A Dynamical Systems Approach to Obstacle Navigation for a Series-ElasticHexapod Robot

Matt Travers, Alex Ansari, and Howie Choset

Abstract— This paper emphasizes reliability in designingcontrollers to enable a blind, modular series-elastic hexapodrobot to autonomously navigate (climb over) obstacles such assteps and curbs. Specifically, we suggest that it is not onlyimportant to limit the complexity of controllers, but also tolimit the number of operating controllers to reduce potentialfailure points. As such, this paper presents a two-tiered controlscheme with a high-level behavioral and a mid-level, modifiedadmittance controller. The behavioral controller, based on adynamical systems approach, is easy to implement – it is model-less, and may be realized using only analog circuitry. At itscore, two oscillators coordinate the phasing of the robot’s legsin open loop for nominal locomotion and climbing behaviors.A simple bistable dynamical system accepts torque feedbackto decide the active mode and smoothly transition betweenbehaviors – an alternating tripod gait for nominal walking anda quadrupedal gait in climbing mode. Finally, the mid-level,modified admittance controller naturally generates discrete mo-tions as a byproduct of regulating compliant interactions. Theproposed controller avoids unnecessary complexities that wouldbe required in switching between discrete motion controllers orprimitives in a state machine. It allows the robot to adapt todifferent obstacle heights while minimizing open parameters.

I. INTRODUCTION

To support real-time operations in unstructured terrains,such as those necessary in urban search and rescue, acontrol scheme should not be overly complex. That is, itis important to limit the number of points of failure andparameters for calibration and performance tuning, whichmay be required on-site. To these ends, this paper proposesan approach that combines open-loop gaits with closed-loopcompliance to allow a blind hexapod robot to autonomouslynavigate (climb over) obstacles using only angular encodersand proprioceptive force feedback.

The proposed scheme is implemented on a hexapod robotcomposed of series elastic actuator (SEA) modules [8] (seeFig. 1). While not particularly fast, the robot’s compliantlegs provide the potential to navigate complex environmentswhere wheeled and stiff-jointed platforms cannot robustlyenter. For search and rescue, the SEA hexapod’s topology isalso beneficial in that 1) the leg SEA modules can withstandforceful impacts; 2) the modules are interchangeable andeasily replaced; and 3) the robot has natural redundancy,capable of locomotion with only 4 legs. We also leveragesensor redundancy for added robustness, using proprioceptivefeedback from joint-torque sensors located in each module.

M. Travers, A. Ansari, and H. Choset are with the RoboticsInstitute at Carnegie Mellon University, Pittsburgh, PA 15213,USA. {mtravers@andrew, aansari1@andrew,choset@cs}.cmu.edu

1 2

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Fig. 1: Series-elastic actuated (SEA) hexapod robot. The legsare numbered beginning from the right-front of the robot’sbody. Each leg has three single degree-of-freedom (DOF)rotational joints (SEA actuated) and two passive elements,one of which has a rubberized “foot.” The proximal jointprovides a yaw DOF, and the medial and distal joints eachprovide a pitch DOF relative to the robot’s body.

To control the SEA hexapod, this paper proposes both ahigh-level behavioral and a mid-level leg compliance con-troller. First, we take a dynamical systems approach to high-level behavior specification that minimizes computationalresources. The proposed controller uses phase oscillatorsto coordinate open-loop desired behaviors, corresponding toa hexapod alternating-tripod gait for nominal locomotionand a quadrupedal gait for climbing. Rather than discretelyswitching between these behaviors when an obstacle isencountered, we design a neuronal model that requires onlyjoint-torque information to transition behaviors. Using thefront legs as antenna, sensed torques are fed back intoa bistable dynamical system that governs the state of asingle parameter that transitions from nominal locomotionto climbing upon obstacle detection. Effectively, we providea single parameter compliant dynamical switch that smoothlytransitions between limit cycles that coordinate different be-haviors. To our knowledge, this strategy for low-dimensional,smooth behavioral switching is unique to this work.

As described in Section II, the proposed behavioral con-troller is similar in spirit and provides many of the benefits ofcentral pattern generators (CPGs) [7], e.g., smooth transitionsand stable limit cycle behavior, but is easier to implement.Each limit cycle behavior evolves according to an individualoscillator rather than CPG networks of oscillators. Hence,there is no need to identify matrices of stabilizing couplingparameters required to converge to desired joint phase offsetsin CPGs. Similarly, joint motions during behavioral transi-tions are intuitive and easy to design, since the dynamics ofthese motions are not tied to the stability properties of anentire network.

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To navigate ledges, curbs, and steps that are too high forthe robot to step over, the SEA hexapod performs severaldiscrete actions for alignment and leg clearance. Ratherthan divide the climbing behavior into smaller, discretemotion phases, i.e. as in a state machine approach, our mid-level, modified admittance controller automatically generatesand phases these discrete motions as the natural resultof regulating limb compliance. A single additional reflexresponse, which modifies the admittance properties of thecontroller during execution, is sufficient for autonomousobstacle navigation.

Augmenting the open-loop quadruped gait in climbing,the mid-level controller ensures the SEA hexapod’s motionsare adaptive to the environment and obstacle geometry.Experiments show the resulting behaviors are more robustand much more effect than simple “canned” motions. Also,by avoiding the need to design, tune, and switch betweenmultiple controllers in climbing, the proposed low-level con-trol scheme improves reliability and eases implementation.

Following this Introduction, Section II provides a back-ground on related works. Section III describes the mid-levelcompliant behavioral controller. Section IV discusses thelow-level admittance controllers. Experimental results are inSection V with conclusions in Section VI.

II. RELATED WORKS

The control approach described in this paper is predicatedon the idea that complex behaviors can be composed of sim-ple primitives. For instance, we generate both walking andobstacle climbing behaviors by layering relatively straight-forward closed-loop policies on top of simple open-loop gaitcycles. While one may question whether open-loop gaitsprovide sufficiently robust base controllers for climbing, wehighlight prior work that motivates our proposed approach.

In particular, [11] cites biological evidence that suggeststhat animals may utilize open-loop strategies during locomo-tion. The authors conduct a series of locomotion experimentsfor Cheeta-cub, a compliant quadruped robot, in whichthey set a record for quadruped trotting speed using onlya simple open-loop gait. The authors highlight the self-stabilizing properties of their robot’s compliant design andemphasize their belief that locomotion (on flat terrain) shouldbe performed “almost blindly”, without sensory feedback orexplicit models. They show the same controller successfullynavigates steps ranging from 7.5%-20% of their robot’sstanding hip height (20% success rate at the 20% stepheight). Here, we build on these concepts, similarly lever-aging our robot’s compliant properties and adding feedbackcontrollers for more aggressive step climbing tasks with stepheights that reach the underside of our robot.

In [11], the open-loop trajectory for Cheeta-cub’s legsis synthesized by a central pattern generator (CPG). CPGsare neural circuits in animals that generate rhythmic signalswithout the aid of the peripheral nervous system. They coor-dinate rhythmic activities in animals, e.g., walking, and havebecome a popular tool in robotic motion control. In robotics,CPGs are typically implemented as networks of coupled

oscillators. When properly designed, these networks yieldrhythmic signals with desired phase offsets to coordinaterobotic gaits and have desirable properties such as phaselocking, mechanical entrainment, and stable limit cycles.However, design of CPGs networks, specification of couplingterms, and incorporation of feedback prove challenging as thefield lacks formal tools and well-established methods [7].1

While open-loop CPG implementations avoid difficultiesin integrating feedback, they no longer benefit from mechan-ical entrainment and merely provide phase coordinated jointsignals. As opposed to using simple sine generators, CPGsstill exhibit stable limit cycles and so may be utilized, asin [11], to provide smooth motion transitions from initialconditions to the desired limit cycle. The gait control schemeimplemented here is similar to such an open-loop CPGscheme in that it coordinates legs based on several (two)nonlinear oscillators. However, as each oscillator coordinatesa separate behavioral phase, we avoid the complicationsinvolved in specifying CPG coupling parameters. Our ap-proach is easy to implement and still provides smooth motiontransitions between behaviors.

In addition to [11], several authors have utilized CPGs inboth open and closed-loop implementations for locomotioncontrol. Most related to our work, [3] presents a bio-inspiredcontroller that switches between different types of gaits ona hexapod. They use a CPG to coordinate the legs andmodulate the CPG with a drive signal that is mapped todifferent sets of CPG parameters for starting, stopping, andtransitioning between different gaits. A parallel controllerprovides feedback to compensate for lateral inclination usinga dynamical system to regulate leg height. The system ex-tends legs in the direction toward which the robot is tiled andfolds others to reduce lateral tilt. Results are demonstratedon a simulated Chiara hexapod.

A more complex approach in [5] combines CPGs, back-bone joint control, leg reflexes, and neural learning to gen-erate adaptive walking and climbing behaviors similar tothose considered here. However, climbing is not blind. Ex-periments use ultrasonic, foot contact, and infrared sensors,to enable a hexapod robot to adapt while climbing steps andcurbs up to 75% of leg length (in the peak, high-frictionindoor case; lower outdoors). As the mentioned in [5], thereare wheel-legged robots, e.g., Rhex [10], that can overcomeobstacles that are higher with respect to leg length, but theserobots tend to have fewer degrees of freedom and thus motioncapabilities that are not comparable.

Furthermore, [9] introduces a new CPG model for hexapodrobots that allows a superposition of discrete and rhythmicprimitives. The authors conduct a simulation study thatshows in certain cases it is possible to superimpose discreteand rhythmic primitives in a manner that leaves amplitudeand frequency constant. They suggest this may facilitatethe design of more complex motions to allow hexapodrobots to accommodate more complicated terrains since jointoffset can be affected without changing the amplitude and

1For a review of CPGs in the context of locomotion control see [7].

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LF

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LH

RHRM

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Walk

LF

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Fig. 2: The step pattern for walking and tripod gaits. Theblack rectangles indicate when each foot is on the ground(L/R for left / right and F/H for front / hind leg).

frequency of rhythmic motion in such cases.In the case of quadrupeds, both [4] and [2] use CPG-based

controllers to facilitate walking over irregular terrain. As with[11], the authors converge on a leg flexion / extension reflexcontroller to account for unexpected bumps and drops, andboth modulate joint stiffness using either a virtual spring-damper system [4] or virtual model control [2]. The CPGcontroller in [4] is provided with sensed rolling motionas feedback to mutually entrain this motion with that ofthe virtual leg spring-dampers. Experimental results in [4]include a robot (Tekken) that can walk over steps up to20% of leg length and slopes of up to 10◦. The controllersin [2] are validated in simulation. Again, simulations showsuccessful walking for steps up to 20% of leg length andslopes of 20% (up to 36.5% with a lower success rate).

Most of these prior works focus on locomotion consid-ering modest terrain variation, with some hexapod results(often in simulation) that rely on complex controllers forclimbing over larger obstacles. This paper focuses on usingfew, relatively straightforward, and low-dimensional controlpolicies to achieve aggressive step climbing behaviors withstep heights approximately equal to the robot’s standingheight (88% of leg length). We demonstrate our resultsexperimentally on a modular SEA robot. We leverage thecompliance in the robot’s modular actuators and show evennoisy torque feedback provided by SEA actuators can beused to regulate effective environmental interactions.2

III. BEHAVIORAL CONTROL

Both the nominal walking as well as climbing behaviorsfor the SEA hexapod are based on open-loop gaits in whichphase oscillators coordinate the desired joint relationships.For nominal walking, an alternating tripod gait is used (see[6], [9]). As shown in Fig. 2, the gait consists of two sets ofthree legs – a pair of ipsilateral fore and hind legs combinedwith the contralateral middle leg. The legs in each of thesetripods move synchronously, while the two tripods remainhalf a gait cycle out of phase. As such, one leg group is in

2We use joint torques as feedback for mid-level leg admittance and high-level compliant behavioral switching controllers.

Fig. 3: Smooth transitioning between the limit cycles of twononlinear oscillators by varying µ. The blue sample trajectoryresults when µ is linearly reduced from 0.5 to −0.5.

swing (in the air) while the remaining three legs are on theground for support.

The climbing behavior is built around a quadrupedalwalking gait (see Fig. 2). This gait requires only the robot’sfore and hind legs and leaves the middle legs free for use ina reactive control scheme that aligns the robot and providessupport to improve climbing reliability. The walking gaitleaves three legs on the ground while the fourth is in swingand enforces a quarter cycle phase lag between legs – lefthind, left fore, right hind, and right fore [6].

Though both the hexapod and quadrupedal gaits can beimplemented in closed-loop using foot switches to detectcontact, based on desired search and rescue applications, wechose to minimize reliance on external sensors not includedon every module. This design uses redundancy to reducesensitivity to sensor failures and ensures sensors are modularand easily replaced. Thus, in both cases, the SEA hexapod’sgaits are implemented in open-loop. In contrast to the typicalapproach adopted in the CPG community, we define staticphase lags for the legs around a single centralized phaseoscillator in each of the two operational modes.

The phase oscillator equations for each the nominal as wellas climbing behaviors are based on Hopf oscillator equations,i.e., in polar coordinates,

rw = (µ− r2w)rw (1)θw = ωw

rc = (−µ− r2c )rc (2)θc = ωc,

where the radii, rw and rc, and phase variables, θw andθc, correspond to the walking and climbing behaviors re-spectively. The bifurcation parameter µ sets the stabilityproperties for each behavior; notice the opposite sign of µin (1) and (2). We use the phase space (r, θ, µ) to providean associated radius, r, and phase, θ, for each behavioras a function of µ. These parameters are used to set thedesired feed-forward joint angles of the robot during real-time operation. For example, considering the SEA hexapodshown in Fig. 1, with joint angles φ ∈ R18, the gait functionsfor walking and climbing f define φg = rg(µ) · fg(θg) + φ0

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for g ∈ {w, c} and where φ0 is the desired nominal stanceat the point rc = rw = µ = 0.

In terms of stability properties for walking and climbing,(1) and (2) are designed so that a dual bifurcation occurs atthe origin. Switching the sign of µ causes the emergence oflimit cycle behavior in one system, and its collapse in theother. That is, when µ is a positive value, (1) has an unstablenode and stable limit cycles at r = {0,±√µ}, while (2) hasa single stable node at r = 0.3 The opposite is true whenthe sign of µ is negative. We use this alternating stabilityproperty as the basis for switching behaviors from climbingto walking, and back again, in our behavioral controller.

The switched stability properties of our behavioral con-troller are presented graphically in Fig. 3. Assume the upperconvex lobe in Fig. 3 corresponds to the stable limit cycle(as a function of µ) of the Hopf oscillator governing thetripod walking gait (controlled by (rw, θw)). The blue sphereindicates a point on the stable limit cycle when µw = 0.5.In Cartesian coordinates, with xi = ri cos(θi) and yi =ri sin(θi), i ∈ {w, c}, the oscillator depicted evolves alonga circle of radius

√0.5 as long as the µ remains fixed.

Assuming that some transition forces the value of µ toapproach zero, the dashed blue line on the surface of theupper lobe in Fig. 3 traces a potential path through thebehavioral phase space, eventually reaching the origin.

Continuing to decrease the value of µ, i.e., µ < 0, thesystem will transition to the stable cycle of (2), whichcoordinates the quadrupedal gait that we use as the basisfor climbing.4 The concave lobe in Fig. 3 depicts the phasespace of stable quadrupedal limit cycles. Assuming that µ isdecreased until it reaches µ = −0.5, the dashed-blue line onthe concave lobe in Fig. 3 traces one potential path throughthe quadrupedal gait phase space. This gait will eventuallysettle into a stable limit cycle with a radius of

√0.5.5 The

red sphere in Fig. 3 indicates a point on this limit cycle.

A. A Compliant Dynamical System to Transition Behaviors

To reliably switch behaviors from walking to climbingand remain robust to sensor noise and failures, we use aneural-inspired model to queue the transition from walkingto climbing. That is, rather than switching based on a singlethreshold sensor value, the behavioral switching is based ona model that integrates torques sensed at the SEA joints. Ifthe integrated sensory signal reaches a threshold, a discrete“firing” event triggers the associated behavioral switch.

For example, Fig. 4 shows experimentally measured torquesignals from the proximal joints on the two front legs of thesystem, τ1p and τ2p respectively (see Fig. 1). The robot isinitially walking in an unconstrained part of the environment,

3Technically, r = 0 corresponds to stable zero amplitude limit cycleunless ω = 0. However, because the joint angles follow r, the robot remainsfixed when r = 0 and ω has no effect.

4Note that rw → 0 as µ → 0. Allowing the radius to become negative,the stable limit cycles for (2) reflect about the Cartesian x, y and µ axesrelative to those of (1). The sample path in Fig 3 demonstrates the designyields smooth transitions through the origin.

5The parameters of (1) and (2) may be chosen differently, in which casethe two lobes would not be symmetric.

0 2 4 6 8 10 12-3

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Fig. 4: Transition from walking to climbing using onlyproprioceptive sensing in the SEA hexapod’s front legs.

but contacts an obstacle with leg 2 at t ≈ 6s. The obstacleexerts a force on the leg that counteracts the propulsive forceof the gait, and subsequently forces τ2p to reverse signs. Fora period of time after t ≈ 6s, leg 1 continues stepping in freespace due to the fact that the robot is not initially alignedwith the obstacle. By continuing the walking gait, the systemnaturally pivots around the leg 2 contact and eventuallyaligns itself so that leg 1 also contacts the obstacle (t ≈ 9).

The reversal of signs in the two torque signals, τ1p andτ2p , caused by the reaction forces imposed by the obstacle,is used as the basis for triggering the dynamic switch inbehaviors between walking and climbing. Specifically, acombined signal of the two torque signals is defined as,τs = max(0, τ1p )−min(0, τ2p ), which discards values of τ1pand τ2p during unconstrained walking. To filter as well asbuild a time history of the measured torques, τs is integratedwith respect to time using an Euler integration scheme,producing τ int

s . A threshold value is defined and used tosignal the point at which the behavior of the system switchesfrom walking to climbing. The value of the threshold isuser defined, and is set high enough to allow the systemto orientate itself to the obstacle before attempting to climb.

When a behavioral transition is enabled, linear dynamicsare used to shape the response of the bifurcation parameter,µ, to ensure smooth transitions. In particular, we definesecond-order linear dynamics

Mµ+Bµ+K(µ− µr) = 0, (3)

where the effective mass, M , damping, B, and springconstant, K, stabilize the system to a desired reference, µr.

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When the neural trigger is fired, e.g., when τ ints = τ int

thresh, thereference parameter, µr ∈ {µr,w > 0, µr,c < 0} discretelyswitches values. The stabilizing dynamics in (3) then causethe value of µ to converge to µr with a step responsegoverned by the parameters M , B, and K.

An advantage of the behavioral controller described in thissection is that it is easy to implement. That is, it requiresonly proprioceptive sensing and minimal computation. Incases of software implementation, the stable limit cycles canbe computed analytically. Also, since Hopf oscillators andµ dynamics can be realized in hardware [1], it is possibleto implement the proposed behavioral control using entirelyanalogue circuitry.

IV. CLIMBING BEHAVIOR ADAPTATION

The behavioral switch discussed in Section III switchesthe SEA hexapod into a climbing behavior, which is coor-dinated around a quadrupedal walking gait. Without addi-tional regulation, the open-loop quadrupedal gait is unableto carry the system over steep obstacle heights reachinghigher than the maximum swing height of the its feet (≈ 10cm). To allow the robot to adapt to obstacles of unknownsizes during climbing, we augment the open-loop gait withcompliant control. Specifically, the SEA hexapod uses amodified admittance controller that runs in parallel with thequadrupedal gait to adapt leg heights to terrain variations.A single compliant reflex response switches the sign of theadmittance controller to swing the robot’s back legs onto theobstacle in the terminal stages of the climbing mode.

For dimensionality reduction, we constrain the verticallyrotating medial and distal joints of each leg such that θim =−θid, i ∈ {1, . . . , 6}. The i superscript indicates the legnumber (see Fig. 1) and the subscripts indicate medial anddistal joints. This constraint ensures the desired orientationof each leg remains normal to the horizontal body planeand enables the height of each leg to be controlled usinga single parameter. Furthermore, we assume the height ofcontralateral legs to be coupled. These assumptions allowthe complaint controller to be reduced to controlling foursignals, q = (q1, q2, q3, q4) representing the height of thefront, middle, and hind leg pairs, as well as an additionalterm that affects the height of the middle legs, respectively.

The dynamics of the admittance controller governing theevolution of q have the form,

Mq q +Bq q +Kq(q − qr) = Fq, (4)

where Mq , Bq , and Kq are an effective mass, damping,and spring constant, and Fq is an effective force acting oneach of the signals. The qr term corresponds to the nominalposition of q, i.e., the reference value of q in the absenceof external forcing. The reference values are set to qr = 0.The effective force term is calculated by mapping measuredjoint torques distributed throughout the different leg pairsinto their respective qi’s.

To successfully climb, the dynamics in (4) need to coordi-nate several different discrete motion sequences at differentphases of the behavior. The components of the forcing term

0 5 10 15 20 25 30-6

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Fig. 5: Sequence of the SEA hexapod encountering anobstacle while walking (t ≈ 6s), transitioning to climb-ing, compliantly controlling its leg heights (q1, q2, q3, q4),to navigate over the obstacle, and transitioning back intowalking (t ≈ 29s).

Fq are defined so that each of the discrete motions happennaturally. For example, in the first phase of climbing, thebody of the system needs to raise high enough to enablethe front legs to swing on top of the obstacle (see picturein Fig. 5 at t = 9s). This motion results from definingF 2q and F 3

q to provide positive feedback, causing them tocounteract ground reaction forces, i.e., causes leg-heightextension, during stance. Specifically F 2

q = −Jv · (τ3 + τ4)and F 3

q = −Jv · (τ5 + τ6), where Jv = [0, 1,−1] andτ3, τ4, τ

5, τ6 ∈ R3 are vectors containing the measured jointtorques in the three joints of legs 3, 4, 5 and 6 respectively(see Fig. 1). Additionally, defining F 1

q to provide negativefeedback, i.e., F 1

q = Jv · (τ1+ τ2), where τ1 and τ2 are thejoint torques measured in legs 1 and 2 respectively, causes thefront legs to comply to the ground reaction forces, which actsto adjust the robot’s body pitch such that it remains parallelto the ground during this portion of the behavior (see the plotof q1,2,3 and the picture of the robot at t = 9s in Fig. 5).

The second phase of the climbing behavior aims to get therobot’s middle legs onto the obstacle. With the robot’s frontlegs on top of the obstacle, the system is primarily bracedby the front and hind legs in this phase. As the magnitudeof F 3

q decreases, the positive feedback policy eventually liftsthe middle legs off the ground. As the middle legs lift, thesystem walks forward until one of the middle legs contactsthe obstacle, as shown in the picture in Fig. 5 at t = 14s. Thecontact helps to stabilize and align the robot to the obstacle.Also, as the reaction forces exerted by the obstacle on themiddle legs grows, the forcing terms F 4

q , defined by F 4q =

Jh · (τ5 − τ6)) with Jh = [1, 0, 0], causes the admittancecontrol policy for q4 to raise the middle legs until they clearthe obstacle (see plot of q4 in Fig. 5). Note that the heightof the middle legs is the sum of q2+ q4, where the height ofthe front and hind legs correspond to q1 and q3, respectively.

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In the third and final phase of the climbing gait, a reflexresponse switches the sign of F 3

q (which governs the heightof the back feet) when the joint torque sensors in the hindlegs indicate touchdown of either hind foot on the top of theobstacle. The reversed compliance property causes the robotto adjust its stance until its body becomes parallel with theplane on top of the obstacle. This reflexive switch providessufficient clearance for the remaining hind foot to robustlystep up and onto the obstacle, as shown in Fig. 5 (t = 24s).

Sensory feedback from touchdown of the robot’s remain-ing hind foot updates the behavioral controller’s referencevalue, µr. The updated reference initiates the transition backto nominal hexapod walking (see t = 29s in Fig. 5).

V. EXPERIMENTAL RESULTS

Experimental trials focused primarily on indoor environ-ments with a variety of obstacles including carpeted stepsof varying heights and toppled desktop computers. A totalof ten experiments were conducted per step height withheights of 17, 20, and 22cm, which correspond to 68%,80%, and 88% of the robot’s effective leg height. In eachcase, we set the robot in a position that was approximatelyone half meter away from the obstacle and orientated sothat the robot would walk toward the obstacle. The robotsuccessfully traversed the obstacle in all cases using the sameparameter values. Sample results are included in Fig. 5 andat https://youtu.be/zJRqJIziJmE.

In addition to the relatively high-friction indoor trials, weconducted preliminary tests of the robot in several differentoutdoor environments (similar to those from [5]). These trialsshow that the robot can successfully navigate over curbs upto 20cm (80% of effective leg height). However, these resultswere unreliable and highly dependent on terrain factors.Dust, dirt, and sand made it difficult for the robot to walkduring its approach, as the stance feet would slide uncontrol-lably during horizontal abduction. Similar problems occurredwhile climbing, as the robot’s feet would slide off the topof curbs. To address these issues, we intend to considernew, more robust foot designs for outdoor environments,as discussed in Section VI. Examples of successful outdoortrials are included in the video https://youtu.be/zJRqJIziJmE.

VI. CONCLUSIONS

This paper suggests that to provide reliable performanceand minimize potential on-site issues such as calibration andtuning, controllers should be few and simple, i.e., avoidingcomplex models and open parameters. To these ends, wedesign a control scheme that enables a hexapod robot toclimb over a variety of steps and ledges. The approachcombines simple limit cycle behaviors (gaits) with a com-pliance policy that automatically generates a sequence ofrequired discrete actions. While the compliant policy requiresintuition to design, experiments show the approach worksreliably, allowing the robot to react flexibly and adaptivelyto the environment. Using the proposed controllers, the

hexapod successfully climbs over different obstacles, e.g.toppled computers cases, steps, and curbs, without precisepositioning or initial conditions. These results do not requireprecise tuning and can be achieved with the same parameters.

We note that the single parameter compliant dynamicalswitch, applied here for low-dimensional control of behav-ioral transitions, is general and not limited to two limitcycles. In future work, we will explore using a similar frame-work to transition between arbitrary numbers of limit-cycle-regulated modes using feedback to modify the evolution ofthe single switching parameter, µ. To improve performancein low-friction environments, future work will also addresshardware design issues pertaining to the SEA hexapod’s feet.Specifically, we intend to experiment with compliant footdesigns with increased surface area. The current feet (rubberhemispheres) provide relatively small contact surfaces thatdo not deform sufficiently and slip over loose debris.

ACKNOWLEDGMENT

The authors would like to thank the Robotics CollaborativeTechnology Alliance (RCTA) program supported through theUnited States Army Research Lab (ARL) for its support ofthis work.

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