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348 IEEE TRANSACTIONS ON ROBOTICS, VOL. 24, NO. 2, APRIL 2008

Experimentally Verified Optimal Serpentine Gaitand Hyperredundancy of a Rigid-Link Snake Robot

Vipul Mehta, Sean Brennan, Member, IEEE, and Farhan Gandhi

Abstract—In this study, we examine, for a six-link snake robot,how an optimal gait might change as a function of the snake–surface interaction model and how the overall locomotion perfor-mance changes under nonoptimal conditions such as joint failure.Simulations are evaluated for three different types of friction mod-els, and it is shown that the gait parameters for serpentine motionare very dependant on the frictional model if minimum powerexpenditure is desired for a given velocity. Experimental investi-gations then motivate a surface interaction model not commonlyused in snake locomotion studies. Using this new model, simulationresults are compared to experiments for nominal and nonnominallocomotion cases including actuator faults. It is shown that thismodel quite accurately predicts locomotion velocities and link pro-files, but that the accuracy of these predictions degrades severelyat speeds where actuator dynamics become significant.

Index Terms—Experimental validation, friction models, gaitanalysis, hyperredundancy, optimization, serpentine robots.

I. INTRODUCTION

IN THE PAST several decades, the need has grown forrobots that can navigate highly complex terrains or blend

in naturally with their surroundings. The result has been an in-creased number of studies on snake robot locomotion. Many ofthese studies have focused primarily on the feasibility, imple-mentation, or novel construction issues in designing snake-likerobots [1]–[12], and/or the path planning considerations uniqueto these systems [13]–[16]. However, there often remain ques-tions of the system efficiency and performance of a snake robotcompared to other locomotion modalities. Among the earliestwork on snake robot locomotion is that of Hirose and Umetamiwho, starting in the early 1970’s and continuing to the present,examined locomotion behavior of snake robots both analyti-cally and experimentally [5], [6]. Ongoing work at the TokyoInstitute of Technology continues to expand Hirose’s resultswith numerous recent publications on snake robots by manyauthors [17]–[21].

When attempting to apply published results to practice, onefinds that many issues remain unresolved. For example, it iscommonly assumed that the robot–ground interaction occurswithout any lateral slip [17], [19], [22]–[26], However, lateral

Manuscript received January 29, 2007; revised October 16, 2007. This pa-per was recommended for publication by Associate Editor S. Ma and EditorH. Arai upon evaluation of the reviewers’ comments. This work was sup-ported by the Defense Advanced Research Projects Agency (DARPA), AwardNBCHC040066, with Dr. J. Main as the Technical Monitor.

The authors are with the Department of Mechanical and Nuclear Engineer-ing, Pennsylvania State University, University Park, PA 16802 USA (e-mail:[email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TRO.2008.915441

slip at the point of ground contact is known to occur for snakerobots [27] but the exact nature of slip is not characterizedwell. Sometimes, a Coulomb friction contact model is assumed[28], [29], sometimes a pure viscous contact model [30], andsometimes combinations of viscous/coulomb friction models[21], [31].

Most snake robots designs use underbelly wheels, rigid links,and serpentine gait for locomotion. For this “most common"design, there are only a few studies comparing simulation-predicted locomotion velocities and gaits with experimentalresults. Experimental comparisons using rolling snakes wererecently presented by Ute and Ono [21], by Ma et al. [32], andby Ye et al. [33]. These studies give results for only a few gaitsettings, not over a wide range of planar gaits and steady-statevelocities.

The study of gait efficiency for serpentine gaits occurring inanimal kingdom suggests efficiencies comparable to other formsof locomotion [34], [35]. However, the efficiency of snake robotlocomotion is reported to be poor [21], [31] to an extent thatsome researchers argue that the use of snake robots is best mo-tivated by a snake’s ability to complete tasks that are intractableto other vehicles, not energy efficiency [27].

There has also been a significant amount of work on thevarious modes of locomotion available to a snake robot on var-ious surfaces [27], [36], [37]. Within this body of work, thereare a number of experimentally demonstrated locomotion stud-ies [12], [24], [37], [38]. However, there is overall little exper-imental confirmation of theory predicted gait behavior underenergy-optimal gait parameters for rigid-link, rolling-contactsnakes with serpentine gait.

The aforementioned issues show a need for a more com-prehensive analysis of the relationship between efficiency oflocomotion and the factors affecting locomotion, namely gaitparameters, surface friction and environmental roughness, ac-tuation modes, actuator dynamics and faults, etc. Further, thisanalysis requires careful comparison between simulations andexperiments. This paper addresses the aforementioned issuesin sections organized as follows. Section II presents the modelformulation as a set of governing differential equations thatallow general describing functions for the snake/surface inter-action. Section III discusses the realization of a serpentine gaitby trajectory control of the joint angles of the snake robot.Section IV discusses power considerations of snake locomo-tion, and frames the objective of locomotion at minimum poweras a min–max problem. Section V examines a distributed contactviscous friction model, and by comparison with previous resultsof others, validates the numerical correctness of the simulationof these equations. Section VI then examines a more appropriate

1552-3098/$25.00 © 2008 IEEE

MEHTA et al.: EXPERIMENTALLY VERIFIED OPTIMAL SERPENTINE GAIT AND HYPERREDUNDANCY OF A RIGID-LINK SNAKE ROBOT 349

Fig. 1. Configuration of the snake robot.

point-contact friction model and how the optimal gait changesunder this friction assumption. Section VII next presents resultsfrom experiments that measure the snake–surface interactionthat led to the development of a new friction model closelymatching these results, and Section VIII presents the simulation-predicted snake performance with this new model. Section IXcompares the simulation results to experiment for a range of op-erational characteristics. Finally, Section X examines the abilityof a snake to locomote under fault conditions and the fidelityof simulations in predicting this locomotion. Conclusions thensummarize the main points of this paper.

II. GOVERNING EQUATIONS OF MOTION

The snake locomotion consisting of a 2-D lateral undulation,i.e., planer serpentine gait with no body lifting is consideredin this paper. To formulate the governing equations of motionfor a snake robot composed of rigid links, the general formu-lation representing a robot with n links and (n − 1) actuatorsconnecting the respective links is considered. The idealized rep-resentation is shown in Fig. 1. We hereafter define the x, y, andθ vectors containing the positions of all links as

x = [xi ]n×1 (1)

y = [yi ]n×1 (2)

θ = [θi ]n×1 (3)

where xi , yi , and θi denote the coordinates and orientationof the center of mass of the ith link. It is assumed that themass, length, and moment of inertia is same for all the linksand are represented by the variables m, 2l, and J (=ml2/3),respectively.

To define the snake robot dynamics, this paper follows thework of Saito et al. [31] with modifications as noted. The equa-tions are written from force and moment balance, starting witha definition of the total friction force vector, given by [fx fy ]′.Using a force balance in x-direction and y-direction (see Fig. 2),one obtains a matrix form of the system equations

mxi = fx + D′gx (4)

myi = fy + D′gy (5)

Fig. 2. Free body diagram of ith link.

where gx and gy are vectors consisting the reaction forces ateach joint (gx = [gxi

](n−1)×1 and gy = [gyi](n−1)×1). In this

paper, the prime symbol ′ is used to denote the transpose of amatrix, so D′ is the transpose of the difference matrix given by

D =

1 −1 0. . .

. . .0 1 −1

(n−1)×n

. (6)

Similarly, by summing moments about the center of mass of eachlink, one obtains the rotational governing equation in matrixform:

Jθ = −lSθA′gx + lCθA

′gy + D′u (7)

where l is the link length, u is a vector composed of terms ui ,representing the torque applied at joint i, and A is the additionmatrix:

A =

1 1 0

. . .. . .

0 1 1

(n−1)×n

. (8)

The matrices Sθ and Cθ are rotation matrices defined as

Sθ = diag(sin θ1 , . . . , sin θn ) (9)

Cθ = diag(cos θ1 , . . . , cos θn ). (10)

One can rearrange these equations to eliminate the couplingforces gx and gy in (4), (5), and (7). Performing this simplifica-tion, and defining the position of the center of mass of the robotas w = 1/n [

∑xi

∑yi ]

′, the following governing differen-tial equation is obtained[

J 00 MI2

] [θw

]+

[Cθ2

0

]−

[L′

E ′

] [fx

fy

]+ τ =

[D′

0

]u

(11)

where

τ = cn (J/m)θ (12)

e = [1 · · · 1]′ (13)

H = ml2[A′(DD′)−1A

](14)


N = l[D′(DD′)−1A

](15)

J = JIn + SθHSθ + CθHCθ (16)

M = nm (the total mass of the robot) (17)

C = SθHCθ − CθHSθ (18)

L = [SθN′ − CθN

′]′ (19)

E =[

e 00 e

](20)

and In is the identity matrix of dimension n. In these equations,J can be viewed as a rotational mass matrix of the robot, whichincludes the link inertia (J) and the reflected inertias in H .Similarly, C represents the forcing function due to rotationalmotion of the links (e.g., centrifugal force) and [L′ E ′]′ givesthe transformation matrix from local link coordinates to globalrobot coordinates.

To this point, the equations of motion are adopted directlyfrom Saito et al. [31]. In their work, the friction formulation wasbased on a distributed contact model. However, this assumptionis not appropriate for the wheeled snake robots that are moreoften used in implementation. Under the assumption of a pointcontact at the wheels, the frictional torque [see (12)] vanishesand the governing equation is simplified to[

J 00 MIn

] [θw

]+

[Cθ2

0

]−

[L′

E ′

] [fx

fy

]=

[D′

0

]u.

(21)Saito’s model is further modified by changing the frictionalmodel, discussed next.

A. Snake to Surface Friction Models

The vectors fx and fy defined in the governing equations(11) and (21) depend on the the snake–surface friction interac-tion. For simple viscous friction models, these vectors for anindividual link are given by[

fxi

fyi

]= −

[cos θi − sin θi

sin θi cos θi

] [ct 00 cn

]

[cos θi − sin θi

sin θi cos θi

] [xi

yi

](22)

where ct and cn are the viscous friction coefficients in the tan-gential and normal directions, respectively.

To represent the forces on all the links in the complete robot,these equations can be put into a matrix form as[

fx

fy

]= −ΩθDf Ω′

θ

[xi

yi

](23)

where

Df =[

ctIn 00 cnIn

]

Ωθ =[

Cθ −Sθ

Sθ Cθ

]. (24)

Later in this paper, experimental results are presented that mo-tivate a friction model combining Coulomb friction and viscous

friction. To obtain this model formulation, the frictional forceshave a static offset and can be represented in a manner similarto viscous friction[

fx

fy

]= −ΩθDf Ω′

θ

[xi

yi

]− ΩθDf0 sgn

(Ω′

θ

[xi

yi

])(25)

where

Df0 =[

ct0 In 00 cn0 In

](26)

and ct0 and cn0 are the Coulomb friction coefficients in thetangential and normal direction, respectively.

III. INPUTS PRODUCING SNAKE-LIKE GAITS: TORQUE VERSUS

POSITION CONTROL

It is known that simulation of snake-like locomotion can bechallenging due to the complex interaction of the control law,body kinematics, and surface contact forces [29], [39]. To per-form a simulation-based analysis of snake dynamics, the equa-tions of motion described previously were represented withinSimulink and solved using MATLAB’s built-in ordinary differ-ential equation (ODE) solvers. The outputs of the simulationare the angular positions of the links (θi) and center of massposition of the entire snake (w).

Previous research [36], [37] has shown that a snake-like gaitcan be obtained by imposing a commanded joint motion profilegiven by

φi = α sin (ωt + (i − 1)β) (27)

where φi is the joint angle for joint i

φi = θi+1 − θi (28)

α is the amplitude of the motion, ω is the frequency, and β isthe phase difference between two consecutive joint angles.

For snake robots, one can choose to control snake motion viajoint torque inputs or joint angle commanded profiles. It is usefulto transform between the two as the joint torque in simulationrelates to power; for example, in the governing equations givenpreviously, (21) and (11), the input to the robot dynamics isa commanded joint torque (u). However, in implementation, itis far easier to operate local controllers at each joint servoingto commanded joint angles. For example, in the experimentaltests reported in this paper, servo motors are used that operatean internal control loop to track a commanded joint angle. It isknown that the joint torque versus joint angle inputs methodsare related and can be decoupled [31] such that the preferredformulation can be chosen.

To appropriately compare simulations to experimental results,the simulation joint torques were varied to track an angle profilegiven by (27) using feedback control. This also facilitates com-parison to the work of others who generally specify referencejoint positions and not torques. The goal of the simulation-basedjoint control strategy is therefore to track a desired referencejoint angle (φ) by outputting a joint torque. The overall feed-back control system structure is represented as in Fig. 3. APID controller was designed to minimize the error between the


Fig. 3. Block diagram showing implementation of the closed-loop controllerto generate joint tracking in the simulations of snake locomotion.

reference joint angles and the calculated joint angles. The con-troller gains are tuned manually such that the simulation closelytracks the given reference trajectory. To confirm that the con-troller had minimal effect on the gait results, comparisons weremade of the gait, joint torque profiles, and system input–outputefficiency under different controller gains. There was little dif-ference observed between controllers as long as the trajectorytracking error (TE) was low. This requirement of close track-ing is discussed further later in this paper as it appears to bean important but rarely discussed consideration in snake robotoperation.

IV. OPTIMAL GAIT ANALYSIS

This section formulates the search for the most efficient mo-tion of the snake robot as an optimization problem. Specifi-cations such as the power input is unquestionably a concernfor an automated robot, yet for many snake robot applications(irrespective of the terrain or whether snake robots are advanta-geous or not), less attention has been given to the performanceof the robot if its gait parameters are varied or actuation oc-curs under nonoptimal conditions. Consistent with the work ofothers [21], [27], [31], the optimal gait and corresponding lo-comotion efficiency in this analysis is defined as the gait thatconsumes minimum power to locomote the center of mass at agiven average velocity. Hence, the optimal gait can be differentfor different average velocities. From (27), the input parametersaffecting gait, and hence, efficiency are α, ω, and β. The outputparameters are the power dissipated and the average forwardpropulsive speed. For brevity, any reference hereafter to thevelocity of the snake is referring specifically to the steady-statevelocity of the center of gravity averaged over at least one cycle.

Mathematically, the task of finding the optimal gait can beformalized as the following minimization problem

min Pin

subject to

vx = vxdesired

αmin ≤ α ≤ αmax

βmin ≤ β ≤ βmax

ωmin ≤ ω ≤ ωmax . (29)

The power input to the system is obtained by taking a dot productof the joint torque (u) and joint velocity (φ), (Pin = uφ). Theterm vx is the speed of the robot in the forward direction, vxdesired

is the desired speed of the robot, αmin , βmin , and ωmin are thelower bounds on the input parameters, and αmax , βmax , andωmax are the upper bounds on the input parameters.

Fig. 4. Variation of forward speed and the power consumed by the robot withthe number of cycles.

In this paper, a graphical technique is used to find the Pareto-optimal front giving the best tradeoff between power consump-tion and forward velocity [40]. To perform the optimal search,the input parameters are varied incrementally in fine steps span-ning the entire parameter domain, e.g., α is varied from 3

(=αmin ) to 90 (=αmax ) in the steps of 3, β is varied from6 (=βmin ) to 180 (=βmax ) in the steps of 6, and ω is var-ied from 0.5 rad/s (=ωmin ) to 5 rad/s (=ωmax ) in the steps of0.5 rad/s. At each parameter triplet (α, β, ω) setting, the snakelocomotion is simulated for eight cycles. The first four cycles aresimulated solely to allow the robot to reach steady state, a dura-tion confirmed by repeated analysis for many different gaits. Asan example, a variation of forward velocity of the robot and theinput power with number of cycles is shown in Fig. 4. The sim-ulation takes approximately two to three cycles to reach steadystate, and the dotted boundary at four cycles indicates the pointafter which steady state is assumed. The remaining four cyclesare used to determine the per cycle average velocity and powerof the snake robot.

After the simulated locomotion has reached steady state, thepower and velocities are recorded and the power is plotted ver-sus the forward speed. In the current settings, it requires 9000function evaluations with approximately total run-time of 25 h.Once all possible parameter combinations are plotted, the re-sulting plot reveals a smooth curve enclosing all the points ofoperation (e.g., Fig. 5). The bottom boundary gives the desiredPareto-optimal front, and the values of α, β, ω that yield pointsclosest to this boundary are hereafter considered the Pareto-optimal gait parameters for that forward speed. It is noted thatfinding the Pareto front using classical methods is a difficultproblem. Other Pareto-optimal methods include weighting fac-tors to describe the Pareto front, or the use of evolutionary algo-rithms [40], [41]. In this paper, we intended to observe the trend


Fig. 5. Distribution of power input with the forward speed and Pareto-optimalfront for the case of viscous friction model with distributed contact. It showsthe maximum speed of around 6 m/s and corresponding power consumption onPareto curve is around 200 J/s.

over entire design space, and therefore, required the completePareto front.

V. VISCOUS FRICTION MODEL WITH DISTRIBUTED CONTACT

To confirm the validity of the model formulation as well asthe validity of the Pareto-optimal search method, the simulation-based analysis of Saito et al. [31] was repeated using the dy-namic equations of Section III. In their work, the frictionalcoefficients (cn and ct) were 10 and 0.1, respectively, the mass(m) and half-length (l) of each link of the robot were taken asunity, and a viscous-only friction model was assumed along adistributed line of contact. Under these assumptions, cn0 = 0and ct0 = 0 in (26), and (11) is used to account for distributedfrictional forces.

The results are shown in Figs. 5, 6, and 7 that, respectively,show the power versus velocity, the Pareto-optimal ω parametersversus α and β, and the relationship between the Pareto-optimalω versus average steady-state forward velocity. These plots in-dicate that optimum values for α increases slightly with ω from15 to 25 and the optimum value for β is constant, around 60.The forward speed varies linearly with the angular speed. Theerror bars are due to the discretization in α and β in the searchspace.

All observations match exactly with results reported by Saitoet al. [31], thereby providing an external validation of boththe simulation of snake dynamics as well as the method of thegraphical optimization procedure. Hereafter, we utilize the samemethod to analyze how the Pareto-optimal gait values changedue to changes in the system model, specifically due to changesin the friction model assumptions [e.g., (11) and (21)].

VI. VISCOUS FRICTION WITH WHEELED CONTACT

The most obvious difference between the previous formula-tion and most snake robot implementations is that most snakerobots are implemented in practice using wheels rather than adistributed underbelly contact; examples include most imple-

Fig. 6. Optimal values of α and β along the Pareto front plotted against ωfor the case of viscous friction model with distributed contact. Within the errorbounds of the measurements, the optimal values of α are increasing linearlyfrom 15 to 25 with ω. The optimal values of β are almost constant around60 for different ω.

Fig. 7. Plot showing the linear variation of the forward speed with ω calculatedalong the Pareto curve for the case of viscous friction model with distributedcontact.

mentations cited in this paper and others such as [27], [37].Wheels help to minimize friction in the forward direction, andon snake robots, are intended to reduce the joint torque andlocomotion power required. Later analysis in this paper, resultsshown in Fig. 8, reveals that wheeled contact require signifi-cantly less locomotion power than the distributed contact case.

In the case of wheeled contact, the frictional torque needed torotate an individual link about the link’s wheel is quite small, sosmall that it can be neglected in the simulation. The governingdifferential equation is therefore given by (21). Because of thisassumption, the optimal gait from the distributed contact modelis not optimal in the wheeled contact case, and therefore, isreinvestigated. To allow comparison with the previous results,the mass, length, and friction parameters were made the sameas in the distributed contact case for all simulations.

Using the same method of parameter search as discussedearlier, the optimal gait was again investigated with the new


Fig. 8. Pareto-optimal front for the case of viscous friction model with wheeledcontact and for the case of viscous friction with distributed contact. It showsthe maximum speed of around 6 m/s and corresponding power consumption onPareto curve is around 160 J/s less than 200 J/s required for the other case.

point-contact friction assumptions. For comparison purposes,only the Pareto curves are shown in Fig. 8. This plot indicatesthat the power required for the same forward velocity with awheeled, point contact model is less than the distributed contactcase. The optimal gait results are presented in Figs. 9 and 10.The results indicate that the optimal values of α remain con-stant at 27 and β at around 78 for the wheeled contact case,as compared to 22 and 60 reported earlier for the distributedcontact case. Thus, the optimal gait determined earlier for dis-tributed contact geometries is indeed different than the optimalgait determined for the case of the wheeled or point contact.The variation of forward speed with angular speed still followsa straight line trend with slope of 0.88 rad/m, which is only6% higher than the slope of 0.83 rad/m reported for distributedcontact. In other words, only a slightly higher frequency of ac-tuation is required for a wheeled snake to achieve the sameforward velocity as a nonwheeled snake with a substantial re-duction in required power. These results agree with expectationsbecause of the elimination of one power dissipating term (fric-tional torque) in the governing equations. Thus, the introductionof wheels for locomotion is clearly beneficial on a power basis.

VII. EXPERIMENTAL FRICTION MEASUREMENTS

Much of the snake robot literature uses either a kinematicmodel (no lateral sliding) or a viscous friction model to describethe snake–ground interaction. However, in the same literature,one can find little if any experimental justification for eitherassumption. This section presents results from experiments ona snake robot that illustrate the friction characteristics expectedin practical operation. These measurements are then fit with afriction model in order to better predict locomotion performanceof an actual snake robot.

The challenges in measuring snake robot surface friction aremany. It is generally difficult in practice to maintain a perfectlyconstant friction surface. Further, it can also be difficult to ob-tain consistent force or velocity measurements from a moving,

Fig. 9. Optimal values of α and β along the Pareto front plotted against ω forthe case of viscous friction model with wheeled contact. The optimal values ofα are remaining constant at around 27 for different ω.The optimal values of βare now constant around 78 for different ω.

Fig. 10. Linear variation of the forward speed with ω calculated along thePareto curve for the case of viscous friction model with wheeled contact.

slithering snake. To overcome such challenges, all experimentalmeasurements of the snake robot in this paper were obtainedwhile the snake slithered on the Pennsylvania State UniversityRolling Roadway Simulator (PURRS), a very large [1.52 m(5 ft) wide by 3.05 m (10 ft) long] seamless treadmill (an ap-paratus consisting of a smoothly moving rubber belt around aperiphery of horizontal cylinders) custom built for vehicle androbot dynamic studies (see Fig. 11). Not only does this testingmethod maintain constant surface characteristics, it also pro-vides a platform with controllable geometry, speed, and surfaceby which the forces produced by a moving snake robot are easilymeasured.

To measure the average tangential friction at each wheel,i.e., the friction in the direction of wheel rotation, the jointangles (φi’s) of all links were fixed at zero position therebyproducing a straight line snake configuration. The inclination ofthe treadmill (γ) was increased very gradually until the robotstarted moving forward due to gravity. The inclination was thenfixed, and the velocity of the treadmill (vt) was increased so that


Fig. 11. PURRS, a controlled large-area custom treadmill surface where thefriction experiments were performed.

Fig. 12. Tangential friction characteristics.

the frictional forces between the wheel and treadmill balancedthe gravitational force. Using force equilibrium, the tangentialforce was then calculated from

Ft = mg sin γ (30)

where g is the acceleration due to gravity. This procedure wasrepeated for γ from 2 (the inclination at which the robot juststarts to move for the first time) to 3 in steps of 0.2.

The resulting tangential friction measurements are shown inFig. 12. It can be seen that the frictional force varies linearlywith the velocity in agreement with the viscous model assumedearlier, but unlike that model, the friction curve clearly has anonzero offset characteristic of coulomb friction. For the exper-imental snake of this study, these combined viscous and coulombcharacteristics can be modeled mathematically as

Ft = 0.83vt + 0.49 (31)

where vt is expressed in meters per second while Ft is measuredin newtons.

The normal friction measurements are also readily obtainedonce the tangential friction characteristics are identified. First,

Fig. 13. Configuration of the robot for normal friction measurements.

Fig. 14. Normal friction characteristics.

the joint angles are kept at some nonzero value λ, in an alter-nating configuration as shown in Fig. 13. Like the procedure fortangential friction measurements, the slope of the treadmill wasincreased until gravitational forces began to move the snakedown the treadmill. The inclination angle was then fixed andthe treadmill velocity was then increased until the snake was instatic equilibrium. The normal friction was then solved using aforce equilibrium equation in the direction of treadmill motion(v), given by

Ft cos λ + Fn sin λ = mg sin γ. (32)

Here, Fn is the frictional force in the normal direction and Ft

is the tangential force. The tangential force is the same as pre-viously measured, and can be calculated using (31) with thetangential velocity (vt = v cos λ). The force Fn is then deter-mined as follows:

Fn =(mg sin λ − Ft cos λ)

sin γ. (33)

Using this method, the measurements of the normal fric-tion characteristics were recorded and plotted in Fig. 14 withvn = v sin λ. Two separate experiments were conducted at twodifferent λ values, one with λ = 7.5 and one with λ = 15. Bothshow similar results that reveal a linear dependence of force onvelocity with a large static offset, again representative of a com-bined viscous and coulomb friction model. These characteristicscan be described mathematically as

Fn = 51.9vn + 2.33 (34)

where vn is expressed in meters per second while Fn is innewtons.

This friction model clearly disagrees with the viscous fric-tion model used in previous sections. In the following section,


TABLE IPHYSICAL PROPERTIES OF THE EXPERIMENTAL SNAKE ROBOT

Fig. 15. Distribution of power input with the forward speed and Pareto-optimalfront for the case of the experimental snake robot.

the optimization analysis is repeated to determine the effect ofcoulomb friction on the Pareto-optimal gait parameters. Theseare compared with the case of purely viscous model.

VIII. OPTIMIZATION FOR EXPERIMENTAL SNAKE ROBOT

The underlying goal of this paper is to develop a mathematicalmodel capable of predicting locomotion and power efficiencyfor an actual snake robot system, and ultimately validate thispaper with an experimental snake robot. The physical proper-ties of this Penn State University robot snake are summarized inTable I. Note that the parameters are notably different than as-sumed in previous sections. With these parameters and with thefriction model as in (25), the Pareto-optimal parameter searchwas repeated. The resulting plots of Pin versus vx , plots showingthe optimal values of α, β, and plots of vx versus ω are shownin Figs. 15, 16, and 17, respectively.

These results indicate that the optimal values of α decreasesfrom 70 to 40 with increasing ω, and β is relatively constantaround 80. The graph of forward speed with angular speedis still linear having slope 7.88 rad/m and a constant offset of0.97 rad/s.

To compare these results with the viscous-only model, theoptimization is repeated assuming the coulomb friction com-ponents to be zero but all other parameters the same. Only thePareto curves comparing these cases are shown in Fig. 18. Forcomparison, the power consumption of a purely rolling robothaving same mass and friction properties (Coulomb + viscous)

Fig. 16. Optimal values of α and β along the Pareto front plotted against ω forthe case of experimental snake robot. The optimal values of α are decreasinglinearly from 70 to 40 with ω. The optimal values of β are now constantaround 80 for different ω.

Fig. 17. Variation of the forward speed with ω calculated along the Paretocurve for the case of experimental snake robot including a least squares linearcurve fit.

as the snake robot is also plotted. This power is calculated nextusing the tangential friction relationship from (31):

Pin,roll =∫ vx

0Ft(v)dv = 0.42v2

x + 0.49vx. (35)

Optimal gait results, neglecting the Coulomb friction componentof the physical snake robot, are presented in Figs. 19 and 20.By comparing the Coulomb and non-Coulomb cases, it is seenthat the power consumed by Coulomb friction is substantial (seeFig. 18), the amplitude of the joint motion α is greater (compareFigs. 16 and 19), and the optimal frequency ω is lower (compareFigs. 17 and 20).

IX. EXPERIMENTAL CONFIRMATION OF GAIT POWER

AND VELOCITY

Assuming that robot acceleration/deceleration over one cycleis negligible (an assumption consistent with [31]), the average


Fig. 18. Simulated distribution of power input with the forward speed andPareto-optimal front for three cases: 1) the experimental snake robot, 2) the samerobot neglecting Coulomb friction, and 3) the same robot without serpentinemotion but wheeled contact with Coulomb friction (a rolling-only case).

Fig. 19. Optimal values of α and β along the Pareto front plotted against ω forthe case of experimental snake robot neglecting Coulomb friction. The optimalvalues of α are almost constant at 30 for different ω. The optimal values of βare constant around 80 for different ω.

Fig. 20. Plot showing the variation of the forward speed with ω calculatedalong the Pareto curve for the case of experimental snake robot. The variationis linear in nature with the slope of the best-fit line passing through the origin is9.30.

power utilized can be obtained from the dot product of the inputtorque [u as in (11) and (21)] and the corresponding joint angularvelocities ([φi ]), a technique also employed in simulation-basedstudies [21].

However, in practice, it is difficult to measure the input powerat the joints either mechanically or electrically. For example, inthis study, an attempt to measure the power input to the snakefor mechanical locomotion was conducted by measuring thevoltage and current demand of each joint servo. It was quicklyfound that the averaged standby current of the servo was roughlyan order of magnitude larger than the average torque demand ofthe servo due to mechanical motion. As a result, the signal-to-noise ratio of this measurement approach was not sufficient tomeasure input power of locomotion from the electrical powerdemands alone.

As an alternative to measure input power, one can measuredissipated power because at steady state, the power input is equalto the power dissipated by frictional forces. Specifically, thepower dissipated is the dot product of frictional forces [see (25)]and the link contact velocity relative to ground ([xi yi ]′). If theground-relative link velocities are measured, and if the frictionalprofile of the snake–surface interaction is well characterized,then the dissipated power can be calculated. These calculationsrequire measurement of the absolute angle (θi) and rotationalvelocity of at least one link, as well as all the interlink relativejoint angles (φi) and velocities.

To measure the robot interlink relative joint angles, rotary sen-sors (optical encoders, 8192 counts per revolution in quadraturemode) were mounted on the snake robot. These sensors are onlycapable of measuring relative angles between links, and there-fore, only relative velocities of motion of each link are known.However, the snake at steady state operates over a long period oftime in only forward velocity such that the lateral velocity of thecenter of gravity is zero on average. As mentioned in earlier sec-tions, experimentally measured joint trajectories closely matchsimulation because these trajectories are maintained by a closed-loop controller. Experimentally, it was also seen that the mea-sured net forward velocity also matched simulation-predictedvalues. Hence, in order to compare average power required forlocomotion between simulations and experiments, first theremust be an agreement between the predicted velocity and gaitgeometry. For this reason, experimental investigation here-after focuses primarily on determining whether the measuredvelocity matches simulation predictions, and whether experi-mentally measured interjoint gait profiles match commandedvalues.

To compare simulation-predicted and experimental veloci-ties, the snake robot is driven on the treadmill surface in alevel configuration. The gait profile is driven by the optimalgait parameters at nine different gait settings that are on thePareto-optimal boundary. The results are presented in Table II.We define an error

verror(%) =vexpt − vpred

vpred× 100. (36)

To characterize how closely the reference joint trajectories weremaintained by the experimental robot, the rms of the error is


TABLE IIGAIT VELOCITY RESULTS FROM THE EXPERIMENTS

Fig. 21. Plot of the experimentally measured speed against the predictedspeed.

calculated. To make the TE independent of amplitude (large am-plitudes cause large TEs), a dimensionless rms value is obtainedby dividing the rms measure by the amplitude commanded gaitmotion (α). Mathematically, this is defined as

TE =Gait rms errorGait amplitude

=

√∑Nk (φk,meas. − φk,ref )

2

Nα(37)

where N is total number of measurements, φk,meas. and φk,refare the measured joint angle and the reference joint angle atan instance k. The table presents this dimensionless rms TE.

Additionally, Fig. 21 shows a plot of predicted speed fromsimulations versus experimentally measured speed. The resultsdemonstrate that at lower speeds, the simulation predictions areclose to the experimental speed. However, discrepancies ariseat higher speeds. The same trend is observed for the TE forthe gait profile. A plot showing the error between the measuredjoint angles and commanded gait profile of joint 4 for ω = 3,4, and 5 rad/s is shown in Fig. 22. The figure demonstrates thatthe servos used in the experimental snake (Hobbico Super TorqCS-70 Metal-Geared Ball Bearing Servo) follow the reference

Fig. 22. Reference gait profile and the gait profile executed by the servos forjoint 3. This figure shows that for ω = 3 rad/s, the reference and actual gaitprofile are very close. But for ω = 4 and 5 rad/s, the reference tracking becomesincreasingly poor. This causes the speed of the robot to be substantially lowerthan expected.

trajectory for ω = 3 rad/s quite well, but failed to closely followthe reference gait profile for ω = 4 and 5 rad/s.

It is hypothesized that the speed of the robot differs drasti-cally from simulation at high speeds because the servos wereunable to follow the joint motion profile for these fast gaits.In practice, it was found that when the actuators had to movequickly, the actual profile of the joint motion was barely sinu-soidal with an amplitude and phase significantly different thanthe desired original reference. To confirm the possible validityof this hypothesis, the measured experimental joint angles wereleast squares fitted to a sinusoid, and the simulation of the snakerobot was repeated except using these degraded measured sinewaves as the commanded trajectory. The simulation predicted aforward speed of 0.28 m/s for ω = 4 rad/s (the measured valuewas 0.24 m/s) and 0.17 m/s for ω = 5 (the measured valuewas 0.18 m/s). Note that these values are much more accuratethan that predicted in the perfect actuation case. In other words,the actuator dynamics and limitations indeed have a very largeimpact on the robot speed.

If the simulation is an accurate representation of the snake lo-comotion, it should accurately predict locomotion behavior notonly at the optimal gait, but also at nonoptimal conditions. Totest this, the simulation and experimental results were comparedat conditions near, but not at, the optimal point. Table III showsthe predicted and experimental speed (both in meters per sec-ond) and % error calculated for different values of α and β nearthe optimum point (66 and 80, respectively) and a fixed ω =2 rad/s. The center point corresponds to the optimal point asreported in the previous section. The table shows that the sametrends are observed in experiments as predicted by the simula-tion, and that the errors between the experimental results andsimulations are all within similar ranges.


TABLE III% ERROR IN THE GAIT SPEED NEAR AN OPTIMAL POINT

TABLE IVINVESTIGATION OF HYPERREDUNDANCY IN A SNAKE ROBOT (α = 66,

β = 80, AND ω = 2 RAD/S)

X. HYPERREDUNDANCY

One often cited advantage of snake robots is their inherenthyperredundancy [10], e.g., having more links and actuatorsthan is necessary for locomotion. The presumed advantage ofredundancy is that, if some of the links are cut or some of thejoint actuators fail, the mechanism can still perform its desiredoperations albeit at a degraded level. This section investigatesthe validity of this claimed property for the current snake robotmodel, and additionally tests the simulation’s ability to matchexperimental results in these extreme conditions.

Although a failure can occur in different ways, we investigatetwo different specific scenarios. In one case, a joint becomes‘free,’ i.e., it is unable to produce any torque. An example ofthis case would be a broken motor shaft or stripped gear drive.The other case is considered when a joint becomes ‘rigid’ as ina bound gear or locked joint bearing. The impact of these fail-ures are investigated for different joints and joint combinations.Results from simulations and experiments are summarized inTable IV. The power input is calculated through the simulationonly.

The results show a very good qualitative and quantitativematch between simulations and experiments, and show interest-ing trends in the predicted forward speed and power demand fordifferent failure conditions. Specifically, the following observa-tions are noted.

1) For the free failure cases, the net forward speed decreasesif this type of failure occurs anywhere in the robot.

2) For the rigid failure in the head or tail link, or failuresrepresented by the 1 + 2 and 4 + 5 cases, the net speedincreases. This increase in velocity after failure is offset bya tremendous increase in the required locomotion power,but even so such bursts of velocity may be useful in afight-or-flight situation. It is also interesting that similarpotential advantages of rigid tail links are noted elsewherein the literature for climbing situations [42] suggesting thatthe study of snake robots with nonuniform link segmentsmight be fruitful in other respects.

3) If more than two links fail, the speed of a six-link robot isnegligible for all practical purposes.

4) If there is any type of failure at the belly of the robot, forexample, the 2, 3, 4, 2 + 3, and 3 + 4 cases, the speeddecreases rapidly. For nearly all belly localized failures,the power intake remains comparable to cases of similarfailures at the head or tail.

The last item is particularly interesting. If the power input isapproximately the same between two different cases, it impliesthat the actuators are reacting to similar external forces betweenthe two cases. However, with certain failure cases, the totaleffort expended by the snake is essentially nullified. This hasobvious implications for distributed joint control techniques orfault-tolerant locomotion. In other words, a snake robot mayflounder wildly even though local path tracking and measuredforces at all but a few of the joint controllers are correct. Thisimplies a need for centralized coordination of the links for caseswhere joint failures could occur.

This analysis also gives some important insight into the basichyperredundancy of snake robots in real-life applications. First,the robot can still move with a reasonable speed even afterloosing some of its link mobility. Secondly, there are somefailure modes where, in fact, the snake may be able to movefaster than the nominal case. For example, if some joints arerigidly locked, the locomotion speed increases. This presents anovel way to achieve sudden burst of speed for situations wherepower is of secondary concern.

XI. CONCLUSION

It is well known that the snake robots are more useful indifficult applications such as pipes, rough terrain, etc., and thatthis is likely the directions in which snake robots might beused in practice. However, the subcase studied in this paper,serpentine gait on a planar surface, is not trivial and is thesituation most studied in the snake robot literature. There is asimple reason for such interest: without analytical understandingbacked by experimental verification for even this most simplecase, a first-principles analysis of more complex situations couldlikely be incorrect.


Using simulations, this study analyzed the effect of variousfriction models on the power-optimal serpentine gait of a six-linksnake robot. Three different types of friction models were an-alyzed: 1) a distributed contact model based on viscous forces,2) a point-contact model based on viscous forces, and 3) a point-contact viscous and Coulomb model. It is shown that the gaitparameters for serpentine motion are very dependant on the fric-tional model if the minimum power expenditure is desired for agiven velocity of locomotion.

Experimental investigations were then conducted to con-firm the assumptions of the dynamic simulation, and thesetests motivated the use of a combined viscous and Coulombsurface interaction model. This was an unexpected result asthis model has not been previously used in the snake locomo-tion literature. Using this new friction description, simulationresults were compared to experiments for nominal and non-nominal locomotion cases including suboptimal gaits and ac-tuator faults. It was seen that this modified simulation modelclosely predicted locomotion velocities and link profiles, butthat these predictions degraded significantly at speeds whereactuator dynamics were significant. Even from a linear approx-imation of the friction model, the results show that the opti-mum gait parameters are different. Optimal gait analysis fordifferent, more practical friction models can be a part of futureinvestigation.

Finally, a hyperredundancy investigation was conductedwhere actuation faults were intentionally introduced. This studyrevealed close agreement between experimentally measured andsimulation predicted locomotion despite many different types ofactuator failures. Additionally, certain types of actuator failureprovided faster locomotion speeds than the nominal case at acost of significantly higher power demand. It was also seenthat actuator failures near the belly of the snake robot severelydegraded the snake’s locomotion capability.

This study on snake robots, while specifically focused onsimple linkages, does provide insight into the validity of manyassumptions that pervade robotics in general. Some of these gen-eral assumptions include: that rolling contact takes place withoutlateral slip, that hyperredundancy generally follows from havingmore actuators than degrees of freedom, and that joint-to-jointfriction effects and actuator limitations can be ignored. Thispaper shows some violation of each of these assumptions, andhence, suggests future avenues of study on other robot systemsmaking similar assumptions.

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Vipul Mehta received the B.Tech. and M.Tech.degrees in mechanical engineering from the IndianInstitute of Technology-Bombay, Mumbai, India, in2005.

He is currently a Graduate Research Assistantin the Mechanical and Nuclear Engineering Depart-ment, Pennsylvania State University, University Park.His current research interests include design opti-mization, control systems, robotics, and automation.

Mr. Mehta was the recipient of the Institute SilverMedal for academically best student in dual degree,mechanical engineering in 2005.

Sean Brennan (S’99-M’03) received two B.S. degreein mechanical engineering and B.S. degree in physicsfrom New Mexico State University, Las Cruces, bothin 1997, and the M.S. and Ph.D. degrees in me-chanical engineering, from the University of Illinois,Urbana-Champaign, in 1999 and 2002, respectively.He joined the Department of Mechanical and Nu-clear Engineering at the Pennsylvania State Univer-sity, University Park, in 2003, and was jointly ap-pointed to the Pennsylvania Transportation Institute,where he is currently an Assistant Professor. His mo-

bile robots are often used for educational outreach. His current research interestsinclude modeling and control of ground vehicles and platforms, particularlytheir interaction with terrain using an approach best characterized as a synergis-tic combination of theory and implementation.

Dr. Brennan was recently a Guest Editor for the IEEE Control SystemsTechnology in 2007. He is the secretary for the International Forum on RoadTransport Technology, and is currently the Vice Chair for the American Soci-ety of Mechanical Engineers (ASME) Automotive and Transportation SystemsTechnical Committee. He is the recipient of the Department’s 2002 OutstandingGraduate Research Award for research unifying the concepts of scalability andcontrol theory, the 2007 Outstanding Teaching Award, the highest recognitionfor tenure-track faculty at Penn State, and the Penn State’s 2006 Quality Im-provement Award for his activity using robotics to teach control systems to K-12students.

Farhan Gandhi received the Ph.D. degree inaerospace engineering from the University ofMaryland, College Park, in 1995.

He is currently a Professor of aerospace engineer-ing and the Deputy Director of the Penn State’s Verti-cal Lift Research Center of Excellence, PennsylvaniaState University, University Park. He is an AssociateEditor of the Journal of Intelligent Material Sys-tems and Structures. His external research sponsorsinclude the National Rotorcraft Technology Center(NRTC), the US Army Research Office, the National

Aeronautics and Space Administration (NASA) Langley Research Center, theUnited Technologies Research Center, Sikorsky Aircraft, Bell Helicopters Tex-tron, Piasecki Aircraft, the National Science Foundation (NSF), the DefenseAdvanced Research Projects Agency (DARPA), the Air Force Office of Scien-tific Research (AFOSR), the Office of Naval Research (ONR), the Center forRotorcraft Innovation (CRI), and NextGen Aeronautics. His current researchinterests include rotary-wing aircraft dynamics, aeroelasticity and configurationdesign, adaptive structures, and morphing aircraft structures.

Prof. Gandhi is the recipient of the US Army Young Investigator Award(1997), the American Helicopter Society’s Francios Xavier Bagnoud Award(1998), and the Popular Mechanics Breakthrough Award (2007).

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