Design, Modeling and Preliminary Control of a Compliant Hexapod Robot

1 2 1 Uluc. Saranli ,∗ Martin Buehler and Daniel E. Koditschek 1Department of Electrical Engineering and Computer Science The University of Michigan, Ann Arbor, MI 48109-2110, USA 2Center for Intelligent Machines McGill University, Montreal, QC H3A 2A7, Canada

Abstract (http://www.eecs.umich.edu/ ulucs/rhex/), that presently — following the six∼ week period of its In this paper, we present the design, modeling and physical construction — runs at roughly one body preliminary control of RHex, an autonomous dynami- length (53 cm) per second for roughly ﬁfteen minutes cally stable hexapod possessing merely six actuated de- over our laboratory ﬂoor with neither power nor grees of freedom (at the hip attachment of each leg). computational tether. We sketch in the conclusion Our design emphasizes mechanical simplicity as well the reasons we expect this robot (or, possibly, a as power and computational autonomy, critical com- more robustly constructed successor) will be able to ponents for legged robotics applications. A compliant run considerably faster (perhaps beyond 2 m/s) over hexapod model, used to build a simulation environment badly broken terrain. closely informed the design and construction of the physical machine and promises to inform, similarly, our future analysis as well. Simulations and experiments show that RHex can achieve dynamically stable walking, running and turning with very simple clock driven open-loop control strategies.

1 Introduction

Robotic mobility over highly broken and unstable terrain requires legged machines. Operation within such unstructured settings requires completely self contained power and computation. Survival beyond Figure 1: The RHex experimental platform. some fleeting “demonstration” requires a rugged and durable design. But no power and computation autonomous robot has yet been built that is agile Our design represents what might be considered the enough to run over the lab floor much less offer any minimal configuration for a legged vehicle that leaves service at, say, an earthquake scene. In this paper room for a reasonably wide range of behaviors. The we report on our first steps toward a machine that central obstacle to high performance autonomous ma- we expect will be able to run, walk and hopefully chines remains, of course, the very limited power den- perform rudimentary manipulation within a field sity of contemporary actuation and energy storage of rubble or similar setting that would defy any technology, and at the most basic level, this design wheeled or (articulated) tracked vehicle and even exercise rehearses the familiar tradeoffs between in- a statically balanced legged machine. We describe creasing performance and debilitating inefficiency as in some detail a design exercise that has resulted the number of motors increases. Hexapodal design in a working physical prototype hexapod, RHex obviously affords simple low torque solutions to static (parked) and quasi-static (walking) operation, but the ∗Supported in part by DARPA/ONR Grant N00014-98-1- 0747 commitment as well to the dynamical regime (running) introduces the need for very high joint torques as well as its potential energy. imposed upon rapidly moving limbs. Evidently, once again, the only recourse is to mechanical energy storage: the effective integration of leg springs into our 2 Design and Modeling mechanical design represents the chief contribution of this paper. In the conclusion we sketch the manner 2.1 Design Concept and Morphology in which our future controls work with this prototype should transform a design necessity into a performance In all robotics applications, mechanical complexity is advantage. As a secondary contribution, an appropri- one of the major sources of failure and considerably ate level of attention to physical modeling details has increases the cost. Our design emphasizes mechanical yielded a sufficiently informative simulation environ- simplicity and thereby promotes robustness. Auton- ment, that the construction stage of this prototype omy, a critical component of our aspiration toward proceeded quite rapidly (six weeks) and with very few real-world tasks in unstructured environments outside surprises. Accordingly, we focus a fair bit of attention the laboratory, imposes very strict design constraints on these modeling details and the manner in which on the hardware and software components. It is of- they have informed our design. ten impossible to achieve with simple modifications to The closest cousins to RHex are arguably the a system otherwise designed for non-autonomous op- hexapods at Case Western [1, 7, 14] resulting from eration. These constraints also justify our preference an outstanding collaboration between neuroscien- for mechanical simplicity, in particular the minimum tists, computer scientists and mechanical engineers [2] amount of actuation and sensing. whose prior work in this area has benefitted us greatly. Our design depicted in Figure 2 consists of a rigid body Their commitment to detailed zoomorphic mimicry re- with six compliant legs, each possessing only one inde- sults in much greater kinematic complexity (36 degrees pendently actuated revolute degree of freedom. The of actuated freedom) and the attendant limitations of attachment points of the legs as well as the joint ori- contemporary commercial actuators incur significant entations are all fixed relative to the body and the performance limitations. In contrast, our work is in- leg compliance is mainly in the unactuated spherical formed by specific principles governing animal loco- degrees of freedom. motion (see the conclusion) but our appeal to what might be termed “functional biomimesis” is always subservient to engineering practice. The imperatives of nimble, robust, autonomous operation drive a design which is morphologically alien to any animal and Þ

our minimalist orientation aﬀords high performance Ý

operation in the dynamical regime with existing com- Ü B

mercial technology. A large and growing literature on a hexapods initiated by Brooks’ Genghis [4] has culmi- i

nated in a very impressive commercial robot capable

i Ö of sustained operation in the surf zone [12]. While we i

aspire to the ruggedness and power autonomy of these Ï

designs, they all rely on static or quasi-static balance, f i while RHex is designed to be a runner. In this regard, our chief inspiration remains the pioneering work of Raibert [15] whose hopping and running machines, al- Figure 2: The Compliant Hexapod Design. though morphologically quite foreign to the present design, inform the future of all dynamically dextrous robotics and represent the real point of departure for This conﬁguration admits an alternating tripod gait our research. The path from Raibert’s runners to our for forward and backward locomotion, as well as other RHex leads through the second author’s Scout class more elaborate (yet to be achieved) behaviors such as quadrapeds [5, 6] that reduce mechanical complexity leaping, stair climbing etc. Moreover, the symmetry of (by the restriction of one actuator per leg) without the design allows identical upside-down operation and abandoning dynamical dexterity. In short, the sin- imposes no restrictions on forward directionality. We gle criterion by which we may best distinguish our explore some of this behavioral repertoire both in sim- machine from the huge surrounding literature is that ulation and experimentally in Section 4 and Section 5, RHex is designed with the ability to manage its kinetic respectively. States on the massless leg result in the following equalities. rb, Rb body position and orientation r˙b translational velocity of body F1 = Fri wb angular velocity of body τθi F2 = Leg states and parameters ρi ai leg attachment point in τ B φi toe position in F3 = . fi ρi cos θi T W vi := [θi,φi,ρi] leg state in spherical coordinates

vi leg state in cartesian coordinates legi stance ﬂag for leg i

Forces and Torques

i Fri radial leg spring force

τθi bend torque in θi direction

F ½

τφ hip torque in φi direction F i ¿

System Parameters Ü F M0 inertia matrix in ¾ B M inertia matrix in W m body mass Figure 3: Analysis of a single leg in the plane defined Controller Parameters by the leg and the y-axis of . u := [tc,ts,φs,φo] control vector B tc period of rotation for a single leg ts duration of slow leg swing The body experiences the negative of the ground re- φs leg sweep angle for slow leg swing action force on the leg, resulting in an effective force φo leg angle offset and torque vectors acting on the center of mass. After projections to , for each leg i =1, ..., 6 we have, Table 1: Notation used throughout the paper B cos θi sin φi sin θi sin φi cos φi − − Fi =  sin θi cos θi 0  . 2.2 The Compliant Hexapod Model cos θi cos φi sin θi cos φi sin φi  − −  Fr Two reference frames, and are defined in Figure i  τθ /ρi  2, the former attachedB to theW hexapod body and the i τφ /(ρi cos θi) latter an inertial frame where the dynamics are formu-  i  τ =(v + a ) F lated. The position and orientation of the rigid body i i i × i are described by r R3 and R SO(3), respec- b b which are the force and torque contributions of a single tively, expressed in ∈. Table 1 details∈ the notation leg to the overall system dynamics. used throughout theW paper. Each leg is assumed to be massless and has complete spherical freedom. The leg state is described in the 2.2.2 Equations of Motion spherical coordinate frame [θ ,φ ,ρ ]T whose origin is i i i The cumulative effect of all the legs on the body is sim- located at ai in the body frame. Note that (rb, Rb), vi ply the sum of the individual contributions, together and fi are related through a simple coordinate trans- with the gravitational force. Note that the legs which formation. are not in contact with the ground have no effect on the body. The force and torque vectors, expressed in , are hence given by W 2.2.1 Analysis of a Single Leg 0 6 FT =  0  + Rb legiFi (1) Our formulation of the equations of motion for the mg Xi=1 hexapod model is based on individually incorporating  −  6 the ground reaction forces at each leg. To this end, it τT = Rb legiτi (2) will suffice to analyze a generic leg parametrized by its Xi=1 attachment and touchdown points, ai and fi, respec- 0 leg i is in flight legi := tively, (see Figure 3). The force and torque balance ½ 1 leg i is in stance The dynamics of a rigid body under this external four variables: tc, ts, φs and φo. The period of both force and torque actuation is governed by the following profiles is tc. In conjunction with ts, it determines equations [10]. the duty factor of each tripod. In a single cycle, both

FT tripods go through their slow and fast swing phases, r¨b = covering φs and 2π φs of the complete rotation, re- m − Mw˙b = J(wb)Mwb + τT spectively. The duration of double support td (where − all six legs are in contact with the ground) is deter- R˙b = J(wb)Rb mined by the duty factors of both tripods. Finally, the φo parameter oﬀsets the motion proﬁle with re- where we have spect to the vertical (see Figure 4). Note that both

0 wz wy proﬁles are monotonically increasing in time; but they T − J( wx wy wz ):= wz 0 wx  can be negated to obtain backward running. − £ ¤ wy wx 0

 − 

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3 Preliminary Control Strategy d

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In our initial simulations as well as the experiments

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space closed loop (“proprioceptive”) but task space ¾

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morphology and emphasize its ability to operate without a sensor-rich environment. Specifically, we present a 4-parameter family of controllers, that yields stable running and turning of the hexapod on flat terrain, Figure 4: The motion profiles for left and right tripods. without explicit enforcement of quasi-static stability. Both controllers generate clock driven desired trajectories for each hip joint, which are then tracked by Control of running behavior is achieved by modify- PD controllers for each individual hip joint. As such, ing these parameters for a particular desired behavior they represent examples near one extreme of possible during locomotion. In Section 4, our simulation stud- control strategies, ranging from purely clock driven ies reveal correlations of these parameters with certain controllers to control laws which are solely functions attributes of running behavior. We demonstrate that of rigid body state. It is evident that neither one of control of average forward running velocity under ac- these extremes is the best approach and a combination tuation and power constraints is possible with these of these should be adopted. The simulations and ex- controller outputs. periments presented in this paper attempt to charac- terize the properties associated with the clock driven approach, which we then hope to complement with 3.2 Turning In Place feedback to explore the aforementioned range. The controller for turning in place employs the same An alternating tripod pattern governs both the run- leg profiles as for running except that contralateral ning and turning controllers, where the legs forming sets of legs rotate in opposite directions. This results the left and right tripods are synchronized with each in the hexapod turning in place in the direction deter- o other and are 180 out of phase with the opposite tri- mined by the rotational polarity of the left and right pod. sets of legs. Note that the tripods are still synchronized internally, maintaining three supporting legs on 3.1 Running the ground. Similar to the control of the forward locomotion speed, the rate of turning depends on the The running controller’s target trajectories for each choice of the particular motion parameters, mainly tc tripod are periodic functions of time, parametrized by and φs. 4 Simulation Studies platform. In our discharge model, we use this curve, together with an approximation of the PWM electron- 4.1 Actuator Model ics driving the DC motors to estimate the duration of autonomous operation. In simulating the compliant hexapod, we also incorpo- First, we derive a method for estimating battery dis- rate a simple model of the hip actuation. This model charge in terms of a continuous time function of vary- imposes realistic limitations on the torque capabilities ing discharge current. This can be accomplished with of the actuators and, together with the battery model of the following section provides a means of estimating dC(t) 1 the power consumption of the overall system. = dt −f(ia(t)) Figure 5(a) portrays the torque-speed curve for the DC motors in the experimental platform. In addition where C(t) is the percent “energy” left in the battery, to these torque limits, the model also incorporates an and f(i) is the battery discharge curve in functional estimation loop for motor voltages v and currents i i i form. During the hexapod operation, all six motors using the commanded leg torques, under the assump- draw current and contribute to the battery discharge tion that the electrical dynamics are negligible relative together. Due to the H-bridge output stages of the to the mechanical dynamics. In consequence, we have motor drives, the motor currents add up, yielding the battery lifetime equation

ii = τφ /(Kτ kg) i t v = i r + K w /k dσ i i a w φi g 1 =0. − Z 6 0 f( ii(σ) ) | | iP=1 where Kτ and Kw are motor constants, kg is the gear reduction ratio of 33:1 at the motor shaft and ra is the Our battery model detects in simulation the zero motor armature resistance. Note that in this simple crossing of this function, which yields the effective life- formulation, the only influence of the actuator model time of the battery. Note that this model does not on the mechanical dynamics is through the limits on take into account more elaborate components such as the maximum available torque. the battery voltage drop as a function of current and discharge time or the effects of ambient temperature.

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×Ô eed ´ÖeÚ»ÑiÒµ ËhafØ x 10 means of integrating the equations of motion for hy- brid dynamical systems with configurable accurate (a) (b) discrete event detection and a 4th order Runge-Kutta integrator [16]. Figure 5: (a) Torque-speed curve for the Maxon The hexapod simulation with SimSect uses the same RE118751 20W DC motor. (b) Battery discharge dimensions and body mass as our experimental plat- curve for the Panasonic 12 V, 2.2 Ah battery. form (see Section 5). However, some of the dynamical parameters used in the simulations, including the leg spring and damping constants, the body inertia matrix 4.2 Battery Model and the ground friction coefficient are not experimentally verified and are likely to be different from their The discharge characteristics of off-the-shelf small bat- actual values. Nevertheless, because our simulations teries are usually given by plots of discharge time vs and robotic platform have the same morphology and constant discharge current. Figure 5(b) is such a dis- mass, we will still be able to do qualitative compar- charge curve for the battery used in our experimental isons of behavior. 4.4 Simulation Results 22 10.5

20 10 9.5 In this section, we verify in simulation that the con- 18 16 9 8.5 trollers of Section 3 are able to produce stable au- 14 8 tonomous running and turning of the hexapod plat- 12

Battery lifetime (min) Battery lifetime (min) 7.5 form. We ﬁrst present simulations with the running 10 7 controller, followed by turning studies. Finally, we use 8 6.5

the battery model of Section 4.2 to estimate the du- 0.2 0.4 0.6 0.5 1 1.5 2

´Öadµ Ø ´×µ c ration of operation of the platform. ×

Figure 7: The battery lifetime a) as a function of tc with φs =0.15rad kept constant. b) as a function of φs with tc =0.6s kept constant.

1.2

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(a) (b) We have built an experimental platform as an instan- tiation of the design concepts of Section 2.1. RHex is an autonomous hexapod robot with compliant legs, Figure 6: (a) Average forward running velocity x˙ as a very close to the model described in Section 2.2. All function of tc and φs over 5s of operation. Remaining the computational and motor control hardware is on controller parameters are chosen as ts = tc/2, φo =0. board, together with two 12 V 2.2 Ah sealed lead-acid (b) Average in-place turning yaw rate α˙ as a function batteries for power autonomous operation. A PC104 of tc and φs over 5s of operation. Remaining con- stack with a 100 MHz Intel 486 microprocessor, to- troller parameters are chosen as ts = tc/2, φo =0. gether with several I/O boards performs all the nec- essary computation and implements the controllers of Section 3. A remote control unit provides the user input for giving higher level commands such as the Figure 6(a) shows the the forward running velocity as running speed, and turning direction, presently via a a function of controller parameters tc and φs. Note joystick. that for these simulations, the remaining two param- Each leg is directly actuated by a Maxon RE118751 eters, ts and φo, have been kept constant. 20W brushed DC motor combined with a Maxon Similar to the forward running controller, the turning 114473 two-stage 33:1 planetary gear [11], delivering controller also presents a controllable range of turning an intermittent stall torque of 6 Nm. Even though speeds. Figure 6(b) shows the average angular body each motor is PWM voltage controlled, additional velocity as a function of the controller parameters. back-EMF compensation in software permits approx- Finally, in Figure 7, we present the predicted lifetime imate motor torque control. The motor angle, and of the on-board battery as a function of controller pa- thus the leg angles, are controlled via 1 kHz PD con- rameters. Our model predicts continuous autonomous trol loops. The software also features several safety running with the low-end Panasonic 12 V 2.2 Ah off- measures, including fault detection for the encoders, the shelf battery for up to 20 minutes, depending on estimation of the rotor temperatures to avoid motor the particular choice of controller parameters. An im- damage, and a watchdog timer which disables the mo- portant detail which is not immediately visible from tors and resets the computer in case of software fail- the graphs is that different controller parameters yield- ures. ing the same speed, result in different battery life- The main body measures 53x20x15 cm. The legs are times. This freedom of choice in the controller param- made from 1 cm diameter Delrin rods and are ”C” eters, given a particular speed, can be used for opti- shaped to provide compliance primarily in the radial mizing the battery lifetime, as well as other measures direction and permit easy clamping to the gear shaft. such as the maximum motor torque to avoid overheat- The leg length is 17.5 cm, measured as the vertical ing. distance from ground to the gear shaft when standing up. The encoder/motor/gear stacks protrude from the stretch, a switch was mounted in the front of the robot, main body and the maximum widths of the front and which was triggered as the robot ran into a Styrofoam back legs amount to 39.4 cm, measured at half the leg panel held at the beginning and the end of the test length. To provide clearance for the rotating front and stretch. The runs over each surface were repeated until back legs, the motors for the middle legs are further ten successful runs were obtained. The average veloc- offset and result in a maximum width of 52 cm. The ity and power consumption for each run was then com- total mass of the robot is 7 kg with each leg contribut- puted with the available data. Moreover, to measure ing only approximately 10 g. energy efficiency we used the “Specific Resistance” [9], ε = P/(mgv), based on the robot’s weight, mg, and its average power consumption, P , at a particular speed, 5.2 Experimental Results v. The robot moved well over these indoor and outdoor surfaces, with only minor velocity variations between carpet 0.45 m/s and 0.55 m/s. The velocity on Linoleum was linoleum lowest due to intermittent slipping, which also causes a grass larger standard deviation of the runs compared to car- gravel 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 pet. The surface irregularities of the outdoor grass and Average forward velocity (m/s) gravel surfaces provided improved traction, and there-

carpet fore average velocities slightly above 0.5 m/s, but also

linoleum resulted in larger variations between the runs. The grass speciﬁc resistance (power consumption) was lowest on gravel carpet with 2.21 (80 W) and highest on gravel with 80 100 120 140 160 180 200 220 240 260 3.74 (140 W). Average power consumption (W)

carpet linoleum 6 Conclusion grass gravel Nimble, robust locomotion over general terrain re- 2 4 6 8 10 12 Average specific resistance mains the sole province of animals, notwithstanding our functional prototype, RHex, nor the generally in- Figure 8: Comparison of average forward velocity and creased recent interest in legged robots. It is surely energetics for different experiments. not evident from the RHex morphology, but our design and longer term controller development effort has been heavily influenced by considerations from biol- In this section, we report a set of experiments assess- ogy. In particular, we believe that systematic appli- ing the performance of our design on different types cation of certain operational principles exhibited by of terrain. In all of these settings, RHex is able to animals will achieve significant increases in RHex per- produce behavioral repertoire of Section 3, achieving formance, and inform the evolution of the underlying stable and fast locomotion at speeds exceeding one mechanical design of future prototypes as well. To body length per second. conclude the paper, we provide a brief sketch of these Figure 8 summarizes the average forward running ve- principles and how they may be applied. locity and energetics for RHex while traversing carpet, Accumulating evidence in the biomechanics literature Linoleum, grass and coarse gravel with the two-stroke suggests that agile locomotion is organized in nature open loop controller of Section 3. For these experi- by recourse to a controlled bouncing gait wherein the ments, we ran the robot over carpet, linoleum, grass “payload”, the mass center, behaves mechanically as and gravel. The carpet and linoleum surfaces were though it were riding on a pogo stick [3]. While Raib- standard office floors found close to the lab. For the ert’s running machines were literally embodied pogo outdoor surfaces, the grass was wet and showed height sticks, more utilitarian robotic devices such as RHex variations of about 2 cm. The gravel patch contained must actively anchor such templates within their alien fairly large gravel pieces between three and eight cm morphology if the animals’ capabilities are ever to be diameter (see Figure 1). successfully engineered [8]. We have previously shown For all the experiments, the robot was driven over a how to anchor a pogo stick template in the more re- test stretch of 2 m. In order to obtain precise tim- lated morphology of a four degree of freedom monopod ing and to synchronize the data logging with the test [17]. The extension of this technique to the far more distant hexapod morphology surely begins with the quadraped that walks, climbs and runs. In Pro- adoption of an alternating tripod gait, but its exact ceedings of the IEEE International Conference On details remain an open question, and the minimalist Robotics and Automation, pages 1707–1712, Leuven, RHex design (only six actuators for a six degree of Belgium, May 1998. freedom payload!) will likely entail additional compro- [6] M. Buehler, A. Cocosco, K. Yamazaki, and mises in its implementation. Moreover, the only well R. Battaglia. Stable open loop walking in quadruped understood pogo stick is the Spring Loaded Inverted robots with stick legs. In Proceedings of the IEEE In- Pendulum [19], a two degree of freedom sagittal plane ternational Conference On Robotics and Automation, template that ignores body attitude and all lateral de- Detroit, Michigan, May 1999. grees of freedom. Recent evidence of a horizontal pogo [7] K. S. Espenscheid, R. D. Quinn, H. J. Chiel, and R. D. stick in sprawled posture animal running [13] and sub- Beer. Biologically-Inspired Hexapod Robot Control. sequent analysis of a proposed lateral leg spring tem- In Proceedings of the 5th International Symposium plate to represent it [18] advance the prospects for on Robotics and Manufacturing: Research, Education developing a spatial pogo stick template in the near and Applications (ISRAM’ 94), Maui, Hawaii, August 1994. future. Much more effort remains before a functionally biomimetic six degree of freedom “payload” controller [8] R. J. Full and D. E. Koditschek. Templates and An- is available, but we believe that the present under- chors: Neuromechanical Hypotheses of Legged Loco- standing of the sagittal plane can already be used to motion on Land. Journal of Experimental Biology, 202:3325–3332, 1999. significantly increase RHex’s running speed, and, as well, to endow our present prototype with an aerial [9] G. Gabrielli and T. H. von Karman. What price phase. speed? Mechanical Engineering, 72(10):775–781, 1950. [10] P. C. Hughes. Spacecraft Attitude Dynamics. John Acknowledgements Wiley & Sons, New York, 1986. [11] Interelectric AG, Sachseln, Switzerland. Maxon Mo- This work was supported in part by DARPA/ONR tor Catalog, 1997/98, www.maxon.com. Grant N00014-98-1-0747. Bob Full consulted on many [12] ISRobotics. Autonomous Legged Underwater Vehicle. of the design decisions and provided numerous tuto- http://www.isr.com/research/ariel.html, 1999. rial explanations of the biomechanics literature. Noah [13] T. M. Kubow and R. J. Full. The role of the me- Cowan offered very helpful advice concerning mod- chanical system in control: A hypothesis of self- eling and simulation. Liana Mitrea made invaluable stabilization in hexapedal runners. Phil. Trans. R. contributions in the mechanical construction of the Soc. Lond., B. 354:849–862, 1999. hexapod platform. [14] G. M. Nelson and R. D. Quinn. Posture Control of a Cockroach-like Robot. In Proceedings of the IEEE In- ternational Conference On Robotics and Automation, References pages 157–162, Leuven, Belgium, May 1998. [1] R. J. Bachman, G. M. Nelson, W. C. Flannigan, [15] M. H. Raibert. Legged robots that balance. MIT Press, R. D. Quinn, J. T. Watson, A. K. Tryba, and R. E. Cambridge MA, 1986. Ritzmann. Construction of a Cockroach-like hexapod [16] U. Saranli. SimSect Programmer’s Manual. The Uni- robot. In Eleventh VPI & SU Symposium on Struc- versity of Michigan, 1999. In preparation. tural Dynamics and Control, pages 647–654, Blacks- [17] U. Saranli, W. J. Schwind, and D. E. Koditschek. To- burg, VA, May 1997. ward the Control of a Multi-Jointed, Monoped Run- [2] R. D. Beer, H. Chiel, R. D. Quinn, and R. E. Ritz- ner. In Proceedings of the IEEE International Confer- mann. Biorobotic Approaches to the Study of Motor ence On Robotics and Automation, Leuven, Belgium, Systems. Current Opinion in Neurobiology, 8:777– May 1998. 782, 1998. [18] J. Schmitt and P. Holmes. Mechanical models for in- [3] R. Blickhan and R. J. Full. Similarity in multilegged sect locomotion I: Dynamics and stability in the hori- locomotion: bouncing like a monopode. Journal of zontal plane. Biological Cybernetics, submitted, 1999. Comparative Physiology, A. 173:509–517, 1993. [19] W. J. Schwind and D. E. Koditschek. Approximat- [4] R. A. Brooks. A Robot That Walks; Emergent Behav- ing the Stance Map of a 2 DOF Monoped Runner. iors from a Carefully Evolved Network. Memo 1091, Journal of Nonlinear Science, to appear. MIT AI Lab, February 1989. [5] M. Buehler, R. Battagila, A. Cocosco, G. Hawker, J. Sarkis, and K. Yamazaki. SCOUT: A simple