using adaptivecontrol.(6) Shaping discrete event dynamics: The abrupt nature

of impact makes it practically impossible to directly con-trol its e:ect on the system state. Even an approximation ofan impulsive Dirac input would demand actuators with toohigh bandwidth (to say nothing of induced vibrations in themechanical structure). An alternate approach can be foundin shaping the system state prior to the impact instant suchthat a desired outcome is assured. Such an approach wastaken in Hurmuzlu (1993) and Chang and Hurmuzlu (1994).In these studies, a set of objective functions in the form of(29) was tailored. Assuming perfect tracking, the authorsderived the expression for the system state immediately be-fore the instant of impact in terms of the parameter vector3.Subsequently, the post-impact state was computed for spe-ci6c values of the parameter vector. The parameter spacewas partitioned into regions according to slippage and con-tact conditions that result from the foot impact. Then, thispartitioning was used to specify controller parameters suchthat the resulting gait pattern has only single support phaseand the feet would not slip as a result of the feet impact. Dunnand Howe (1994) developed conditions in terms of motionand structural parameters such that they minimize/eliminatethe velocity jumps due to ground impact and limb switch-ing. Thus, in their case, the objective of the shaping was to

remove the e:ect of the impact altogether. Miura and Shi-moyama (1984) used a feedforward input that modi6es themotion at the end of each step from measurements infor-mations. Grizzle et al. (1999, 2001) and Werstervelt et al.(2003) have also used a similar approach. They shape thestate before the impact, so that at the next step the state re-sides in the zero dynamics. Doing so they create a periodicgait that corresponds to the zero dynamics de6ned from aset of output functions. Piiroinen and Dankowicz (2002) lo-cally stabilize a passive walk with a speci6c strategy, seeSection 5.1.(7) Stability and periodic motions: Stability of the over-

all gait is often overlooked in locomotion studies. Typically,controllers have been developed, and few gait cycles havebeen shown to demonstrate that the biped “walks” with thegiven controller. A thorough analysis of the nonlinear dy-namics of a planar, 6ve-element biped (Hurmuzlu, 1993)reveals a rich set of stable, periodic motions that do notnecessarily conform to the classical period one locomotion.Tracking errors in the control action may lead to stable gaitpatterns that are di:erent than the ones that are intendedby the objective functions. One way to overcome this diL-culty is to partition the parameter space such that one wouldchoose speci6c values that lead to a desired gait pattern.This approach is taken in Hurmuzlu (1993) and Chang andHurmuzlu (1994).(8) Other specialized control schemes: Several investi-

gators (Grishin & Formal’sky, 1990; Grishin et al., 1994;Beletskii, 1975; Chudinov, 1980, 1984; Katoh & Mori,1994; Lavrovskii, 1979, 1980). used simpli6ed modelswithout impact and constructed periodic trajectories byconcatenation of orbits obtained from individually con-trolled segments of the gait cycle. This approach is quitesimilar in spirit to the Kobrinskii (1965) method that isused the existence of trajectories of the impact damper andthe impacting inverted pendulum (see Fig. 1). Blajer andSchielen (1992) compute a nonlinear feedforward torquecorresponding to a “non-impacting” reference walk and usePD motion and PI force feedback to stabilize around thereference trajectory. Fuzzy logic control was used (Shih,Gruver, & Zhu, 1991) to develop a force controller thatregulates ground reaction forces in swaying actions of anexperimental biped. A group of investigators changed theparameters in the objective functions such that the desiredmotion is adapted to changing terrain conditions (Igarashi &Nogai, 1992; Shih & Klein, 1993; Zheng & Sheng, 1990).Zheng (1989) used an acceleration compensation methodto eliminate external disturbances from the motion of anexperimental eight joint robot. Kuo (1999) derives numer-ically an impact Poincar*e map that represents the walkingcycle, and proposes a linear state feedback that stabilizesthis cycle. Clearly, this is conceptually completely di:er-ent from the works described above (see item (1) Linearcontrol) since the design is based on a linearization of thePoincar*e map itself and not of the continuous dynamics onone step.

1660 Y. Hurmuzlu et al. / Automatica 40 (2004) 1647–1664

5.2.2. DES stabilizationAs we have shown in Section 4.2, walking corresponds to

a particular sequence of activations of the modes of the DESassociated to the biped seen as a complementarity mechani-cal system. Such a sequence can be seen as an invariant setof the DES dynamics, see (21a). These control techniquesaim at stabilizing this invariant set in the sense that the robotshould ultimately be able to recover from falls and restartwalking (Fujiwara et al., 2002). Notice that this approachdoes not emphasize the low-level details of the walk (walk-ing speed, steps length, etc.). An interesting approach inthe area is the zero moment point (ZMP) method proposed6rst by Vukobratovic and his co-workers (Vukobratovic &Juricic, 1969; Vukobratovic et al., 1990). The reader can re-fer to Goswami (1999) and Wieber (2002) and referenceswithin for a detailed discussion regarding the real meaningof ZMP, and to Garcia et al. (2002) for a complete de-scription of equivalent stability concepts. Several di:erent,but equivalent, de6nitions of the ZMP are given (Hemami& Farnsworth, 1977; Takanishi, Ishida, Yamaziki, & Kato,1985; Arakawa & Fukuda, 1997; Hirai, Hirose, & Kenada,1998). The simplest one is (Hemami & Farnsworth, 1977)the point where the vertical reaction force intersects theground, i.e. the center of pressure. The ZMP stability crite-rion states that the biped will not fall down as long as theZMP remains inside the convex hull of the foot-support. Inthese studies, the authors impose the motion of the lowerlimb kinematics from human kinematic data, which theyterm synergies. This way, the ZMP criterion is used to switchbetween low-level controllers (which satisfy some objec-tive functions like trajectory tracking), so as to stabilize theDES orbit in (21a) (Park, 2001), and possibly avoid ob-stacles (Yagi & Lumelsky, 2000). The ZMP method wasalso applied with other controllers that are not based onprescribing human data (Borovac, Vukobratovic, & Surla,1989; Fukuda, Komota, & Arakawa, 1997; Shih, Gruver, &Lee, 1993; Mitobe et al., 2000; Park, 2001; Huang et al.,2001; Vanel & Gorce, 1997). One of the best example ofthe high degree of eLciency that such control approachesare able to attain are the bipeds constructed by Honda (Hiraiet al., 1998; Hirai, 1997; Pratt, 2000), whose control mainlyrely on a suitable combination of local linear controller withhigh-level (or logical) conditions. In Pratt, Chew, Torres,Dilworth, and Pratt (2001), an intuitive approach for mak-ing some bipedal machines walk is proposed. It is based onthe so-called virtual model control. The need for both low-and high-level control together with on-line desired trajec-tories planning is explained in El Ha6 and Gorce (1999)and Vanel and Gorce (1997) where only the supervisoryaspects are studied. Wieber (2002) proposes a quite inter-esting study, starting from (22). A general criterion for theDES path (21a) stability (equivalently, its invariance) is es-tablished, and the link with Lyapunov functions is made(excepting impacts). Stability margin (roughly, the distancefrom the actual trajectory to the boundary of an admissibleset of trajectories, outside of which the robot falls down

(Seo & Yoon, 1995) can be derived. Such studies are ofprimary importance for characterizing the stability of reha-bilitated paraplegics (Popovic et al., 2000).

6. Conclusions and directions for future research

This survey is devoted to the problem of modeling andcontrol of a class of non-smooth nonlinear mechanical sys-tems, namely bipedal robots. It is proposed to recast thesedynamical systems in the framework of mechanical sys-tems subject to complementarity conditions. Unilateral con-straints that represent possible detachment of the feet fromthe ground and Coulomb friction model can be written thisway. In the language of Full and Koditschek (1999), this isa suitable template. Such a point of view possesses severaladvantages:

(1) It provides a uni6ed approach for mathematical, nu-merical and control investigations. This is a quite im-portant point since numerical studies are mandatory inany mechanical and/or control design.

(2) This framework encompasses all the models whichhave been used to study locomotion in the control androbotics literature.

(3) Though we restrict ourselves to rigid body con-tact/impact models, lumped Dexibilities can easily beintroduced, both at the contact or in the structure it-self (Dexible joints). Introducing Dexibilities may benecessary (Pratt, 2000; Pratt & Williamson, 1995),and is physiologically sound (Gunther & Blickman,2002). This will, however, make the control problemharder to solve and may be ine:ective in walking (Karet al., 2003). It may also create serious diLculties inthe analysis, especially with multiple contacts (Paoli &Schatzman, 2002).

(4) Such models have proved to predict quite well the mo-tion as several experimental validations available in theliterature show (Abadie, 2000).

(5) As shown in this survey, the proposed modeling ap-proach allows one to clarify which stability tools onemay use to characterize the stability of a bipedal robot.

(6) Finally, it is the opinion of the authors that one impor-tant development is still missing in the 6eld of bipeddesign: a concise and suLciently general theoreticalanalysis framework, based on realistic models, that al-lows the designer to derive stable controllers takinginto account the hybrid dynamics in their entirety.

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Yildirim Hurmuzlu received his Ph.D.degree in Mechanical Engineering fromDrexel University. Since 1987, he has beenat the Southern Methodist University, Dal-las, Texas, where he is a Professor andChairman of the Department of Mechani-cal Engineering. His research focuses onnonlinear dynamical systems and Control,with emphasis on robotics, biomechanics,and vibration control. He has publishedmore than 60 articles in these areas. Dr.Hurmuzlu is the associate Editor of theASME Transactions on Dynamics Systems,Measurement and Control.

Frank G*enot was born in 1970 inZweibrNucken (Germany). He graduatedfrom the Ecole Nationale Superieured’Informatique et de Mathematiques Ap-pliqu*ees de Grenoble (France) in 1993.He got the Ph.D. degree from the Insti-tut National Polytechnique de Grenoble inComputer Science in January 1998. SinceSeptember 2000, he has been an INRIAResearcher in the MACS research projectat INRIA Rocquencourt (France). His mainresearch interests include modelling andsimulation issues of systems with unilateralconstraints, in Mechanics and Finance, andStructural Control.

Bernard Brogliato got his Ph.D. from theInstitut National Polytechnique de Grenoblein January 1991. He is presently workingfor the French National Institute in Com-puter Science and Control (INRIA), in theBipop project. His scienti6c interests are innon-smooth dynamical systems, modelling,stability and control. He is a member ofthe Euromech Non Linear Oscillations Con-ference committee (ENOCC), reviewer forMathematical Reviews and the ASME Ap-plied Mechanics Reviews, and is AssociateEditor for Automatica since October 1999.