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Theoretical and experimental study on a compliant flipper-leg during terrestrial locomotion

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2016 Bioinspir. Biomim. 11 056005

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Bioinspir. Biomim. 11 (2016) 056005 doi:10.1088/1748-3190/11/5/056005

PAPER

Theoretical and experimental study on a compliant flipper-leg duringterrestrial locomotion

TaoFang1, YouchengZhou1, ShikunLi2,MinXu1, Haiyi Liang2,Weihua Li3 and ShiwuZhang1

1 Department of PrecisionMachinery and Precision Instrumentation, University of Science andTechnology of China,Hefei, Anhui,230026, People’s Republic of China

2 Department ofModernMechanics, University of Science andTechnology of China,Hefei, Anhui, 230026, People’s Republic of China3 School ofMechanical,Materials andMechatronic Engineering, University ofWollongong,Wollongong, NSW2522, Australia

E-mail: swzhang@ustc.edu.cn

Keywords: compliant flipper-leg, amphibious robot,mechanical analysis, single-leg platform, four-leg-robot platform

AbstractAn amphibious robot with straight compliant flipper-legs can conquer various amphibiousenvironments. The robot can rotate itsflipper-legs and utilize their large deflection towalk on roughterrain, and it can oscillate the straight flipper-legs to propel itself underwater. This paper focuses onthe dynamics of the compliant straight flipper-legs during terrestrial locomotion bymodeling itsdeformation dynamically with large deflection theory and simulating it to investigate the parametersof locomotion such as trajectory, velocity, and propulsion. To validate the theoreticalmodel ofdynamic locomotion, a single-leg experimental platform is used to explore theflipper-legs inmotionwith various structural and kinematic parameters. Furthermore, a robotic platformmountingwithfour compliant flipper-legs is also developed and used to experiment with locomotion. Thetrajectories of the rotating axle of the compliant flipper-leg during locomotionwere approximatelycoincidental in simulation and in experiments. The speed of locomotion and cost of transport duringlocomotionwere explored and analyzed. The performance of different types of compliant flipper-legsduring locomotion shows that varying the degrees of stiffness will have a significant effect on theirlocomotion. The dynamicmodel and analysis of the compliant flipper-leg for terrestrial locomotionfacilitates the ability of amphibious robots to conquer complex environments.

1. Introduction

Amphibious robots perform important roles in manycivilian and military applications such as navigationshores, clearing mines and mapping terrain [1].However, different types of amphibious robots havebeen developed for different roles, such as salaman-der-like robot [2, 3], ACM series robots [4], and turtle-like robots [5]. To improve their performance interrestrial and underwater tasks, amphibious robotsneed to be stable and able to adapt to various terrainsand water environments. Nowadays, researchers areproposing approaches that will enable robots tosmoothly and efficiently switch between differentoperational modes on terrestrial and aquatic environ-ments. We have developed an actively transformableflipper-leg capable of swimming under water andwalking on terrain, and an amphibious robot—AmphiHex-I where the mechanism is a propulsion

unit that has been assembled and implemented [6–8].AmphiHex-I has six flipper-legs with embedded steelplates and cables that can transform the flipper-legsbetween straight flippers and curved legs by looseningor tightening the cables. However, this active transfor-mation needs extra motors to move the cables, whichincreases the complexity of the control strategy anddriving modules. Hence, a simpler mechanism oftransformable propulsion for the amphibious robot isneeded forfield applications.

Many other robots, a s well as AmphiHex-I use softstructures for the propulsion unit and the connectoror body, such as the arms of an octopus-like robot orcompliant joints in artificial fingers, softworms [9–11]. Inspired by these soft structures, we developed acompliant flipper-leg that can act as the propulsionunit. This compliant flipper-leg is designed as astraight platemade from an elasticmaterial that can betransformed passively [12]. This flipper-leg can propel

RECEIVED

13April 2016

REVISED

16 June 2016

ACCEPTED FOR PUBLICATION

20 July 2016

PUBLISHED

17August 2016

© 2016 IOPPublishing Ltd

the robot underwater like turtle’s flippers, and alsobend to a curved shape for terrestrial locomotionwhen normal and tangential forces exerted by differ-ent terrains applied. Thus, this compliant flipper-leghas a similar function as AmphiHex-I, and thus has asimpler structure, a simplified driving module, and aneasier control strategy.

This flipper-leg has a two-fold compliance: (1) itcan be used as a flipper and oscillate to propel under-water, and (2) it can bend into a curved leg duringlocomotion on different terrain. Compliant flippersmounted on robots as propulsion units to mimic ani-mal flippers has been applied and verified in manyunderwater robots [13–16]. However, the ability ofstraight compliant flipper-legs to propel an amphi-bious robot on different terrain still needs fur-ther work.

Exploring the interaction between the propulsionunit and the environment help understand thedynamics of the robot and guide the design and thecontrol of the field robot [17–20]. The dynamics of acompliant flipper-leg must be examined in order tounderstand their propulsive ability on different ter-rains. However, modeling the dynamics of a com-pliant flipper-leg means simplifying its deformationand the complex interaction between the flipper-legand terrain. This deformation is difficult to model dueto its nonlinearity and multi- dimensionality. Ariberthas contributed a great deal to the analysis and appli-cation of compliant legs [21] by focusing on compliantlegs thatmainly consist of rigid and soft parts that havetheoretically proved to be stable, controllable, and effi-cient due to compliance [22]. Since a compliant legmade from soft material, as a pseudo rigid bodymodel(PRBM), has proved a successful way of modeling acompliant beamwith large deflections [23–25], PRBMregards a compliant beam as a combination of tworigid bars and a torque spring set at a suitable position,while neglecting any nonlinearity of the material inlarge deformations. As with PRBM, PRBM 3R dividesa compliant beam into three rotating springs and fourbars that results in a more accurate but more complexmodel than PRBM [26]. A rolling spring loaded inver-ted pendulum is suitable for curved-leg, like that ofRHex, because it divides the leg into a straight rigidbar, a curved rigid bar, and a rotating spring [27, 28].However, it is hard to derive an explicit solutionbecause these theoretical models are too complex, sofinite element analysis is a good choice to covermechanical problems that include a credible non-linearity and the complex interaction of parts witheach other [29, 30].

In this paper we aim to explore the feasibility andperformance of a compliant flipper-leg during loco-motion on terrain by theoretical modeling andexperimental validation. We used large deflectionequations once used to verify PRBM, to model thecompliant leg, and set up a finite element model tosimulate the entire motion of the leg to obtain more

parameters. With these methods, we can exploremovements such as forward speed, fluctuations inheight, propulsion efficiency, and the design of thecompliant legs. The theoretical results were validatedby experiments with moving platform, and movingrobots. This study facilitates implementing amphi-bious robots to conquer various complexenvironments.

The reminder of the paper is organized as follows.Section 2 introduces the leg locomotion platform, thetheoretical model of the compliant flipper-leg, and therobot locomotion platform. Section 3 presents theperformance of propulsion from simulation andexperiment, the special design of the compliant legs,and section 4 presents the conclusion.

2. Experimental system andmechanicalmodel

In order to explore the feasibility and performance ofstraight compliant leg moving over terrain, we devel-oped a single-leg locomotion platform, a theoreticalmodel of the compliant leg, and a robot locomotionplatform to conduct a theoretical and experimentalanalysis.

2.1. Single-leg locomotion platformA single-leg locomotion platform was developed toexplore the movement of compliant flipper-legs.Figure 1(a) shows a simplified draft of the locomotionplatform; it consists of a horizontal cylindrical slide,two vertical cylindrical slides, a vertical displacementsensor, and a locomotion unit. The cylindrical slidesprovide horizontal and vertical translational freedomsfor the locomotion unit; the propulsion unit alsoincludes a driving unit and a compliant flipper-leg.The leg is made from polyurethane, which is widelyused as a super elastic material. The driving unitcontains a MAXON motor to drive the flipper-leg, agear box, and a torque sensor, and the coordinate is setat the center of the flipper-leg’s rotation. Figure 1(b) isa series of photographs of the leg during terrestrialmotion. The red pointer at the front of the drive unit isused tomeasure the angle of rotation of the flipper-leg.The MAXONmotor is controlled to vary the speed ofrotation in a clockwise direction, and a weight is usedto adjust the payload of the driving unit. The loadapplied vertically to the compliant leg is 2 kg, and12.36 kg applied horizontally. The vertical load meansthe weight of a driving unit capable of moving in avertical direction, while the horizontal load includestwo vertical slide ways, two horizontal slide ways, andthe driving unit and sensors that can move in ahorizontal direction. The leg is 310 mm long x 55 mmwide x 15 mm deep; the leg weighs 76.7 g and Young’smodule is 25 Mpa.

The process by which the compliant leg is pro-pelled on the locomotion platform is shown in

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Bioinspir. Biomim. 11 (2016) 056005 TFang et al

figure 1(b). Here the leg is straight (see figures 1(b)(1)),but when the driving unit starts the leg bends into acurved shape under the driving load (see figure 1(b)(2)), while the driving unit is pushed forward by theflipper-leg. When the flipper-leg leaves the ground thedriving unit drops down freely (see figures 1(b)(3) and(b)(4)). During locomotion, the flipper-leg bends con-tinuously, an action that determines howwell the pro-pulsive unit canmove itself along the platform.

2.2.Model of compliantflipper-legIn PRBM or similar models, transforming part of theleg to be analyzed is simplified into bars of fixedlengths with torsion springs, but the length of that partof the flipper-leg that contacts the ground varies,which means the length of the bent part also varies;this means that modeling the leg as two bars with fixedlengths and rotation spring in a fixed position is verydifficult, so the transforming part of the leg is regarded

Figure 1. Single-leg locomotion platform. (a)Draft of the locomotion platform. (b) Snapshots of the compliant leg during one cycle inlocomotion.

Figure 2.Diagram of large deflection cantilever’smechanicalmodel.

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Bioinspir. Biomim. 11 (2016) 056005 TFang et al

as a cantilever capable of large deflection, as shown infigure 2.

Thus, we can model the deflection of the leg withan implicit method that can be incorporated into threelarge deflection equations as shown below, which arederived from elastic theories and used to validatepseud rigid bodymodel [23, 24]

[ ( ) ( )]( )

òq

q q

=

´+ F - + F +

ql F EI

M EIF

21

2d

cos cos 2,

1

o

2 00

02

0

0

( )[ ( ) ( )]

( )

òq q

q q

=

´+ F - + F +

qa

F EI

M EIF

1

2 2

cos d

cos cos 2,

2

o

0 0

02

0

0

( )[ ( ) ( )]

( )

òq q

q q

=

´+ F - + F +

qb

F EI

M EIF

1

2 2

sin d

cos cos 2,

3

o

0 0

02

0

0

where a and b denote the x-direction and y-directiondisplacements. θ0 denotes the angle of tangentialvector at the end of the flipper-leg. F0 and M0 denotethe force and moment applied to the rotating tip,

respectively. j is the direction of F0. l2 denotes thelength of the bent compliant flipper-leg. E is Young’smodulus of the leg and I is themoment of inertia of theleg’s cross section. Here we classified six independentvariables into two groups, where the variables in thefirst group are a, b, and θ0, which stand for the postureand position of the end of the compliant leg; thevariables in the second group include F0, M0 and j,which stand for the load applied to the compliant leg.In these equations there are six independent variables,so more equations are needed as boundary conditionsto obtain a solution, albeit the three equations could besolved with boundary conditions by numericalmethod because they are implicit.

2.3.Model of single-leg locomotion platformAsmentioned above, since the number of the indepen-dent variables is six, more equations are needed toobtain a solution for leg deformation. Here weconsidered the locomotion platform of a singlecompliant flipper-leg to model the interactionsbetween the flipper-leg, the driving unit, and theground. To obtain the boundary conditions duringlocomotion the motion of the compliant flipper-legwas divided into three phases, as shown in figure 3; thebending phase, the lifting phase, and the flying phase.In the bending phase the compliant leg begins to bend,and the driving unit is always in contact with theground. In the lifting phase the driving unit begins toleave the ground while the surface beneath the leg is

Figure 3.Three phases of themotion during one cycle of locomotion.

Figure 4.Diagram of the systembefore the driving unit leaves the ground (bending phase).

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Bioinspir. Biomim. 11 (2016) 056005 TFang et al

still in contact with the ground. In the flying phase thedriving unit and the leg both leave the ground. Thismotion is similar but it is not strictly a free fallingmotion due to friction.

Figure 4 is a mechanical diagram of the locomo-tion platform before the driving unit leaves theground, that is, the bending phase. Here the drivingunit remains stationary until Fy is large enough to liftthe driving unit up and enter the lifting phase.

We considered that the leg bends slowly during thebending phase, so it can be regarded as a quasi-staticprocess. In the figure, F ,x F ,y and M are the x-axialload, the y-axial load, and the moment from motorapplied to the upper end of the leg, respectively. FA

and fA are the normal force and friction force betweenthe sliders and vertical slide ways. The coefficient offriction of FA and fA is m ,s andG stands for the gravityof the driving unit. Thus, the dynamical equations ofthe driving unit are as shownbelow:

( )+ = +F N G f2 , 4y m A

( )+ =F h f R M2 , 5A m

( )+ =f F F2 6m A x.

The compliant flipper-leg can be divided into twoparts, the bending part of the leg and the straight partof the leg part II in figure 4 so that three large deflec-tion equations can be applied. The straight part con-tacts the ground, while the bending part denotes thetransforming part of the leg. The friction force f and

normal force N between the leg and the ground arealmost equal to Fx and F ,y and the moments are bal-ance for the bending part are as listed below:

( ) ( )+ + =N a l fb M0.5 , 71

where l1 is the length of the straight part, and thecoefficient of friction between the leg and the round isdefined as md.

As analyzed before, the driving unit leaves theground when the y-axial component of F0 is largerthan the weight of the unit, and then the lifting phasebegins. In this phase the dynamical model of the driv-ing unit can bemodeled using the following equations:

( )- =F F mx2 , 8x A

( )- + - =f F mg my2 , 9A y

wherem is the weight of the payload, x and y are the xaxial and y axial accelerations, respectively. Thebending of the leg is still regarded as a quasi-staticprocess if the speed of rotation is low. The driving unitenters the flying phase when the length of the straightpart becomes zero.

These equations are difficult to be solved due tothe integral and second derivative. To solveequations (1)–(3) where the integral is involved, asearchingmethod was used to look for solution withingiven errors. As for equations (8) to (9) that containeda second derivative, we used an iterative algorithm forthis calculation.

Figure 5. Four-leg-robot locomotion platform.

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Bioinspir. Biomim. 11 (2016) 056005 TFang et al

2.4. Robot locomotion platformA platform for a robot with four legs has also beendeveloped to explore the performance of the compli-ant flipper-leg as a propulsion unit for a completerobot. This platform has four yellow compliant legsmade from polyurethane, as shown in figure 5. Theframe of this robot is 385×620×110 mm, and itweighs 11.3 kg. Yellow sponges are fixed to the bottomof the robot to absorb the shock when it collides withthe floor to protect the mechanical structures, circuitboards and motors. When the robot is stationary onthe floor, point C is 70 mm above the floor. Each leg isdriven by aMAXONmotor with an Elmo driving unit.The frame of the robot is assembled by 2020 alumi-num profiles. The compliant leg is designed to be asimple rectangular block 315 mm long×55 mmwide×15 mm thick. Compared to RHex [31, 32] andAQUA [33] who possess six legs, the robot platformhas only four flipper-legs with limited gaits. However,it is sufficient to obtain the locomotion performanceof the compliant flipper-leg during the robot walkingon terrains with four flipper-legs rotating synchro-nously. When the robot is turned on, the legs bend themotor drives the robot up and then propels it forward.In the experiment, the forward speeds, fluctuations inheight, and the cost of transport (COT)were recordedto evaluate how well the robot performed. COT is anon-dimensional value to evaluate the consumptionof energy as the animals propel themselves along, orthe robots are transported [34]. COT is also widelyused to study the cost of robot’s energy duringlocomotion [35, 36]. Here COT is defined as shownbelow:

( )=P

WVCOT . 11i

Pi is the input power provided by the motor, whichfinally transforms to mechanical energy of robot anddissipates to air as fuel caused by material’s viscosity.W is the gravity of the system, and V is the averageforward speed of the robot. COT is calculated for anentire stride, which means the legs experience motionin all three phases, in a period defined as T. Since theoutput power of the motors changes during locomo-tion, Pi is defined as the mean output power of themotors that we record at a discrete time in a strideperiodT:

( ) ( )åw=PT

M t t , 12T

i i

where M(t) means the output torque of the motor attime Dt t, and i means the interval of time betweentwo recorded torque values. By combiningequations (11) and (12), COT is finally formed asshownbelow:

( )( )

w=

å M t t

WVTCOT . 13T i

To calculate COT, we must obtain the robot’s for-ward speed, and the motor’s output torque and period

of stride. We recorded the leg bending process androbot movement with a camera, so its forward speedand period time can be calculated. We obtained themotor’s output torque by recording the current flow-ing through its coil, based on the relationship wherethe output torque is proportional to the current.Before commencing this experiment, a whiteboardwith standardized grids, called amask board, was erec-ted along a red line marked on the rubber floor. Wethen used a camera to photograph themask board andthen use it as background. We then remove the maskboard and allowed the robot to walk along the red line.This red line is to ensure that the mask board and oneside of robot’s leg would be in the same vertical plane,and thus eliminate any error caused by the visual angleof the camera. The camera also recorded the move-ment of the robot as a videowith 30 frames per second.After finishing the experiment, we extracted everyframe from the video, and overlaid them with thebackground photo. By reading the position of the out-put axle of the motor marked as point C in figure 5(a)on the mask board, we obtained the trajectories andlength of stride at while the motor was at differentspeeds of rotation. The period of stride T is calculatedfrom when the motor begins to rotate to when therobot falls down on the floor. Then its forward speedcould be calculated using the length of stride and per-iod of time. Every experiment was repeated three timesunder the same condition in order to obtain anyerrors.

3. Theoretical and experimental results

With the platforms and models now developed, wecan examine and explore the locomotion of thecompliant flipper-leg under various conditions.

3.1. Results of single-leg platformFigure 6 shows the trajectories of the compliantflipper-leg at 0.5 rad s−1 speed of rotation in thesimulations and experiments, respectively. Thestraight yellow bar represents the initial posture andposition of the compliant flipper-leg, and the blackdots represent the center of the motor’s output axle.The trajectory can be divided into part I, part II andpart III. The result shows that the two trajectoriescoincided well for part I and the simulated length ofstride was almost the same as in the experiment.However, the trajectory from the simulation washigher than the experiment for part II, while part IIIwas close to falling freely. Both trajectories actually fellquickly, but in the simulation the driving unit fellfaster than the other one. This inconsistency betweenthe two trajectories from the simulation and experi-ment occurred because (1)Polyurethane is not an ideallinear material because its elastic modulus decreaseswith a large strain. When the driving unit moves intopart II, the trajectory from experiment is lower than

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Bioinspir. Biomim. 11 (2016) 056005 TFang et al

that from simulation in which material nonlinearity isignored. (2) The inertial force of the legs was ignoredbecause the weight of the leg is small and we assumedthat the movement of legs would be quasi-static; (3)the coefficient between rubber material such as poly-urethane and wooden ground may not be a constantvalue, an average measured value was used for thecoefficient of friction. (4) Beside, friction and gapsbetween driving unit and vertical slide ways causecreep phenomenon in experiment, which leads to acreep phenomenon, further to a fluctuation and alower trajectory. However, in simulation the coeffi-cients of static friction and kinetic friction are set to bethe same. Therefore, the creep phenomenon does notoccur in simulation. Despite the small difference in thetrajectory between the simulation and the experiment,we concluded that the mechanical model can basicallyreflect the locomotion dynamics of the compliantflipper-leg.

Based on the mechanical analysis of a single-leglocomotion platform, the trajectories, forward speedVx and fluctuating height H of the driving unit werecalculated with respect to five different rotation speedsω ofmotor, as shown infigure 7.

Figure 7(a) shows the trajectories of the leg rota-tion axle during locomotion when the leg rotates atspeeds of 0.5 rad s−1, 1 rad s−1, 2 rad s−1, 3 rad s−1 and4 rad s−1, respectively. When the driving unit raiseditself from the floor, the trajectories at lower rotatingspeeds of 0.5, 1 and 2 rad s−1 were similar. The trajec-tories at 0.5, 1 and 2 rad s−1 increased slowly until theywere almost coincident at the first half part. Thesethree trajectories also had a similar length of stridebecause the trajectory at 3 rad s−1 rises quickly andwith a shorter length stride than those at lower rotat-ing speeds. However, the trajectory at a rotating speedof 4 rad s−1 appears to be different because it rises veryquickly and then drops down quickly; this was the

shortest stride length in the trajectories at all fivespeeds. Figures 7(b) and (c) presents the forwardspeed, the COT, and fluctuations in the height of thelocomotion platform, respectively. Here, the forwardspeed Vx increased from 0.5 to 3 rad s−1 and thendecreased at 4 rad s−1. The fluctuations in heightdecreased at 0.5 to 3 rad s−1 and then increased at4 rad s−1. The highest forward speed and lowest fluc-tuations in height are both at 3 rad s−1. The dynamicsof COT shows that it generally became larger at higherrotating speeds but experienced its lowest value at2 rad s−1. Moreover, COT changed slowly from 0.5 to3 rad s−1, and then went up quickly from 3 to4 rad s−1. So, if the motor is allowed to rotate faster, ahigher forward speed can be obtained without sacrifi-cing energy efficiency, and if the motor rotates veryquickly, forward speed decreases and the energy effi-ciency drops quickly.

It can be concluded from the above that themotions at 0.5, 1, 2 and 3 rad s−1 were regular, butwhenωwas 4 rad s−1, themotion was out of the order.The difference between trajectories at 4 rad s−1 andother rotating speeds is caused by the severe slippage.The theoretical analysis is based on assumption thatthe leg would not slip on ground for simplicity. How-ever, the slippage actually happens when the flipper-leg rotates fast. And the slippage leads to a smaller a(displacement of driving unit along x-axial) and a lar-ger l1 (length of the straight part of the leg), and furthera smaller component of F0 in x-direction. As a result,the driving unit moves forward with a lower speed anda shorter stride length. Besides, COT that stands forenergy efficiency increases quickly from 3 to 4 rad s−1

because severe slippagemakesmore energy dissipate.

3.2. Results of four-leg-robot platformThe robot locomotion platform allows the locomotionof the compliant flipper-leg to be examined

Figure 6.Trajectories of the compliant flipper-leg.

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Bioinspir. Biomim. 11 (2016) 056005 TFang et al

experimentally. As mentioned before, we selectedvariable rotation speeds in the experiments. Afterconsidering the capability of the motors, a range ofspeed from 0.5 to 4 rad s−1 with just one stride waschosen for recording purposes. Their trajectories areshown in figure 8(a). All the trajectories are similar at0.5, 1, 2 and 3 rad s−1, but when the motor reached arotating speed of 4 rad s−1, the trajectory is obviously

higher than others when x is around 0.1 m and thelength of stride becomes shorter. So at low speeds, thespeed at which the motor rotates is not an importantfactor of trajectory because it only begins to differ athigher speeds. The video shows that at low speeds, thelegs of the robot did not slip on the ground, but oncethe speed of rotation increased, so too did slippage. Infigure 8(c), fluctuations in the height of the robot

Figure 7. Simulation results of the single-leg locomotion platform.

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Bioinspir. Biomim. 11 (2016) 056005 TFang et al

increased slowly before reaching 3 rad s−1, but from 3to 4 rad s−1, the height fluctuated faster due toslippage. Figure 8(b) shows that the robot’s forwardspeed varied with COT at five speeds ranging from 0.5to 4 rad s−1. The chart representing forward speedshows that the forward speed Vx is strictly linear, andthus the forward speed of robots propelled by

compliant legs on terrain is approximately propor-tional to the rotation speed of the motor. Theproportional factor was 75.6 mm rad−1 s−1, so themaximum forward speed can reach 0.3 m s−1 at4 rad s−1, which is 0.48 body lengths per second.Another chart in figure 8(b) shows the COT where AsCOT increased with almost all rotating speeds the

Figure 8.Experimental result of the robot locomotion platform.

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Bioinspir. Biomim. 11 (2016) 056005 TFang et al

compliant leg experienced a subsequent decrease in itspower efficiency. As with the fluctuations in height,COT rose faster from 3 to 4 rad s−1 because moreenergy dissipated in dynamical friction between the legand the ground. Though a higher rotation speed leadsto a higher forward speed, slippage occurs at thissituation, which leads to a sharp decline of powerefficiency.

3.3. Special design of compliantflipper-legsNow that the mechanical model has been developed,the design of the complaint flipper-legs can beexamined in order to obtain a high locomotionperformance. By observing animals with compliantappendages, almost all legs have a slope from the rootto the end tip, and the tip of the root is thicker than thetip at the end, just like leg I in figure 9(a). One typicalexample is the pectoral fins of mudskippers, anamphibious fish who excels to locomote in muddysubstrates and swimming underwater with its compli-ant pectoral fins [37, 38]. In order to validate theadvantages of a flipper-leg with such a shape, we

constructed two different legs; one leg has the samethickness along its length, while the other was thinnerat the root tip and thicker at the end tip. These two legsare called legs II and III, respectively, as shown infigure 9(a). Thus, we now have three leg shapes withdifferent slopes, so the locomotion on terrain can nowbe compared.

We calculated the trajectories Vx, and COT basedon a single-leg platform with three different legs, andpresented the results in figures 9(a) and (b). The figureshows that leg I has the highest forward speed and thelowest COT, so a leg shaped like leg I was better thanthe other two. That leg I had the best performancecoincides with the cases in nature, where for instance,fish are stiffer in their anterior region than at their pos-terior region, so leg I was better able to reproduce thekinematics offish swimming freely [39]. The trajectoryof leg III is lowest and has an obvious valley that wascaused by energy accumulation and releasing as thecompliant leg was bending. Since leg III has a thinnerroot tip and a strong end tip, when the leg begins tobend, more energy is stored in the root which causes

Figure 9. Special design of the complaint flipper-legs.

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Bioinspir. Biomim. 11 (2016) 056005 TFang et al

more the root to bendmore, and that results in a lowertrajectory. The stored energy is released when it entersthe flying phase because the end tip is strong enough,and it has bent less and thus can generate more pro-pulsive force, which let the driving unit go upobviously and then a valley appears.

3.4.DiscussionIn the above section, we presented the results fromsimulations and experiments for a single-leg platformand experiments for four-leg-robot platform, includ-ing their trajectories, forward speed, fluctuations inheight, and COT versus various output rotation speedof themotor unit.

It is necessary to point out why two different plat-forms were used to explore the performance of thecompliant legs, and compare the results of two differ-ent platforms. The description of the single-leg plat-form in section 2.1 indicates that the horizontal load,including the vertical slide ways and some other parts,weight almost 12.36 kg, which is much larger than the2 kg vertical load. There was no situationwhere a com-pliant leg was used in a robot, even though a a multi-ple-leg platform was necessary. However, a four-leg-robot platform is difficult to model because the twofront legs have a different mechanical motion to thetwo back legs. We therefore used a single-leg platformto simulate a compliant flipper-leg, and a four-leg-robot platform to examine how this kind of flipper-legwould perform.

Although these two platformswere built for differ-ent purposes, there are still some interesting resultsfrom a comparison of locomotion. When the motionof the driving unit in the single-leg platform finishes itslifting phase and enters the flying phase, the shift ofmotion is more obvious than in four-leg-robot plat-form because the weight of vertical slide ways in thesingle-leg platform is enough to generate considerableinertia which causes the driving unit to quickly decele-rate, and the leg slip on the ground easily. For that rea-son, the driving unit’s length of stride decreases at arotation speed of 4 rad s−1 compared to slower speeds.Alternatively, forward speed increases when speed ofrotation of the robot’s locomotion platform increasedfrom0.5 to 4 rad s−1.

4. Conclusions

The purpose of this paper is to present a theoreticaland experimental analysis of a compliant straightflipper-leg during terrestrial locomotion. Through theresults from a single-leg and a four-leg-robot locomo-tion platform, we found that the best forward speed islinked to a certain rotation speed of the motor; if themotor rotates too fast, slippage between the leg and theground occurs. To analyze propulsive efficiency, wecalculated COT and found that a lower speed ofrotation is better because once the legs slip on the

floor, COT increases quickly. The locomotion perfor-mance of a four-leg robot reveals that these compliantlegs can be applied to robots. Moreover, we comparedthree kinds of legs with different slopes and found thatleg I performed best at forward speed and propulsiveefficiency. Leg I has a slope where its thicknessdecreases from the root tip to the end tip, which issimilar to animal appendages in nature, such as thepectoral fins ofmudskippers.

The methods to obtain the experimental trajec-tories of compliant flipper-legs and analyze the dyna-mical behavior of compliant flipper-legs can be used tooptimize the structure of the amphibious robot, andprovide advice for the controlling strategies of themulti-flipper-leg robot with versatile gaits. Futurework will focus on a theoretical and experimentalexploration of compliant flipper-legs walking on otherterrain such as soil and sand.

Acknowledgments

This research is supported byNationalNatural ScienceFoundation of China (No. 51375468, 11272303), ARCDiscovery Grant (DP150102636) and the Fundamen-tal Research Funds for the Central Universities ofChina (WK2090050026).

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