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The International Journal of

http://ijr.sagepub.com/content/9/2/62The online version of this article can be found at:

DOI: 10.1177/027836499000900206

1990 9: 62The International Journal of Robotics Research Tad McGeer

Passive Dynamic Walking

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Passive DynamicWalking

Tad McGeerSchool of Engineering ScienceSimon Fraser UniversityBurnaby, British Columbia, Canada V5A 1S6

Abstract

There exists a class of two-legged machines for which walkingis a natural dynamic mode. Once started on a shallow slope,a machine of this class will settle into a steady gait quitecomparable to human walking, without active control or en-

ergy input. Interpretation and analysis of the physics arestraightforward; the walking cycle, its stability, and its sensi-tivity to parameter variations are easily calculated. Experi-ments with a test machine verify that the passive walkingeffect can be readily exploited in practice. The dynamics aremost clearly demonstrated by a machinepowered only bygravity, but they can be combined easily with active energyinput to produce efficient and dextrous walking over a broadrange of terrain.

1. Static vs.

Dynamic Walking

Research on legged locomotion is motivated partly byfundamental curiousity about its mechanics, andpartly by the practical utility ofmachines capable oftraversing uneven surfaces. Increasing general interestin robotics over recent years has coincided with the

appearance ofa wide variety of legged machines. Abrief classification will indicate where our own workfits in. First one should distinguishbetween static anddynamic machines. The former maintain static equi-librium

throughouttheir motion. This

requiresat least

four legs and, more commonly, six. It also imposes aspeed restriction, since cyclic accelerations must belimited in order to minimize inertial effects. Outstand-

ing examples of static walkers are the Odex series(Russell 1983) and the Adaptive Suspension Vehicle(Waldron 1986). Dynamic machines, on the otherhand, are more like people; they can have fewer legsthan static machines, and are potentially faster.

2. Dynamics vs. Control

Our interest is in dynamic walking machines, whichfor our purposes can be classified according to the roleofactive control in generating the gait. At one end ofthe spectrum is the biped of Mita et al.

(1984),whose

motion is generated entirely by linear feedback con-trol. At the end of one step, joint angles are com-manded corresponding to the end of the next step,and the controller attempts to null the errors. There isno explicit specification ofthe trajectory betweenthese end conditions. Yamada, Furusho, and Sano

( 1985) took an approach that also relies on feedback,but in their machine it is used to track a fully specifiedtrajectory rather than just to close the gap betweenstart and end positions. Meanwhile the stance leg is leftfree to rotate as an inverted pendulum, which, as weshall

discuss,is a

keyelement of

passive walking.Sim-

ilar techniques are used in biped walkers by Takanishiet al. (1985), Lee and Liao (1988), and Zheng, Shen,and Sias ( 1988).By contrast the bipeds of Miura and Shimoyama

(1984) generate their gait by feedforward rather thanfeedback; joint torque schedules are precalculated andplayed back on command. Again the stance leg is leftfree. However, the &dquo;feedforward&dquo; gait is unstable, sosmall feedback corrections are added to maintain the

walking cycle. Most significantly, these are not appliedcontinuously (i.e., for tracking ofthe nominal trajec-

tory).Instead the &dquo;feedforward&dquo; step is treated as a

processor whose output (the end-of-step state) varieswith the input (the start-of-step state). Thus the feed-back controller responds to an error in tracking bymodifying initial conditions for subsequent steps, andso over several steps the error is eliminated. In this

paper you will see analysis ofa similar process. Raibert(1986) has developed comparable concepts but with amore pure implementation, and applied them withgreat success to running machines having from one tofour legs. All ofthese machines use active control in some

form togenerate

the locomotion

pattern. Theycan be

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Fig. 1. A bipedal toy thatwalks passively down shal-low inclines. reprinted from(McMahon 1984).J

ordered according to the style of implementation,ranging from continuous active feedback to once-per-step adjustment ofan actively generated but neverthe-less fixed cycle. This paper discusses a machine at theextreme end of the spectrum: gravity and inertia alonegenerate the locomotion pattern, which we thereforecall &dquo;passive walking.&dquo;

3. Motivation for Passive Walking

The practical motivation for studying passive walkingis, first, that it makes for mechanical simplicity andrelatively high efficiency. (The specific resistance ofour test biped is -=0.025 in a human-like walk.) Sec-ond, control ofspeed and direction is simplified whenone doesn’t have to worry about the details ofgenerat-ing a substrate motion. Moreover, the simplicity pro-motes understanding. Consider an analogy with thedevelopment ofpowered flight. The Wrights put theirinitial efforts into studying gliders, as did their prede-cessors Cayley and Lilienthal. Once they had a reason-

Fig. 2. General arrangementof a 2D biped. It includeslegs of arbitrary mass andinertia, semicircular feet,and a point mass at the hip.

able grasp ofaerodynamics and control, adding a po-werplant was only a small change. (In fact, their enginewasn’t very good even for its day, but their otherstrengths ensured their success.) Adding power to apassive walker involves a comparably minor modifica-tion (McGeer 1988). Actually passive walkers existed long before contem-

porary research machines. My interest was stimulated

by the bipedal toy shown in Figure 1; it walks all byitselfdown shallow slopes, while rocking sideways tolift its swing foot clear of the ground. A similar quad-ruped toy walks on level ground while being pulled bya dangling weight. I learned ofsuch toys through apaper by Mochon and McMahon (1980), who showedhow walking could be generated, at least in large mea-sure, by passive interaction ofgravity and inertia.Their &dquo;ballistic walking&dquo; model is especially helpfulfor understanding knee flexion, which is discussedtoward the end ofthe paper.

Our discussion here is based on the model shown in

Figure 2, which is no more than two stiff legs pin-jointed at the hip. It can be regarded as a two-dimen-sional version ofthe toy; its dynamics in the longitu-dinal plane are similar, but it doesn’t rock sideways.This simplifies the motion, but it left us with newproblems in building a test machine: how to keep the



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Fig. 3. Biped used for exper-iments on two-dimensional

gravity-powered walking.The outer legs are connectedby a crossbar and alternatelike crutches with the center

leg. The feet are semicircularand have roughened rubbertreads. Toe-stubbing is pre-

vented by small motors that

fold the feet sideways duringthe swing phase. Apart fromthat, this machine, like thetoy, walks in a naturallystable limit cycle requiringno active control. Leg lengthis 50 cm, and weight is3.5 kg.

motion two-dimensional, and how to clear the swingfoot. Figure 3 is a photo of our solution. Two-dimen-

sionality is enforced by building the outer leg as a pairof crutches. Foot clearance is by either oftwomethods. Occasionally we use a checkerboard patternof tiles, which in effect retract the ground beneath theswing feet. However, usually it is more convenient toshorten the legs; thus the machine can lift its feet viasmall motors driving leadscrews.The discussion here begins with two elementary

models to illustrate the energeticsand dynamics ofpassive walking. Next follow analyses of cyclic walkingand stability for the general model ofFigure 2, andcomparison with experimental results. Then comes a

survey of parametric effects, and finally some com-ments on extensions of the model.

4. Reinventing the Wheel

Imagine an ideal wagon wheel that can roll smoothlyand steadily along a level surface, maintaining any

speedwithout loss of

energy.Its

rollingseems

quitedifferent from walking (and on the whole more sensi-ble !), but in fact rolling can be transformed to walkingby a simple metamorphosis.

4.1. The Rimless Wheel

Following Margaria (1976), remove the rim from thewagon wheel as in Figure 4, leaving, in effect, a set of

legs. Unlike the original wheel, this device cannot rollsteadily on a level surface; instead it loses some speedeach time a new leg hits the ground. We treat each ofthese collisions as inelastic and impulsive. In that casethe wheel conserves angular momentum about the

impact point, and the loss in speed can be calculatedas follows. Immediately before the collision the angularmomentum is

(Note that the wheel’s radius of gyration is normalized

by leg length l.) Immediately after the collision, theangular momentum is simply

Equating these implies that

All ofour analysis is cast in dimensionless terms, withmass m, leg length I, and time~ providing the base



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65

units. To define what amounts to a dimensionless

pendulum frequency (7),

Then

It follows from (3) that over a series ofk steps

Hence on a level surface the rimless wheel will decel-

erate exponentially. However, on a downhill grade,say with slope y, the wheel can recoup its losses and soestablish a steady rolling cycle. The equilibrium speedcan be calculated from the differential equation forrotation about the stance foot. Over the range of anglesused in walking, linearization for small angles is en-tirely justified, so the equation can be written as

Here 0 is measured from the surface normal; thus thewheel has an unstable equilibrium at 0 = - y.One &dquo;step&dquo; begins with 0 = &horbar;ao, and rolling is

cyclic if the initial speed, say Qo, repeats from onestep to the next. Repetition of.oo implies that eachstep must end with rotational speed ilo/i7 (6). Thus thesteady step has the following initial and final states:

Both Qo and the steady step period To can be evaluatedby applying these boundary conditions to the equationof motion (7). The results are

Fig. 4. Removing the rimfrom a wagon wheel allows a

simple illustration ofwalk-

ing energetics. On a levelsurface a rimless wheelwould grind to a halt; but onan incline it can establish a

steady rolling cycle some-

what comparable to walkingof a passive biped (see Fig.6). The speed, here in unitsof g , is a function of theslope, the inter-leg angle2ao, and the radius of gyra-tion r gyr about the hub.

For small oea the dimensionlessforward speed is then

Figure 4 includes plots ofthis speed as a function ofslope for various rimless wheels. These may be com-pared with speed versus slope in passive walking ofabiped, which is plotted in Figure 6. The plots are qual-itatively similar and quantitatively comparable. As anexample, take ao = 0.3, which is typical ofhumanwalking. Our test machine achieves this stride on aslope of2. %, and its forward speed is then 0.46 mls,or 0.21 in units of ig-l. This speed would be matched

bya rimless wheel

having rgyr =0.5.

Of course the wheel need not always roll at its steadyspeed. However, it will converge to that speed follow-

ing a perturbation. In fact, small perturbations decayaccording to

We call this decay the &dquo;speed mode,&dquo; which also ap-pears in bipedal walking. Incidentally, it is interestingto observe that convergence on a downhill slope (12)is twice as fast as deceleration on the level (6).



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Fig. 5. Walks like a biped;rolls like a wheel. It’s a

&dquo;synthetic wheel,&dquo;

made

with two legs, two semicircu-lar feet, and a pin joint atthe hip. On each step the freefoot swings forward to syn-thesize a continuous rim. Ifsupport is transferred from

trailing to leading foot whenthe leg angles are equal andopposite, then the walkingcycle can continue withoutloss ofenergy. The period ofthe cycle (here normalizedby thependulumfrequency ofthe swing leg) is independentof the step length.

4.2. The Synthetic Wheel

In view of the poor energetics that result, it seems thatrim removal is not a very progressive modification ofan ordinary wheel. But perhaps improvement mightbe realized by making cuts elsewhere. In particular,imaging splitting the rim halfway between each spoke.Then discard all but two ofthe spokes, leaving a pairof legs with big semicircular feet as shown in Figure 5.Put a pin joint at the hip, and ask the following ques-tion : could the dynamics be such that while one leg isrolling along the ground, the other swings forward injust the right way to pick up the motion where thefirst leg leaves off?

In fact, one can devise a solution quite easily. Figure5 shows a cycle that will continue indefinitelyon levelground. The legs start with opposite angles ± GO andequal rotational speeds SZo, as they did in the originalwagon wheel. The appropriate value for ao depends on

~20, as we will show in a moment. Since the stance leg(subscripted C for &dquo;contact&dquo;) is a section of wheel, itrolls along the ground at constant speed. (Here wepresume that the wheel has a large point mass at thehip; otherwise motion ofthe swing leg would disturbthe steady rolling.) The hip (like the hub ofan ordi-nary wheel) therefore translates steadily parallel to theground, and so the second leg (F for &dquo;free&dquo;) swings asan unforced pendulum. Then following the paths

shown in Figure 5 the legs will reach angles ± ao withspeeds again equal to no. At that instant, support canroll seamlessly from one rim to the next. Thus a con-tinuous rim has been synthesizedfrom two small pieces.Of course something must be done to clear the free

leg as it swings forward, but we shall deal with thatproblem later. Also the cycle works only if the stepangle ao is correct for the speed Qo; the relation is de-rived by matching boundary conditions. First, by in-spection of the stance trajectory in Figure 5, no mustsatisfy

To is determined by the swing leg, which behaves as apendulum and so follows a sinusoidal trajectory. Itsformula is

The sinusoid passes through AOF = -afl, with speed

go, when

Thus the step period for a synthetic wheel is about i ofthe period for a full pendulum swing. Notice that thisperiod is independent of a.~ . The speed (13) is then

Thus to change the speed ofa synthetic wheel, changeao (i.e., the length ofthe step), while To remains constant-determined solely by the leg’s inertial propertiesand gravity g.Of course the synthetic wheel is contrived for conve-

nient analysis, but the results have broader applica-tion. Our test machine has similar behavior. Figure 6shows that its step period is fairly insensitive to steplength; moreover io -~ 2.8 in units of Iïfi and co, -1.39, so COFTO ~ 3.9, which is quite close to the valuegiven by (15). However, (15) is not so accurate forhuman walking; my own pendulum period, measured

by standing on a ledge and dangling one leg, is ~ 1.4 s,



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Fig. 6. Step period To (inunits of~, here 0.226 s)and angle ao in passivewalking of our test biped.Bars show experimental data

from trials such as thatplotted in Fig. 7. Continuouscurves show analytical re-

sults, with uncertainty bandscarried forward from mea-surement of the biped’s

parameters as listed on the

plot. The analysis matchesexperiment poorly ifnoallowance is made for rollingfriction (Tc), but reasonableconsistency is realized if thefriction is taken to be0.007 mgl. With this level of

friction, walking is calcu-lated to be unstable on slopesless than 0.8%.

and my step period varies from --~=0.55 s at low speedto ~ 0.54 s at high speed. Thus the ratio of step periodto pendulum period is -~ 0.36 to ~ 0.39, as opposed to

0.65 for the wheel (15). I have significantly more lifein my stride than either the wheel or our test biped!tWhile Figure 5 shows that a synthetic wheel is possi-

ble in theory, can its cycle actually be realized in prac-tice ? This boils down to an issue of stability, and itturns out that the synthetic wheel is just as stable as anordinary wheel. That is, both have neutrally stable&dquo;speed modes&dquo;; in the case of a synthetic wheel, achange in speed leads to a new ao according to (16). Inaddition, the synthetic wheel has two other &dquo;step-to-step&dquo; modes that prove to be stable; these are discussedin section 8.

5. Steady Walking of a General 2D Biped

The biped ofFigure 2 is similar to the synthetic wheelbut allows for broader variation ofparameters. Adjust-able parameters include

1. Foot radius R

2. Leg radius of gyration r gyr

3. Leg center ofmass height c4. Fore/aft center ofmass offset w5. Hip mass fraction mH

We will examine the effects of varying each parameter.Our calculations ofgait and stability rely on step-to-

step (S-to-S) analysis, which is explained as follows. Atthe start ofa step, say the k~’, the legs have equal andopposite angles ±ak and rotational speeds ilCk, 5~~.The motion proceeds roughly as for the syntheticwheel, with appropriate arrangements to prevent toe-

stubbing at mid-stride. The step ends when leg anglesare again equal and opposite, but in general different(say ak+l) from ak. At that point the swing foot hitsthe ground, and the leg speeds change instantaneouslyto their initial values for the following step 92ck+,,QPk+l’ We will formulate equations relating(ak, n ck, QFk) to (ak+l, i2ck+,, QFk+l)’ We will thenuse these to find naturally repeating initial conditions(i.e., passive cycles) and to examine how perturbationsevolve from step to step.

5.1. Start- to End-of-Step Equations

During the step the machine is supported on one foot,and its state is specified by the two leg angles Bc, 8~,and the speeds S~c, S2F. Ingeneral the equations ofmotion are nonlinear in these variables, but since inwalking the legs remain near the vertical and thespeeds remain small (

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68

d8 is the rotation from the surface normal. For small

y, 8.8SE can be written as

8.8SE = 8.8w + by (19)

By way ofexplanation you can imagine that if the feetwere points (i.e., zero foot radius) then the equilib-rium position would be legs-vertical, which means thatboth elements of b would be -1. However with non-

zero foot radius the stance leg would have to rotatepast the vertical to put the overall mass center over the

contact point, so hi < -1. Also moving the legs’ masscenters fore and aft from the leg axes (i.e., nonzero w,Fig. 2) would make for nonvertical equilibrium evenon level ground, and this is handled by 8.8w.

Putting the gravitational torque (18) into the equa-tions ofmotion (17) leaves a fourth-order linear sys-tem. This can be solved to jump from the start-of-stepto any later time, via a 4 X 4 transition matrix D:

The elements of D for a given Mo, K, and Tk are calcu-lated by standard methods for linear systems. Fortransition to the end-of-step, Tk must be chosen so thatthe elements of ð8(L&dquo;J are equal and opposite. Thusdefine

Then

5.2. Support Transfer

When (23) is satisfied, foot strike occurs, which as forthe rimless wheel we treat as an inelastic collision. In

this case two conditions apply:

1. Conservation of angular momentum ofthewhole machine about the point ofcollision, asfor the rimless wheel (3).

2. Conservation of angular momentum of the

trailing leg about the hip.

These are expressed mathematically as

where&dquo;-&dquo;

and &dquo;+&dquo; respectively denote pre- and post-support transfer. The inertia matrices M- and M+

depend on the leg angles at foot strike, as discussed inthe appendix [eqs. (65), (68)]. The ordering of il maybe confusing, since the stance and swing legs exchangeroles at support transfer. We adopt the conventionthat the first element of 0 refers to the post-transferstance leg. Hence the pre-transfer indexing must beflipped:

From (24), then, the initial speeds for step k + 1 are

5.3. &dquo;S-to-S&dquo; Equations

Combining the start- to end-of-step equation (20) withthe foot strike conditions [eqs. (22), (23), (26)] pro-duces the S-to-S equations. It is convenient to break Din (20) into 2 X 2 submatrices, so the system iswritten as

Bear in mind that while formulation ofthis set has

been simple, evaluation is not quite so straightforward.Given initial conditions (ak, ilk), the time ik (whichdetermines D) at which the leg angles are next equaland opposite must be determined. Then 0~+1 (27)



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69

Fig. 7. Step period and anglemeasured at heel strike in

trials of the test biped. Themachine was started byhand on a 5.5-m ramp in-

clined at 2.5% downhill, and

after afew steps settled intoa fairly steady gait. Dots

show the mean values, andbars the scatter recorded

over six trials. Lengths arenormalized by leg length I.and periods by Jïïg. &dquo;C&dquo;denotes start-of stance on thecenter leg; &dquo;0 &dquo; on the outerlegs.

determines A (26), which in turn allows evaluation of~k+1 (28).

5.4. Solution for the Walking Cyclez

For cyclic walking, initial conditions must repeat fromstep to step:

Imposing these conditions on S-to-S equations (27)and (28) leads to a compact solution for the walkingcycle. First solve for no using (28):

Then substitute for no in (27); the end result can bewritten as follows. Define

Then (27) can be written as

The last line follows from (19). This is the steady-cyclecondition, with two equations in three variables a.~,z~, and y. Usually we specify aa and solve for the othertwo. Equation (33) then has either two solutions ornone. If two, then one cycle has (OF TO < ~c, and is in-

variably unstable. The other corresponds to a synthetic-wheel-like cycle with coFTO between n and 3n/2. Thisis the solution ofinterest. Thus we use the synthetic-wheel estimate (15) to start a Newton’s method searchfor To, and if a solution exists, then convergence tofive significant figures requires about five iterations.

6. Steady Walking of the Test Machine

The analytical solution for steady walking is comparedwith experiment in Figure 6. Experimental data wereobtained by multiple trials as shown in Figure 7; ineach trial our test biped was started by hand from thetop of a ramp, and after a few steps it settled into the

steady gait appropriate for the slope in use. (Notice,

incidentally, that the rate ofconvergence is a measureof stability; we will pursue this analytically in the nextsection.) Several trials were done on each slope, andmeans and standard deviations for ao and To were

calculated over all data except those from the first few

steps ofeach trial.Our predictive ability to compare with analysis was

limited by uncertainty in the machine’s parameters.Each leg’s center-of-mass height c was measured toabout 1 mm by balancing each leg on a knife edge; wto about 0.5 mm by hanging each leg freely from thehip; and r~ to about 2.5 mm by timing pendulum

swings. The center and outer legs were ballasted tomatch within these tolerances and differed in mass byonly a few grams (i.e., -= 0.001 1 m).When these parameters were put into (33) we found

(as Figure 6 indicates) that the observed cadence wasslower than predicted. We suspected that the discrep-ancy was caused by rolling friction on the machine’srubber-soled feet. Hence we added a constant to thefirst element ofT in (17), which modified aBw in (19).

A value of -0.007 brought the analytical results intoline with observation. In dimensional terms this is

equivalent to 25 gm-force applied at the hip. We could



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70

not devise an independent measurement of rollingfriction, but this figure is credible.However, after adjusting friction for a reasonable fit,

we still found a discrepancy in that the observed stepperiod is less sensitive to speed than the analysis indi-cates. We therefore suspected that the inelastic modelfor foot strike was imprecise, and in fact there was abit ofbouncing at high speed. But in any case theresidual discrepancy is small, so our current S-to-Sequations (27) and (28) apparently form a sound basisfor further investigation, in particular of stability and

parametric variations.Here the efhciency of biped walking should benoted. For comparison with other modes of transport,we measure efficiency by specific resistance:

For a vehicle powered by descending a slope, SR isjust equal to y, or about 2.5% for our machine usingao = 0.3, which is comfortable for humans. This figurewould be terrible for a car, but it is very good in com-

parison with other rough-terrain vehicles such as mul-tilegged crawlers and bulldozers (Waldron 1984).

7. Linearized Step-to-Step Equations

A cyclic solution is a necessary but not quite sufficientcondition for practical passive walking. The walkingcycle must also be stable. Stability can be assessed by

linearizing eqs. (27) and (28) for small perturbationson the steady gait. The transition matrixD and thesupport transfer matrix A can be approximated for this

purpose as follows:

After substituting these into (27) and (28) and mani-

Table 1. S-to-S Eigenvectors of the Test Machine on a2.5% Slope

pulating to collect terms, we are left with the following

approximate form ofthe S-to-S equations:

The formula for S is given in the appendix [eqs. (69),(70), and (72)]. The important point here is that S isthe transition matrix of a standard linear difference

equation, so the eigenvalues of its upper 3 X 3 blockindicate stability [the equation for (zk - io ) is ancil-

lary]. If all have magnitude less than unity, then thewalking cycle is stable; the smaller the magnitude, thefaster the recovery from a disturbance. (If, however,the walking cycle is unstable, then linearized S-to-S

equations are helpful for design of a stabilizing controllaw. This was in essence the approach of Miura and

Shimoyama (1984). See also McGeer (1989) for activestabilization of a passive cycle.)

8. S-to-S Modes of the Test Machine

Results of stability analysis for our test biped are listedin Table 1. Similar results are found over a wide rangeof parameter variations, so the modes can be consid-ered typical for passive walking.The &dquo;speed mode&dquo; is analogous to the transient

behavior of a rimless wheel, with z corresponding to

q2 in (12). The eigenvalue is linked to energy dissipa-tion at support transfer. Thus for the synthetic wheel,z = 1 in this mode (i.e., stability is neutral with respectto speed change), as we noted earlier.



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The &dquo;swing mode&dquo; is so named because the eigen-vector is dominated by L2F. The eigenvalue of thismode is usually small, and in fact reduces to zero inthe synthetic wheel, which means that a &dquo;swing&dquo; per-turbation is eliminated immediately at the first sup-port transfer. Physically this occurs because the post-transfer speed of the synthetic wheel’s legs isdetermined entirely by the momentum ofthe large hipmass; the pretransfer OF is irrelevant.

Finally, the &dquo;totter mode&dquo; is distinguished by anegative eigenvalue, which means that perturbationsalternate in sign from one step to the next. Again thesynthetic wheel offers an easily understood specialcase. Since the wheel cannot dissipate energy, it retainsits initial rolling speed for all time. If the initial stepangle is not appropriate to the initial speed as specifiedby (13), then the angle must accommodate throughsome sort oftransient, which turns out to be the tottermode. The eigenvalue can be found analytically bygeneralizing the swing trajectory formula (14) for ak #0!o:1

,

Meanwhile the generalized stance trajectory [cf. ( 13)) is

Now differentiate eqs. (37) and (38) with respect to ak:

Solving for the derivatives and evaluating at fo gives

It follows from (41 ) that perturbations converge expo-nentially according to

In summary then, and with some oversimplificationfor clarity, the &dquo;speed&dquo; mode in passive walking is amonotonic convergence to the speed appropriate forthe slope in use. The &dquo;swing&dquo; mode is a rapid adjust-ment ofthe swing motion to a normal walking pattern.The &dquo;totter&dquo; mode is an oscillatory attempt to match

step length with forward speed. An arbitrary perturba-tion will excite all three modes simultaneously. Ofthethree, the totter mode differs most between the syn-thetic wheel and our test biped (z = -0.2 vs. -0.83),and our parametric surveys indicate that it bears

watching. We will show examples ofparameter choicesthat make the totter mode unstable.

9. Larger Perturbations

The preceding analysis applies only for small perturba-tions, but an obvious question is, &dquo;How small issmall?&dquo; There is a limit: if the machine were started

just by standing the legs upright and letting go, itwould fall over rather than walk! Thus starting requiresa bit more care. In general, the machine could be re-leased with arbitrary leg angles and speeds, and itwould be nice to know the &dquo;convergence zone&dquo; in thefour-dimensional state space. Unfortunately the

boundary of a 4D volume is rather difficult to map, sowe have restricted attention to finding the maximumtolerable perturbation in the initial 92c or ’2F. This isdone by serial evaluation of the nonlinear S-to-S equa-tions (27) and (28). The results show that ’2F can varyover wide limits (e.g., < 0 to > +150% ofthe cyclicvalue). However, 92c is sensitive; many passive walkerscan only tolerate an error ofa few percent in the start-

ing speed. Still this is not so exacting as it sounds;manual starting (by various techniques) is quite natu-ral, and success is achievable with little or no practice.Furthermore, the boundaries of the convergence zone

are sharp, and ifthe machine starts even barely inside



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the edge it settles to the steady cycle very nearly assuggested by the small-perturbation analysis. An alternative to the convergence-zone measure of

robustness is resistance to jostling. Thus we calculatedtransients produced by disrupting the steady gait witha horizontal impulse, applied at the hip just as the legspassed through A0 = 0. As an intuitive standard ofreference, we compared the walker with a similar ma-chine resting with legs locked at :t: ao. It turns out thata passive walker can tolerate a useful fraction ofthe

impulse required to topple the locked machine; the

level varies widely with choice of machine parameters,but 25% (forward or backward) is representative. Thecalculation has to be done carefully, since bands oftolerable and intolerable impulse magnitudes are in-terspersed. Twenty-five percent is a typical upperbound for the first tolerable band. More aggressivejostling would have to be countered by active control.The jostling calculation was particularly helpful in

dispelling the inclination to associate robustness withrapid convergence from small perturbations. In fact, abiped with a slowly convergent or even slightly unsta-ble totter mode may well tolerate a stronger midstride

jostle than a machine with better totter stability. Thusa practical biped designer might be willing to accept arequirement for &dquo;weak&dquo; active stabilization of thesteady cycle in return for better resistance to knock-downs.

10. Effect of Parameter Variations

We now embark on a brief survey ofthe effect ofde-

sign variables on walking performance. Our purpose istwofold: first, to outline the designer’s options, andsecond, to illustrate that walking cycles can be foundover a wide range ofvariations on the 2D biped theme.

10.1. Scale

Before discussing variation of dimensionless parame-ters we should note the effect ofscale (i.e., changing

Fig. 8. Step period and slopefor passive walking of bipedshaving various foot sizes.The step angle is specified tobe 0.3, which is typical inhuman walking, and the

mass center and inertia are

held constant while R is

varied. The slope (and so theresistance) is zero with R =

I, as for the synthetic wheel.

m, I, or g). Quite simply, changing m scales the forcesbut doesn’t change the gait. Changing I scales the stepperiod by 1/.Ji, and the speed by ~-1. Changing g scalesboth period and speed by ~. A noteworthy conse-quence was experienced by the lunar astronauts in

1 /6th Earth’s gravity. Apparently they had a sensationofbeing in slow motion, and indeed walking couldachieve only 40% ofnormal speed. Rather than acceptthat, they hopped instead (McMahon 1984).

10.2. Foot Radius

Figure 8 shows an example ofthe effect of foot radiuson steady walking. Most notable is the improvementin efficiency as the foot changes from a point to asection ofwheel. Thus with R = 0 we have a bipedthat, like the rimless wheel, needs a relatively steepslope, and with R = 1 we have a synthetic wheel rollingon the level.

Figure 9 shows the locus ofS-to-S eigenvalues as afunction of R. The speed eigenvalue increases fromz= 0.2 with a point foot to z = 1 (i.e., neutral speedstability) with a wheel; this is associated with the im-provement in efhciency and is consistent with ( 12).Meanwhile the totter and swing modes are well sepa-rated with a point foot, but coupled with mid-sized



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Fig. 9. Locus of the threestep-to-step eigenvalues forthe walking cycles plotted inFig. 8, with foot radius asthe parameter. In this exam-

ple one eigenvalue (for the&dquo;totter mode’J lies outside

the unit circle ifthe feet arevery small; this indicatesthat passive walking is un-stable. However, with other

choicesfor c and r gyr’ walk-ing is stable even on pointfeet.

feet so that the two eigenvalues form a complex pair.Then with larger foot radius they separate again, andR = 1 puts the totter eigenvalue near z = -1. [A truesynthetic wheel has z = -0.2 (43), but its hip mass ismuch larger than specified in this example.] The big-footed bipeds in this example thus have relativelyweak totter stability, but it still turns out that their

resistance to jostling is better than with small feet.We should mention that the semicircular foot is amathematical convenience rather than a physical ne-cessity ; doubtless other arrangements are feasible. Forexample, a flat foot could be used on which supportwould transfer impulsively from heel to toe at mid-stride. Walking would be less efficient than with acurved foot, but apart from that we would expect asimilar passive gait. For comparison with a circularfoot, the key feature is translation ofthe support pointduring stance, which is 2 aoR. Thus a human, withheel-to-ball distance of ~ 0.21 and a typical stride

angle of ao = 0.3, has an &dquo;equivalent radius&dquo; of = 0.3.

10.3. Leg Inertia and Height ofthe Mass Center

Leg radius ofgyration and center-of-mass height havesimilar effects, as illustrated in Figures 10 and 11.Increasing r~, or raising c with r gyr held constant,lengths the pendulum period and so slows the cadence( 15). If the mass center is too close to the hip, then the

Fig. 10. Variation of thewalking cycle with c, theheight of the leg’s masscenter. lzl is the magnitude ofthe totter mode eigenvalue;here it indicates that passivewalking is unstable forbipeds with c too low or too

high. (The kinks appearwhere z branches into the

complex plane, as in Fig. 9)The high-c problem arisesbecause ofexcessively longswing-pendulum periods,while the low-c problem iscaused by inefficient supporttransfer. The latter can beremedied by adding mass at

the hip. In this example, mvis zero.

swing leg can’t come forward in time to break the fallofthe stance leg, and the walking cycle vanishes. Onthe other hand, lowering rgyr or c causes a different

problem: support transfer becomes inefficient accord-

ing to (5). [Note that ao in (5) is the angle subtendedby the feet, at the overall mass center.] If the efficiencyis too poor, then the cycle becomes unstable or evenvanishes entirely, as in the example of Figure 11.However, the situation can be retrieved by addingmass at the hip, which raises the overall mass centerand thus improves efficiency.. A human has about 70%of body mass above the hip; this is sufficient to makesupport transfer efficient regardless of leg properties.

10.4. Hip Mass

Figure 12 illustrates the effect ofadding point mass atthe hip. Efficiency improves, and it turns out thatjostling resistance improves as well. Still more advan-tage can be gained if the mass is in the form ofanextended torso. For example, the torso can be held ina backward recline, reacting against the stance leg; thereaction provides a braking torque that allows a steepdescent. Analysis of this scheme and other roles ofthetorso is reported by McGeer (1988).



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Fig. I]. Leg inertia and chave similar effects on thewalking cycle, as indicatedby comparing this plot withFig. 10. The effects here arealso mediated by the swingpendulumfrequency and bythe efficiency of support

transfer.

Fig. J 2. Our test biped isjust a pair of legs, but passivewalking also works whilecarrying a &dquo;payload &dquo; at thehip. Actually the added massimproves e,fficiency of thewalking cycle.

10.5. Hip Damping and Mass Offset

The last few examples have indicated that passivewalking is robust with respect to parametric variations.However, it is not universally tolerant, and the hipjoint is particularly sensitive. As shown in Figure 13,introducing only a small amount of friction makes the

cyclic solution vanish. Consequently, in our test bipedwe use ball bearings on the hip axle, and these keepthe friction acceptably low. However, alternative me-

Fig. 13. Passive walking isforgiving ofmost parametervariations, but even a smallamount of friction in the hipjoint can destroy the cycle.However, walking can berestored by moving the legs’mass centers backward from

the leg axes (w < 0, Fig. 18)as long as the friction is onlymoderate. For this examplewe have used a viscous

model for friction, which ismeasured by the dampingratio for pendulum oscilla-tions of the swing leg.

chanical arrangements, for instance those made bybone and cartilage, may not be quite as good. Fortu-nately, compensation can be made by lateral offset (w)ofthe legs’ mass centers from the axes. Thus, as Figure13 indicates, if the joint has significant friction, thenthe designer should shift the leg mass backward fromthe line between hip and foot center ofcurvature (i.e.,

w

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Fig. 14. Experiments con-firm the powerful effect ofmass offset w on the walkingcycle. (w was adjusted byshifting the feet relative tothe legs.) With each settingwe did a series of trials as in

Fig.7. ao

and Toare well

matched by calculations ifTc is taken to be - 0.007,but then walks that were in

fact sustained for the fulllength ofthree six-foot tablesare calculated to be unstable.

Taking Tc to be zero gives abetter match to the observed

stability, but leaves relativelylarge discrepancies in ao andTo. We suspect that our

support transfer model isimprecise, but in any casethe sensitivity to w is clear.

investigation we calculated walking cycles with legs ofdifferent mass. The steady-walking solution (33)doesn’t apply with mismatched legs, so instead we didserial calculations with S-to-S equations (27) and (28)until a steady gait emerged. With a small mismatch,the cycle repeats in two steps (as would be expected).But with larger mismatch, the stable cycle repeats infour steps, as plotted in Figure 15. Here, then, is anexample of frequency jumping, which is not uncom-mon in nonlinear systems.

11. Fully Passive Walking

We mentioned earlier that our 2D, stiff-legged modelallows simple analytical treatment of walking, but itforces resort to inelegant methods ofclearing the swingfoot. Fully passive foot clearance would be preferable;we will discuss two options.

11.1. Rocking

The toy ofFigure 1, and its quadrupedal cousins, clear

swing feet by lateral rocking. The frontal view shows

Fig. 15’. The stable cyclecalculated for a biped havinglegs with a 1 D% mass mis-match. me is the mass of thestance leg. Each step is

similar to that of a syntheticwheel, but the cycle repeatsaver four steps rather thanone. (With smaller mismatchthe cycle repeats in twosteps

the key design feature, which is that the feet are ap-proximately circular in lateral as well as longitudinalsection, and have approximately coincidental centersof curvature. The center ofcurvature is above the

center of gravity, which makes lateral rocking a pen-dulum oscillation.We have analyzed the combination ofrocking with

2D syntheticwheel dynamics, under the approxima-tion of zero yaw (i.e., that the hip axle remains normalto the direction of motion). This is exactly true for

quadrupedal toys and seems fairly accurate for thebiped. The rocking frequency must be tuned to theswing frequency, such that halfa rocking cycle is com-pleted in one step. Hence the ratio ofswing to rockingfrequencies, from (15), should be

If this condition is satisfied, then rocking and swingingnaturally phase-lock at the first support transfer, andthereafter walking is non-dissipative. However, ifthecondition is not satisfied, then the phasing must be&dquo;reset&dquo; on each step; this entails an energy loss. If the

mistuning is too large, then there is no cycle at all. An additional problem arises if the feet have non-zero lateral separation. Then support transfer bleedsenergy out ofthe rocking motion via the rimless-wheelmechanism (3). But it turns out that, at least in the



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zero yaw approximation, only a small amount ofen-ergy can be regained from forward motion (i.e., bydescending a steeper hill). Hence passive walking can-not be sustained unless the foot spacing is quite small.These constraints of frequency tuning and lateral

foot spacing reduce the designer’s options, and more-over produce a machine that is inevitably rather tippyfrom side to side. Hence rocking is unattractive for apractical biped, but it remains a wonderful device forthe toy.

11.2. Knees

Of course we humans rock as we walk, but only forlateral balance; for foot clearance we rely on kneeflexion. Mochon and McMahon (1980) demonstratedthat this motion might well be passive. The key resultofthe study is that if a straight stance leg and knee-jointed swing leg are given initial conditions in anappropriate range, then they will shift ballistically fromstart- to

end-of-step angles. Recentlywe have found

that support transfer can then regenerate the start-of-

step conditions, thus producing a closed passive cycle. A full discussion of passive walking with knees mustbe left for a future report, but here we will review one

example as shown in Figure 16. The model is still a2D biped, but with pin-jointed knees and mechanicalstops, as in the human knee, to prevent hyperexten-sion. As in ballistic walking, the stance knee is specifiedto remain locked against the stop throughout the step,while the swing knee is initially free and flexes pas-sively as plotted in Figure 16. Then it re-extends andhits the stop. (As always we treat the collision as im-

pulsive and inelastic.) Thereafter both knees remainlocked until foot strike.

Parametric studies of this knee-jointed model haveproduced these results:

1. Passive cycles are found over parameter rangesmore limited than those in stiff-legged walking,but still quite broad.

2. With proper choice of parameters, naturallyarising torques keep the stance and swing kneeslocked through the appropriate phases of the

cycle.

Fig. 16. Our test biped hasrigid legs, but passive walk-ing also works with knees.The parameters here are

quoted according to the con-vention of Fig. 18 and aresimilar to those of a human(including mH = O.b7c5).(A) Flexure ofthe swingknee takes care of foot clear-ance (althoughjust barely).Meanwhile a naturally gen-erated torque holds the

stance knee locked agair¡ -t amechanical stop. (EK is theangle from the locked knee to

the foot’s center of curva-ture.) (B) The motion isobviously similar to human

walking, but it is slower; theperiod of this cycle would be=0.85 sfor a biped with 1-m

leg length. There is also afaster cycle, but we prefer theslower because it turns out to

be stable and more efficient.

3. A passive cycle with knee joints is possibleonly if the foot is displaced forward relative tothe leg, as shown in Figure 16. The implicationis that humans might have difficulty if theirfeet didn’t stick out in front of their legs.

4. A large fraction ofthe swing leg’s kinetic en-ergy is dissipated at knee lock, and therefore



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knee-jointed walking is less efhcient than stiff-legged walking. To keep the penalty small, theswing leg’s energy must be small comparedwith the total energy ofmotion. This impliesthat most of the mass should be above the hip.

On the last point it might be argued that foot clear-ance with knees seems inefficient, whereas active re-traction can in principle consume no energy at all.However, in practice economical food retraction maybe hard to achieve. Our test biped has a &dquo;fundamen-tal&dquo; dissipation (from Figure 6) of = 0.2 J per step,while the tiny motors that lift its feet waste - 3 J perstep! Some goes into friction in the leadscrews, andsome into switching transistors. At this rate, passiveknee flexion is a bargain, and moreover a small activeintervention at the right time may reduce the knee-locking loss substantially..

12. Achon on Passive Walking

Having surveyed the physics of passive walking, ournext objective must be exploitation ofthe effect tomake machines with some practical capability. We seedevelopments proceeding as follows:

1. &dquo;Powered&dquo; passive walking on shallow up-and downhill slopes in 2D

2. Step-to-step gait modulation over unevenly-spaced footholds in 2D

3. Walking on steep slopes, stairs, and 2D roughterrain

4. Lateral balance and steering

We are presently building a machine to test the firsttwo developments. It will be similar to the first biped,but with 1-m leg length and a torso. Power will beprovided by pushing with the trailing leg as it leavesthe ground. (This is analogous to plantar flexion inhumans, but the implementation is quite different.)Gait modulation is to be done by applying torquesbetween the legs and torso. McGeer (1988) presentsthe analytical basis for this machine, as well as alterna-tive options for applying power and control. S-to-Streatment is a key principle; thus control laws look atthe state ofthe machine only once per step. This ap-

proach proves attractively simple, both mathematicallyand mechanically.We have yet to proceed to the next problem of

walking with steep changes in elevation. One promis-ing strategy is to seek passive fore/aft leg swinging thatcan proceed in synchrony with actively cycled varia-tion in leg length. This approach works on shallowslopes, with the length adjustments serving as a sourceof energy (McGeer 1988). However, further resultsawait reformulation ofequations of motion (17) and

(18), since the linearization used here is invalid on

steep slopes.Finally, we expect that lateral balance will have tobe done actively. One possibility is to control as whenstanding still (i.e., by leaning in appropriate direc-tions). Another scheme more in line with the &dquo;S-to-Sprinciple&dquo; is once-per-step adjustment oflateral footplacement; this has been analyzed by Townsend(1985). Turning, at least at low rates, can be done bythe same mechanism.

Much of this discussion and development strategycan also apply to running. Running is of practicalinterest because walking has a fundamental speed limit

oforder ig-l-, at higher speeds centrifugal effect wouldlift the stance leg off the ground. It turns out that byadding a torsional spring at the hip and translationalsprings in each leg, the model ofFigure 2 becomes apassive runner, with the same features of simplicity,efficiency., and ease of control that make passive walk-ing attractive. Passive running has been described byMcGeer (1989), and Thompson and Raibert (1989)have independently found similar behavior with both

monoped and biped models. Investigation thus far hasbeen limited to shallow slopes; as in walking, we haveyet to do the steep-terrain and 3D investigations.We hope that passive models will provide muchinsight into the dynamics and control oflegged ma-

chines, but ultimately we must admit that this is onlythe easy part ofthe design problem. Legged locomo-tion is not competitive on smooth terrain; thereforepractical machines must be capable of finding foot-holds and planning paths through difficult surround-ings. Yet contemporary robots are hard pressed even topick their way along paved highways and across flatfloors! Thus it appears that the demands ofa leggedautomaton must stimulate research for some years tocome.



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Fig. 17. Notation for anN-link, two-dimensional

open chain with rolling sup-port.

Appendix A. Dynamics of an N-Link Chain

Although we are interested in a machine with onlytwo rigid links, there is little extra effort involved ingeneralizing to N links, and the more general result isneeded for a knee-jointed machine like that in Figure16. The dynamics can be expressed in N second-order

equations ofthe form

where Hn is the angular momentum and Tn the torqueabout the nth joint. In the case of n = 1, the &dquo;joint&dquo; is

the instantaneous point ofcontact. Simplification willbe afforded by subtracting equations for successivejoints, so that (45) becomes

The torque is produced by gravity, and amounts to

Figures 17 and 18 provide the necessary notation. Then

where - is the vector from joint n to the mass centeroflink n, and I’ the vector from joint n to joint n + 1.For link 1

x, and §i are unit vectors fixed in link 1, as illustratedby Figure 18.For walking analysis we linearize the torque for

small ~6, y. Thus from (18) and (19) we have

where (Kâ8w)11 is the value of (48) at e8 = 0, y = 0;(Kb), is the derivative of (48) with respect to y, evalu-ated at A0= 0, y = 0; and K&dquo;m is the derivative of (48)with respect to 6m, evaluated at !:J.8 = 0, y = 0. For atwo-link chain with a point hip mass mH, the resultsare as follows:

Note that according to the convention of Figure 17, abiped with matched legs has



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79

Fig. 18. Notation for individ-ual links of the chain.

Now to work on the angular momentum terms in the

equations ofmotion (46). Hn is

Therefore

Equation (58) should be recognized as the angularmomentum about a stationarypoint that is instanta-neously coincident with joint n. Therefore, for the

equationsofmotion

(46) only Cnand

V,are differen-

tiated :

The kinematics of the chain are such that

where the an vector is directed into the page ofFigure17. This is differentiated for (60) as follows:

The last term comes from differentiating ’11 (49), andaccounts for rolling ofthe contact point. Substitutinginto (60) leaves (after substantial but straightforwardsimplification)

I



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80

In matrix terms, as in (17), this can be expressed as

In sti$=Iegged walking the rotational speeds are small,so the Q2 term (centrifugal effect) can be neglected.

(Thisapproximation, however, is not valid for knee-

jointed walking as in Figure 16). M is the inertia ma-trix ; for a two-link biped

In equation ofmotion (17), M is linearized; thus Mois found by evaluating (65) with ð.8 = 0. However, forsupport transferM+ (24) is evaluatedwith ð.8 = Aa (22).Support transfer also involves the pre-transfer inertia

matrix M-, which is calculated as follows. Equation(59) still holds for the angular momentum, but {6i) isno longer correct for the velocity, since prior totransfer the chain is rolling about linkN rather thanlink 1. Thus recasting {61 ) gives

Insertingthis into

(59)and

collectingterms leaves

For our machine, in matrix form, this is

Appendix B. Step-to-Step Transition Matrix

The S-to-S transition (36) has been derived byMcGeer (19 8); here we present only the final formula.

where

With the torque given by (18), the time derivative ofDis



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Appendix C. Explanation of Symbols

Roman

b derivative ofstatic equilibrium w.r.t. slope (19), (53)c Figs. 2, 18D start- to end-of-step transition matrix (20)

D~, submatrices of start- to

Don, end-of-step transition matrix

D~, (27)~ (28)Dan.D’ (32)F index exchanger (25)g gravitational acceleration, Fig. 2H angular momentumI 2 X 2 identity matrixK stance stiffness matrix (54)1 leg length, Figs. 2, 18/ joint-to-joint vector (50), Fig. 18

M inertia matrix (65), (68)Mo MfbrA0=0(17),(65)m mass, Figs. 2, 17mH hip mass fractionR foot radius, Fig. 2r joint-to-mass center vector (49), Fig. 17

r,,

radius ofgyrationS step-to-step transition matrix (36), (69)T torqueY linear velocityw offset from leg axis to leg mass center (Fig. 2)

8, ) unit vectors, Figs. 17, 18z eigenvalue of step-to-step equations (36)

Greek

a leg angle at support transfer (22), Figs. 4, 5y slope, Fig. 2

A8 angle from surface normal, Fig. 2

n coefficient of restitution (5) A8sE static equilibrium position (19)9w static equilibrium on level ground (52) A support transfer matrix (26) A support transfer vector (21 )a pendulum frequency (4)T dimensionless time t.fiïiQ angular speedcoF swing pendulum frequencycoR lateral rocking frequency

Sub- and superscripts+ immediately after support transfer- immediately before support transfer

0 steady cycle conditionsC stance legF swing legg gravitational effectk step index

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McGeer, T. 1988. Stability and control oftwo-dimensionalbiped walking. Technical report CSS-IS TR 88-01. SimonFraser University, Centre for Systems Science, Burnaby,B.C.

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Thompson, C. and Raibert, M. (in press). Passive dynamicrunning. In Hayward, V., and Khatib, O. (eds.): Int. Sym-posium of Experimental Robotics. New York: Springer-Verlag.

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mcgeer 1990

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