Adaptive and Resilient Soft Tensegrity RobotsJohn Rieffel1,∗ and Jean-Baptiste Mouret2,3,4,∗,†

1 Union College, Schenectady, NY 12308, USA2 Inria Nancy Grand - Est, Villers-les-Nancy, F-54600, France3 CNRS, Loria, UMR 7503, Vandœuvre-les-Nancy, F-54500, France4 Universite de Lorraine, Loria, UMR 7503, Vandœuvre-ls-Nancy, F-54500, France

∗ J.R. and J.-B. M. contributed equally to this work

†To whom correspondence should be addressed; E-mail:rieffelj@union.edu

Preprint — February 21, 2018

Living organisms intertwine soft (e.g., muscle) andhard (e.g., bones) materials, giving them an intrinsicflexibility and resiliency often lacking in conventionalrigid robots. The emerging field of soft robotics seeksto harness these same properties in order to createresilient machines. The nature of soft materials, how-ever, presents considerable challenges to aspects ofdesign, construction, and control – and up until now,the vast majority of gaits for soft robots have beenhand-designed through empirical trial-and-error. Thismanuscript describes an easy-to-assemble tensegrity-based soft robot capable of highly dynamic locomotivegaits and demonstrating structural and behavioral re-silience in the face of physical damage. Enabling thisis the use of a machine learning algorithm able to dis-cover effective gaits with a minimal number of physi-cal trials. These results lend further credence to soft-robotic approaches that seek to harness the interac-tion of complex material dynamics in order to generatea wealth of dynamical behaviors.

Introduction

Unlike machines, animals exhibit a tremendous amount of re-silience, due in part to their intertwining of soft tissues and rigidskeletons. In nature, this suppleness leads to several compellingbehaviors which exploit the dynamics of soft systems. Octopi, forexample, are able to adaptively shape their limbs with “joints” inorder to perform efficient grasping (Sumbre et al., 2005). Jel-lyfish exploit their inherent elasticity in order to passively re-cover energy during swimming (Gemmell et al., 2013). Mand-uca sexta caterpillars have a mid-gut which acts like a “visceral-locomotory piston” – sliding forward ahead of the surroundingsoft tissues, shifting the animal’s center of mass forward well be-fore any visible exterior change (Simon et al., 2010).

Taking inspiration from the natural world, the field of softrobotics seeks to address some of the constraints of conventionalrigid robots through the use of compliant, flexible, and elasticmaterials (Lipson, 2014; Wehner et al., 2016). Trimmer et al., forinstance, construct soft robots from silicone rubber, using shape-memory alloy (SMA) micro-coil actuation, which can slowlycrawl in controlled fashion (Trimmer, 2008) or roll in an un-controlled ballistic fashion (Lin et al., 2011). Similarly, research

Fig. 1. Concept of our soft tensegrity robot. Tensegrity struc-tures are combinations of rigid elements (struts) joined at theirendpoints by tensile elements (spring or cables) that are kept sta-ble by the interplay of pre-stress forces. A. The first tensegritystructures appeared in art, with the sculptures of Kenneth Snel-son (Skelton and de Oliveira, 2009; Snelson, 2012). B. Theyhave been subsequently used in architecture, for instance for theKurilpa bridge (Brisbane, Australia). C. More recently, tensegrityhas been found to be a good model of the mechanotransduc-tion of living cells (Wang et al., 2001). D. Our tensegrity robot isbased on carbon struts and springs. It is is actuated by 3 vibrators(glued to 3 of the struts) whose frequency is automatically tunedby a trial-and-error learning algorithm (Methods). E. Thanks tothe tensegrity structure and to the compliance of the springs, ourrobot will keep its integrity when deformed and spring back into itsinitial form. A video is available in supplementary materials (VideoS1 — https://youtu.be/SuLQDhrk9tQ).

by Whitesides et al. uses pneumatic inflation to produce slow, dy-namically stable crawling motions (Shepherd et al., 2011) as wellas fast, but less controlled tentacle-like grippers (Martinez et al.,2013), combustion-driven jumpers (Bartlett et al., 2015) and aself-contained microfluidic “octobot” (Wehner et al., 2016).

Despite their advantages, soft-material robots are difficult tocontrol by conventional means (Lipson, 2014; Shepherd et al.,2011). They are by their very nature high dimensional dynamicsystems with an essentially infinite number of degrees of free-dom. The elasticity and deformability which provide their appealcome at the cost of resonances and tight dynamic coupling be-

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tween components (Trimmer, 2008), properties which are oftenavoided, or at least suppressed, in conventional engineering ap-proaches to robotic design. This complexity precludes the useof many of the traditional kinematic and inverse-dynamics ap-proaches to robotic control (Craig, 1989).

As a result, up until now, the locomotive gaits of most softrobots have been developed by hand through empirical trial-and-error (Shepherd et al., 2011). This process can be both chal-lenging and time consuming, particularly when seeking to fullyexploit the dynamical complexity of soft mechanisms. Impor-tantly, this manual process also prevents these robots from adapt-ing their control strategy when the context changes, for instancewhen they encounter an unexpected type of terrain, or when theyare physically damaged.

In this work, we describe a new class of soft robot based upona tensegrity structure driven by vibration. Like many other softrobots, this tensegrity robot is resilient, and can resist damagewhen perturbed or crushed. Unlike other soft robots, however,this particular modular tensegrity robot is easy to build, easy tocontrol, and thanks to a data-efficient reinforcement learning al-gorithm (Cully et al., 2015), it can autonomously discover how tomove, and quickly relearn and adapt its behavior when damaged.

Vibration is an increasingly popular method of sensor-free ma-nipulation and control for automated systems (Berretty et al.,2001). Rezik et al., for instance, developed a vibration-drivenplanar manipulator (Reznik and Canny, 1998) able to performlarge-scale distributed planar control of small parts (Rezniket al., 2000). In mobile robotics, stick-and-slip frictional mo-tion (Joe, 2015; Vartholomeos and Papadopoulos, 2006) drivenby paired vibrating motors has been used in a variety of mo-bile robots (Parshakova et al., 2016; Rubenstein et al., 2012).Often, these approaches use empirically-derived hand-tuned fre-quencies to generate motion, using linear interpolation of theirtwo motor speeds in order to smoothly generate a range of behav-iors. One weakness of vibration-based approaches to locomotionis that vibration of this type leads to unpredictable motion, evenwhen assuming perfectly consistent surfaces (Joe, 2015), whichpresents a challenge to modeling and simulation.

Tensegrities are relatively simple mechanical systems, consist-ing of a number of rigid elements (struts) joined at their end-points by tensile elements (cables or springs), and kept stablethrough a synergistic interplay of pre-stress forces (Fig. 1 A-C).Beyond engineering, properties of tensegrity have been demon-strated at all scales of the natural world, ranging from the tendi-nous network of the human hand (Valero-Cuevas et al., 2007)to the mechanotransduction of living cells (Wang et al., 2001).At every size, tensegrity structures exhibit two interesting fea-tures (Skelton and de Oliveira, 2009; Snelson, 2012): they havean impressive strength-to-weight ratio, and they are structurallyrobust and stable in the face of deformation. Moreover, unlikemany other soft robots, tensegrity structures are inherently mod-ular (consisting of only struts and springs) and are therefore rela-tively easy to construct. They are simple enough to be baby toysand featured in books for children activities (Ceceri, 2015), whilecomplex enough to serve as the basis for the next generation ofNASA’s planetary rovers (Caluwaerts et al., 2014).

The most common control method for tensegrity robots is toslowly change the lengths of the struts and/or cables, causinglarge-scale, quasi-static (rather than dynamic) structural defor-mations, which, in turn, make the robot move via tumbling androlling (Caluwaerts et al., 2014; Koizumi et al., 2012). As theyassume that the structure is relatively stiff throughout locomo-tion, such control strategies are not suitable for more compliant

soft tensegrity robots. In addition, they lead to slow locomotionspeeds.

Lately, researchers have begun investigating more dynamicalmethods of tensegrity robot control. Bliss et al. have used centralpattern generators (CPGs) to produce resonance entrainment ofsimulated non-mobile tensegrity structures (Bliss et al., 2012).Mirletz et al. have used CPGs to produce goal-directed behaviorin simulated tensegrity-spine-based robots (Mirletz et al., 2015).These efforts, however valuable, were all produced in simulatedenvironments, and have not yet been successfully transferred intoreal-world robots. As Mirletz et al point out (Bliss et al., 2012),the dynamic behavior of tensegrities is highly dependent upon thesubstrate they interact with – this means that results developed insimulated environments cannot necessarily be simply transferredto real robots (in Evolutionary Robotics, this is known as the “Re-ality Gap” (Jakobi et al., 1995; Koos et al., 2013)).

More recently, Bohm and Zimmermann developed atensegrity-inspired robot actuated by an single oscillating elec-tromagnet (Bohm and Zimmermann, 2013). Although this robotwas not a pure tensegrity (it rigidly connected multiple linearstruts), it was, compellingly, able to change between forward andbackward locomotion by changing the frequency of the oscilla-tor. Vibration has been proposed as a means of controlling muchsofter robots as well (Berger et al., 2015).

Here we explore the hypothesis that the inherent resonance anddynamical complexity of real-world soft tensegrity robots can bebeneficially harnessed (rather than suppressed), and that, if prop-erly excited (Oppenheim and Williams, 2001), it can resonate sothat the robot performs step-like patterns that enable it to loco-mote. To test this hypothesis and demonstrate the potential ofsoft tensegrity robots, we designed a pocked-size, soft tensegrityrobot whose parameters were tuned to maximize resonance, andwhose goal is to locomote as fast as possible across flat terrain.To find the right vibrational frequencies, we equipped the robotwith a data-efficient trial-and-error algorithm, which also allowsit to adapt when needed.

Results

Our soft tensegrity robot (Fig. 1D-E) is based upon a canoni-cal six-bar tensegrity shape consisting of equal length compositestruts connected via 24 identical helical springs, with four springsemanating from each strut end. Unlike most tensegrity structures,which seek to maximize stiffness by, among other things, usingnearly inelastic cables (Oppenheim and Williams, 2001), here wereplace the cables with very elastic springs, with spring constantschosen with the goal of producing suitably low natural frequen-cies of the structure, with corresponding large displacements –in other words, to maximize suppleness. This allows the pocket-sized robot to maintain its structural shape under normal opera-tion, and yet be easily compressed flat in one’s hand. A variablespeed motor coupled to offset masses was then attached to threeof the struts in order to excite the natural frequencies of the struc-ture. The motor and weight were chosen to be in a range consis-tent with preliminary models. Because of the difficulty in model-ing real-world interactions of these tensegrity robots, as well asthe fact that we use a real-world optimization process describedbelow, variability in the exact placement of each motor on a strutis allowed.

Like many robots, the tensegrity robot needs to use differentgaits in order to achieve lomocotion, depending on terrain. In ourcase, these gaits are determined by the speeds of the three vi-

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A B

Fig. 2. Evaluation of the learning algorithm. A. Locomotionspeed after each of the 30 trials. The light zones represent the25th and 75th percentiles. B. Locomotion speed after 30 trials.The central mark is the median, the edges of the box are the25th and 75th percentiles (inter-quartile range – IQR), the whiskerscorresponds to the range [25%−1.5×IQR, 75%+1.5×IQR], andpoints outside of the whiskers are considered to be outliers (thiscorresponds to the “interquartile rule”). Each condition is testedwith 20 independent runs of the algorithms.

bratory motors. As the exact properties of the terrain are seldomknown a priori, and because hand-designing gaits is time con-suming (not to mention impossible when the robot is in remoteor hazardous environments) this robot finds effective motor fre-quencies by using a trial-and-error learning algorithm whose goalis to maximize the locomotion speed.

Earlier work of ours (Khazanov et al., 2013, 2014) used in-teractive trial-and-error as well as automated hill climbing tech-niques to find optimal gaits for a tensegrity robot. These gaits,could in turn, be incorporated into a simple state machine for di-rectional control. However, these techniques required hundredsof physical trials that were time consuming and produced signif-icant wear on the physical robot. More importantly, the interac-tive procedure required a human in the loop, whereas we envi-sion robots that can adapt autonomously to new situations (e.g. adamage or a new terrain). The work described in this paper, byautomating the optimization process while minimizing the num-ber of physical trials required, substantially reduces the amountof human interaction required, and is an important step towardfull autonomy.

Here, as a substantial improvement upon these earlier time-intensive methods, we employ a Bayesian optimization algorithm(Cully et al., 2015; Ghahramani, 2015; Shahriari et al., 2016),which is a mathematical optimizer designed to find the maximumof a performance function with as few trials as possible.

Conceptually, Bayesian optimization fits a probabilistic model(in this case a Gaussian process (Rasmussen and Williams,2006), see Methods) that maps motor speeds to locomotionspeed. Because the model is probabilistic, the algorithm can notonly predict which motor speeds are the most likely to be good,but also associate it to a confidence level. Bayesian optimizationexploits this model to select the next trial by balancing exploita-tion – selecting motor speeds that are likely to make the robotmove faster – and exploration – trying combinations of motorspeeds that have not been tried so far (Methods). As an additionalbenefit, this algorithm can take into account that observations are

A B

missing spring

Fig. 3. Experiments with a damaged robot. A. Damaged robot.A spring is disconnected from the robot of Fig. 1. B. Locomo-tion speed after 30 trials. The central mark is the median,the edges of the box are the 25th and 75th percentiles (inter-quartile range – IQR), the whiskers corresponds to the range[25% − 1.5 × IQR, 75% + 1.5 × IQR], and points outside of thewhiskers are considered to be outliers (this corresponds to the“interquartile rule”). Each condition is tested with 20 independentruns of the algorithms.

by nature uncertain.The Bayesian optimization algorithm usually starts with a con-

stant prior for the expected observation (e.g., the expected speedis 10 cm/s) and a few randomly chosen trials to initialize themodel. For this robot, however, common sense, along with pre-liminary modeling, suggests that speeds near the motor maxi-mums are more likely to produce successful gaits, and that near-zero motor speeds are not expected to make the robot move. Thisinsight was substantiated in preliminary experiments: many ef-fective gaits were produced by high motor speeds, both forwardand backward. Therefore, to speed up learning, we use a non-linear prior model as follows: (i) if the three motor speeds areclose to 0, then we should expect a locomotion speed close to 0and (ii) if all the motors are close to full speed (in any direction),then we should expect the maximum locomotion speed (Methodsand Fig. 5D). Thanks to this prior, the Bayesian optimization al-gorithm does not need any random sample points to seed the priorand instead starts with promising solutions. In spite of this priorlearning is still needed, because many combinations of motors atfull speeds make the robot tumble or rotate on itself, resultingin low performance; in addition, subtle changes to motor speedscan have dramatic effects upon the resulting robot gait.

We first evaluate the effectiveness of the learning algorithm(Fig. 2). The performance function is the locomotion speed,measured over 3 seconds, in any direction. If the robot turnstoo much, that is if the yaw exceeds a threshold, the evaluationis stopped (Methods). The covered distance is measured withan external motion capture system (Methods), although similarmeasurements can be obtained with an onboard visual odometrysystem (Cully et al., 2015; Davison et al., 2007). We comparethree algorithms: random search, Bayesian optimization withoutprior (using 10 random points to initialize the algorithm), andBayesian optimization with prior. Each algorithm is allowed totest 30 different motor combinations (resulting in 90 seconds oflearning for each experiment) and is independently run 20 times

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Weight (W) 89gBody length 13cmBest locomotion speed (S) [after learning] 15cm.s−1 (1.15 body lengths per second)Current drawn (full speed): 700mAPower drawn at 5V (P) 3.5W

Cost of transport at maximum speed (COT) 262 (COT , PWS )

Table 1. Best locomotion speed, power consumption, and cost of transport for the gait that corresponds to the maximum speed at 5V.For reference, a cost of transport (COT) of 262 is comparable to the COT of a mouse (more than 100), but much higher than a car ora motorcycle (around 3) (Tucker, 1970).

to gather statistics. The results show that the best locomotionspeeds are obtained with the prior-based Bayesian optimization(11.5cm/s, 5th and 95th percentiles [8.1, 13.7]), followed by theprior-free Bayesian optimization (6.3cm/s[5.5, 12.4]). The worstresults are obtained with the random search (5.4cm/s[3.5, 9.9]).The absolute best locomotion speed (15cm.s−1 was found withBayesian optimization (Table 1) and corresponds to 1.15 bodylengths per second. Overall, these experiments demonstrate thatthe prior-based Bayesian optimization is an effective way to auto-matically discover a gait in only 30 trials with this robot. Videosof a a typical gait is available as supplementary material (VideoS1).

We then investigate our hypothesis that the interplay between aflexible tensegrity structure and vibration is the key for effectivelocomotion. To do so, we designed a rigid replica of our robotthat does not contain any springs: the carbon fiber struts are heldin place with small carbon fiber rods (Fig. 4A). All the dimen-sions, strut positions, and motor positions are the same as for thetensegrity version (Fig. 1D-E). We used the same learning algo-rithm as for the tensegrity robot and the same prior, since we havethe same intuitions about good control policies for the rigid robotas for the soft one. We replicated the learning experiment 20times. The results (Fig. 4B) show that the rigid replica moves atabout 60% of the speed of the tensegrity robot (7.1cm/s[5.6, 9.3]vs 11.5cm/s[8.1, 13.7]), which suggests that the flexibility of thetensegrity structure plays a critical role in its effective locomo-tion. In addition, we measured the amplitude of movement alongthe vertical axis for the end of 4 struts, both with the soft tenseg-rity robot and the rigid replica; we repeated this measure with 50random gaits in both cases. These measurements (Fig. 4C) showthat the markers move at least twice more when the structure isflexible (2.3[1.5, 4.8] cm vs 0.99[0.61, 2.1] cm), which demon-strates that the structure amplifies the movements induced by thevibrators.

In addition to being deformable, tensegrity structures oftenmaintain most of their shape when a link (a spring or a strut) ismissing, leading to relatively smooth failure modes. We evaluatethe ability of our robot to operate after such damage by removinga spring (Fig. 3A). As the shape of the robot is changed, we re-launch the learning algorithms. The experiments reveal that suc-cessful, straight gaits can be found in 30 trials, although they aresignificantly lower-performing than gaits obtained with the intactrobot (11.5cm/s[8.1, 13.7] versus 6.5cm/s[5.6, 8.2] Fig. 3B).

During all the reported experiments, we evaluated 20 × 30 ×3 = 1800 different gaits on the intact robot, 20 × 30 = 600gaits on the rigid robot (20 replicates, 30 trials for each repli-cate, and 3 treatments), and 20 × 30 = 600 gaits on the dam-aged robot. We can use these points to draw a picture of thesearch space that does not depend on the learning algorithm(Fig. 5). Since the search space is too high-dimensional to be

easily visualized (3 dimensions + performance, resulting in a4D plot), we compute performance profiles (Mouret and Clune,2015; Reuillon et al., 2015): for each combination of 2 motorspeeds

{v1, v2

}, we report the best performance measured re-

gardless of the speed of the third motor (Methods). The perfor-mance profiles (Fig. 5A) for the intact robot reveal that thereare two high-performing regions, roughly positioned around{− 100%, 100%,−100%

}and

{− 100%,−100%, 100%

}and

that the first region ({− 100%, 100%,−100%

}) is where most

high-performing solutions can be found. This finding is consis-tent with the prior given to the learning algorithm (Fig. 5D),which models that the best performance should be obtained witha combination of −100% and +100% values. It should be em-phasized that the best gaits do not correspond to the most extremevalues for the motor speeds: the most reliable optima is around{−90%, 100%,−90%

}, mostly because too extreme values tend

to make the robot tumble. The best solutions for the rigid robotsare also found in the corners, that is, for combinations of +100%and−100% motor speeds, but the measurements suggest that theoptimum might be different from the one obtained with the in-tact robot (more data would be needed to conclude). The datafor the damaged robot show more clearly that the best solutionsare around

{−100%,−100%, 100%

}, which corresponds to the

second optimum found for the intact robot (the lowest perform-ing one).

The performance profiles thus demonstrate that the priorknowledge given to the learning algorithm is consistent with thethree different robots (intact, rigid, and damaged), which sug-gests that it might be helpful in other situations (e.g., differentdamage conditions). They also demonstrate that gaits that workthe best on the intact robot do not work on the damaged robot(Fig. 5 A versus C, second column): this shows that the learningalgorithm is needed to adapt the gait if the robot is damaged.

DiscussionSoft tensegrity robots are highly resilient, easy to assemble withthe current technology, and made with inexpensive materials. Insummary, vibratory soft tensegrity robots recast most of the com-plexity of soft robotics – building and actuating soft structures –into a much simpler class of robots – easy to build and to ac-tuate – while keeping many of the attractive properties of softrobots – e.g., resilience, deformability. Thanks to the learning al-gorithm, our prototype can achieve locomotion speeds of morethan 10 cm/s (more 1 body length per second) and learn newgaits in fewer than 30 trials, which allows it to adapt to damage ornew situations. To our knowledge, this places it among the fastestsoft robots. Our soft tensegrity robots achieve this speed becausethey uniquely harness the flexibility and the resonance of tenseg-rity structures. Discovering methods of exploiting flexibility and

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A B C

Fig. 4. Experiments with a rigid robot (with priors). A. Rigid replica of our soft tensegrity robot. This robot is identical tothe robot of Fig. 1, except that it contains no spring: the carbon fiber struts are held in place with small carbon fiber rods. All thedimensions, strut positions, and motor positions are the same as for the tensegrity version. B. Locomotion speed after 30 trials forthe intact (Fig. 1) and the rigid robot (A). Each condition is tested with 20 independent runs of the algorithm (Bayesian optimizationwith priors). C. Maximum amplitude of the markers for random gaits. In each case, we captured the vertical position of the 4markers for 50 random gaits of 3 seconds. We report the maximum height minus the minimum height (over the 4 markers). For thebox plots, the central mark is the median, the edges of the box are the 25th and 75th percentiles, the whiskers extend to the mostextreme data points not considered outliers, and outliers are plotted individually.

resonance in this manner opens new research avenues for futuretensegrity structures, in particular when mechanical design canbe coupled with machine learning algorithms that automaticallyidentify how to control the resonances.

Although our soft tensegrity robots also to large extent benefitfrom anisotropic friction, our effort is distinct other vibration-based robots such as Kilobot (Rubenstein et al., 2012) andRatChair (Parshakova et al., 2016) in several important ways.First, because of the nature of the structure, opposing pairsof vibrating motors aren’t effective - and as our results show,small changes to our robot’s motor speeds can have large andnon-linear effects upon its behavior. This renders the linear-interpolation approach of that work ineffective. As a conse-quence, rather than relying upon hand-tuning, we instead employBayesian Optimization in order to determine the most effectivevibrational frequencies in a minimum number of physical trials.

Another distinction is that our soft tensegrity robot’s intrinsicresonance is tuned to respond the vibratory input of their actua-tors. The benefit of this tuned resonance is particularly noticeablewhen the performance of the soft tensegrity robot is compared tothe rigid mock-tensegrity robot described in experiments (Fig.4). These soft tensegrity robots also stand in contrast to othermore rigid tensegrity robots (Caluwaerts et al., 2014; Koizumiet al., 2012), which generally try to suppress their resonance.Harnessing flexibility and resonance opens new research avenuesfor future soft robots, in particular when mechanical design canbe coupled with machine learning algorithms that automaticallyidentify how to control the resonances.

One of the more thought-provoking illustrations of the poten-tial of soft tensegrity robots is best observed on the supplemen-tary video, at slow speed: once properly tuned by the learningalgorithm, the vibrations induce large, visible deformations ofthe structures that create a step-like pattern for the “feet” at theend of the rigid struts (more quantitative results can be seen on

Fig. 4-C). These step-like patterns have the potential to allowtensegrity robots to step over small irregularities of the groundlike a walking robot. Importantly, these patterns are made possi-ble by the mix of soft and rigid elements in the same structure:they are likely to be much harder to induce and control both witha fully soft robot and with a fully rigid robot. A promising re-search avenue is to focus on how to control the movement of thefeet explicitly and make steps that are little disturbed as possibleby the irregularities of the floor.

An added benefit of vibrational locomotion for soft robotics isthat, although our current robot is tethered, it could in principlebe easy to power soft tensegrity robots with an embedded bat-tery, by contrast with the many fluid-actuated soft robots (Lipson,2014; Shepherd et al., 2011), which need innovative ways to storeenergy (Wehner et al., 2016). Nevertheless, soft tensegrity robotscould excite their structure by other means; for instance, a fly-wheel that is rapidly decelerated could help the robot to achievefast movements (Romanishin et al., 2013), or high-amplitude,low-frequency oscillations could be generated by moving a pen-dulum inside the structure (Chase and Pandya, 2012).

Early work of ours on mobile tensegrities (Khazanov et al.,2013, 2014) used a rather simple interactive hill-climber in or-der to discover effective locomotive gaits, however this type ofsimplistic stochastic search was suboptimal. While there may belittle qualitative difference between our earlier gaits and those de-scribed here, there are profound differences in terms of the timeand data efficiency of this Bayesian Optimization approach. Mostsignificantly, the hill-climber places no emphasis on reducing thenumber of physical trials performed, and as a consequence re-quired hundreds of trials and hours of experimentation beforediscovering effective gaits. These repeated physical trials put un-necessary wear on the robot, and required a substantial amount ofhuman effort in resetting the robot between trials. Furthermore,the OpenCV-based optical tracking of the robot was rudimentary

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B - rigid robot

C - damaged robot

A - intact robot

D - prior knowledge

Fig. 5. Performance profiles for all the conditions. These performance profiles show the performance potential of each combinationof 2 motor speeds (the third motor is considered as a “free variable”). Three plots are required to get a comprehensive picture of theperformance space: v1 vs v2, v1 vs v3, and v2 vs v3. A. Intact robot (Fig. 1D). The profiles are computed with 1800 policy evaluations(20 replicates × 30 trials × 3 sets of experiments – with prior, without prior, random search). B. Rigid robot (Fig 4A). The profilesare computed with 600 policy evaluations (30 trials × 20 replicates). C. Damaged robot (Fig. 3). The profiles are computed with 600policy evaluations (30 trials × 20 replicates). D. Prior knowledge. Prior knowledge used to guide the learning algorithm (Methods).

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and lacked the spatial precision required of more effective algo-rithms. The Bayesian Optimization approach we have used here,along with the high precision Optitrack system, profoundly re-duces the number of physical trials and the corresponding wearon the robot, thereby increasing its capacity for faster and moreautonomous resilience and adaptivity.

We purposely designed the robot so that the search space isas small as possible, which, in turn, makes it more likely forthe robot to be capable of adapting in a few trials. Put differ-ently, one of the main strength of vibration-based locomotion isto make the search problem as simple as possible. Although, inprinciple, a variety of optimization techniques (e.g., simulatedannealing (Kirkpatrick et al., 1983)) might have been used, thereare compelling reasons why our adaptation algorithm is based onBayesian optimization, namely because (i) it is a principled ap-proach to optimize an unknown cost/reward function when onlya few dozen of samples are possible (Shahriari et al., 2016) (bycontrast, the simulated annealing algorithm relies on the statis-tical properties of the search space, which are valid only with alarge number of samples (Kirkpatrick et al., 1983)), (ii) it canincorporate prior knowledge in a theoretically sound way (in-cluding trusting real samples more than prior information) (Cullyet al., 2015), and (iii) it takes into account account the acquisitionnoise (Rasmussen and Williams, 2006). For instance, Bayesianoptimization is the current method of choice for optimizingthe hyper-parameters of neural networks (Shahriari et al., 2016;Snoek et al., 2012), because evaluating the learning abilities ofa neural network is both noisy and time-intensive. The downsideof Bayesian optimization is a relatively high computational cost:the next sample is chosen by optimizing the acquisition func-tion, which typically requires using a costly, non-linear optimizerlike DIRECT (Finkel, 2003) or CMA-ES (Hansen et al., 2003)(our implementation uses CMA-ES, see Methods). Put differ-ently, Bayesian optimization trades data with computation, whichmakes it data-efficient, but computationally costly. As we mostlycare about data-efficiency, we neglect this cost in this work, butit could be an issue on some low-power embedded computers.

Most black-box optimization (e.g. CMA-ES (Hansen et al.,2003)) and direct policy search algorithms (e.g policy gradients(Peters and Schaal, 2006)), could substitute Bayesian optimiza-tion as an adaptation algorithm by directly optimizing the reward(instead of first modeling it with Gaussian process). While theywould not need time-intensive optimizations to select the nextsample to acquire, these algorithms are tailored for at least athousand evaluation (e.g. 104 to 105 evaluations in benchmarksof 2D functions for black-box optimizers (Hansen et al., 2010)),are not designed to incorporate priors on the reward function, andare, at best, only tolerant to noisy functions. As a consequence,while algorithms like CMA-ES could work as an adaptation al-gorithm, they appear to be a sub-optimal choice for online adap-tation when only a few dozen of evaluations are possible.

Traditional Bayesian optimization uses a constant mean as aprior (Calandra et al., 2014; Lizotte et al., 2007), that is, the onlyprior knowledge is the expectation of the cost/reward. By con-trast, we show here that it is effective to introduce some basicintuitions about the system as a non-constant prior on the rewardfunction. We thus increase the data-efficiency while keeping thelearning algorithm theoretically consistent. Cully et al. (Cullyet al., 2015) also used a non-constant prior; however, (i) theygenerated it using a physics simulator, which is especially chal-lenging for a vibrating tensegrity robot, and (ii) they only com-puted this prior for a discrete set of potential solutions, which,in turn, constrain Bayesian optimization to search only in this

set. Here we follow a more continuous approach as our prior isa continuous function, and we show that relevant priors can bedefined without needing a physics simulator. The more generalproblem of how to generate “ideal” priors is far from trivial. Intu-itively, priors should come from a meta-learning process (Lemkeet al., 2015), for instance, an evolution-like process (Kirschnerand Gerhart, 2006), which would search for priors that wouldwork well in as many situations as possible (i.e., instincts). Ef-fectively implementing such a process remains an open grandchallenge in machine learning and is sometimes called “meta-learning” (Lemke et al., 2015).

Putting all these attractive features altogether, soft tenseg-rity robots combine simplicity, flexibility, performance, and re-siliency, which makes this new class of robots a promising build-ing block for future soft robots.

Of course, additional work is needed to have a more completetheory of the “optimal suppleness” of soft robots. Intuitively, toomuch suppleness would absorb the energy transmitted by the vi-brator and prevent effective gaits; but, at the other end of thespectrum, a rigid robot cannot generate the form changes that arenecessary for the most interesting gaits. This may be what we ob-served when we damaged the robot: by making the structure lessconstrained, the shape of the robot may have become looser, and“softer”, which impacted the maximum locomotion speed (alter-natively, the removal of a spring might have prevented the trans-mission of some oscillations or some resonance modes). Never-theless, for every kind of suppleness that we tried, the Bayesianoptimization algorithm was always capable of finding some ef-fective gaits, which means that the “optimal softness” does notneed to be known a priori in order to discover effective locomo-tion. In this regard, trial-and-error approaches like the ones usedhere provide a valuable ability to respond and adapt to changesonline in a rather robust manner, much like living systems (Cullyet al., 2015).

Several exciting open questions remain. So far, we have onlydemonstrated the effectiveness of this technique on a single sub-strate rather than across an entire range of environments. A com-pelling question we look forward to exploring in future work,for instance, is the extent to which the locomotive gaits we havediscovered are robust and self-stabilizing in the face of externalperturbations and changes in the substrate. Of course, the generalproblem of robust locomotion of any robot, much less soft robots,across multiple substrates and environments remains a relativelyopen topic. Recent work has, for instance, explored hand-pickedstrategies for the quasi-static locomotion of a cable-actuatedtensegrity on inclined surfaces (Chen et al., 2017). Our own abil-ity to harness tensegrity vibration in order to induce large-scaleand dynamic structure offers a compelling and promising methodof discovering much more dynamic gaits for these environments.Indeed, our robot design is already capable of interesting behav-ioral diversity, including several unique rolling behaviors, whichmight be beneficial across environments - however we were un-able to explore these more deeply due to the tethered nature ofthis design. Nonetheless, the speed with which our algorithmcan learn effective gaits, especially when damaged, provides aglimpse into how future soft robots could adapt to new and unex-pected environments in situ, with no pre-existing knowledge orexperience of that environment.

This leads to the recognition that the the present prototype,while more than sufficient to demonstrate the claims of this pa-per, is not yet fully autonomous : it relies on a tether for power,uses an external motion capture to evaluate its performance (lo-comotion speed), and uses an offboard computer for the the learn-

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Fig. 6. An untethered version of the tensegrity robot. Thisnew robot, still under development will allow for more interest-ing dynamical behaviors such as rolling, as well as complex en-vironments. This could in principle allow for completely on-boardlearning as well.

ing algorithm. We are in the process of designing a fully wirelessand autonomous tensegrity robot, as illustrated by Figure 6. Thisnext generation of robot will be capable of substantially moredynamical behaviors, such as rolling and jumping, and more ca-pable of exploring complex environments. Evaluating the per-formance of locomotion techniques using on-board processingcould in principle by achieved either with accelerometers or withan embedded camera paired with a visual odometry algorithm,(Cully et al., 2015; Forster et al., 2014), but the vibrations andthe fast movements of the struts are likely to disturb many visualalgorithms. Additionally, the modular nature of this wireless strutdesign means that we could explore an entire range of tensegrityrobot morphologies, including those with considerably more thansix struts.

Overall, our soft tensegrity robots move thanks to the com-plex interactions between the actuators (vibrators), the structure(springs and struts), and the environment (the ground). This kindof emergent behavior is central in the embodied intelligence the-ory (Pfeifer et al., 2007), which suggests that we will achieve bet-ter and more life-like robots if we encourage such deep couplingsbetween the body and the “mind” – here, the controller. However,as demonstrated in the present work, trial-and-error learning al-gorithms offer a strongly viable approach to discovering theseemergent behaviors.

Material and Methods

Robot

The tensegrity used is defined by six equal length compositestruts which are connected to each other via 24 identical helicalsprings, with four springs emanating from each strut end. Thisfollows the geometry described as TR-6 by Skelton (Skelton andde Oliveira, 2009). Few actual machining operations are requiredto produce the tensegrity. The six 9.4 cm long composite struts

are cut from 6.35 mm square graphite composite tubes (Good-winds). The three 12mm vibrational motors (Precision Micro-drives Model 312-107) were mounted to the flat outer surface ofthe struts using hot melt adhesive. Both ends of each strut werethen tapped for 10-24 nylon screws fitted with nylon washers.The hooked ends of the helical springs (Century Spring StockNo. 5368) were attached directly to holes drilled through the ny-lon washers. The motors were connected via thin gauge mag-net wire to Serial Motor Controllers (Pololu Qik 2s9v1 Dual Se-rial Motor Controller) connected in turn to a USB Serial Adapter(SparkFun FTDI Basic Breakout board)

The specific spring constants were chosen in order to producerelatively low natural frequencies and correspondingly large dis-placements of the structure while at the same time limiting esti-mated static deflection to 5% of strut length. In order to determinethis, a single strut was modeled as being connected to four linearsprings at each end, equally spaced around the radius, each at a45◦ angle. Limiting static deflection to 5% of strut length resultsin a spring constant value of 0.209 N/cm. Subsequently, the en-tire 6-bar structure was modeled by assuming that one strut wasbe anchored in place and then using matrix structural analysisto determine the natural frequencies. The vibrational motor wasthen chosen that was capable of generating sufficient centrifugalforce at a suitable range of frequencies. Details of the modelingand design are provided in (Khazanov et al., 2013).

Control policy

Each policy is defined by three PWM values that determine theinput voltage of the 3 vibrating motors (χ = [v1, v2, v3]), whichcan take values between 0 (full speed, backward) and 1 (fullspeed, forward); 0.5 corresponds to a speed of 0, that is, to nomovement.

Performance function

Each controller is tested for 3 seconds, then the Euclidean dis-tance between the starting point and the end point is recorded.The performance function is the distance (in cm/s) divided by 3.If during the 3 second evaluation period the yaw of the robot ex-ceeds 1 radian, the evaluation is stopped and the recorded dis-tance is the distance between the starting point and the pointreached by the robot when it exceeded the yaw limit.

The policies are evaluated externally with a motion trackingsystem (Optitrack Prime 13 / 8 cameras), but the same measure-ments can be obtained with an embedded camera connected to avisual odometry system (Cully et al., 2015; Davison et al., 2007).

Profile plots

We use the profile plots to depict the search space and the priorused by the learning algorithm (Fig. 5). For each pair of di-mensions, we discretize the motor speeds into 25 bins. For eachbin, we compute pprofile(v1, v2) = maxv3 p(v1, v2, v3), wherep(v1, v2, v3) is the performance of the robot for motor speedsv1, v2, v3 and pprofile(v1, v2) is the performance reported in theprofile. To get a comprehensive pictures, we need three plots:pprofile(v1, v2), pprofile(v1, v3), and pprofile(v2, v3).

Learning algorithm

Our learning algorithm allows the robot to discover by trial-and-error the best rotation speeds for its three motors. It essentially

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implements a variant of Bayesian optimization, which is a state-of-the-art optimization algorithm designed to maximize expen-sive performance functions (a.k.a. cost functions) whose gradi-ent cannot be evaluated analytically (Ghahramani, 2015; Shahri-ari et al., 2016). Like other model-based optimization algorithms(e.g., surrogate-based algorithms (Booker et al., 1999; Forresterand Keane, 2009; Jin, 2011), kriging (Simpson et al., 1998), orDACE (Jones et al., 1998; Sacks et al., 1989)), Bayesian opti-mization models the objective function with a regression method,uses this model to select the next point to acquire, then updatesthe model, etc. until the algorithm has exhausted its budget offunction evaluations.

Here a Gaussian process models the objective function (Ras-mussen and Williams, 2006), which is a common choice forBayesian optimization (Brochu et al., 2010; Calandra et al., 2014;Ghahramani, 2015; Lizotte et al., 2007; Shahriari et al., 2016).For an unknown cost function f , a Gaussian process defines theprobability distribution of the possible values f(x) for each pointx. These probability distributions are Gaussian, and are thereforedefined by a mean (µ) and a variance (σ2). However, µ and σ2

can be different for each x; a Gaussian process therefore definesa probability distribution over functions:

P (f(x)|x) = N (µ(x), σ2(x)) (1)

where N denotes the standard normal distribution.At iteration t, if the performance [P1, · · · , Pt] = P1:t of the

points [χ1, · · · , χt] = χ1:t has already been evaluated, thenµt(x) and σ2

t (x) are fitted as follows (Rasmussen and Williams,2006):

µt(x) = kᵀK−1P1:t

σ2t (x) = k(x,x) + σ2

noise − kᵀK−1kwhere:

K =

k(χ1, χ1) · · · k(χ1, χt)...

. . ....

k(χt, χ1) · · · k(χt, χt)

+ σ2noiseI

k =[k(x, χ1) k(x, χ2) · · · k(x, χt)

](2)

The matrix K is called the covariance matrix. It is based ona kernel function k(x1,x2) which defines how samples influ-ence each other. Kernel functions are classically variants of theEuclidean distance. Here we use the exponential kernel (Brochuet al., 2010; Cully et al., 2015; Rasmussen and Williams, 2006;Shahriari et al., 2016):

k(x1,x2) = exp(− 1

β2||x1 − x2||2

)(3)

because this is the most common kernel in Bayesian optimizationand we did not see any reason to choose a different one (Brochuet al., 2010; Shahriari et al., 2016). We fixed β to 0.15.

An interesting feature of Gaussian processes is that they caneasily incorporate a prior µp(x) for the mean function, whichhelps to guide the optimization process to zones that are knownto be promising:

µt(x) = µp(x) + kᵀK−1(P1:t − µp(χ1:t)) (4)

In our implementation, the prior is a second Gaussian processdefined by hand-picked points (see the “prior” section below).

To select the next χ to test (χt+1), Bayesian optimization max-imizes an acquisition function, a function that reflects the needto balance exploration – improving the model in the less known

parts of the search space – and exploitation – favoring parts thatthe model predicts as promising. Numerous acquisition functionshave been proposed (e.g., probability of improvement, the ex-pected improvement, or the Upper Confidence Bound (UCB)(Brochu et al., 2010; Calandra et al., 2014; Shahriari et al.,2016)); we chose UCB because it provided the best results in sev-eral previous studies (Brochu et al., 2010; Calandra et al., 2014)and because of its simplicity. The equation for UCB is:

χt+1 = argmaxx

(µt(x) + κσt(x)) (5)

where κ is a user-defined parameter that tunes the tradeoff be-tween exploration and exploitation. We chose κ = 0.2.

Prior for the learning algorithm

The learning algorithm is guided by a prior that captures the ideathat the highest-performing gaits are likely to be a combinationof motors at full speed (in forward or in reverse). In our imple-mentation, it is implemented with a Gaussian process defined by9 hand-picked points and whose variance is ignored (equation 2).The kernel function is the exponential kernel (equation 3), withβ = 0.15.

The 9 hand-picked points (χ1, · · · , χ9) are as follows (Fig 5-D):

χ1 = [−100%,−100%,−100%]χ2 = [−100%,−100%,+100%]χ3 = [−100%, 100%,−100%]χ4 = [−100%,+100%,+100%]χ5 = [+100%,−100%,−100%]χ6 = [+100%,+100%,+100%]χ7 = [+100%,−100%,−100%]χ8 = [+100%,−100%,+100%]χ9 = [0%, 0%, 0%];P (χ1), · · · , P (χ8) = 0.3;P (χ9) = 0

Statistics

For all experiments, we report the 5th and 95th percentiles. Weused a two-tailed Mann-Whitney U test for all statistical tests.For the box plots, the central mark is the median, the edges of thebox are the 25th and 75th percentiles (inter-quartile range – IQR),the whiskers corresponds to the range [25%−1.5×IQR, 75%+1.5× IQR], and points outside of the whiskers are considered tobe outliers (this corresponds to the “interquartile rule”). For eachbox plot, the result of the Mann-Whitney U test (two-tailed) isindicated with stars: * means p ≤ 0.05, ** means p ≤ 0.01, ***means p ≤ 0.001, and **** means p ≤ 0.0001.

Computer code

http://members.loria.fr/JBMouret/src/limbo-tensegrity.tar.gz ; this code will be re-leased with an open-source license on Github for the finalpublication.

Data availability

http://members.loria.fr/JBMouret/data/tensegrity.tar.gz ; these data will be released on Dryadfor the final publication.

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Acknowledgments

This project received funding from the European ResearchCouncil (ERC) under the European Union’s Horizon 2020research and innovation programme (Project: ResiBots,grant agreement No 637972); J.R. received funding for a1-month visit at University of Lorraine, France. The authorswould like to thank Dorian Goepp for his invaluable help theResiBots team for their comments on this manuscript. Theauthors would also like to thank Bill Keat for his help andinsight into the design of the robot, and all the undergradu-ates of Union College’s Evolutionary Robotics Group.

The pictures of Fig.1 are distributed under a CreativeCommons License. A: c©Jonathan Rieke. B: c©MargaretDonald. C: c©National Institute of Dental and CraniofacialResearch, National Institutes of Health (NIH).

Author Contributions

J.R. designed the robot; J.R. and J.-B.M. designed thestudy, performed the experiments, analyzed the data, andwrote the paper.

Supplementary information

Video S1

Presentation of our soft tensegrity robot. The videoshows the soft tensegrity robot in action: how it can loco-mote and how it can learn to compensate when damaged.The video is available online one: https://youtu.be/SuLQDhrk9tQ

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