Dynamic locomotion with four an six-legge robots M Buehler, U Saranli 2, D Papaopoulos an D Koitschek 2 Centre for Intelligent Machines, Ambulatory Robotics Laboratory, McGill University 2 Department of Electrical Engineering an Computer Science, University of Michigan http://wwwcimmcgillca/~arlweb http://wwweecsumicheu/~ulucs/rhex/ Abstract Stable an robust autonomous ynamic locomotion is emonstrate experimentally in a four an a six-legge robot The Scout II quarupe runs on flat groun in a bouning gait, an was motivate by an effort to unerstan the minimal mechanical esign an control complexity for ynamically stable locomotion The RHex hexapo runs ynamically in a tripo gait over flat an baly broken terrain Its esign an control was motivate by a collaboration of roboticists, biologists, an mathematicians, in an attempt to capture specific biomechanical locomotion principles Both robots share some basic features: Compliant legs, each with only one actuate egree of freeom, an reliance on (task space) open loop controllers Introuction Designers of statically stable autonomous legge robots in the past have pai careful attention to minimize negative work by minimizing vertical boy movements uring locomotion This require complex leg esigns with at least three egrees of freeom per leg, more if an ankle/foot combination is require The resulting cost, mechanical complexity, an low reliability make it ifficult for these robots to be profitably eploye in real worl tasks In contrast, ynamic locomotion with compliant legs permits not only higher spees an the potential for rastically improve mobility compare to statically stable machines, but at the same time permits these improvements with greatly simplifie leg mechanics With compliant legs, instantaneously controlle boy motion can no longer be achieve, an energy efficient locomotion must utilize intermittent storage an release of energy in the passive leg compliances It is remarkable that espite their mechanical simplicity, outstaning ynamic mobility is obtaine in both machines escribe in this paper, base on very simple (task space) open loop controllers In the Scout II quarupe we have attempte to emonstrate the limits of mechanical simplicity, while still obtaining a range of useful ynamic mobility Even with only one actuator per leg, we obtaine full mobility in the plane on flat groun, an running spees of up to 2 m/s with a bouning gait [7] These preliminary results an ongoing research suggest that further spee an mobility improvements, incluing compliant walking, leaping, an rough terrain hanling are within reach The extension of the basic engineering esign principles of Scout II to the funamentally ifferent hexapeal running of RHex is base on insights from biomechanics, whose careful consieration excees the scope of this paper In a paper ocumenting the performance of cockroach locomotion in a setting similar to our recreation in Figure, R J Full et al, state Simple feeforwar motor output may be effective in negotiation of rough terrain when use in concert with a mechanical system that stabilizes passively Dynamic stability an a conservative motor program may allow manylegge, sprawle posture animals to miss-step an collie with obstacles, but suffer little loss in performance Rapi isturbance rejection may be an emergent property of the mechanical system" In particular, Full's vieo of a Blaberus cockroach racing seemingly effortlessly over a rough surface, shown at an interisciplinary meeting [6] motivate an initiate the evelopment of RHex Though morphologically quite istinct from its biological counterparts, RHex emulates the basic principles of insect locomotion as articulate by Full The robot s sprawle posture with properly esigne compliant legs affors strong passive stability properties, even on baly broken terrain These stability properties, combine with a rugge mechanical esign forgiving to obstacle collisions permits controllers base on open loop ( clocke ) leg trajectories to negotiate a large variety of terrains The Scout project was supporte by IRIS (A Feeral Network of Centers of Excellence of Canaa) an NSERC (The National Science an Engineering Research Council of Canaa) The RHex project was supporte by DARPA (The US Defense Avance Research Projects Agency) uner grant number DARPA/ONR N4-98--747 Portions of this paper have appeare in the Proc of the International Conference on Robotics an Automation, 2 [7,9]
Report Documentation Page Form Approve OMB No 74-88 Public reporting buren for the collection of information is estimate to average hour per response, incluing the time for reviewing instructions, searching existing ata sources, gathering an maintaining the ata neee, an completing an reviewing the collection of information Sen comments regaring this buren estimate or any other aspect of this collection of information, incluing suggestions for reucing this buren, to Washington Heaquarters Services, Directorate for Information Operations an Reports, 25 Jefferson Davis Highway, Suite 24, Arlington VA 2222-432 Responents shoul be aware that notwithstaning any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it oes not isplay a currently vali OMB control number REPORT DATE 2 2 REPORT TYPE 3 DATES COVERED - 4 TITLE AND SUBTITLE Dynamic Locomotion with four an six-legge robots 5a CONTRACT NUMBER 5b GRANT NUMBER 5c PROGRAM ELEMENT NUMBER 6 AUTHOR(S) 5 PROJECT NUMBER 5e TASK NUMBER 5f WORK UNIT NUMBER 7 PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Defense Avance Research Projects Agency,37 North Fairfax Drive,Arlington,VA,2223-74 8 PERFORMING ORGANIZATION REPORT NUMBER 9 SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) SPONSOR/MONITOR S ACRONYM(S) 2 DISTRIBUTION/AVAILABILITY STATEMENT Approve for public release; istribution unlimite 3 SUPPLEMENTARY NOTES The original ocument contains color images 4 ABSTRACT see report 5 SUBJECT TERMS SPONSOR/MONITOR S REPORT NUMBER(S) 6 SECURITY CLASSIFICATION OF: 7 LIMITATION OF ABSTRACT a REPORT unclassifie b ABSTRACT unclassifie c THIS PAGE unclassifie 8 NUMBER OF PAGES 6 9a NAME OF RESPONSIBLE PERSON Stanar Form 298 (Rev 8-98) Prescribe by ANSI St Z39-8
2 Scout II Quarupe Control The bouning controller accomplishes running at a esire forwar spee, x&, by placing each leg at the esire angle, φ, XCG γ = x T / 2 + k = arctan l φ = γ S 2 x ( x x XCG XCG θ ) + a 2, () Figure : Scout II Scout II, shown in Fig, has a main boy an four compliant legs The boy contains all elements for autonomous operation, incluing computing, I/O, sensing, actuation, an batteries Each leg is a passive prismatic joint with compliance an rotates in the sagittal plane, actuate at the hip by one motor Without leg articulation, toe clearance uring the swing phase can be achieve with any running gait that inclues a flight phase, for example, pronking, trotting an bouning We have chosen the bouning gait (Fig 2) since it permits a smooth transition from a bouning walking gait, the subject of current research Figure : Illustration of a bouning gait an applying a leg torque τ = kv ( x x ) uring stance This controller is motivate by the foot placement algorithm in Raibert's three-part controller [8] The key ifferences in our controller are necessitate by the absence of a linear leg thrusting actuator, an thus the lack of a irect means to a energy to the vertical (boy pitching) ynamics First, the offset term, a, in (), iverts some forwar energy to the vertical ynamics in each step This reuce forwar energy (the robot slows own) is then compensate uring stance phase via the explicit velocity control There is no explicit control of the boy pitch oscillation - front an back leg controllers are inepenent They only rely on the iniviual leg states, an make no use of an overall boy state Computer simulations show that this controller, espite its simplicity, succees not only in stable velocity control, but also in tracking rapi set point changes in forwar velocity, as shown in Fig 4 Forw ar spee (m/s) 5 5-5 2 4 6 8 2 time (sec) Figure 3: Step changes in forwar velocities controlle by the hip actuator torque Figure 2: Scout II moel The sagittal plane moel, shown in Fig 3, is a four egree-of-freeom system in each single stance phase, an a five egree-of-freeom system uring flight, with only two hip torque control inputs An open loop version of this controller is an attempt to emonstrate the simplest form of compliant quarupe running control without any explicit feeback control of boy oscillation an forwar spee It simply commans a constant esire hip torque, τ, uring stance an a constant esire leg angle, φ, controlle uring flight via a set point PD algorithm With two values for front an back legs, this controller is etermine by only four parameters
Fig 5 shows a Working Moel 2D [4] simulation of the open loop controller, with fixe values of touchown leg angles (8 o for the back legs an 22 o for the front legs) an stance torques (4 Nm for the back legs an Nm for the front legs) The result is steay running with 2 m/s forwar spee with boy oscillation with an amplitue of 65 o an a perio of 29 s Boy pitch (eg) Forw ar spee (m/s) 5-5 - 2 3 4 5 6 7 8 9 time (s) 5 5-5 2 3 4 5 6 7 8 9 time (s) Figure 4: Boy pitch an forwar velocity uring running with the open loop controller Thus, surprisingly, compliant quarupe running is possible without explicit feeback control of forwar spee or stance time The isavantage of this controller is that each particular spee requires the selection of the appropriate touchown leg angles an stance torques However, this coul be implemente in a straightforwar fashion as a lookup table, an coul serve as a potentially robot-saving backup controller in case of sensor failure Experiments As suggeste by the simulations, it is possible to achieve a steay bouning gait by choosing a suitable set of constant motor torques uring stance an leg touchown angles uring flight Even though there is no active control of the boy roll ynamics in the experimental four-legge robot, the amping in the leg springs was sufficient for passive roll stability We have implemente the open loop controller on Scout II A back torque of 35 Nm per leg an a front torque of Nm per leg was use A touchown angle of 22 o with respect to the vertical for the front legs an 8 o for the back legs was commane for the flight phases A slip prevention torque limit (escribe in [7] an omitte here for brevity) was implemente in simulation an experiments The only ifference in the experimental slip prevention function is that it ealt with each of the two front an back legs inepenently Both simulation an experimental runs starte at zero spee an accelerate until steay state spees were achieve While the first two to three secons transition phase is ifferent in simulation an experiment, the remaining operating time is comparable Both spees reach a steay value of about 2 m/s The large experimental spee fluctuations in Fig 6 are primarily an artifact of our spee calculation, base on the hip angular velocities, which suffers ue to the combine backlash of the gear an the belt transmission of several egrees Forwar spee (m/s)(experimental) Forwar spee (m/s)(simulate) 5 5 5 5 2 3 4 5 6 time (sec) 2 3 4 5 6 time (sec) Figure 5: Forwar velocity Top: Experiment Bottom: Simulation Turning while running is accomplishe via a simple moification to the open loop bouning controller The iea is to apply ifferential torques to the left an right sies of the legs uring the stance phases Implementation of the turning algorithm resulte in rapi turns as illustrate in Fig 7 252 s 368 s 44 s 488 s Figure 6: Turning experiment
3 RHex Hexapo the running an turning controllers, where the legs forming the left an right tripos are synchronize with each other an are 8 out of phase with the opposite tripo, as shown in Fig 9 Figure 7: RHex RHex, shown in Fig 8, has a main boy an six compliant legs As in Scout II, the boy contains all elements for autonomous operation, incluing computing, I/O, sensing, actuation, an batteries Unlike most hexapoal robots built to ate, RHex has compliant legs, an was built to be a runner Each leg rotates in the sagittal plane, actuate at the hip by one motor Since a bouning type walking gait is not feasible with six legs, RHex walks with a compliant tripo gait, an eliminates any toe clearance problems by rotating the legs in a full circle Control Since the present prototype robot has no external sensors by which its boy coorinates may be estimate, we have use joint space close loop ( proprioceptive ) but task space open loop control strategies These are tailore to emonstrate the intrinsic stability properties of the compliant hexapo morphology an emphasize its ability to operate without a sensor-rich environment Specifically, we present a four-parameter family of controllers that yiels stable running an turning of the hexapo on flat terrain, without explicit enforcement of quasistatic stability All controllers generate perioic esire trajectories for each hip joint, which are then enforce by six local PD controllers, one for each hip actuator As such, they represent examples near one extreme of possible control strategies, which range from purely open-loop controllers to control laws that are solely functions of the leg an rigi boy state It is evient that neither one of these extremes is the best approach an a combination of these shoul be aopte An alternating tripo pattern governs both Figure 9: Motion profiles for left an right tripos The running controller's target trajectories for each tripo are perioic functions of time, parametrize by four variables: t c, t s, φ s an φ o The perio of both profiles is t c In conjunction with t s, it etermines the uty factor of each tripo In a single cycle, both tripos go through their slow an fast phases, covering φ s an 2π - φ s of the complete rotation, respectively The uration of ouble support t, when all six legs are in contact with the groun, is etermine by the uty factors of both tripos Finally, the φ o parameter offsets the motion profile with respect to the vertical Note that both profiles are monotonically increasing in time; but they can be negate to obtain backwar running Simulations (Fig ) emonstrate that control of average forwar running velocity is possible with these controller outputs Figure : Simulation of forwar boy velocity We have evelope two ifferent controllers for two qualitatively ifferent turning moes: turning in place
an turning uring running The controller for turning in place employs the same leg profiles as for running except that contralateral sets of legs rotate in opposite irections This results in the hexapo turning in place Note that the tripos are still synchronize internally, maintaining three supporting legs on the groun Similar to the control of forwar spee, the rate of turning epens on the choice of the particular motion parameters, mainly t c an φ s In contrast, we achieve turning uring forwar locomotion by introucing ifferential perturbations to the forwar running controller parameters for contralateral legs In this scheme, t c is still constraine to be ientical for all legs, which amits ifferentials in the remaining profile parameters, φ o an t s, while φ s remains unchange Two new gain parameters, t s an φ o are introuce Consequently, turning in +x (right) irection is achieve by using u l = [t c ; t s + t s ; φ s ; φ o + φ o ] an u r = [t c ; t s - t s ; φ s ; φ o - φ o ] for the legs on the left an right sies, respectively Experiments We have implemente the open loop controller on the RHex prototype Extensive testing emonstrate that RHex was able to negotiate a variety of challenging obstacle courses, with obstacles well exceeing the robot s groun clearance, all with fixe (unchange) open loop control trajectories, an with only minor velocity variations between 45 m/s an 55 m/s Detaile statistical performance ocumentation over all the terrains will be the subject of a forthcoming publication On flat groun (carpet), the forwar spee (average over ten runs) is, as preicte by the simulation, slightly above 5 m/s, or about one boy length/s On this surface, the average total electrical power consumption is 8 W As simulation stuy ha preicte as well, steering is possible, even though the leg actuation is limite to motion in the sagittal plane only, via ifferential motion between left an right legs We selecte control parameters that resulte in turns in place an robot spees up to about 4 m/s The maximum forwar velocity is reuce uring turning, because the ifferential leg motion precipitates the onset of the spee limiting vertical boy oscillations The maximum yaw angular velocities increase almost linearly with forwar velocity up to 9 ra/s at 39 m/s Interestingly, the resulting turn raius is almost constant with approximately 2 m Turning in place provies the highest yaw angular velocity of 7 ra/s One particular rough terrain experiment was an attempt to evaluate RHex's performance in a similar environment to that negotiate by a Blaberus cockroach in [2] Our efforts at re-creating such a surface at RHex's scale are shown in Figure To our surprise, RHex was able to traverse this surface with ranom height variations of up to 232 cm (6% leg length) with relative ease at an average velocity of 42 m/s (average over ten successful runs) Figure : Locomotion on rough terrain Accumulating evience in the biomechanics literature suggests that agile locomotion is organize in nature by recourse to a controlle bouncing gait wherein the payloa", the mass center, behaves mechanically as though it were riing on a pogo stick [] While Raibert's running machines were literally emboie pogo sticks, more utilitarian robotic evices such as RHex must actively anchor such templates within their alien morphology if the animals' capabilities are ever to be successfully engineere [3] A previous publication showe how to anchor a pogo stick template in the more relate morphology of a four egree of freeom monopo [] The extension of this technique to the far more istant hexapo morphology surely begins with the aoption of an alternating tripo gait, but its exact etails remain an open question, an the minimalist RHex esign (only six actuators for a six egree of freeom payloa!) will likely entail aitional compromises in its implementation Moreover, the only well unerstoo pogo stick is the Spring Loae Inverte Penulum [2], a two-egree of freeom sagittal plane template that ignores boy attitue an all lateral egrees of freeom Recent evience of a horizontal pogo stick in sprawle posture animal running [5] an subsequent analysis of a propose lateral leg spring template to represent it [] avance the prospects for eveloping a spatial pogo stick template in the near future Much more effort remains before a functionally biomimetic six egree of freeom payloa controller is available, but we believe that
the present unerstaning of the sagittal plane can alreay be use to significantly increase RHex's running spee, an, as well, to enow our present prototype with an aerial phase J Schmitt an P Holmes, Mechanical moels for insect locomotion I: Dynamics an stability in the horizontal plane, Biol Cybernetics, submitte, 999 2 W J Schwin an D E Koitschek, Approximating the Stance Map of a 2 DOF Monope Runner, J Nonlinear Science, to appear Acknowlegments The Scout project was supporte by IRIS (A Feeral Network of Centers of Excellence of Canaa) an NSERC (The National Science an Engineering Research Council of Canaa) The RHex project was supporte by DARPA (The US Defense Avance Research Projects Agency) uner grant number DARPA/ONR N4-98--747 We also acknowlege the generous an talente help of L Mitrea, G Hawker, D McMorie, an S Obai Close collaboration with R J Full of UCB has provie many of the biomechanical insights which motivate the esign an control of RHex Bibliography R Blickhan an R J Full, Similarity in multilegge locomotion: bouncing like a monopoe, J Comparative Physiology, vol A 73, pp 59-57, 993 2 R J Full, K Autumn, J I Chung, an A Ahn, Rapi negotiation of rough terrain by the eath-hea cockroach, American Zoologist, vol 38, pp 8A, 998 3 R J Full an D E Koitschek, Templates an Anchors: Neuromechanical Hypotheses of Legge Locomotion on Lan, Journal of Experimental Biology, vol 22, pp 3325-3332, 999 4 Knowlege Revolution Working Moel 2D User s Guie San Mateo, CA, 996 5 T M Kubow an R J Full, The role of the mechanical system in control: A hypothesis of selfstabilization in hexapeal runners, Phil Trans R Soc Lon, vol B 354, pp 849-862, 999 6 NSF Institute for Mathematics an Its Applications Spring 998 Workshop on Animal Locomotion an Robotics June -5 998, wwwimaumneu/ynsys/spring/ynsyshtml 7 D Papaopoulos an M Buehler, "Stable Running in a Quarupe Robot with Compliant Legs," IEEE Int Conf Robotics an Automation, April 2 8 M H Raibert Legge Robots That Balance MIT Press, Cambrige, MA, 986 9 U Saranli, M Buehler, an D E Koitschek, Design, Moeling an Preliminary Control of a Compliant Hexapo Robot, IEEE Int Conf Robotics an Automation, April 2 U Saranli, W J Schwin, an D E Koitschek, Towar the Control of a Multi-Jointe, Monope Runner, IEEE Int Conf Robotics an Automation, Leuven, Belgium, May 998