The effect of limb kinematics on the speed of a legged robot on granular media
Chen Li, Paul B. Umbanhowar, Haldun Komsuoglu, Daniel I. Goldman
TThe effect of limb kinematics on the speed of a legged robot on granular media
Chen Li , Paul B. Umbanhowar , Haldun Komsuoglu , and Daniel I. Goldman ∗ School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Department of Mechanical Engineering, Northwestern University, Evanston, IL, 60208, USA Department of Electrical and Systems Engineering,University of Pennsylvania, Philadelphia, PA 19104, USA ∗ Corresponding author. E-mail: [email protected]
Achieving effective locomotion on diverse ter-restrial substrates can require subtle changes oflimb kinematics. Biologically inspired leggedrobots (physical models of organisms) have shownimpressive mobility on hard ground but sufferperformance loss on unconsolidated granular ma-terials like sand. Because comprehensive limb-ground interaction models are lacking, optimalgaits on complex yielding terrain have been de-termined empirically. To develop predictive mod-els for legged devices and to provide hypothe-ses for biological locomotors, we systematicallystudy the performance of SandBot, a small leggedrobot, on granular media as a function of gaitparameters. High performance occurs only in asmall region of parameter space. A previously in-troduced kinematic model of the robot combinedwith a new anisotropic granular penetration forcelaw predicts the speed. Performance on granularmedia is maximized when gait parameters mini-mize body acceleration and limb interference, andutilize solidification features of granular media.
INTRODUCTION
To move effectively over a wide range of terrestrialterrain requires generation of propulsive forces throughappropriate muscle function and limb kinematics [1,2]. Most biological locomotion studies have focusedon steady rhythmic locomotion on hard, flat, non-slipground. On these surfaces kinematic (gait) parameterslike limb frequency, stride length, stance and swing dura-tions, and duty factor can change as organisms walk, run,hop and gallop [1]. There have been fewer biological stud-ies of gait parameter modulation on non-rigid and non-flat ground, although it is clear that gait parameters aremodulated as the substrate changes during challenges likeclimbing [3, 4], running on elastic/damped substrates [5],transitioning from running to swimming [6], and runningon different preparations of granular media [7]. Evensubtle kinematic changes in gait can lead to major dif-ferences in limb function [8]. A major challenge is to de-velop models of limb interaction with complex substratesand to develop hypotheses for how organisms vary gaitparameters in response to substrate changes. The RHex class of model locomotors (robots) hasproved useful to test hypotheses of limb use in biolog-ical organisms on hard ground [9] and recently on morecomplex ground with few footholds [10] or the ability toflow [11]. These hexapedal devices model the dynami-cally stable locomotion of a cockroach and were the firstlegged machines to achieve autonomous locomotion atspeeds exceeding one body length/s. In these devices,complexity in limb motion is pared down to a few biologi-cally relevant parameters controlling intra-cycle “stance”and “swing” phases of 1-dof rotating limbs (referred toas“gait” parameters hereafter; see detailed description inMethods and Results). When these gait parameters areappropriately adjusted, RHex shows performance compa-rable in speed and stability to organisms on a diversityof terrain [12]. However, because of the scarcity of exist-ing models of limb interaction with complex substrates,adjustment of the gait parameters is typically done em-pirically [13, 14].Sand, a granular medium [15], is of particular interestfor studies examining the effects of limb kinematics on lo-comotor performance on yielding terrain. In a previousstudy [11] we found that minor changes in the limb kine-matics of a small RHex-class robot, SandBot (Fig. 1a),produced major changes in its locomotor mode and per-formance (speed) on a granular medium, poppy seeds(see Section 2.2). This sensitivity occurs, in part, be-cause forced granular media remain solid below the yieldstress, but can flow like a fluid when the yield stress isexceeded [16]. We tested SandBot on granular media ofdifferent yielding properties (set by granular volume frac-tion; see Section 2.1) at various limb frequencies but withthe other gait parameters fixed. While there is no fun-damental theory at the level of fluid mechanics that ac-counts for the physics of the solid-fluid transition of gran-ular media or the dynamics of the fluidized regime, em-pirical models of granular penetration force have proveduseful to predict SandBot’s speed [11]. SandBot’s propul-sion is determined by factors that control this transitionduring limb-ground interaction (limb penetration depth,limb speed, body mass, grain friction, volume fraction,etc.). Using a simplified equation describing the granu-lar penetration force, we developed a kinematic model toexplain the locomotion of SandBot (see Section 2.2).In this study, we advance our understanding of theeffects of limb kinematics on locomotor performance by a r X i v : . [ phy s i c s . b i o - ph ] M a r testing SandBot with varying gait parameters on sand offixed yield strength and at fixed limb frequency. We findthat robot speed depends sensitively on limb kinemat-ics; while the original model qualitatively captures thissensitivity, the penetration force used in the model andother assumptions need to be modified to explain someimportant features. Our study not only reveals the spe-cific optimal kinematics for SandBot on granular media,but also advances our understanding of how in general toachieve effective legged locomotion on complex terrain. BACKGROUND AND REVIEW OF PREVIOUSSTUDY
To understand the effect of limb kinematics addressedhere, we first summarize the mechanism of SandBot lo-comotion on granular media (called rotary walking) dis-covered in our previous study [11]. In this section, wediscuss the physics of granular media that controls thelimb penetration depth (which governs locomotion per-formance) and then review our previous experiments andkinematic model.
Physics of Limb-Granular Media Interaction
The physics that controls locomotor performance isthe relative magnitude of the penetration resistance force(originating in the granular media) and the sum of theexternal forces (weight, inertial forces). When these bal-ance, the granular media solidifies, allowing the robot tobe supported at a fixed limb penetration depth.The previous SandBot study [11] revealed that as thelimb (or any simple intruder) vertically penetrates intothe medium, the penetration force scales with z , thedepth of the intruder below the surface [17], as F p ( z ) = k ( φ ) z , where φ is the volume fraction, the ratio of thesolid volume of the granular media to the volume that itoccupies (for natural dry sand, 0 . < φ < . k ( φ ), characterizes the penetration resistanceand increases with φ . In this paper we keep φ fixed at ap-proximately the critical packing state [16, 18, 19] (whichis close to the as-poured volume fraction) where granularmedia neither globally dilate nor compact in response toshear (see Section 3). Review of Previous Observations and Model
In the previous study of SandBot [11], we fixed intra-cycle limb kinematics (by using gait parameters thatproduce consistent motion on granular media, see Sec-tion 3) and measured SandBot’s average speed v x onpoppy seeds as a function of volume fraction and the cycle-averaged limb frequency ω. We observed a sensi-tive dependence of v x on both φ and ω , and developed akinematic model which explained this dependence andrevealed two distinct locomotor modes determined bywhether the granular media solidifies during limb-groundinteraction.Our kinematic model describes the limb-ground in-teraction of SandBot by considering the motion of justa single limb (Fig. 1b). SandBot has six approxi-mately c-shaped limbs (c-legs) divided into two alternat-ing tripods. C-legs in the same tripod rotate in synchronyand each c-leg rotates about a horizontal axis normal tothe robot body. We simplify the multi-leg ground inter- FIG. 1. Mechanism of SandBot locomotion on granular me-dia. (a) SandBot, a six-legged insect inspired robot, moveswith an alternating tripod gait. The three arrows indicatethe limbs of one tripod. (b) Schematic of single-leg represen-tation of SandBot with mass m = 1 / h = 2 . R = 3 .
55 cm, arc span 225 degree). Legangle θ is measured clockwise about the axle and betweenthe downward vertical and a diameter through the axle. Legdepth z = 2 R cos θ − h . (c) Magnitude of penetration force F p (blue curve) relative to force required for upward motion, m ( g + a ) , (red lines) determines the locomotor mode. Theforce required for quasi-static movement ( mg ) is shown forreference. When F p and m ( g + a ) intersect rotary walkingoccurs. When F p and m ( g + a ) do not intersect (dashed redcurve and above), the robot swims. (d) Schematic of rotarywalking. The granular material flows in the intervals [ α i , β i ](red arrow) and [ β f , α f ] (blue arrow) where F p < m ( g + a ) andthe c-leg rotates about the axle (red and blue circles). Thematerial is a solid in the interval [ β i , β f ] (gray sector; linewith arrows in (c)) where F p exceeds m ( g + a ) and the c-legrotates about its center (green circle and arrow), lifting andpropelling body forward by step length s = R (sin β f − sin β i ).The c-leg is above the ground in the interval [ α f , α i + 2 π ].Note that [ α i , α f ] in (c) is symmetric to vertical ( θ = 0) as aresult of assuming the force is isotropic (see Section 4.2). action of each tripod to that of a single limb carrying1 / m (2 . i.e. F p ( z ) = kz. Theprevious study [11] showed that the simple intruder ap-proximation gave approximately the same results as amore realistic treatment in which penetration force wasintegrated over the submerged leading surface of a c-leg.In this study we use the simple intruder approximation.We make the approximation that SandBot’s body is instationary contact with the surface at the onset forwardmotion in each cycle [20], and define the c-leg’s angularposition, θ, as the clockwise angular displacement fromthe configuration where the center of curvature of the c-leg is directly beneath the axle, see Fig. 1b. During afull rotation, as θ changes from − π to π , the c-leg ini-tially contacts the ground at θ = α i and loses groundcontact at θ = α i . Because leg depth can be approxi-mated as z = 2 R cos θ − h when the body is in contactwith the surface [21], penetration force can be written as F p ( θ ) = 2 Rk (cid:2) cos θ − h R (cid:3) (blue curve in Fig. 1c), where R = 3 .
55 cm and h = 2 . i.e. distance from c-leg axle tounderside of body) respectively.Of prime importance in determining SandBot’s per-formance is the magnitude of the penetration force F p ( θ )relative to the sum of the forces required to the sup-port the body weight and accelerate the body upward m ( g + a ) (red curve in Fig. 1c), where g is the accel-eration due to gravity and a the acceleration [11]. Therelevant acceleration is given by the jump in robot speedwhen the granular media solidifies, Rω, divided by thecharacteristic response time of the c-leg interacting withthe granular media, ∆ t ( φ ) , i.e. a = Rω/ ∆ t. Two distinctlocomotor modes are possible depending on whether ornot F p ( θ = 0) > m ( g + a ):1. Rotary walking – movement with solidification (seeFig.1b-d): As the c-leg rotates into the ground afterinitial leg-ground contact at θ = α i , the penetra-tion force increases with increasing depth. In therotary walking regime the material beneath the c-leg solidifies and leg penetration stops at an angle θ = β i when F p ( β i ) = m ( g + a ) , see Fig. 1c,d. Sincethe frictional force between the c-leg and granu-lar material is insufficient for the leg to roll, thec-leg instead rotates about its center of curvature(green circle and arrow) lifting and advancing therobot in the process. Rotary walking continuesuntil θ = β f , beyond which the c-leg again pene-trates through the material since F p ( θ ) < m ( g + a )and the body is again in contact with the ground(blue circle and arrow). Rotary walking thus oc-curs over a finite range of leg angle β i < θ < β f or[ β i , β f ] (horizontal arrow in Fig. 1c and gray sec- tor in Fig. 1d) where β i and β f are determinedby F p ( β i,f ) = 2 Rk (cos β i,f − h R ) = m ( g + a ) . Fora given [ β i , β f ], Fig. 1d shows that the robot ad-vances a distance s = R (sin β f − sin β i ) , where wecall s the step size. During one complete gait cy-cle of period T, each alternating tripod advancesthe robot by s, giving an average robot speed of v x = 2 s/T = sω/π .2. Swimming – movement without solidification:When F p (0) < m ( g + a ) (Fig. 1c, dashed red curve),the granular material beneath the penetrating c-legnever solidifies and rotary walking does not occur, i.e. β i = β f = 0 . Instead, the limb constantly slipsthrough the surrounding fluidized granular mate-rial, similar to a swimmer’s arm in water, and therobot advances slowly ( v x < k and ∆ t , together with limb fre-quency ω , determine the relative magnitudes of F p and m ( g + a ) and consequently control which locomotor modethe robot operates in. Reducing k (by decreasing φ )and/or increasing ω reduces the rotary walking range;in other words, the less compact the granular material isand/or the faster the limbs rotate, the deeper the c-legshave to penetrate before the granular material solidifiesand rotary walking begins, and the more susceptible therobot is to entering the slow swimming mode. This sim-ple kinematic model captures the observed sensitive de-pendence of v x on φ and ω , with k ( φ ) and ∆ t ( φ ) as twofitting parameters.In summary, our previous study of SandBot [11]showed that to locomote effectively on granular media,limbs kinematics that access the solid phase of granularmedia should be employed. METHODS AND RESULTS
The limb kinematics of each tripod during one cy-cle are parameterized by three “gait parameters”, seeFig. 2(a,b). The kinematics of both tripods are periodic(with period T ) and offset by half a period T / ω f and ω s .During hard ground locomotion in the RHex-class ofRobots (and for animal locomotion in general), ”swing”and ”stance” phases typically correspond to off-ground FIG. 2. SandBot’s intra-cycle limb kinematics and affectsits speed on granular media. (a) Each leg rotation is com-posed of a fast phase (orange) and a slow phase (green). θ s and θ define the angular extent and center of the slowphase respectively. (b) Leg angle θ as a function of timeduring one cycle (normalized to T ). θ ( t ) of the other tri-pod is shifted by T / d c is theduty cycle of the slow phase, i.e. fraction of the periodspent in the slow phase. (c) Instantaneous speed of Sand-Bot on granular media with hard ground clock signal (HGK: { θ s , θ , d c } = { . , . , . } ; red) and soft ground clocksignal (SGK: { θ s , θ , d c } = { . , − . , . } ; blue). WithHGK SandBot moves slowly ( v x ≈ v x ≈ and ground-contact phases, respectively. But becauseduring locomotion on granular media this correspondenceis not necessarily true, we simply call them fast and slowphases. In practice the fast and slow phases are implicitlydefined by the triplet { θ s , θ , d c } where θ s is the angularextent of the slow phase, θ is the angular location ofthe center of the slow phase, and d c is the duty cycleof the slow phase (the fraction of the period in the slowphase). Specifying the cycle averaged limb frequency ω fully determines the motion of the limbs in the robotframe. By definition, ω s = θ s T d c , ω f = π − θ s T (1 − d c ) , and ω = πT . Typically, gait parameters are set so that ω s < ω <ω f , but the reverse is possible when θ s becomes largeenough and/or d c small enough.In the first tests of SandBot on granular media(Fig. 2c), we found that kinematics tuned for rapid stablebouncing motion on hard ground (HGK: { θ s , θ , d c } = { . , . , . } ) produced little motion on granular me-dia (red curve). Empirical adjustment to soft groundkinematics (SGK: { θ s , θ , d c } = { . , − . , . } ) re-stored effective (walking) locomotion on granular media(blue curve). In the previous study [11], we used SGK to test v x ( φ, ω ). Now armed with the understanding of howSandBot moves on granular media gained from this work,we set out to determine the effects of limb kinematics indetail. We set φ = 0 .
605 and ω = 8 rad/s, and measureSandBot’s average speed on granular media as we sys-tematically vary gait parameters, i.e. v x = v x ( θ s , θ , d c ).We pick ω = 8 rad/s because at this intermediate fre-quency SandBot displays both rotary walking and swim-ming as the clock signal is varied. We pick φ = 0 .
605 toremove the effect of local volume fraction change whichcauses a premature transition from rotary walking toswimming [22] and adds to the complexity of the prob-lem.We first test the effect of the extent and location ofthe slow phase for fixed d c = 0 .
5, measuring speed v x = v x ( θ s , θ ). We vary the parameters between 0 ≤ θ s ≤ − ≤ θ ≤
2, which are the limits set by the robot’scontroller. We choose d c = 0.5 because it is close to the d c values of both HGK and SGK. This gave us an easy FIG. 3. Average speed v x of SandBot on granular media( φ = 0 . θ s , θ ) for d c = 0 . ω =8 rad/s. (a) The experimental data has a localized regionof high speeds with peak v x ≈ { θ s , θ } = { − . } . . Circles show that v x for SGK (red) and HGK (blue)matches data for d c = 0 . d c values, see Section 4.3. Inset: original data from whichmain figure is interpolated. (b) Predicted v x ( θ s , θ ) from thekinematic model with F p = kz captures the single peak butpredicts a lower speed for SGK than for HGK contrary toobservation and fails to account for the observed peak in speedat θ = − . way to project HGK ( d c = 0 .
56) and SGK ( d c = 0 . v x = v x ( θ s , θ ) plot ( d c = 0 . d c near d c = 0 . v x = v x ( θ s , θ ) (Fig. 3a) show a sin-gle sharp peak in speed near { θ s , θ } = { − } . Highspeeds only occur within a small island of − < θ < θ s > . θ is varied away from the peak, and is less so when θ s is varied away from the peak; this is also evident incross sections through the peak (blue circles in Figs. 3aand Fig. 6a, respectively). Ignoring the effect of d c , theSGK parameters (blue dot) lie close to the peak whilethe HGK parameters (red dot) are in the low speed re-gion. The optimal gait parameters which we found forSandBot locomotion on poppy seeds at φ = 0 .
605 and ω = 8 rad/s are: { θ s , θ , d c } = { . , − . , . } . Thesegait parameters generate about 20% higher speed thanthe previously used SGK parameters.Variation of the duty cycle at fixed { θ s , θ } = { . , − . } also has a substantial influence on speed.Data (blue circles in Fig. 6b) show a well defined peakat d c ≈ .
5. Speed drops off relatively slowly for d c > d c < . DISCUSSIONApplication of Model to Slow Phase Extent andLocation Variation
To apply our kinematic model to SandBot locomotionwith varied limb kinematics, we must consider the effectsof variable limb kinematics during limb-ground interac-tion. Depending on the gait parameters during groundcontact, the limbs could be rotating in the fast phase, inthe slow phase, or in a combination of both. In our pre-vious study, the kinematic model ignored limb frequencyvariability during ground contact and only considered therobot limb rotating at the constant cycle averaged limbfrequency ω .However, as limb kinematics change, the variability oflimb frequency in ground contact needs to be taken intoaccount. For our test of v x = v x ( θ s , θ ) within 0 ≤ θ s ≤ − ≤ θ ≤ d c = 0 . ω f >> ω s so that onlythe slow phase can possibly achieve rotary walking, asfast limb rotation results in swimming. In this case ω s (instead of ω ) controls acceleration a and thus determinesthe rotary walking range [23].For fixed d c , varying [ θ s , θ ] changes the extent andlocation of the slow phase; the angular extremes of theslow phase are θ i,f = θ ± θ s , see Fig. 5a. Varying θ s also changes ω s which controls the rotary walking range. Therefore varying [ θ s , θ ] affects where the slow phaseoverlaps with the rotary walking range. The step length s is given by s = R (sin ψ f − sin ψ i ) where ψ i = max( β i , θ i )and ψ f = min( β f , θ f ) if there is overlap or s = 0 if thereis no overlap. The larger the overlap, the further therobot moves forward in a cycle.As shown in Section 2.3, the rotary walking range[ β i , β f ] is given by solving the equation F p ( θ ) =2 Rk (cos β i,f − h R ) = m ( g + a ), with a given by a = mω s ∆ t .We can evaluate how [ θ i , θ f ] overlaps with [ β i , β f ] to de-termine s , and calculate the robot speed using v x = sT = sωπ . For fixed ω , speed v x scales with step length s .Figure 3b shows the model prediction of v x using fit-ting parameters k = 210 N/m, ∆ t = 0 .
37 s. Compar-ing prediction with observation (Fig. 3a), the model cap-tures the peak and predicts similar magnitudes of speeds.However the predicted peak is symmetric about θ = 0while the observed peak is symmetric about θ = − . . Anisotropic Penetration Force Law
If the penetration force of the granular material in-creased like kz as assumed in the model, we would ex-pect θ = 0 as this value would give the largest overlapbetween the slow phase and the rotary walking rangeas determined by the material strength (see scheme inFig. 5a). To investigate why the robot performs best with θ = − . ω = 0 .
35 s − (thehorizontal rotation axis of the c-leg was positioned thesame distance h = 2 . β = − .
75; pen-etration force during rotation peaks before the intruderreaches the maximum depth. We confirmed that themeasured anisotropy in the penetration force is intrin-sic to our granular medium and is not an artifact of theparticular shape of the c-leg by additionally rotating arectangular bar and a sphere into granular media at thesame hip height: both objects exhibited a peak force at β = − . θ = 0). Inrotational intrusion, however, the direction of intrusionis constantly changing; the direction of the force coneshould change as well and correlate with the instanta- FIG. 4. Asymmetry of v x with respect to θ = 0 is due toanisotropic penetration force during limb rotation into gran-ular media. (a) For all θ s at d c = 0 . , v x ( θ, θ ) (Fig. 2a) ismaximal (dashed vertical black line) at θ = − . θ s = 1 . θ = − . d c = 0 .
8. (b) Vertical penetration force F p (solid blue curve)during c-leg rotation into poppy seeds reaches maximum at β = − .
75 (dashed black line) and is asymmetric to θ = 0.Inset: force measurement schematic. Solid blue curve in (a) isprediction from the model with anisotropic penetration force. neous direction of intrusion.We hypothesize that the force during rotational intru-sion is maximal at β = − .
75 because for larger anglespart of the cone reaches the surface and/or terminates onthe horizontal walls of the container and can no longersupport the entire grain contact network, thus reducingthe maximal yield force. We also note that the angle atwhich maximal force is developed is close to the angle ofrepose 0 .
52 that we measure for the poppy seeds. Thisangle is the same as the internal slip angle in cohesionlessgranular material [16] which plays an important role inthe formation of the grain contact network, supportingthe plausibility of our speculation.To account for the measured angular offset in peakforce from vertical (Figs. 2c and 3a), we modify the orig- inal penetration force law in our model to F p ( θ ) = 2 Rk (cid:48) { cos [ b ( θ − β )] + 1 } for F >
0, where β = − .
75, and k (cid:48) and b are newfit parameters. Following the same procedure describedin Section 4.1, we find the robot speed by calculating[ β i , β f ], the overlap between [ θ i , θ f ] and [ β i , β f ], and thestep size.Figure 5c shows v x predicted by a fit to the model usingthe anisotropic penetration force law (fitting parameters k (cid:48) = 65 N/m, ∆ t = 0 . b = 0 . θ = − . i.e. asymmetry to θ = 0). For fixed θ s , speed is maximalwhen the center of the slow phase corresponds with thecenter of the rotary walking range (Fig. 5a). If θ is differ-ent from β = − .
75, the overlap of the slow phase and
FIG. 5. Overlap of the slow phase and the rotary walk-ing interval predicted by the anisotropic penetration forcelaw better predicts v x ( θ s , θ ). (a) Overlap of the slow phase[ θ i , θ f ] and the rotary walking range [ β i , β f ] determines steplength s and thus speed v x . For the configuration shown, s = R (sin β f − sin θ i ). θ and β are centers of the slowphase and the rotary walking range, respectively. (b) InSGK the slow phase (centered at θ = − .
50) overlapsnearly completely with the rotary walking range (centered at β = − . θ = 0 .
13) which has little overlap. (c) Pre-diction of v x ( θ s , θ ) from the the model with the anisotropicpenetration force for d c = 0 . v x with respect to θ = 0 and predicts higher speed for SGKthan for HGK. Fitting parameters: k (cid:48) = 65 N/m, ∆ t = 0 . b = 0 . the rotary walking range decreases, which reduces steplength and thus speed. In accord with observation, SGK(red dot) lies near the peak while HGK (blue dot) lies ina region of low speeds. Figure 5a,b shows that SGK hashigher speed than HGK because the overlap between theslow phase and the rotary walking range (gray sector inFig. 5a) is significantly larger.At fixed θ = β , increasing the extent of the slowphase (increasing θ s ) from zero initially increases speedas the extent of the slow phase increases within the ro-tary walking envelope (see Figs. 2a and 6a). However, ω s increases with θ s which increases the acceleration andreduces the rotary walking range. For sufficient extent(near θ s = 1 . θ s reduce the rotary walking range. Rotary walking isnot possible for θ s ≥ πd c ∆ tω (cid:16) k (cid:48) m − gR (cid:17) as the material isnever strong enough to both support and accelerate therobot. In Fig. 6a the experimental speed is noticeablylower than the model prediction at the largest θ s = 2.As we discuss below in regards to variation in d c , this re-duction is a apparently the result of tripod overlap (bothtripods simultaneously in ground contact) which occursfor a greater portion of the slow phase for larger θ s . Effect of Duty Cycle
While the model prediction of v x ( d c ) (blue curve inFig. 6b) matches the magnitudes of speeds at intermedi-ate d c ≈ .
5, it does not quantitatively match the shapeof measured speed vs d c . Below d c ≈ .
5, the model pre-dicts that v x ( d c ) increases monotonically with increasing d c . This trend is in accord with the experimental ob-servations; however the model prediction is consistentlyhigher than measured speed. Above d c ≈ .
5, the modelpredicts that v x ( d c ) is independent of d c , but the mea-sured speed decreases with increasing d c and is lower thanthe model prediction. We now discuss possible reasonsfor the discrepancies at low d c < . . < d c < . d c > . d c is small so that ω s and thus a becomelarge enough to ensure that the robot is in the swimmingmode (i.e. movement without solidification). Here themodel assumes rotary walking and predicts zero speed.In experiment, the robot can still advance slowly at eachstep due to thrust forces from continuously slipping limbsgenerated by frictional drag [25] and/or inertial move-ment of material. This thrust competes with frictionfrom belly drag, and as in [11], we find that these re-sult in low average speed of v x ≈ d c (regionIII) is a result of tripod overlap and can be readily under- stood. When there is no tripod overlap (only one tripodwith ground contact at any given time) each tripod ad-vances the robot a distance s for a total displacement of2 s per period. This is the case for d c ≤ .
5. However inthe limit of d c = 1 both tripods are simultaneously in theslow phase as the duration of the fast phase is zero. Thesimultaneous slow phases generate a total displacementof just s [26] instead of 2 s without tripod overlap. Asa result, the predicted speed at d c = 1 must be halved(i.e. v x = sT = sω π ). Lacking a way to quantify the tri- FIG. 6. Comparison of data and anisotropic force model pre-dictions for v x ( θ s ) and v x ( d c ) . (a) Measured v x ( θ s , θ = − . θ s due to limb overlap. (b) Measured v x ( d c ) (blue cir-cles) for { θ s = 1 . , θ = − . } is maximum at d c ≈ . d c ≈ . d c (III).Model prediction with tripod overlap included (dashed blueline) better matches the data in region III. pod overlap effect, we assume that the reduction of steplength from 2 s to s is linear with d c for d c > .
5; thedata is in good agreement with this prediction (dashedblue curve in Fig. 5). This reduction in speed is a purelykinematic effect that is inherent to the rotary walkinggait at high d c .Two plausible mechanisms explain the model’s over-estimate of speed at intermediate d c (region II): holedigging and uneven weight distribution. Lateral obser-vations of the robot kinematics at low d c show that therapid motion of the c-leg during the slow phase throwssignificant numbers of particles out of the limb’s pathwhich creates a depression. For a deep enough hole, ro-tary walking is impossible due to the reduced penetrationdepth of the leg below the now lower surface of the de-pression. The second mechanism concerns the model’sassumption of uniform weight distribution between thethree legs of the tripod. In the d c range where the robotadvances slower than predicted, observations show thatthe robot rotates in the horizontal plane. Rotation oc-curs in this transition region between pure swimming andpure rotary walking because the side of the robot withtwo c-legs in ground contact undergoes rotary walkingwhile the opposite side is in the swimming mode. Dueto the increased gravitational and inertial forces on thesingle c-leg, the penetration forces are never sufficient toachieve ground solidification under the leg even at themaximum penetration depth. CONCLUSIONS
We have built upon our previous experiments andmodels of a legged robot, SandBot, to explore howchanges in limb kinematics affect locomotion on granularmedia. We found that even when moving on controlledgranular media of fixed volume fraction at fixed cycle-averaged limb frequency, speed remains sensitive to vari-ations in gait parameters that control angular extent, an-gular location, and temporal duty factor of the slow phaseof the limb cycle. We showed that the assumptions in apreviously introduced model (which accurately predictedspeed as a function of limb frequency and volume frac-tion) had to be modified to incorporate an anisotropicpenetration force during rotational intrusion into gran-ular media as well as changes in acceleration of the legas gait parameters were varied. With these modifica-tions the model was able to capture speed as a functionof angular extent and angular location. The model alsoindicates that as duty cycle is changed, effects due to si-multaneous limb pairs (tripods) in ground contact, rapidlimb impact into sand, and unequal weight distributionon limbs become important.Our experiments and modified model explain why gaitparameters that allow the robot to rapidly bounce overhard ground lead to loss of performance on granular me- dia. They demonstrate how the angular extent and lo-cation of the slow phase must be adjusted to optimizeinteraction with granular media by minimizing inertialforce and limb interference, and maximizing the use ofsolid properties of granular media. Further studies ofSandBot guided by our kinematic model should revealhow physical parameters of both robot (mass distribu-tion, limb compliance, limb shape, belly shape) and theenvironment (grain friction, density, incline angle, grav-ity) control the solid-fluid transition and thus affect thelimb-ground interaction and performance. However, ad-vances are required in theory and experimental charac-terization of complex media. Otherwise we must continueto rely on empirical force laws specific to particular ge-ometries, kinematics and granular media.The existence of a speed optimum in gait parameterspace implies that control of limb kinematics is criticalto move effectively on granular media, whether activelythrough sensory feedback, and/or passively through me-chanical feedback. Future work should compare theseresults to investigations of gait optimization on hardground [13]. The differences in limb kinematics on sandcompared to hard ground are intriguing because on hardground performance is optimized by making the robotbounce. However, this carries with it the risk of yaw,pitch and roll instability due to mismanaged kinetic en-ergy. On granular media such instabilities appear rare;instead most gait parameters (see Fig. 3a) result in littleor no forward movement due to mismanaged fluidizationof the ground. Thus, our results could have a practicalbenefit as they suggest strategies for improving the per-formance of current machines [27–29] on variable terrainvia new limb and foot designs and control strategies.Finally, an enormous number of organisms contendwith sand [30], moving on the surface (or even swim-ming within it [31]). While the observed phenomenaand proposed locomotion modes (e.g. rotary walking)appear specific to SandBot and its c-shaped limbs, theunderlying principles could apply to locomotion of or-ganisms on yielding substrates. For example, our re-cent work on terrestrial hatchling sea turtle locomotiondemonstrates that their effective movement on sand pro-ceeds through solidification of the granular medium [32].Integrated studies of biological organisms and physicalmodels can provide hypotheses [33] for passive and ac-tive neuro-mechanical [34] control strategies as well asbetter understanding of energetics [35] for movement oncomplex terrain. [1] R. McNeill Alexander.
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Acknowledgements
We thank Daniel Koditschek, Ryan Maladen, YangDing, Nick Gravish, and Predrag Cvitanovi´c for help- ful discussion. This work was supported by the Bur-roughs Wellcome Fund (D.I.G., C.L., and P.B.U.), theArmy Research Laboratory (ARL) Micro AutonomousSystems and Technology (MAST) Collaborative Technol-ogy Alliance (CTA) under cooperative agreement num-ber W911NF-08-2-0004 (D.I.G. and P.B.U.), and the Na-tional Science Foundation (H.K.).
Citation Information
Chen Li, Paul B. Umbanhowar, Haldun Komsuoglu,Daniel I. Goldman, The effect of limb kinematics on thespeed of a legged robot on granular media,
ExperimentalMechanics50