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Inverted Loaded ground Spring hard incorporates biol the called on of RHex smallest diversity running a the organisms in [18]. observed ical is dynamics class like 1a), RHex pogo-stick inspire the biologically the of series robots, (Fig. successful hexapedal a SandBot in kg) study, 2.3 (mass we robot The Discussion and Results on locomotion bet- for of paradigms possibility control terrains. the and physics complex and design the media robotic of granular ter understanding within sophisticated motion more of a both granu- to for on points substra which need that to characteristics, sensitive find gait remarkably we and is preparation terrain, locomotion natural the (and of media rapidl lar SandBot’s range and nimbly wide Despite move a to ability over [17]. [18]) sand RHex fractio predecessor desert volume its of with me- states typical granular packing on ranges different SandBot, in device, per prepared legged the dia small explore a systematically of we formance Here differ terrains. locomotio mimic complex during that on produced materials strength flowing material deformable 1 varying ent [15, of devices states scale be create laboratory can using media controlled achieved granular precisely het be environments, more Unlike real-world can theory. erogeneous physics and experiment collective fundamen- of interplay the that through of enough understanding simple flow. are tal o during mud, materials debris, possible granular like materials are snow, gas-lik complex other behaviors [13], to fluid-like [15] compared Yet, while glass-like [12], even stress and yield critical a media granular locomotion, robots, office.’ PNAS Abbreviations: the to directly submitted was paper ’This placeholder footnote interest of Conflict c 07b h ainlAaeyo cecso h USA the of Sciences of Academy National The by 2007 ‡ ailI Goldman I. Daniel , PNAS su Date Issue t fPnslai,Piaepi,P 19104, PA Philadelphia, Pennsylvania, of ity Volume ∗ su Number Issue [14], e [19] d erties ]to 6] the og- 1 te – n n y r 6 - - - , , and tracked vehicles like Packbot [25]. is effectively constant for φ above a critical volume fraction SandBot moves using an alternating tripod gait in which φc(ω), but is close to zero for φ below φc(ω). For fixed ω, two sets of three approximately c-shaped legs rotate syn- φc(ω) separates volume fraction into two regimes: the “walk- chronously and π out of phase. A clock signal (Fig. 1c), ing” regime (φ ≥ φc, vx >> 0) and the “swimming” regime defined by three gait parameters (see Materials and Meth- (φ<φc(ω), vx ≈ 2 cm/s). See Movies S4 and S5 for examples ods), prescribes the angular trajectory of each tripod. The of rotary walking and swimming modes. c-legs distribute contact [26] over their surface and allow the The rotary walking mode is dominant at low ω and high robot to move effectively on a variety of terrain. On rigid, no- φ. In this mode, a tripod limb penetrates down and backward slip ground SandBot’s limb trajectories are tuned to create a into the ground until the granular yield stress exceeds the limb bouncing locomotion [18] that generates speeds up to 2 body transmitted inertial, gravitational, and frictional stresses at a lengths/s (≈ 60 cm/s). We tested this clock signal on granular depth d(ω,φ). At this point, rather than rolling forward like media, but found that the robot instead of bouncing adopts a wheel, the c-leg abruptly stops translating relative to the a swimming gait in which the legs always slip backward rel- grains and begins slipping tangentially in the circular depres- ative to the stationary grain bed and for which performance sion surrounding it; at the same time, the center of rotation is reduced by a factor of 30 to ≈ 2 cm/s (see Supporting In- moves from the axle to the now stationary center of curvature formation Movie S1). We surmised that this was due to an (see Fig. 3a). The simultaneous halt in both vertical and interval of double stance (both tripods in simultaneous con- horizontal leg motion is apparently due to the large reduction tact with the ground), which is useful on hard ground during in friction forces which occurs when the weight of the robot is bouncing gaits but apparently causes tripod interference on supported by the limbs rather than the underside of the body. granular media. Changing the gait parameters to remove the The ensuing rotary motion propels the axle and consequently double stance allowed SandBot to move (see Movie S2) in the the rest of the robot body along a circular trajectory in the granular media at speeds up to 1 body length/s (≈ 30 cm/s) x − z plane with speed Rω, where R = 3.55 cm is the c-leg in a rotary walking gait that resembles the pendular gait of radius. The forward body motion ends when, depending on φ the robot on hard ground [27] but with important kinematic and ω, either the second tripod begins to lift the robot or the differences (discussed below). No amount of gait parameter underside of the robot contacts the ground. adjustment produced rapid bouncing locomotion on granular With increased ω limbs penetrate further as the requisite media. We hypothesize that the ≈ 50% decrease in top speed force to rapidly accelerate the robot body to the finite limb relative to hard ground is associated with the inability of the speed (Rω) increases. As the penetration depth approaches robot to undergo the arial phases associated with the bounc- its maximum 2R − h, where h = 2.5 cm is the height of the ing gait [28]. axle above the flat underside of the robot, the walking step In the desert, animals and man-made devices can en- size goes to zero since there is no longer a point in the cycle counter granular media ranging in volume fraction from φ = where the limb ceases its motion relative to the grain bed. 0.55 to φ = 0.64 [17], and some desert adapted animals (like Any subsequent forward motion is due solely to thrust forces lizards) can traverse a range of granular media with little loss generated by the swimming-like relative translational motion in performance [16]. To test the robot performance on con- of the limb though the grains. Note that φc(ω) increases with trolled volume fraction granular media, we employ a 2.5 m ω, and that the transition from rotary walking to swimming long fluidized bed trackway (Fig. 1b) [29], which allows the is sharper in vx for higher ω and smoother for lower ω. The flow of air through a bed of granular media, in this case much slower swimming mode occurs for all volume fractions ∼ 1 mm poppy seeds. With initial fluidization followed by for ω ≥ 28 rad/s. repeated pulses of air [30], we prepare controlled volume frac- Plotting the average robot speed as a function of limb fre- tion states with different penetration properties [31]. In this quency (Fig. 2a) shows how the robot suffers performance study, we test the performance (forward speed vx) of SandBot loss as its legs rotate more rapidly. For fixed φ, vx increases with varied leg-shaft angular frequency (ω) for volume fraction sub-linearly with ω to a maximal speed vx∗ at a critical limb (φ) states ranging from loosely to closely packed (φ = 0.580 frequency ωc, above which vx quickly decreases to ≈ 2 cm/s to φ = 0.633). We chose forward speed as a metric of perfor- (swimming) [32]. The robot suffers partial performance loss mance since it could be readily measured by video imaging. (mixed walking and swimming) below ωc and near total per- We hypothesized that limb frequency would be important to formance loss (pure swimming) above ωc. Performance loss robot locomotion since the substrate yield strength increases for φ ' 0.6 is more sudden (∆ω ≈ 1 rad/s) compared to with volume fraction and the yield stress × robot limb area performance loss for φ / 0.6. Both ωc and vx∗ display tran- divided by the robot mass × velocity is proportional to the sitions at φ ≈ 0.6 (Figs. 2b,c). The transition at φ ≈ 0.6 maximum limb frequency for efficient locomotion. for the rapidly running robot is noteworthy since it has been We find that robot performance (speed) is remarkably sen- observed that granular media undergo a transition in quasi- sitive to φ (see Movie S3). For example, at ω = 16 rad/s the static penetration properties at φ ≈ 0.6 [31]. robot speed vx(t) shows a change in average speed vx of nearly Starting with the observed kinematics of rotary walking a factor of five as φ changes by just 5 % (Figs. 1d,e). For the with circular slipping, we constructed a straightforward two- closely packed state (φ = 0.633), vx ≈ 20 cm/s with 5 cm/s parameter model that captures the essential elements deter- oscillations during each tripod rotation, whereas for a more mining granular locomotion for our legged device and agrees loosely packed state (φ = 0.600), vx ≈ 2 cm/s with 1 cm/s well with the data (dashed lines in Fig. 2a). The model, which oscillations in velocity. incorporates simplified kinematics and granular penetration This sensitivity to volume fraction is shown in the aver- forces while still agreeing well with a more realistic treatment age robot speed vs. volume fraction (Fig. 1e). For fixed ω, vx (for a more detailed discussion of the model, see Materials
2 www.pnas.org — — Footline Author and Methods), indicates that reduction of step length through we assume the two tripods do not simultaneously contact the increased penetration depth is the cause of the sub-linear in- ground; however, in soft ground this is not the case, which crease in vx with ω and the rapid loss of performance above consequently reduces the effective step size per period from ωc. The model assumes that the two tripods act indepen- 2s to a lesser value. The fit value of ∆t is sensitive to this dently, that the motion of each tripod can be understood by variation; reducing the step size (and thus the speed) in the examining the motion of a single c-leg supporting a mass m experimental data by just 13 % decreases ∆t to 0.2 s while k equal to one-third of SandBot’s total mass, and that the un- is increased by less than 10 %. derside of the robot rests on the surface at the beginning of Our model indicates that for deep penetration the walking limb/ground contact. step length is sensitive to penetration depth (e.g. Fig. 3b). Using the geometry of rotary walking (see Fig. 3a), the As the walking step size goes to zero with increasing ω or de- walking step size per c-leg rotation is s = 2 R2 − (d + h − R)2, creasing φ, the fraction of the ground contact time that the leg where d is the maximum depth of the lowestp point on the leg. slips through the grains (swimming) goes to one. Swimming After the robot has advanced a distance s, the body again in granular media differs from swimming in simple fluids as contacts the ground and the c-leg moves upward. Since dur- the friction dominated thrust and drag forces are largely rate ing each clock signal period there are two leg rotations (one independent at slower speeds [33, 35]. When thrust exceeds for each tripod), the average horizontal velocity is 2s × limb drag and using constant acceleration kinematics, the robot ωs frequency or vx = π . The maximum limb penetration depth advances a distance proportional to the net force divided by d is thus the key model component as it controls the step ω2 per leg rotation, and, consequently, speed is proportional 1 length (see Fig. 3b) and consequently the speed. Maximum to ω− . This explains the weak dependence of vx on ω in the limb penetration depth is determined by balancing the vertical swimming mode. The increase in robot speed with decreas- acceleration of the robot center of mass ma with the sum of ing ω is bounded by the condition that the robot speed in a the vertical granular penetration force kz [33] and the gravita- reference frame at rest with respect to the ground cannot ex- tional force mg, where g is the acceleration due to gravity, and ceed the horizontal leg speed in a reference frame at rest with k(φ) is a constant characterizing the penetration resistance of respect to the robot’s center of mass. This condition ensures the granular material of volume fraction φ. the existence of and eventual transition to a walking mode as At small ω, ma ≈ 0 = Fi = mg − kd so d = mg/k, ω is decreased. which is the minimum penetrationP depth. For finite ω the The transition from walking to swimming appears gradual penetration depth is greater since an additional force must be for φ / 0.6 since the penetration depth increases slowly with ω 2 supplied by the ground to accelerate the robot body to the at small ω (Rω/∆t ≪ g) and the ω− contribution to the per leg speed Rω when the c-leg stops translating in the material. cycle displacement from swimming is relatively large (see e.g. Taking a = ∆v/∆t, with ∆v = Rω − 0 and ∆t the charac- the data at ω = 12 rad/s in Fig. 3b). However, for φ ' 0.6, teristic elastic response time of the limb and grain bed, gives the transition is abrupt. This sharp transition occurs because the acceleration magnitude a = Rω/∆t. The direction of the the step size is reduced sufficiently that the legs encounter ma- acceleration depends on the position of the c-leg. To keep the terial disturbed by the previous step; we hypothesize that the model simple we approximate the vertical component of the disturbed material has lower φ and k. At higher φ, the volume acceleration with its magnitude. Equating the vertical forces fraction of the disturbed ground is significantly less than the Rω with mass × acceleration (see Fig. 3c), −m ∆t = mg − kd, bulk which increases penetration and consequently greatly re- m Rω gives c-leg penetration d = k ( ∆t + g) with average horizon- duces s. This is not the case for the transition from walking to 2 swimming at lower φ (and low ω) where the volume fraction 2Rω m ω g h tal velocity vx = π 1 − k(φ) ( ∆t + R )+ R − 1 . Fits of the disturbed material is largely unchanged relative to its r h i to this model are indicated by dashed lines in Fig. 2b. The initial value. For the robot to avoid disturbed ground it must expression captures the sub-linear increase in vx with ω at advance a distance R on each step, i.e. s ≥ R, or in terms √3 fixed k(φ), the increase in speed at fixed ω as the material of the penetration depth, d ≤ ( 2 + 1)R − h = 5.0 cm (green strengthens (increasing k with increasing φ), and transition dashed lines in Fig. 3b,c). The disturbed ground hypothesis is to zero rotary walking velocity when ω is sufficiently large. supported by calculations of the step size derived from the av- The expression for vx is determined by the two fit param- erage velocity 2s = 2πvx/ω which show a critical step size near eters k and ∆t. The parameter k characterizing the pene- s/R = 1 at the walking/swimming transition (Fig. 3d). The tration resistance increases monotonically with φ from 170 to somewhat smaller value of s/R ≈ 0.9 evident in the figure can 220 N/m and varies rapidly below φ ≈ 0.6 and less rapidly be understood by recognizing that for s slightly smaller than above. Its average value of ≈ 200 N/m corresponds to a shear R the majority of the c-leg still encounters undisturbed mate- stress per unit depth of α ≈ 470 kN/m3 (using leg area=wR rial. Signatures of the walking/swimming transition are also where w is the leg width) which is in good agreement with evident in lateral views of the robot kinematics (see Movies penetration experiments we performed on poppy seeds that S3-S5). yield α = 300 and 480 kN/m3 for φ = 0.580 and 0.622 respec- At higher ω in the swimming mode, limbs moves with tively and is comparable to previous measurements of slow sufficient speed to fling material out of their path and form penetration into glass beads [34] where α ≈ 250 kN/m3. In a depression which reduces thrust because the limbs are not contrast, ∆t varies little with φ and has an average value of as deeply immersed on subsequent passes through the mate- 0.4 s compared to the robot’s measured hard ground oscilla- rial. However, as limb speed increases further, thrust forces tion period of 0.2 s when supported on a single tripod. The becomes rate dependent and increase because the inertia im- differences in ∆t can be understood as follows. In our model parted to the displaced grains is proportional to ω2. Between strokes, the excavated depression refills at a rate dependent
Footline Author PNAS Issue Date Volume Issue Number 3 on the difference between the local surface angle and the an- substrate volume fraction. gle of repose [36] and the depression size. Investigating the Integrated motor encoders record the position and current (and competition between these different processes at high ω and thus torque) of SandBot’s motors vs. time. Comparison of the measured and prescribed angular trajectories for both sets of tim- their consequences for locomotion could be relevant to under- ing parameters show a high degree of fidelity with an error of a standing how to avoid becoming stranded or to free a stranded few percent. Therefore, SandBot’s change in performance between device. HGCS and SGCS timing comes from the physics of the substrate interaction. Trackway volume fraction control To systematically test Sand- Conclusions Bot’s performance vs. substrate volume fraction, we employ a 2.5 m Our studies are the first to systematically investigate the per- long, 0.5 m wide fluidized bed trackway with a porous plastic (Porex) flow distributor (thickness 0.64 cm, average pore size 90 formance of a legged robot on granular media, varying both µm). Four 300 L/min leaf blowers (Toro) provide the requisite air properties of the medium (volume fraction) and properties of flow. Poppy seeds are chosen as the granular media because they are the robot (limb frequency). Our experiments reveal how pre- similar in size to natural sand [42] and are of low enough density to carious it can be to move on such complex material: changes be fluidized. The air flow across the fluidized bed is measured with in φ of less than one percent result in either rapid motion or an anemometer (Omega Engineering FMA-900-V) and is uniform to within 10 percent. failure to move, and slight kinematic changes have a similar A computer controlled fluidization protocol sets the volume effect. A kinematic model captures the speed dependence of fraction and thus the mechanical properties of the granular me- SandBot on granular material as a function of φ and ω. The dia. A continuous air flow initially fluidizes the granular media in model reveals that the sublinear dependence of speed on ω the bubbling regime. The flow is slowly turned off leaving the gran- and the rapid failure for sufficiently small φ and/or large ω are ular media in a loosely packed state (φ = 0.580). Short air pulses (ON/OFF time = 0.1/1 s) pack the material [30]. Increasing the consequences of increasing limb penetration with decreasing φ number of pulses increases φ up to a maximum of φ = 0.633. Vol- and/or increasing ω, and changes to local φ due to penetration ume fraction is calculated by dividing the total grain mass by the and removal of limbs. While detailed studies of impact and bed volume and the intrinsic poppy seed density. The mass is mea- penetration of simple rigid objects exist [37, 33], further ad- sured with a precision scale (Setra). The density of the granular media is measured by means of displacement in water. In exper- vances in performance (including increases in efficiency) and iment, since the horizontal area of the fluidized bed trackway is design of limb geometry will require a more detailed under- fixed, volume fraction is set by controlling the height of the granu- standing of the physics associated with penetration, drag, and lar media (e.g. volume = area × height). crater formation and collapse, especially their dependence on Kinematics measurements To characterize SandBot’s motion, φ. Better understanding of this physics can guide development we record simultaneous dorsal and lateral views with synchronized high speed video cameras (AOS Switzerland) at 100 frames/s. The of theory of interaction with complex media advanced enough center of mass (dorsal landmark) and the axles of the right-side to predict limb design [38] and control [39] strategies, similar front and rear motors (lateral landmarks) are marked with reflec- to the well-developed models of aerial and aquatic craft. Anal- tive material (WhiteOut). A rail-pulley system allows the robot’s ysis of physical models such as SandBot can also inform loco- power and communication cables to follow the robot as it moves to minimize the drag from the cables. For each trial, we prepare the motion biology in understanding how animals appear to move trackway with the desired volume fraction and place the robot on effortlessly across a diversity of complex substrates [40, 26]. the prepared granular media at the far end of the trackway with Such devices will begin to have capabilities comparable to or- both tripods in the same standing position. An LED on the robot ganisms; these capabilities could be used for more efficient synchronizes the video and robot motor encoder data. After each and capable exploration of challenging terrestrial (e.g. rub- trial, MATLAB (The MathWorks) is used to obtain landmark co- ordinates from the video frames and calculate vx(t). Three trials ble and disasters sites) and extra-terrestrial (e.g. the Moon were run for each combination of (φ, ω) that was tested. and Mars) environments. Detailed discussion of rotary walking locomotion model The model presented in the main body of the manuscript simplifies the underlying physics while capturing the essential features determin- Materials and Methods ing robot speed. Here we describe a more complete model (which Limb kinematics SandBot’s six motors are controlled by a clock sig- lacks a simple expression for vx) and compare its predictions to nal to follow the same prescribed kinematic path during each rota- those of the simple model. The exact expression for the vertical tion and, as shown in previous work on RHex, changes in these kine- acceleration component of the body when the limbs gain purchase matics have substantial effects on robot locomotor performance [41]. 2 2(h+z) h+z The controlling clock signal consists of a fast phase and a slow phase is maz = ma sin θ = ma R − R instead of the approx- r “ ” with respective angular frequencies. The fast phase corresponds imation maz = ma used in the simple model. Using the exact to the swing phase, and the slow phase corresponds to the stance expression, the vertical granular force necessary for walking still phase. A set of three gait parameters uniquely determines the clock has the same peak value of m(a + g) but decreases to mg when the signal configuration: θs, the angular span of the slow phase; θ0, the limb is at its lowest point. leg-shaft angle of the center of the slow phase; and dc, the duty The second approximation we employed in the simple model is cycle of the slow phase. Specifying the cycle average limb angular that the grain force on the leg is kz. This expression is only strictly frequency ω fully determines the limb motion. valid for a flat bottomed vertically penetrating intruder [33]. Since In pilot experiments we tested two sets of clock signals: a hard the leg is a circular arc, the leg-grain contact area and the vertical ground clock signal (HGCS) with (θs = 0.85 rad, θ0 = 0.13 rad, component of the grain force are functions of limb depth and leg- dc = 0.56) which generates a fast bouncing gait (60 cm/s) on hard shaft angle. Generalizing kz to a local isotropic yield stress given ground [18] but very slow (∼ 1 cm/s) motion on granular media, by αz [12], the vertical force on a small segment of the limb Rdψ and a soft ground clock signal (SGCS) with (θs = 1.1 rad, θ0 = -0.5 in length at depth z is dFz = wαzRdψ cos ψ, where w is the limb rad, dc = 0.45) which produces unstable motion on hard ground width and ψ the angular position of the segment with respect to but regular motion on granular media. These experiments showed a vertical line passing through the axle. The total vertical compo- that the locomotor capacity of SandBot is sensitive to the clock sig- ψmax nent of the force acting on the leg is then Rwα ψmin z cos ψdψ. nal. Careful observation of limb kinematics revealed that the hard − ground clock signal fails on granular media because of the simul- Substituting z(ψ) = R(cos ψ − 1) + d and integratingR gives Fz = ψmax R taneous stance phase of two tripods. In this study, we use SGCS Rwα (ψ + cos ψ sin ψ) + (d − R) sin ψ , where ψmax = 2 ψmin and explore robot performance as a function of limb frequency and h i−
4 www.pnas.org — — Footline Author 1 cos− (1 − d/R) and ψmin = ψmax when the leg tip is above the force to support the robot’s mass and accelerate it. Fig. 4b presents 1 d+h fits to the experimental data of the average speed vx vs. ω for the center of the c-leg and cos− R − 1 +∆ξ when it is below the full and simple models for vx ≤ v∗ at each φ. The fits and fit pa- center of the c-leg. ∆ξ is the“ angular extent” of the limb beyond π x rameters for the simple (∆t = 0.4 s, α = 470 kN/m3) and full (e.g ∆ξ = 0 for a semi-circular limb). 3 Fig. 4a shows that the full model using realistic parameters (∆t = 0.2 s, α = 330 kN/m ) models are in good agreement when shares the same essential physics as the simple model. For a given the step size is less than the critical value s = R. material strength (blue curves), the penetration depth increases with increasing ω (intersection of blue and red curves) until ei- ACKNOWLEDGEMENTS. We thank Daniel Cohen, Andrew Slat- ther the step size is reduced below the critical value (vertical green ton and Adam Kamor for helpful discussion. This work was sup- dashed line) or the granular force no longer provides the required ported by the Burroughs Wellcome Fund (D.I.G., C.L., P.B.U).
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Schroter, M, Nagle, S, Radin, C, Swinney, HL (2007) Phase transition in a static granular system. Europhysics Letters 78:44004–44007. 11. Terzaghi, K (1943) Theoretical soil mechanics (New York: Wiley), p 510. 32. We varied ω with 1 rad/s resolution near the transition to locate the 12. Nedderman, R (1992) Statics and kinematics of granular materials (Cam- exact critical volume fraction ωc. bridge University Press), p 352. 33. Albert, R, Pfeifer, MA, Barabasi, AL, Schiffer, P (1999) Slow drag in a 13. Heil, P, Rericha, EC, Goldman, DI, Swinney, HL (2004) Mach cone in a granular medium. Physical Review Letters 82:205–208. shallow granular fluid. Physical Review E 70:060301–060304. 34. Stone, MB et al. (2004) Local jamming via penetration of a granular 14. van Zon, JS et al. (2004) Crucial role of sidewalls in velocity distributions medium. Physical Review E 70:041301–041310. in quasi-two-dimensional granular gases. Physical Review E 70:040301– 040304. 35. Wieghardt, K (1975) Experiments in granular flow. Annual Review of Fluid Mechanics 7:89–114. 15. Goldman, DI, Swinney, HL (2006) Signatures of glass formation in a fluidized bed of hard spheres. Physical Review Letters 96:174302–174305. 36. de Vet, S, de Bruyn, J (2007) Shape of impact craters in granular media. Physical Review E 76:041306–041311. 16. Goldman, DI, Korff, WL, Wehner, M, Berns, MS, Full, RJ (2006) The mechanism of rapid running in weak sand. Integrative and Comparative 37. Goldman, DI, Umbanhowar, P (2008) Scaling and dynamics of sphere and Biology 46:E50–E50. disk impact into granular media. Physical Review E 77:021308–021321. 17. (2008) personal communication from Professor Michel Louge based on 38. Santos, D, Spenko, M, Parness, A, Kim, S, Cutkosky, M (2007) Di- field observations in the Sahara desert. rectional adhesion for climbing: theoretical and practical considerations. Journal of Adhesion Science and Technology 21:1317–1341. 18. Saranli, U, Buehler, M, Koditschek, DE (2001) Rhex: A simple and highly mobile hexapod robot. International Journal of Robotics Research 39. Hodgins, JK, Raibert, MH (1991) Adjusting step length for rough terrain 20:616–631. locomotion. Ieee Transactions on Robotics and Automation 7:289–298. 19. Koditschek, DE, Full, RJ, Buehler, M (2004) Mechanical aspects of 40. Goldman, DI, Chen, TS, Dudek, DM, Full, RJ (2006) Dynamics of rapid legged locomotion control. Arthropod Structure and Development 33:251– vertical climbing in cockroaches reveals a template. Journal of Experi- 272. mental Biology 209:2990–3000. 20. Blickhan, R, Full, RJ (1993) Similarity in multilegged locomotion: bounc- 41. Weingarten, J, Groff, RE, Buehler, M, Koditschek, DE (2004) Automated ing like a monopode. J. Comp. Physiol. A 173:509-517. Gait Adaptation for Legged Robots pp 2153–2158. 21. Holmes, P, Full, RJ, Koditschek, D, Guckenheimer, J (2006) The dynam- 42. Bagnold, RA (1954) The Physics of Blown Sand and Desert Dunes ics of legged locomotion: Models, analyses, and challenges. Siam Review (Methuen and Co. Ltd.), p 265. 48:207–304.
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6 www.pnas.org — — Footline Author List of Figures 1 Locomotion of a legged robot on granular media is sensitive to substrate packing and limb frequency. (a) The six- legged robot, SandBot, moves with an alternating tripod gait (alternate triplets of limbs rotate π out of phase); arrows indicate members of one tripod. (b) Pulses of air through the bottom of the fluidized bed trackway control the initial volume fraction φ of the granular substrate; air is turned off before the robot begins to move. (c) Tripod leg-shaft angle θ vs. time is controlled to follow a prescribed trajectory with two phases: a slow stance phase and a fast swing phase. Overlapping trajectories from trials with φ = 0.633 (red) and φ = 0.600 (blue) at ω = 16 rad/s demonstrate that the controller maintains the desired kinematics independent of material state. (d) Identical tripod trajectories produce different motion for φ = 0.600 and φ = 0.633. (e) For given limb frequency (ω = 4, 8, 12, 16, 20, 24, and 30 rad/s) the robot speed is remarkably sensitive to φ. Red and blue circles show thecorrespondingstatesin(c)and(d)...... 8 2 Average robot speed vs. limb frequency. (a) For a given volume fraction φ, vx increases sub-linearly with ω to a maximal average speed vx∗ at a critical limb frequency ωc above which the robot swims (vx ∼ 2 cm/s). The solid lines and symbols are for φ = 0.580, 0.590, 0.600, 0.611, 0.616, 0.622, and 0.633. The dashed lines are fits from a simplified model discussed in the text. (b)(c) The dependence of ωc and vx∗ on φ shows transitions at φ ≈ 0.6 (dashedlines)...... 9 3 (a) Schematic of a single robot leg during a step in granular media. After reaching penetration depth d, the leg rotates about its center and propels the robot forward a step length s. The solid shape denotes the initial stage of the rotational motion and the dashed shape indicates when the limb begins to withdraw from the material (end of forward body motion). (b) Step length vs. penetration depth (blue) with critical step size (green dashed horizontal) and critical penetration depth (green dashed vertical) indicating where the robot begins to encounter ground disturbed by the previous step. (c) Granular penetration force for k = 1.75, 2.00, 2.25, 2.50, 2.75×105 N/m (blue) and force required to initiate rotary walking for ω = 0, 8, 16, 24, 32 rad/s (red) vs. penetration depth using simplified walking model with ∆t = 0.2 s. The penetration depth at constant φ is determined by the intersections of the corresponding blue line with the red lines. Beyond the critical depth (green dashed line) limbs encounter disturbed material and move to lower blue lines. (d) Step length as a function of ω derived from 2s = 2πv/ω reveals the condition for the onset of swimming as s/R ≈ 1...... 10 4 2 2 4 (a) Non-dimensionalized granular force (blue curves) for α × 10− = 2.50, 2.75, 3.00, 3.25, and 3.50 g cm− s− and the required force to initiate rotary walking, az/g + 1 (red curves) for ω = 0, 5, 10, 15, 20, 25 and 30 rad/s for the full model as a function of limb penetration depth with a 225◦ c-leg arc angle and ∆t = 0.15 s. The intersection of the red and blue curves determines the penetration depth of the limb and consequently the step size. At constant material strength (blue) d increases with increasing ω, while at constant ω increasing material strength reduces d. The vertical green dashed line indicates the critical penetration depth beyond which the leg encounters material disturbed by the previous step. (b) Comparison of vx vs. ω for simple (red dotted curve) and full (blue solid curve) models. Models are fit to the measured robot speed (symbols) for vx ≤ vx∗ . The green dashed line indicates vx∗ = Rω/π or equivalently s = R. In both figures h = 2.5 cm, R = 3.55 cm, w = 1.2 cm and m =767g...... 11
Footline Author PNAS Issue Date Volume Issue Number 7 Fig. 1. Locomotion of a legged robot on granular media is sensitive to substrate packing and limb frequency. (a) The six-legged robot, SandBot, moves with an alternating tripod gait (alternate triplets of limbs rotate π out of phase); arrows indicate members of one tripod. (b) Pulses of air through the bottom of the fluidized bed trackway control the initial volume fraction φ of the granular substrate; air is turned off before the robot begins to move. (c) Tripod leg-shaft angle θ vs. time is controlled to follow a prescribed trajectory with two phases: a slow stance phase and a fast swing phase. Overlapping trajectories from trials with φ = 0.633 (red) and φ = 0.600 (blue) at ω = 16 rad/s demonstrate that the controller maintains the desired kinematics independent of material state. (d) Identical tripod trajectories produce different motion for φ = 0.600 and φ = 0.633. (e) For given limb frequency (ω = 4, 8, 12, 16, 20, 24, and 30 rad/s) the robot speed is remarkably sensitive to φ. Red and blue circles show the corresponding states in (c) and (d).
8 www.pnas.org — — Footline Author Fig. 2. Average robot speed vs. limb frequency. (a) For a given volume fraction φ, vx increases sub-linearly with ω to a maximal average speed vx∗ at a critical limb frequency ωc above which the robot swims (vx ∼ 2 cm/s). The solid lines and symbols are for φ = 0.580, 0.590, 0.600, 0.611, 0.616, 0.622, and 0.633. The dashed lines are fits from a simplified model discussed in the text. (b)(c) The dependence of ωc and vx∗ on φ shows transitions at φ ≈ 0.6 (dashed lines).
Footline Author PNAS Issue Date Volume Issue Number 9 Fig. 3. (a) Schematic of a single robot leg during a step in granular media. After reaching penetration depth d, the leg rotates about its center and propels the robot forward a step length s. The solid shape denotes the initial stage of the rotational motion and the dashed shape indicates when the limb begins to withdraw from the material (end of forward body motion). (b) Step length vs. penetration depth (blue) with critical step size (green dashed horizontal) and critical penetration depth (green dashed vertical) indicating where the robot begins to encounter ground disturbed by the previous step. (c) Granular penetration force for k = 1.75, 2.00, 2.25, 2.50, 2.75×105 N/m (blue) and force required to initiate rotary walking for ω = 0, 8, 16, 24, 32 rad/s (red) vs. penetration depth using simplified walking model with ∆t = 0.2 s. The penetration depth at constant φ is determined by the intersections of the corresponding blue line with the red lines. Beyond the critical depth (green dashed line) limbs encounter disturbed material and move to lower blue lines. (d) Step length as a function of ω derived from 2s = 2πv/ω reveals the condition for the onset of swimming as s/R ≈ 1.
10 www.pnas.org — — Footline Author 4 2 2 Fig. 4. (a) Non-dimensionalized granular force (blue curves) for α × 10− = 2.50, 2.75, 3.00, 3.25, and 3.50 g cm− s− and the required force to initiate rotary walking, az/g + 1 (red curves) for ω = 0, 5, 10, 15, 20, 25 and 30 rad/s for the full model as a function of limb penetration depth with a 225◦ c-leg arc angle and ∆t = 0.15 s. The intersection of the red and blue curves determines the penetration depth of the limb and consequently the step size. At constant material strength (blue) d increases with increasing ω, while at constant ω increasing material strength reduces d. The vertical green dashed line indicates the critical penetration depth beyond which the leg encounters material disturbed by the previous step. (b) Comparison of vx vs. ω for simple (red dotted curve) and full (blue solid curve) models. Models are fit to the measured robot speed (symbols) for vx ≤ vx∗ . The green dashed line indicates vx∗ = Rω/π or equivalently s = R. In both figures h = 2.5 cm, R = 3.55 cm, w = 1.2 cm and m = 767 g.
Footline Author PNAS Issue Date Volume Issue Number 11