RHex: A Biologically Inspired Hexapod Runner1
R. Altendorfer2 , N. Moore3 , H. Komsuo?lu2 , M. Buehler3 , g H.B. Brown Jr.4 , D. McMordie3 , U. Saranli2 , R. Full5 , D.E. Koditschek2
Abstract RHex is an untethered, compliant leg hexapod robot that travels at better than one body length per second over terrain few other robots can negotiate at all. Inspired by biomechanics insights into arthropod locomotion, RHex uses a clock excited alternating tripod gait to walk and run in a highly maneuverable and robust manner. We present empirical data establishing that RHex exhibits a dynamical (“bouncing”) gait — its mass center moves in a manner well approximated by trajectories from a Spring Loaded Inverted Pendulum (SLIP) — characteristic of a large and diverse group of running animals, when its central clock, body mass, and leg sti?nesses are appropriately tuned. The SLIP template can function as a useful control guide in developing more complex autonomous locomotion behaviors such as registration via visual servoing, local exploration via visual odometry, obstacle avoidance, and, eventually, global mapping and localization.
Animals inspire our intuition that legs may be necessary for satisfactory exploration of the highly broken and unstable landscapes prevailing on other planets. In this view, legged
machines that capture some measure of animal mobility a?ord the best hope against the many inhospitable conditions extra-terrestrial settings impose. We have derived substantial inspiration from cockroach running in the de- Figure 1: (www.RHex.net). sign of a prototype robot, RHex, that breaks
This work was supported in part by DARPA/SPAWAR under contract N66001-00-C-8026. Portions of the material reported here were ?rst presented in a conference paper appearing in the collection (Altendorfer et al., 2000). 2 Arti?cial Intelligence Laboratory, University of Michigan Ann Arbor, MI 48109. 3 Ambulatory Robotics Laboratory, Dept. of Mech. Engineering, McGill University Montr?al, Qu?bec, e e Canada H2A 2A7. 4 The Robotics Institute, Carnegie Mellon University Pittsburgh, PA 15213. 5 Dept. of Integrative Biology, University of California at Berkeley Berkeley, CA94720.
new ground in arti?cial legged locomotion (Saranli et al., 2000; 2001). Our breadbox sized, compliant leg hexapod, RHex (Fig. 1), travels at speeds better than one body length per second over terrain that few other robots can negotiate at all. The large and growing ?eld of legged robotics includes a diverse array of laboratory vehicles inspired by arthropod locomotion, as well as at least one commercial crab-inspired machine capable of sustained operation in the surf zone (iRobot, 1999). Yet, morphology notwithstanding, RHex arguably bears a closer relationship to Raibert’s pioneering monopods (Raibert, 1986). For the hexapod literature has heretofore concerned static or quasi-static walking machines, while RHex can walk and run. In this paper we present initial evidence establishing that RHex can “bounce” along its way as if it were indeed built like a pair of Raibert’s pogo sticks, alternating in a 50% duty factor with no aerial phase. We will ?rst review the biological inspiration motivating that observation, noting that cockroaches bounce along in the same manner, running, according to biomechanics, without ever leaving the ground. We end with somewhat more speculative remarks concerning its utility, imagining the bene?ts for stability and maneuverability.
In considering the astonishing performance of the death-head cockroach, Blaberus discoidalis, over badly broken terrain, one of us (Full) has opined: “Simple feedforward motor output may be e?ective in negotiation of rough terrain when used in concert with a mechanical system that stabilizes passively. Dynamic stability and a conservative motor program may allow many-legged, sprawled posture animals to miss-step and collide with obstacles, but su?er little loss in performance. Rapid disturbance rejection may be an emergent property of the mechanical system.” (Full et al., 1998) While the materials and morphology displayed in Fig. 1 will not evoke in all viewers the image of a cockroach, their coordination in the generation of body motion is very strongly inspired by the following observations that underlie the previous quote. Cockroach legs are (i) arranged in a sprawled posture a?ording (ii) signi?cant overall compliance (Blickhan and Full, 1993) that may be (iii) excited by a strongly stereotypical “clock” reference signal (Kubow et al.) with the apparent (and surprising) consequence of a (iv) mechanically self-stabilized gait (Kubow and Full, 1999; Schmitt and Holmes, 2
2000). Thus, although RHex’s body shape, leg design and materials are quite dissimilar from those of a cockroach, their coordination in the phasing of stance and swing are arranged so as to achieve a similar dynamical e?ect. The legs are attached in a (i) sprawled posture, spreading out so as to confer static stability at rest and low speed operation. Further, while each of RHex’s 6 legs has only one actuated degree of freedom (at the hip), in contrast to the multiply actuated leg joints of a cockroach they incorporate (ii) a carefully tuned passive radial compliance. Finally, RHex exhibits a large suite of stable locomotion behaviors when each alternating tripod of hip motors is forced via a local proportional-derivative (PD) control law to track (iii) the identical (or, for the opposed tripod of legs, a 180? out of phase) copy of a feedforward reference signal (Saranli et al., 2000). We presume (but cannot yet formally prove) that this feedforward stimulus excites a limit cycle (iv) in the coupled leg-body mechanism. We will present empirical evidence below that the projection of this putative limit cycle onto the mechanism’s center of mass coordinates yields the motion of a speci?c pogo stick — the spring loaded inverted pendulum (SLIP) — when the controller and mechanical parameters are properly tuned. Philosophically, RHex’s design exempli?es our notion of biological inspiration in robotics, namely, that novel concept transfer from biology can increase performance in machines. This represents a cautionary approach that eschews “biology by default” (Ritzmann et al., 2000) or even morphologically identi?ed design in favor of identi?cation and mimicry of selected functional components. In the long run, we aim to articulate broad principles with mathematically precise formulations of biomechanical observed fact and then translate these into speci?c design practices. Meanwhile, pending that formal understanding, empiricism and intuition about biological function remains subservient to engineering practice in our work.
Accumulating evidence in the biomechanics literature suggests that agile locomotion is organized in nature by recourse to a controlled bouncing gait wherein the “payload,” the mass center, behaves mechanically as though it were riding on a SLIP (Blickhan and Full, 1993; Full and Farley, 2000). Given its ubiquity, we surmise that this speci?c form
of dynamic locomotion must confer signi?cant bene?ts upon animal runners. There is, indeed, clear evidence in some animal species that bouncing stores and returns energy (Biewener and Baudinette, 1995). For robots, storing periodically in a spring potential some portion of their fore-aft kinetic energy seems like an e?ective resolution to the inevitably constraining actuator torque and power limits. Limbs must move both quickly and forcefully in the high performance regimes we seek, but actuators cannot operate both at high speeds and at high torque levels. A careful exposition of the power and energy implications attendant upon a bouncing gait lies well beyond the scope of the present article. We explore instead the bene?ts of imposing this hierarchical organization on the control and stabilization architectures of complex, high degree of freedom mechanisms. That exploration centers on the notion of the “template and anchor” hierarchy (Full and Koditschek, 1999). A template is a low dimensional model of a mechanism operating within a speci?ed environment that is capable of expressing a speci?c task as the limit set of a suitably tuned dynamical system involving some controlled (robot) and uncontrolled (environment) degrees of freedom. To “anchor” this low dimensional model in a more physically realistic higher degree of freedom representation of the robot and its environment, we seek controllers whose closed loops result in a low dimensional attracting invariant submanifold on which the restriction dynamics is a copy of the template. Raibert’s monopods were literal embodiments of the SLIP (Raibert, 1986). His bipeds functioned as alternating pogo sticks and his quadrupeds were constituted as virtual alternating pogo sticks. By recourse to such decompositions, he found it possible to impose relatively simple and largely decoupled stride-to-stride gait control laws over the within-stride regulation loops. One of us (Buehler), has pursued a more minimalist version of the Raibert quadrapeds in the form of passively compliant telescoping legs driven at each hip by only one actuator per leg in a design that might be seen as the immediate forebear of RHex (Buehler et al., 1998; Buehler et al., 1999; Papadopoulos and Buehler, 2000). Biomechanical evidence for the seeming ubiquity of a SLIP template naturally suggests the possibility of pogo stick based control for more diverse leg morphologies. For example, a simulation study (Saranli et al., 1998) demonstrated an approximate embedding of the
SLIP template in a planar 4 DOF leg with ankle, knee and hip joints (AKH), similar in morphology to a human leg. Raibert’s machines were tethered to the wall by power cables. One central advance that RHex introduces is, of course, power autonomy. We believe that our insistence on one actuator only per leg — a remedy against the longstanding problem in robotics and automation of poor force and power density properties in commercially available actuators (Hollerbach et al., 1992) — represents the key enabler in this development. But the absence of any actuation beyond the torsional hip motors signi?cantly complicates the decomposition methods outlined just above. Following the insect example, we constitute RHex’s gait from the alternation of two tripods that each act (relatively out of phase) as virtual monopod pogo sticks. However, in contrast to Raibert’s mechanisms, none of RHex’s legs has the literal morphology of the SLIP — they bend rather than telescope at the knee. Still more importantly, there is no direct a?ordance over the radial compliance in any of the physical limbs — all energy and phase manipulations throughout the machine must be introduced via hip torques. In summary, RHex’s essential di?erence from Raibert’s machines rests in its greater number of legs placed in a sprawled tripod and in the absence of radially oriented actuators in the shanks. The virtue of the ?rst change lies in the greater ease of traversal over broken terrain exploiting passive dynamic stability; vertical balance is far less of an issue (although pitching oscillations are still far from negligible and have the e?ect of destabilizing or at least badly perturbing the gait). The virtue of the second change lies in pulling out the wall plug. Can we, nevertheless, realize the supposed bene?ts of the SLIP template in this very di?erent anchor?
Finding SLIP in RHex’s Motion
In this section we discuss evidence suggesting the presence of certain “sweet spots” in the RHex parameter space, wherein the SLIP emerges naturally. It is still unclear whether this respects the formal “anchored template” paradigm wherein the lower dimensional dynamics actually appears as an attracting invariant dynamical submanifold. However, experimental evidence revealing the template behavior in steady state from various dif5
ferent initial conditions suggests there are, indeed, operating regimes where the system trajectories are attracted to the low dimensional SLIP template dynamics. Further evidence for the SLIP template comes from numerical studies (Saranli, 2000). To determine whether RHex passively anchors a SLIP, the ground reaction forces produced by RHex during locomotion were measured during 92 trials using two six-component force plates6 . The force and torque signals were ampli?ed and each channel was recorded at 1000Hz by an analog to digital converter. Each trial was also recorded by a high speed video camera. As discussed in detail in (Altendorfer et al., 2000), of the two di?erent leg designs tested, only the passive four bar linkage displayed in Fig. 1, had su?ciently low sti?ness to permit operating in the SLIP regime. Moreover, in order to further reduce the natural frequency of the e?ective vertical spring-mass system de?ned by the body-legs (down to the maximal stride frequency permitted by the limited motor power) we found it necessary to increase the payload
[J] 100 [N] 0 20 [N] 0 ?20 1 [N] 0 ?1 11.6 11.4 11.2 9.8
mass incrementally from 7.83kg to 9.47kg to 11.12kg to 11.94kg. In this highest mass regime we observed a transition from sti? virtual inverted pendulum (IP) to the desired SLIP reported
Potential grav. energy
17 16 2
9.4 9 0 0.1 time [s] 0.2
Fore/aft kinetic energy
0 0.1 0.2 time [s] 0.3 0.4
1.5 1 0.5
Figure 2: Comparison of sample RHex data with SLIP predictions.
below. There were 14 trials in this collection. First, in order to help visualize what the ?tting study establishes, we qualitatively compare force and energy data of a SLIP trajectory with short aerial phase and sample RHex data (Fig. 2). In particular, all time trajectories approximately enjoy the same phase relationships. Indeed, comparing SLIP and RHex data of the ?rst and fourth row, the vertical force peaks at mid-step when the COM is at its lowest height — a necessary
Biomechanics Force Platform, Advanced Mechanical Technology, Inc., Newton, MA.
and characteristic property of SLIP equilibrium gaits. Also, comparing the fourth and the ?fth row, gravitational potential energy and fore/aft kinetic energy of the COM ?uctuate in phase. This “in-phase” relationship between vertical height and horizontal kinetic energy also represents a key feature of SLIP locomotion, in contrast to inverted pendulum (IP) vaulting.
t 0 0.03 y(t) 0.02 0.135 0.01 0.13 0 0.5 y′(t) 0 0.3 z′(t) y′(t) 0.2 0.1 0.3 2 0 y′′(t) z′′(t) y′′(t) ?2 ?4 ?6 0.145 0.145 0.14 q(t) z(t) z(t) 0.14 0.135 0.13 0 0.01 0.02 y(t) 0.03 0 0.02 0.04 t 0.06 0.135 0.13 0.144 0 0.005 0.01 0.015 0.02 0.025 y(t) 0 0.01 0.02 0.03 t 0.04 0.05 5 0 0 10 0.54 0.52 0.5 0 6 1.5 1 0.5 0 0.152 0.15 0.148 0.146 z′′(t) q(t) 4 2 0 0.154 0.152 0.15 0.148 0.146 0.144 0.1 z′(t) 0.02 0.04 0.06 0 0.02 t 0.04 0.06 hexapod SLIP 0.145 0.14 z(t) y(t) 0.02 0.01 0 0.01 0.02 t 0.03 0.04 0.05 0 0.01 0.02 t 0.03 0.04 0.05 0.152 0.15 0.148 0.146 0.144 0.2 z(t) hexapod SLIP
(a) Worst ?tting trial with rms error ?L2 = 21.6%.
(b) Best ?tting trial with rms error ?L2 = 0.3%.
Figure 3: Worst and best SLIP ?ts: Dotted lines represent experimental data; solid lines represent ?tted SLIP trajectories. This ?gure is taken from (Altendorfer et al., 2000).
The SLIP template imposes a very particular set of relationships — those speci?ed by the Lagrangian mechanics of a single point mass prismatic-revolute (i.e., polar coordinate) kinematic chain — between the ground reaction forces, motion of the COM, and system energies. Denoting by b the position vector (relative to some inertial frame) in the saggital plane of the center of mass in cartesian coordinates, this relationship may be speci?ed as ¨ = ?Dφ(||b||)/||b|| ? g, where φ is a spring potential normalized by mass and g denotes b the acceleration due to gravity. Ruina has pointed out (Ruina) that any convex curve supports in a neighborhood of its vertical minimum at least one time varying trajectory generated by some SLIP. RHex’s saggital plane COM might well lie along a convex curve without establishing what we are interested in testing: whether its actual time trajectory 7
along this curve can be readily generated by some SLIP model. To this end, given a COM trajectory fragment, the COM position, and acceleration were used to ?t via linear regression a Hooke spring law with unknown spring length qr0 and sti?ness κ (note this corresponds to setting φ(q) = (κ/2m)(q ? qr0 )2 in the equations of motion above). As an illustration of the ?tting results, the best and worst SLIP ?ts are presented in Fig. 3. In all cases, we report rms error as a percentage of the rms value of the data trajectory (arithmetic mean of positions and velocities) itself. Thus, Fig. 3(a) shows that this worst trial yielded rms errors of 21.6%, while Fig. 3(b) shows that the best trial yielded an rms error of less than 1%. On average, the ensemble of 14 trials generated a mean rms error of 7%, with mean spring constant of κ = 6100N/m. To get a bigger data sample, we ran a large number of simulations in SimSect in the same parameter range and obtained very similar results. We conclude that the SLIP dynamical system describes these data quite well.
Hierarchy promotes the use of few parameters to control complex systems with many degrees of freedom. In this light, as we understand matters, the emergence of a passively anchored SLIP in RHex is most fortunate. The pogo-stick can function as a useful control guide in developing more complex autonomous locomotion behaviors such as registration via visual servoing, local exploration via visual odometry, obstacle avoidance, and, eventually, global mapping and localization. In the longer term, we propose to work with the anchored SLIP in RHex in analogy to the manner in which the simple two-bead template has been exploited in prior work by some of the authors and colleagues with simpler dynamically dexterous robots. Namely, as we shape behavior via manipulation of gains-in-the-loop (Burridge et al., 1999), we hope to develop a formal programming language with semantics in the world of dynamical attractors (Klavins and Koditschek, 2000). Examples of the virtues of such decompositions and hierarchies at work in functioning robots include a series of batting machines that anchored a “Raibert vertical” template (Koditschek and Buehler, 1991) in a one degree of freedom paddle robot (operating into a 8
two degree of freedom environment) (Buehler et al., 1990) and a three degree of freedom paddle robot (operating into a three degree of freedom environment) (Rizzi et al., 1992). This same idea is used to control a recently reported brachiating robot (Nakanishi et al., 2000). All of these robots were conceived as laboratory demonstration machines. In contrast, RHex’s intrinsic robustness, stability and maneuverability suggest that there might be signi?cant impact on applications if we succeed in transferring these ideas into the present setting. Even now, prior to the imposition of more sophisticated hierarchical controllers, RHex’s intrinsic characteristics — simple, rugged, light, autonomous, down-sizeable, reliable quasi-static operation, demonstrated dynamical capability (e.g., great jumping potential in reduced gravity environments) — all make RHex an ideal candidate for a planetary explorer.
This work is supported by DARPA/ONR under contract N00014-98-1-0747 and DARPA/ SPAWAR under contract N66001-00-C-8026. We thank Rodger Kram and Claire Farley for the use of the force platform, Irv Scher for collaboration at an early stage of this project and William Schwind for the insight and the analytical tools he provided.
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