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Biped robots: Correlations between technological design and dynamic behavior

Biped robots: Correlations between technological design and dynamic behavior
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  Control Engineering Practice 7 (1999) 401 —  411 Biped robots: Correlations between technologicaldesign and dynamic behavior P. Sardain * , M. Rostami, E. Thomas, G. Bessonnet  Poitiers Uni v ersity, Mechanics of Solids Laboratory SP2MI, BP 179, 86960 Futuroscope Cedex, France Received 23 March 1998; accepted 30 September 1998 Abstract The authors of the paper have collaborated in a joint project involving four French control, mechanics and computer-sciencelaboratories. In the paper, various mechanical architectures of biped robots are examined in detail, showing that their walkingcapabilities are closely linked to the kinematic characteristics of the mechanical structure. Then, it is shown that the geometrical andinertial parameters of the mechanical systems strongly affect the gait. In particular, the influence of the biped inertia on the lateralstability of the system, as well as the conditions of the existence of passive pendular gaits during the swing phase, are computationallyanalyzed. Extending the ideas previously developed, some characteristics of the mechanical architecture and design of the BIP projectcan be clearly justified. It turns out that a kinematic structure with 15 degrees of freedom is necessary in order for the biped robot todevelop anthropomorphic gaits. Furthermore, as an anthropometric mass distribution can improve the walking abilities of the robot,special transmitters have been designed in order to help to fulfil this requirement.  1999  Elsevier Science  ¸ td .  All rights reserved .  Keywords:  Biped robots; Dynamics of walking; Equilibrium; Natural walking; Mechanical design 1. Introduction: some ideas on the design of a mechanicalbiped robot In the field of mobile robotics, the legged robots adapteasily to varying types of ground surfaces. In the caseof biped robots, a second quality (theoretically) is theirability to move in environments marked by a great num-ber of constraints. In addition, if these bipeds have an-thropometriccharacteristics,they can also be designed toperform various tasks in environments especially plan-ned for human biped activities. Therefore, in terms of potential applications, anthropometric biped robots ap-pear to be promising. However, a human walker caninspire researchers to an even greater extent than whathas been presented up to this point: indeed, a humanbeing cannot only keep his balance on a multitude of ground surfaces, in the presence of obstacles, he also hasthe ability to adjust his postures and equilibrium swiftly,efficiently and (most often!) with elegance. Thus, byconstructing a biped robot with anthropomorphiccharacteristics, researchers are both inspired by and * Corresponding author. Fax: # 33 549 49 65 04; e-mail: sardain @ provided with a rich reference in order to obtain the bestperformances.In this paper, the authors reply from a mechanicalviewpoint to specialists in control who are inquiringabout biped demonstrators that would be able to vali-date their theoretical approaches experimentally. Hereare some questions.   The number of joints drastically increases the geomet-ric, kinematic and dynamic model complexity, andconsequently there is a temptation to limit the modelto one that is planar.Is it possible to test walking in thesagittal plane with a demonstrator under proper con-ditions? The answer is ‘no’, for a really planar demon-strator is technically quasi-impossible to design.   In order to make on-line processing possible, the dy-namic model is often limited to a simple invertedpendulum. Is this simplified model sufficient? Theanswer is ‘yes’ for lateral motion in the frontal plane,and ‘no’ for walking forward in the sagittal plane.Indeed, in this plane, the swing of the leg and theheel-strike have predominant dynamic effects. So themodel reduced to an inverted pendulum for the carry-ing leg and a double pendulum for the swinging legseems to be minimal. 0967-0661/99/$  —   see front matter    1999 Elsevier Science Ltd. All rights reserved.PII: S0 9 67 -0 6 6 1 (9 8 ) 0 0 1 65 - 8    A simplified approach to the quite extensive problemof control would lie in stabilizing cyclic joint trajecto-ries by means of impulsive controls released at the startof each cycle. Does a mechanical structure with ananthropomorphic kinematic chain (i.e., with a knee)exist, capable of moving passively during the singlesupport phase? The answer is ‘yes’ and ‘no’. Yes, sucha structure exists, but it does not possess the requiredanthropomorphic functions. Indeed, it will be submit-ted to either a counter-flexing of the knee or an excess-ive heel impact speed.The line of argument allowing the questions to be an-swered in such a way will be developed in the followingsections. Section 2 will examine biped demonstratorsinvolving 1 —  12 internal degrees of freedom. As a con-clusiondrawn from these observations,it willappearthaton the one hand, the double support phase is well con-trolled, but that on the other hand, some undesirableeffects are encountered during the single support phase,which can be decomposed into two problems. Witha planar model, however complex it might be, the prob-lem of the lateral equilibrium around the edge of the foothas not been solved, and consequently, the demonstratorbehaves badly. A second problem occurs at the end of the single support phase when the foot of the swingingleg collides with the ground; impact is very difficult tocontrol with the commonly used position-trackingalgorithms.Section 3 will study the two previous problems, thatis to say the lateral equilibrium and the heel-strike,from the viewpoint of dynamics. It will appear thata heavy, high vertical-sized trunk would facilitate thelateral control. Considering a sagittal model with fourlinks, initial conditions will be sought in terms of jointvelocities and positions that generate the leg swing,during the unipodal phase, with two requirements:without energy consumption, and with symmetricalrepositioning of the foot at heel-strike. The study willshow that an anthropometric mechanical structure isable to move passively during the single supportphase, but it does not possess the required anthropo-morphic gait characteristics, particularly a smoothheel-strike.Section 4 will present the mechanical architecture of the biped designed for the  BIP  project (Espiau, 1997),derived closely from observations and results describedin Sections 2 and 3. Firstly, a simple line of argument willshow that 15 active joints are required to ensure thelocomotion of a biped with anthropomorphic gaits, inorder for it to walk and change direction on differenttypes of surfaces. Secondly, light and compact transmis-sion systems will be described. They will make it possibleto drive the 15 active joints of the biped with efficiency,while ensuring reversibility. Finally, the positioning of transmitters and driving motors on the skeleton will beanalyzed, in order to obtain an anthropometric distribu-tion of masses. 2. Analysis of some biped mechanical architectures Several biped robots have been designed during theselast 20 yr. They have allowed researchersto establish andexperimentally validate various control laws. Regardlessof what these laws are, the gaits being generated arebased essentially on mechanical architectures. So, anexamination of the latter is essential to this paper. Bipedrobots are walkers in most cases (jumpers are eithermonopod or quadruped robots). In the class of walkers,a distinction is established between two categories: first,‘dynamic’ walkers whose center of pressure (or zero mo-ment point) is situated inside the polygon that circum-scribes the one or two feet in contact with the ground;and second, ‘purely dynamic’ walkers that have no feet,and whose ZMP coincides with the leg contact point onthe ground during the single support phase, or followsalong the segment linking the two contact points duringthe double support phase.McGeer (1990) has designed a ‘purely dynamic’ walkerwith 1 theoretical degree of freedom. This biped (Fig. 1)does not possess a pelvis; it has only two stiff legs,articulated one with the other. It shows in theory onlya single degree of freedom. In practice, these two legs aretelescopic, so as to allow the swinging leg to rock back-wards and forwards without colliding with the groundsurface. Therefore, this biped has a total of 3 dof. There isno motion in the lateral plane. In the sagittal plane, thebiped moves like a compass made of 2 links. Thanks tothe extreme simplicity of the kinematic model, McGeerhas been able to complete a perfectly symmetrical mech-anical construction with respect to the sagittal plane,thus obtaining a demonstrator presenting both a trueand real planar architecture. The main articulation be-tween the two legs is not actuated. Since this biped isactivated uniquely by gravitational forces, it can onlywalk down a sloping surface.Grishin et al. (1994) have designed a ‘purely dynamic’walker with 2 dof. It is composed (Fig. 2) of a pelvis onwhich are two articulated telescopic straight legs. Of  Fig. 1. Biped robot with 1 internal dof.402  P. Sardain et al. /  Control Engineering Practice 7 (1999) 401 —  411  Fig. 2. Biped robot with 2 dof. these 4 dof (2 rotations plus 2 translations), it is necessaryto subtract 2 constraints since, thanks to two mechanicalsystems, the total length of the two legs is constant, andthe pelvis is brought into alignment with the bisector of the two legs. Thus the number of degrees of real mobilityis 2. The legs are equipped with passive feet which areperpendicular to the sagittal plane, in order to avoidfalling over laterally. The motion in the sagittal plane isthat of a compass with 3 links. This biped walks alonga straight line on a flat surface. This demonstrator is notprovided with a real 2D architecture and it is, as men-tioned by its authors, more or less destabilized by heel-strike impacts.Similarly, Kajita et al. (1992) have designed a ‘purelydynamic’ walker with 4 dof. The authors report that thebiped can take some steps along a straight line on a flatsurface before becoming unbalanced sideways by a roll-ing oscillation that is created by each heel-strike impact.Furusho and Sano (1990) have designed a ‘dynamic’walker with 8 dof. It is composed (Fig. 3) of a pelvis andtwo legs. The hip and knee have a single dof in flexing-stretching. The ankle possesses 1 dof in flexing —  stretch-ing, plus 1 dof in lateral rotation. The lateral motion isthat of an inverted pendulum; the robot moves in itsfrontal plane as a single rigid solid without mobility. Thislateral deviation allows for equilibrium. The motion inthe sagittal plane involves 7 links and 6 joints. The ratioof the mass of leg to the total mass is 30%. ‘BLR-G2’walks along a straight line on flat ground. Inventors haveobserved that disruptive impacts are provoked when theheels regain their support base. Fig. 3. Biped robot with 4 dof.Fig. 4. Biped robot with 9 dof. Takanishi et al. (1990) have designed several walkerswith trunks. The most recent one is a ‘dynamic’ walkerwith 9 dof. The locomotive system of this biped is purelysagittal, Fig. 4 showing how the 6 axes of the ankles,knees and hips are parallel. A trunk is articulated on thepelvis by 3 perpendicular axes. In the lateral plane, onlythe trunk is mobile, which creates an inverted pendulumon the pelvis. The motion in the sagittal plane involves8 links and 7 joints. The ratio of the mass of the leg to thetotal mass is 26%. The ratio of the mass of the trunk tothe total mass is 30%. The motions required for thetrunk, to maintain balance, have amplitudes larger thanthose of a human walker. The authors have noticed thatimpacts with the ground are disruptive.Shih et al. (1993) have designed a ‘dynamic’ walkerwith12 dof. It is composed of a pelvis and two legs whosehips have 3 dof. Thanks to the vertical axis dof at thehips, this walker can potentially change direction, but itcurrently walks only along a straight line. The pelvis is of small vertical dimension, so this biped looks as if it doesnot have a trunk. The ratio of the mass of the leg to thetotal mass is 37%.Honda Motor Co. publicly presented a humanoid ro-bot with two legs and two arms, in December 1996. Thebiped shown in Fig. 5 is the Honda robot in its first stage,without arms, as described by the Managing Director of the Honda R & D Center in (Hirai, 1997). This biped hasno trunk, but a very long vertical pelvis that looks likea pelvis-trunk (the arms of the complete humanoid robotare connected in the upper part of this pelvis-trunk). It isa ‘dynamic’ walker with 12 dof: the hips have 3 dof, theknees 1 and the ankles 2. The motion in the lateral planeinvolves 5 links and 4 dof, while the motion in the sagittalplane involves 7 links and 6 dof. The ratio of the mass of the leg to the total mass is obviously very low (approxim-ately between 5 and 10%). It is not possible to give theratio of the mass of the trunk to the total mass since thisbiped robot has no trunk (the pelvis-trunk is a uniquebody).If the movementof the armsinfluencesthe balancedynamics, then the ratio of the mass of the two arms to  P. Sardain et al. /  Control Engineering Practice 7 (1999) 401 —  411  403  Fig. 5. Biped robot with 12 dof. the total mass is very small. This biped robot is able towalk up and down staircases and on a known slope of about 10%. It can turn, and can walk sideways andbackwards.In reviewing all the aforementioned observations, itappears that most of the demonstrators are well control-led during the transfer of the weight from one foot to theother in the double support phase. On the other hand,the pseudo-sagittal demonstrators show lateral instabil-ity, while all the demonstrators, regardless of how theyare constructed, are subjected to sizeable impacts at theend of the swinging phase. These problems are closelylinked to the mechanical architecture of the system, in asfar as its intrinsic dynamic behavior can dramaticallyaffect the control functions. As the results presented inthe next section will show, the ratio of the mass of thetrunk to the total mass and the ratio of the mass of leg tothetotalmass havea noticeableinfluence on the dynamicbehavior of biped robots. 3. Two pendular problems related to biped walking During the single support phase of walking, a biped isquite unstable. First of all, it must control and maintainits lateral equilibrium. A second important conditionrequiredto generatesafe walking is related to the dynam-ics of the swing phase, and especially concerns heel-strikes, which may have destabilizing effects in cases of hard impact. This final condition is closely related toboth the initial momentum of the biped and its intrinsicinertia characteristics. In connection with these consid-erations, two specific problems are examined in the fol-lowing subsections. The first one is related to thelateral instability of the biped when considered as aninverted pendulum. The second problem deals with theswing phase when studied as a free motion of pendulartype. 3.1. Passi v e lateral instability of a biped  This subsection considers a standing biped that isdestabilized by a loss of contact at the level of one foot(Fig. 6). At the beginning of the motion that results, it canbe assumed that the biped performs like an invertedpendulum. In this case, it is well known that the higherthe center of mass is, the more slowly the initial motionstarts. Consequently, when the center of mass is higher,the biped has more time to analyse and adjust its motionin order to recover its equilibrium.The brief study which follows is aimed at estimatingprecisely the effect of the relative internal position of thecenter of mass on the elapsed time until the moment atwhich the biped reaches its limit position, beyond whichthe equilibrium is hard to recover.The mechanical system schematically represented inFig. 6 is assumed to be passive and stiff as it begins torotate around the sagittal axis O z , which coincides withtheinternal edge of the foot remainingin contact withtheground. The rotation is described by the angular coordi-nate   , which varies from its initial value     to the finalvalue    .The equation of mechanical energy conservation canbe set as  I  Q  # mgh " cste (1)where ‘ h ’ is the height of   G ,  h " a  cos(  ),  a " OG ; ‘ I ’stands for the total moment of inertia about the axis Oz;‘ m ’ is the total mass. In terms of initial conditions    ' 0,  Q  " 0,   Q , can be expressed as  Q "   2 mgaI  (cos    ! cos   ). (2) Fig. 6. Upright biped before loss of lateral equilibrium.404  P. Sardain et al. /  Control Engineering Practice 7 (1999) 401 —  411  Then, the time required to move from the initial position G   of   G  to a critical position  G   is determined by theelliptic integral  t , t (   ) ! t (   ) "    I 2 mga        d    cos    ! cos   . (3)In the computations that follow, the angle     locates thelimiting position of the center of mass  G  on the vertical of the left foot.Here, two bipeds are considered, one without a trunk,and the other with a trunk on the pelvis. Their commoncharacteristics are: leg mass " 17 kg, pelvis mass " 14 kg, between feet  e " 0.18 m, foot width  l " 0.06m. Biped without a trunk : total mass  m " 48 kg, moment of inertia  I " 20 kg m  ,  h " 0.575 m,  a " 0.578 m.The square root equals 0.1917 and the integral 2.234;consequently   t " 0.43 s for a deviation    " 15 ° . Biped with a trunk : total mass  m " 90 kg, moment of inertia  I " 98 kg m  ,  h " 0.932 m,  a " 0.934 m.The square root equals 0.2438 and the integral 2.222;consequently   t " 0.54 s for a deviation    " 9 ° .Thus, the biped with a trunk reaches its final de-sequilibriumposition after a lapse of time longer than thecorresponding time of the biped with a lower center of mass (0.54 s instead of 0.43 s). Moreover, the biped witha trunk is less tilted in its critical position (9 °  only insteadof 15 ° ), which is attained with a rate of rotation only half the rate of the biped without a trunk (0.36 rd/s versus0.77 rd/s).The design of BIP takes the previous result into ac-count: the ratio of the mass of the trunk to the total massequals 44%, as explained in Section 4. 3.2. The pendular problem of the swing phase The objective of this study is to examine to what extentandunderwhat conditionstheswingingphase ofwalkingcan be considered as a motion during which the mechan-ical energy of the system is kept constant. This problemhas been mentioned in (McMahon, 1984) in the study of human locomotion. McMahon’s argument is based onthe observation that the mechanical energy of the walkervaries little during the swing phase. He presents anexample termed a ‘ballistic walking model’, since duringthe swinging phase the total mechanical energy of thesystem is conserved. McGeer (1990) develops the ideathrough the concept of passive walking, tested ona mechanical biped of compass type. This idea is ad-vanced in (Espiau and Goswami, 1994) and (Goswamiet al., 1996) where the existence of limiting cycles thatregulate passive walking on sloping ground is clearlyestablished. Fig. 7. Parameters of the swinging phase. Following the example of McMahon, the study of thepassive swing phase of an anthropomorphic biped is of great interest. The system selected here is a planar bipedwhich is reduced to 4 links (Fig. 7). It has been assumedthat the support leg is extended (the knee is stiff) andworks as an inverted pendulum supporting the free leg,which moves as a double pendulum. The motion is de-scribed by the rotations at three joints which have beenassumed to be free: the ankle of the support leg, and thehip and knee of the swinging leg. The trunk is submittedin this model to an active torque on the axis of thehip, allowing it to keep a constant orientation duringthe motion. The presence of this active joint allows oneto consider the motion as pseudo-pendular, and quasi-conservative. 3.2.1. Initial and   fi nal constraints The only specified conditions concern the position andthe speed of the contact point  C  of the swinging leg’s footat the moment of toe-off. These are defined as OC  . X  #¸" 0,  OC  . ½  " 0 » ( C  ). X  " 0, » ( C  ). ½  " 0 (4)where ¸ represents an unknown step length; » ( C  ) is thevelocity vector of   C   in the reference frame ( O ;  X  , ½  );the symbol ‘.’ stands for the scalar dot product.Relationships (4) can be fully stated in terms of initialvalues of the 6 phase parametersof motion,as the genericfunctions g  ( q   ,  q   ,  q   ,  q R   ,  q R   ,  q R   ) " 0,  k " 1, 2 , 4 (5)where the superscript ‘ i ’ designates the initial values of  joint coordinates and velocities. At the end of the swing-ing phase, heel-strike conditions are defined under  P. Sardain et al. /  Control Engineering Practice 7 (1999) 401 —  411  405
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