Research Article Influence of -Curve Spring Stiffness on Running...

Research ArticleInfluence of (J)-Curve Spring Stiffness on Running Speeds ofSegmented Legs during High-Speed Locomotion

Runxiao Wang Wentao Zhao Shujun Li and Shunqi Zhang

School of Mechanical Engineering Northwestern Polytechnical University Xirsquoan Shaanxi 710072 China

Correspondence should be addressed to Wentao Zhao semper zhaowentaohotmailcom

Received 12 May 2016 Revised 22 September 2016 Accepted 1 November 2016

Academic Editor Young-Hui Chang

Copyright copy 2016 Runxiao Wang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Both the linear leg spring model and the two-segment leg model with constant spring stiffness have been broadly used as templatemodels to investigate bouncing gaits for legged robots with compliant legs In addition to these two models the other stiffness legspringmodels developed using inspiration frombiological characteristic have the potential to improve high-speed running capacityof spring-legged robots In this paper we investigate the effects of ldquoJrdquo-curve spring stiffness inspired by biological materials on run-ning speeds of segmented legs during high-speed locomotion Mathematical formulation of the relationship between the virtual legforce and the virtual leg compression is establishedWhen the SLIPmodel and the two-segment legmodel with constant spring stiff-ness and with ldquoJrdquo-curve spring stiffness have the same dimensionless reference stiffness the two-segment leg model with ldquoJrdquo-curvespring stiffness reveals that (1) both the largest tolerated range of running speeds and the tolerated maximum running speed arefound and (2) at fast running speed from 25 to 4092m sminus1 both the tolerated range of landing angle and the stability region are thelargest It is suggested that the two-segment leg model with ldquoJrdquo-curve spring stiffness is more advantageous for high-speed runningcompared with the SLIP model and with constant spring stiffness

1 Introduction

Owing to the elastic elements (muscles tendons ligamentsand other soft tissues) of legged systems in fast animallocomotion spring-like leg behavior is discovered to representbouncing gaits like running hopping and trotting [1 2]On the one hand the leg compliance can reduce the impactof the ground contact phase [3] and recycle kinetic energyby using of elastic strain energy storage and release [4]which can lead to low energy costs For instance due tostorage and utilization of elastic strain energy the reductionrate of the required muscle work is up to 40 percent in atrotting or galloping horse [5] On the other hand compliantlegged systems exhibiting self-stability in response to inter-nal (speed variations) and external perturbations (groundsurface irregularities) can simplify the dynamic control ofbouncing motion [6] Therefore the elastic elements play animportant role in fast animal locomotion [7]

According to spring-like leg behavior of bouncy gaitsbiomechanists [8] propose a simple spring-mass model todescribe animal and human locomotion which is also called

spring-loaded inverted pendulum (SLIP) consisting of apoint mass representing the body and a massless linearspring describing the leg For the purpose of achieving stablerunning in the SLIP model an increase in leg stiffnessis required for increased running velocities when angle ofattack is unchanged [9] At present the SLIP model isextensively applied to the study of bouncing gaits for leggedrobots including one-legged hopping robots bipedal robotsquadruped robots and hexapod robots Sato and Buehlerpropose a hopping robot with one leg based on the SLIPmodel [10]

However biological limbs are not the telescopic linearleg model rather they are made up of multiple joints theircompliance is situated at the joint level [11 12] Accordingto this concept Rummel and Seyfarth [6] study the effectsof the two-segment leg model with constant spring stiffnesson running stability during low-speed running and resultsfrom simulations and experiments show that adjustment ofjoint stiffness is required to support stable running at differentspeeds Furthermore this segmented leg supports self-stablerunning at an enlarged range of speeds (lower minimum

Hindawi Publishing CorporationApplied Bionics and BiomechanicsVolume 2016 Article ID 1453713 11 pageshttpdxdoiorg10115520161453713

2 Applied Bionics and Biomechanics

m

k

l0

(a) Model parameters

DecompressionCompression

Stance phase Flight phaseFlight phase

Forward

yap

exi

y

x

mg

0

Fleg

1205720

Step (i)

yap

exi

+1

(b) Locomotion phases and transition conditions

Figure 1 SLIP model during a running cycle

speed) compared with the SLIP model At present the two-segment leg model with constant spring stiffness is widelyapplied to the design of compliant legged robots such as theintersegmental joint configuration of ldquoJenaHopperrdquo designedby Rummel et al [13]

It is well known that the SLIPmodel and the two-segmentleg model with constant spring stiffness can be seen as aneffective tool to research on bouncing gaits for legged robotsIn addition to these two models Karssen and Wisse [14] alsopropose the running model with nonlinear leg spring andstudy the effect of the nonlinear leg springs on disturbancerejection behavior Some results of this optimization revealthat the push (push forward and backward) disturbancesrejection with the optimal nonlinear leg spring is much betterthan with the optimal linear spring What is more it can beseen from the size of the basin of attraction that the rangeof running speeds of the optimal nonliner spring is largelyincreased compared with the optimal linear spring But thisrunningmodel is analyzed in the low-speed case At present anumber of prominent high-speed spring-legged robots suchas bipedal robot MABEL [15] and Cheetah-cub quadrupedrobot [16] are developed Here it is interesting to note thata speed record of 306m sminus1 is implemented on MABEL byadjusting its effective leg stiffness [17]

Although there are many running robots and runningmodels at present we focus only on the two-segment legmodel in this study This is because it is the most reducedleg configuration in spite of the low complexity of this two-segment leg model it is still suitable to solve the questionof how leg segmentation and joint stiffness influence thestability of running at different speeds [6] On the otherhand so far the potential effects of joint stiffness on runningspeeds during high-speed locomotion remain an unresolvedissue Therefore in the present paper we propose the two-segment leg model with ldquoJrdquo-curve spring stiffness Here ldquoJrdquo-curve spring stiffness is inspired by biological materials [18]Subsequently we employ this proposed model to investigate

the potential role of ldquoJrdquo-curve spring stiffness on runningspeeds during high-speed locomotion Here in this paperrunning models focus only on the SLIP model the two-segment leg model with constant spring stiffness and withldquoJrdquo-curve spring stiffness This is to develop a deeper under-standing of the benefits and drawbacks of the two-segmentleg model with ldquoJrdquo-curve spring stiffness compared with theother two models regarding high-speed running capacity

In this study we not only hope that the two-segment legmodel with ldquoJrdquo-curve spring stiffness will show the largestrange of running speed for self-stable high-speed running inall threemodels but also expect that results of ourworkwill beregarded as a promising concept for the design of bioinspiredhigh-speed robots

2 Methods

21 SLIP Model As shown in Figure 1(a) the SLIP model ismodeled as a point mass 119898 attached to a massless spring legwith linear stiffness of 119896 and rest length of 1198970 This model forrunning can be represented as the flight and stance phasesalternatively which is shown in Figure 1(b) During the flightphase the system dynamics is determined by the point-massgravity which results in a ballistic trajectory of the pointmassand then the equation of motion can be expressed as

Fflight = 119898g (1)

where Fflight is the total force vector during flight and g =[0 minus119892]T denotes the gravitational acceleration vector Inaddition considering that the total energy 119864total is assumedto be conserved during flight 119864total is

119864total =12119898V02 + 119898119892119910apex119894 (2)

where V0 is the constant horizontal speed of the point massand 119910apex119894 denotes the apex height (the point mass has themaximum vertical height at the beginning of the flight phase

Applied Bionics and Biomechanics 3

Hip joint

Upper leg

Intersegmental joint

Cable pulley

Lower leg

Spring properties

m

R

c1205730

120573

Figure 2 The configuration of the two-segment leg model

of running) Afterwards the transition from flight to stancephase occurs when the spring leg strikes the ground at a givenangle of attack 1205720 This transition event can be formulated by

119897 (119905) = 1198970 (3)

where 119897(119905) which is the function of time 119905 denotes thecurrent spring leg length The next moment the stancephase starts With the tip of foot regarded as pivot withoutslipping the stance phase can be split into the compressionand decompression subphases Here the transition betweenthe above-described two subphases takes place when the legreaches its maximum compression Note that leg stiffness is afixed value of 119896 and the direction of leg force is from the tipof foot to the point mass during these two subphases Thusthe motion equations of stance phase are written as

Fstance = 119898g + Fleg119865leg = 119896 (1198970 minus 119897 (119905))

(4)

where Fleg is the leg force vector and Fstance denotes the totalforce vector Finally when the tip of foot leaves the groundthe flight phase starts and then the point mass reaches thesubsequent apex variable119910apex119894+1 Consequently a step calleda cycle can be defined as the movement between 119910apex119894 and119910apex119894+1

22 Two-Segment LegModel The following section describesthe configuration of the two-segment leg model and itsdynamics of running with spring-like legs As illustratedin Figure 2 the two-segment leg model is described by apoint mass 119898 attached to a rotating segmented leg and thissegmented leg is represented by massless upper and lower leglinked by the intersegmental joint with the joint angle of 1205730at rest and the radius of cable pulley of 119877 What is more thismodel can be considered as hip movement actively and kneemovement passively Note that knee joint elasticity originatesfrom spring compliance and spring property has a significantinfluence on running stability Thus the two-segment legmodel can be divided into the two-segment leg model with

constant spring stiffness and with nonlinear spring stiffnessin terms of the linear or nonlinear characteristics of tensilespring and its joint stiffness 119888(Δ120573) is

119888 (Δ120573) =119865spring119877Δ120573

Δ120573 = 1205730 minus 120573(5)

where 120573 denotes instantaneous joint angle Δ120573 represents theamount of joint flexion and 119865spring is the tensile force of thespring In addition it is evident that both upper leg of length1198971 and lower leg of length 1198972 affect the dynamics of runningTherefore to make the analysis of the mathematical modeleasily the two segment lengths are defined as 1198971 = 1198972 = 119871in order to facilitate the comparison between the SLIP modeland the two-segment leg model rest length of the virtual legis also defined as 1198970

Figure 3 illustrates the two-segment leg model during arunning period It is worth noting that the two-segment legmodel can be conceived as an equivalent SLIP model whoseleg stiffness is nonlinear Thus similar to the SLIP model thetwo-segment legmodel for running is also composed of flightand stance phases and stance phase can also be divided intothe compression and decompression subphases Similarlythe total energy transition conditions the total force duringflight and the direction of leg force of the stance phase areidentical with those of the SLIP model respectively

221 Two-Segment Leg Model with Constant Spring StiffnessAs can be seen in Figures 2 and 3 if tensile spring has aconstant stiffness the system can be regarded as the two-segment leg model with constant spring stiffness reported in[6] and the equation of motion for this running system isgiven by the following equation (based on [6])

Fstance = 119898g + Fleg

119865leg =radic11989712 + 11989722 minus 211989711198972 cos120573

11989711198972119888 (120573 minus 1205730)sin120573

(6)

222 Two-Segment Leg Model with ldquoJrdquo-Curve Spring StiffnessIn this section we illustrate a nonlinear ldquoJrdquo-curve springforce-elongation relationship of the proposed model and itsdynamics of running A schematic diagram of a joint oflarge mammals presented in [19] describes a knee or anankle joint configuration shown in this schematic diagramis similar to the two-segment leg model Here owing to thesimilar role of elastic elements muscles and tendons shownin this schematic diagram can be considered as tensile springof the two-segment leg model it is very interesting to notethat muscles [20] tendons [21] and muscle-tendon complex[22] force-elongation curves resemble a ldquoJrdquo-shape and StevenVogle also notices that biological tissues (ligaments skin etc)force-length curves resemble a ldquoJrdquo-shape [23] Thereby weimitate a joint configuration of largemammals and adopt ldquoJrdquo-curve spring stiffness inspired by biological materials [18] toestablish the two-segment leg model with ldquoJrdquo-curve spring




Forward

y

x

mg

0

F leg

1205720

Step (i)

yap

exi

yap

exi

+1

Figure 3 Two-segment leg model during a running period

Table 1 Five points representing ldquoJrdquo curve

Elongation 0 25 50 75 100Force 0 4 17 48 100

stiffness By doing this we hope that the proposed two-segment leg model with ldquoJrdquo-curve spring stiffness will becapable of realizing high-speed locomotion resulting in theimitation of biological high-speed running capacity Herefive points shown in Table 1 presented in [18] can be used todescribe ldquoJrdquo-curve spring properties and then mathematicalformulation of the relationship between the ldquoJrdquo-curve springforce 119865119904(119897119904) and the ldquoJrdquo-curve spring elongation 119897119904 is obtainedby using of the interpolation method and this equation isrepresented by

119865119904 (119897119904) = 1198864 (119897119904)4 + 1198863 (119897119904)3 + 1198862 (119897119904)2 + 1198861119897119904 + 1198860 (7)

where 1198864 1198863 1198862 1198861 and 1198860 denote the undetermined coeffi-cients subsequently substituting these five points into (7) canyield

1198864 = minus16119865max

25 (119897max)4

1198863 =48119865max

25 (119897max)3

1198862 = minus11119865max

25 (119897max)2

1198861 =4119865max25119897max

1198860 = 0

(8)

where 119897max represents the maximum elongation of ldquoJrdquo-curvespring and 119865max denotes the ldquoJrdquo-curve spring force corre-sponding to 119897max

The virtual leg force vector Fleg(Δl) relates gravitationalforce vector 119898g to the total force vector Fstance during thestance phase with

Fstance = 119898g + Fleg (Δl) (9)

in [6] the virtual leg force 119865leg(Δ119897) is a function of the virtualleg compression Δ119897 with

119865leg (Δ119897) =(1198970 minus Δ119897) 1205911198712 sin (1205730 minus Δ120573)

(10)

where the joint torque 120591 can be determined according to119865119904(119897119904)with

120591 = 119865119904 (119897119904) 119877 (11)

the relationship between the amount of the joint flexion Δ120573and 119897119904 can be given by

Δ120573 = (180∘119897119904)(119877120587) (12)

Δ119897 can be then represented by

Δ119897 = 1198970 minus 119871radic2 minus 2 cos (1205730 minus Δ120573) (13)

Seeing that the maximum compression of the virtual legrarely exceeds 30 of the rest virtual leg length among run-ning animals [24] both the springrsquos maximum compressionin the SLIP model and the maximum compression of thevirtual leg in the two-segment leg model are defined as 031198970in this study and then the corresponding maximum amountof the joint flexion Δ12057330 in the two-segment leg model isformulated by

Δ12057330 = 1205730 minus arccos21198712 minus (071198970)221198712 (14)

Finally the radius of cable pulley 119877 can be obtained based onthe amount of the joint flexion Δ12057330 with

119877 = 180∘119897max120587Δ12057330 (15)


23 Analysis Methods The steps-to-fall analysis and the apexreturn map which are reported in [6 9] are adopted toanalyze the running dynamics as represented by the above-mentioned three models In the following section we brieflyreview these two methods The steps-to-fall analysis canrecord the maximum number of steps to fall for a givensystem parameters (119898 1198970 V0 etc) and simulation calculationis not stopped until the number of predefined steps is reachedHere the maximum number of successive steps is definedas 50 [6] Although the steps-to-fall analysis can provide amethod for counting the number of successive steps thesystem might still fall after a finite threshold Therefore inorder to solve this question the apex return map is adopted

The second approach the apex return map can identifythe fixed point Furthermore the stability of the fixed point119910lowast = 119910apex119894+1 = 119910apex119894 can be described by

119904 = 119889119910apex119894+1119889119910apex119894

100381610038161003816100381610038161003816100381610038161003816119910lowast (16)

here if 119904 is smaller than 1 this conditionmeans that the systemis stable

24 Simulation Parameters Setup For the purpose of facili-tating the comparison of all three models simulation param-eters can be defined as follows (1) Seeing that a representativevalue for humans running is leg compression at 10 of rest leglength [25] the reference stiffness 11989610 can be defined by thefollowing equation (based on [6])

11989610 =1198651011989710 (17)

where 11989710 denotes a reference leg compression at 10 ofrest leg length and 11986510 is the corresponding leg force (2) adimensionless reference stiffness can be expressed by thefollowing equation (based on [26])

= 1198961198970119898119892 (18)

(3) the dimensionless reference stiffness 10 can be given bythe following equation (based on [6])

10 =119896101198970119898119892 (19)

three models parameters 119898 = 80 kg 1198970 = 1m 30∘ le1205720 le 90∘ and 0 le 10 le 50 and the initial apex heightof 1m are defined [6] Additionally in the two-segment legmodels the small (1205730 = 115∘) and large nominal joint angles(1205730 = 165∘) are adopted to analyze the effects of differentnominal joint angles on running speeds respectively inour model (1205730 = 115∘ and 165∘) the maximum speed ofstable running is 92 and 40m sminus1 respectively Thus twospeed ranges (5 to 92m sminus1 and 5 to 40m sminus1) are utilized toinvestigate the advantages and disadvantages of our modelcompared with the other two models with respect to high-speed running capacity respectively Here the speed of5m sminus1 is the minimum speed of stable running in our modelat 1205730 = 115∘

0 5 10 15 20 25 30

SLIP model

87 88 89 9 91 92 9308509

0951

Normalized virtual leg compression Δll0 ()

0123456789

Nor

mal

ized

virt

ual l

eg fo

rce

165 deg140 deg115 deg

115 deg140 deg

165 deg

Fle

gF10

Figure 4 Normalized force-compression relationships of the SLIPmodel and the two-segment leg model Solid curves are the pro-posed model and dashed ones denote the two-segment leg modelwith constant spring stiffness The same nominal joint angle 1205730is adopted to facilitate the comparison of normalized force-lengthrelationships between the two-segment leg model with ldquoJrdquo-curvespring stiffness and with constant spring stiffness with the fact thatred green and purple curves represent 1205730 = 115∘ 1205730 = 140∘ and1205730 = 165∘ respectively

3 Results

31 Normalized Force-Length Relationships of the Two-Segment Leg Model with ldquoJrdquo-Curve Spring Stiffness In thissection we not only analyze normalized force-compressionrelationships of the two-segment leg model with ldquoJrdquo-curvespring stiffness but also investigate the effects of differentnominal joint angles 1205730 on these relationships For easeof understanding we divide the range of Δ1198971198970 into twosubintervals Here the first one is 0 ⩽ Δ1198971198970 lt 10 and thesecond is 10 ⩽ Δ1198971198970 ⩽ 30 For 0 ⩽ Δ1198971198970 lt 10 results shownin Figure 4 reveal that large variations in Δ1198971198970 result in smallchanges in leg force119865leg11986510 In contrast for 10 ⩽ Δ1198971198970 ⩽ 30large changes in Δ1198971198970 lead to large variations in leg forceFurthermore at the beginning of the second subinterval agiven variations in Δ1198971198970 lead to increasingly large changes inleg force with increasing virtual leg compression Next weconsider roles of nominal joint anglesThe larger the nominaljoint angle the slightly smaller the leg force for a given Δ1198971198970in the first subinterval In the second subinterval the smallerthe nominal joint angle the greater the rise in leg stiffness ata given virtual leg compression

32 Comparison of Normalized Force-Length Relationshipsbetween the Two-Segment Leg Model with ldquoJrdquo-Curve SpringStiffness and the Other Two Models Considering that in[6] comparison of normalized force-length relationshipsbetween the SLIP model and the two-segment leg modelwith constant spring stiffness has already been discussedthis section focuses on comparison of relationships betweenthe two-segment leg model with ldquoJrdquo-curve spring stiffnessand the other two models In the first subinterval leg forceof the proposed model is smaller than those of the othertwo models for a given Δ1198971198970 and leg stiffness of our model


becomes increasingly small with increasing 1205730 In the secondsubinterval for a givenΔ1198971198970 ourmodel has themaximum legforce in all three models and leg force becomes increasinglylarge with increasing 1205730 in our model

33 Stability Analysis at Different Running Speeds V0 andNominal Joint Angles 1205730 In this section we concentrate onthe effects of a running speed and the dimensionless referencestiffness on each other Again to gain better insights intoadvantages and disadvantages of our model compared withthe other twomodels regarding high-speed running capacitywe also analyze the effects of these speeds on angle of attackand the stability region in all threemodels respectively Herefor ease of understanding some typical examples which arethe regions of stable running at different running velocities(V0 = 7msminus1 V0 = 29msminus1 and V0 = 36msminus1) and nominaljoint angles (1205730 = 115∘ and 1205730 = 165∘) are used to analyzeself-stabilizing behavior of the above-represented three mod-els according to the normalized force-compression relation-ships depicted in Figure 4 Interestingly the speed of 29m sminus1which is the maximum running speed of the cheetah is thehighest running speed recorded from land animal [27]

In all three models the proposed model has the toleratedmaximum range of the dimensionless reference stiffness ata running speed from 7 to 4092m sminus1 the smaller thenominal joint angles the larger the stability regions forgiven combinations of V0 and 10 as shown in Figure 5(a)In other words we can see from Figure 5(a) that for agiven dimensionless reference stiffness the proposed modelexhibits not only the largest tolerated range of running speedsbut also the tolerated maximum running speed in all threemodels for example for a given 10 = 412698 our modelat 1205730 = 115∘ is capable of accomplishing stable runningbehavior at a speed range from 5 to 76m sminus1 Howeverat the same dimensionless reference stiffness the toleratedspeed range is from 5 to 27m sminus1 in the SLIP model worsestill in the two-segment leg model with constant springstiffness this range is only from 5 to 19m sminus1 In additiona tolerated minimum 10 is required in order to guaranteethe stability of running in a region (one of the twelve regionsin Figure 6) Interestingly although the values of these 10become increasingly large with increasing running speedsthis tolerated minimum 10 in our model is smaller thanthose of the other two models for a given speed from 7 to4092m sminus1 the smaller the nominal joint angle the smallerthe rate of increment in this tolerated minimum 10

At fast running speed form 25 to 4092m sminus1 our modelhas the maximum range in angle of attack (1205721 and 1205722) Forinstance at high running speed (29m sminus1) 1205721 and 1205722 (seeFigure 5(b)) are 1143 and 762∘ in our model at 1205730 = 115∘respectively In contrast 1205721 = 1205722 = 38∘ in the SLIP modeland 1205721 = 1205722 = 0∘ in the two-segment leg model with constantspring stiffness are found Additionally the range of landingangle becomes smaller with increased 1205730 for example in ourmodel at V0 = 40msminus1 and 1205730 = 115∘ 1205721 = 952∘ and 1205722 =762∘ are found However at the same speed both 1205721 and 1205722at 1205730 = 165∘ are only 095∘ As for landing angle sensitivityall three models are sensitive for angle of attack variations

because of 1205721 = 1205722 but our model is sensitive for angle ofattack variations at higher speeds Again when our model issensitive for landing angel variations it is necessary to stablerunning at higher velocities for decreased 1205730

At fast running speed form 17 to 4092m sminus1 regionsof stable running of the proposed model are larger thanthose of the other two models as illustrated in Figure 5(c)For instance a region representing the number of successfulrunning steps shown in Figure 6 is made up of the 64 times 64equidistant grid for a given running speed of 29m sminus1 thestability region consists of the 247 equidistant grids in ourmodel (1205730 = 115∘) However at the same running speedthe region of stable running of the SLIP model is only the16 worse still there is no stable region in the two-segmentleg model with constant spring stiffness In contrast at lowrunning speed from 5 to 10m sminus1 our model demonstratesthe minimum stability region in all three models Again inthe two-segment leg model an increase in nominal jointangle leads to a decrease of the stability region duringhigh-speed locomotion For instance at high running speed(36m sminus1) the stability region of ourmodel is decreased fromthe 205 at 1205730 = 115∘ to the 10 equidistant grids at 1205730 = 165∘

34 Return Map of the Apex Height In the following sectionwe analyze stability of the proposed model by using of thissingle apex returnmap For a given total energy119864 = 344248 Jand system parameters (1205720 = 50∘ and 10 = 1875) resultsfrom the effects of different nominal joint angles on stablefixed points (the intersections of between three curves andthe diagonal) are shown in Figure 7(a) From this figure wecan see that the small nominal joint angle results in the highvalue of fixed point Here all curves show that our model canaccomplish periodic running patterns at the running speedof 29m sminus1

Figure 7(b) shows the effects of another factormdashdifferentrunning velocitiesmdashon stable fixed points with systemparameters (1205720 = 47∘ 10 = 238 and 1205730 = 115∘)

We can obtain that the lower the running speeds thehigher the values of fixed points and the high runningvelocities (31m sminus1) result in the small basin of attractioncontaining all apex heights

4 Discussion

In this paper we discuss the effects of the two-segment legwith ldquoJrdquo-curve spring stiffness on running speeds duringhigh-speed running Two methods the steps-to-fall analysisand the apex return map [6 9] can be adopted to exploithigh-speed running capacity of the model

Compared with the other two models during fast run-ning it reveals that (1) system can provide the larger regionsof stable running (2) the tolerated range of 10 for self-stable running is even larger for a given running speed (3)the proposed model shows the larger tolerated speed rangeand running speeds In addition when the proposedmodel isstable running the small nominal joint angle can lead to thelow tolerated minimum 10 with the large tolerated speedrange and running speed


The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]

The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]

Running speed 0 (ms) two-segment leg model at 1205730 = 165∘Running speed 0 (ms) two-segment leg model at 1205730 = 115∘908070605040302010

10

20

30

40

50

10

20

30

40

50

15

25

35

45

5 10 15 20 25 30 35 40

SLIP model

Two-segment leg model with ldquoJrdquo-curve spring stiffnessTwo-segment leg model with constant spring stiffness

SLIP model


(a)

1205722 in the SLIP model


1205722 in the two-segment leg model with ldquoJrdquo-curve spring stiffness

1205721 in the two-segment leg model with constant spring stiffness


1205721 in the two-segment leg model with ldquoJrdquo-curve spring stiffness1205722 in the SLIP model






The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

0

5

10

15

20

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]

0

5

10

15

20

25

30

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]

10 15 20 25 30 35 405Running speed 0 (ms) two-segment leg model at 1205730 = 165∘

20 30 40 50 60 70 80 9010Running speed 0 (ms) two-segment leg model at 1205730 = 115∘

(b)

SLIP model


SLIP model


The n

umbe

r of t

he eq

uidi

stan

t grid

The n

umbe

r of t

he eq

uidi

stan

t grid

Running speed 0 (ms) two-segment leg model at 1205730 = 115∘908070605040302010 10 15 20 25 30 35 405

Running speed 0 (ms) two-segment leg model at 1205730 = 165∘

0

50

100

150

200

250

300

[indi

catin

g th

e sta

bilit

y re

gion

]

0

100

200

300

400

500

[indi

catin

g th

e sta

bilit

y re

gion

]

(c)

Figure 5 Properties of regions of stable running for given combinations of 10 and 1205720 at different running speeds in the SLIP model thetwo-segment leg model with constant spring stiffness and with ldquoJrdquo-curve spring stiffness respectively Here the speed range of stable runningfor a given 10 from 0 to 50 is shown in (a) and (b) 1205721 denotes the difference between the minimum and maximum landing angles and 1205722is the maximum tolerated range of landing angle on a region of stable running


0

10

20

30

40

50

Two-segment leg model with ldquoJrdquo-curve spring stiffness = 115∘

Two-segment leg model with ldquoJrdquo-curve spring stiffness1205730 = 165∘

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

Two-segment leg model with constant spring stiffness 1205730 = 115∘


SLIP model

Angel of attack 1205720 (deg) 0 = 7msminus1





Angel of attack 1205720 (deg) 0 = 29 msminus1



Angel of attack 1205720 (deg) 0 = 29 msminus1 Angel of attack 1205720 (deg) 0 = 29 msminus1



1205730

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

0

10

20

30

40

50

0

10

20

30

40

50

0

10

20

30

40

50

Figure 6 Regions of stable running for given combinations of 10 and 1205720 in the two-segment leg model with ldquoJrdquo-curve spring stiffness (thefirst row 1205730 = 115∘ and the second row 1205730 = 165∘) the two-segment leg model with constant spring stiffness (the first and second figures ofthe third row 1205730 = 115∘ and the first and second figures of the last row 1205730 = 165∘) and the SLIP model (the third and fourth figures of thethird column)


1 12 14 16 1808Apex height at step i (yapexi) (m)

08

1

12

14

16

18Ap

ex h

eigh

t at s

tepi+1

(yap

exi

+1) (

m)

115 deg75 deg95 deg

(a) Return maps function 119910apex119894+1(119910apex119894) at different 1205730

08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)


Fixed point

31ms27ms29ms

(b) Return maps function 119910apex119894+1(119910apex119894) at different V0

Figure 7 Return maps of the apex height 119910apex119894+1(119910apex119894) of a single step in the two-segment leg model with ldquoJrdquo-curve spring stiffness

These characteristics mentioned above are due to thetwo-segment leg structural configuration and ldquoJrdquo-curvespring stiffness properties resulting in the nonlinear force-compression relationships depicted in Figure 4 In additionowing to the passive compliance identified in the proposedmodel this elastic two-segment leg configuration can takeadvantage of simple control strategies to guarantee the steady-state running behavior with little or even no sensory infor-mation We believe that our model can be seen as a templateto analyze the high-speed dynamic locomotion for animalsand robots However this two-segment leg model with aldquoJrdquo-shape force-elongation curve cannot mimic adjustmentof leg stiffness in fast animal locomotion This is becauselimb stiffness is adapted to running speed for exampleHobara et al [28] reveal through experiments that in humansleg stiffness is adjusted to different hopping frequencieswith the fact that there are two different force-elongationcurves during compression and decompression Hence inorder to adequately mimic limb compliant locomotion theseaspects should be taken into account in the design andimplementation of robotic system

Additionally in the two-segment leg model with ldquoJrdquo-curve spring stiffness the running is simulated across smoothand level terrain Yet the terrain is not the same in thereal world where the ground surface irregularities must betaken into account for running robots and running modelsTherefore in order to achieve stable running of the proposedmodel on uneven terrain it is necessary to adjust apex heightwith adequate ground clearance in response to disturbancesin ground height Currently for a given total system energythe return map of the apex height 119910apex119894+1(119910apex119894) can beadopted to analyze running stability when the system isperturbed by a change in ground height Furthermore asthe proposed model is conservative the total system energyis distributed to the vertical energy and the forward kineticenergy by the adjustment of angle of attack in other words

1812 14 16108Apex height at step i (yapexi) (m)

08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)

1205720 = 49∘

1205720 = 50∘1205720 = 51∘

1205720 = 52∘1205720 = 53∘

Figure 8 Return maps function 119910apex119894+1(119910apex119894) at different 1205720

different angles of attack can be used to control a desiredapex height or correspondingly the forward speed Herethe return map of the apex height is adopted to investigatethe potential effects of different landing angles on the apexheight in our model (V0 = 29msminus1 10 = 40 and 1205730 =115∘) Results shown in Figure 8 reveal that the apex height119910apex119894+1 is dependent on the preceding apex height 119910apex119894 andthe selected angle of attack 1205720 What is more for a givenapex height 119910apex119894 both the value of fixed point and thesubsequent apex height 119910apex119894+1 become increasingly smallwith increasing 1205720 and the apex height is sensitive for angleof attack variations This means that a higher apex heightcan be implemented by tuning a smaller angle of attackwith the fact that only one step or a few more steps are


required to achieve higher steady-state heights in our modelAgain it is interesting to note that Seyfarth and Geyer [29]introduce a generalized return map and derive an optimalcontrol strategy of the apex height in the SLIP model Thismethod could be utilized to study the control strategy of theapex height of the proposed model which will be subject offurther investigations

Running at High Speed In Figure 5(a) the SLIP model indi-cates that the tolerated minimum leg stiffness has to increasewith running speed to achieve stable runningThis is in agree-ment with a simulation study in which increasing leg stiffnessis required to guarantee the stability of a galloping quadrupedwhose single leg consists of a linear spring attached to aprismatic leg when running speed is increasingly high [30]In the two-segment leg model with constant spring stiffnessan increase in the tolerated minimum leg stiffness is alsonecessary to accomplish stable running behavior at higherspeeds [6] This finding is supported by robotic trials Herehopping robot with one leg which is made up of a segmentedleg with constant spring stiffness becomes unstable withincreasing speed [31] Therefore leg stiffness is sensitivefor speed variation in these two models The situation isnot all the same in the two-segment leg model with ldquoJrdquo-curve spring stiffness Here in our model the amount ofincrement in the toleratedminimumdimensionless referencestiffness for the same amount of increment in running speedis largely decreased compared with the other two modelsThis characteristic of our model (leg stiffness is insensitivefor speed variation) enlarges tolerated range of speeds Onthe other hand for a given running speed the proposedmodel has the minimum value of leg stiffness to achieve self-stable running in all three models as depicted in Figure 5(a)For instance at high running speed (29m sminus1) the toleratedminimum dimensionless reference stiffness in our model at1205730 = 115∘ is 1825 compared with 4444 in linear leg modeland no stability solutions in the two-segment leg model withconstant spring stiffness As a result in all three models fora given nominal joint angle and leg stiffness the proposedmodel at high speed has not only the maximum speedrange of stable locomotion but also the tolerated maximumrunning speed This means that in all three models the two-segment leg model with ldquoJrdquo-curve spring stiffness is mostadvantageous for high-speed running

Practically several researchers have designed theirrobotic legs similar to our model configuration andleg stiffness For example the robotic leg proposed bySchmiedeler and Waldron [32] used seven parameters toaccomplish biomimetic feature With these parameterswe here in this paper have calculated their virtual legforce-compression relationship as shown in Figure 9Obviously this curve is similar to the curves as representedby our model depicted in Figure 4 leading to the followingcommon characteristics (1) the virtual leg spring is capableof being soft at the beginning of touch down which canproduce a relatively small impact force [32] (2) it becomesstiffer with increased compression resulting in a large forcewhich can prevent the leg to fold over on itself [32] With thefacts that their prototype leg exhibits the potential for fast

0

50

100

150

200

250

0 01006 008004 012 014 016002

Virt

ual l

eg fo

rce

Virtual leg compression Δl (m)

Fle

g(N

)

Figure 9 Virtual leg force-compression relationships of KOLTquadruped robot

locomotion in experiment [32] and then the correspondingKOLT quadruped robot can implement gallop and high-speed turning [33] we conclude that our simulation resultshave been confirmed in practice

5 Conclusion

In this paper we have presented the two-segment leg modelwith ldquoJrdquo-curve spring stiffness and have analyzed the effects ofldquoJrdquo-curve spring stiffness on running speeds of the proposedmodel during high-speed running According to simulationresults of all threemodels for a given dimensionless referencestiffness we have demonstrated that (1) our model has notonly the largest tolerated speed range but also the toleratedmaximum running speed with the fact that the smallerthe nominal joint angle the better the running capacitymentioned above (2) at fast running speed from 25 to4092m sminus1 our model has not only the largest range of angleof attack but the largest region of stable running It has alreadybeen successfully applied to quadruped robot such as KOLTfor high-speed running

Next we will investigate the effects of a broad ratio ofleg length and nominal joint angle range on high-speedrunning performance respectively Again wewill analyze thewhole stability zone of segmented leg in order to gain betterinsights into advantages and disadvantages of our model atfast running

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (Grant no 51475373)

References

[1] R M Alexander ldquoTendon elasticity and muscle functionrdquoComparative Biochemistry and Physiology A Molecular andIntegrative Physiology vol 133 no 4 pp 1001ndash1011 2002


[2] Y Blum S W Lipfert J Rummel and A Seyfarth ldquoSwing legcontrol in human runningrdquoBioinspiration and Biomimetics vol5 no 2 Article ID 026006 2010

[3] R M Alexander ldquoThree uses for springs in legged locomotionrdquoThe International Journal of Robotics Research vol 9 no 2 pp53ndash61 1990

[4] A Seyfarth HGeyer andHHerr ldquoSwing-leg retraction a sim-ple control model for stable runningrdquo Journal of ExperimentalBiology vol 206 no 15 pp 2547ndash2555 2003

[5] A A Biewener Animal Locomotion Oxford University PressOxford UK 2003

[6] J Rummel and A Seyfarth ldquoStable running with segmentedlegsrdquo International Journal of Robotics Research vol 27 no 8pp 919ndash934 2008

[7] M F Bobbert and L J R Casius ldquoSpring-like leg behaviourmusculoskeletal mechanics and control in maximum and sub-maximum height human hoppingrdquo Philosophical Transactionsof the Royal Society B Biological Sciences vol 366 no 1570 pp1516ndash1529 2011

[8] RM Alexander and A S Jayes ldquoVertical movement in walkingand runningrdquo Journal of Zoology vol 185 no 1 pp 27ndash40 1978

[9] A Seyfarth H Geyer M Gunther and R Blickhan ldquoAmovement criterion for runningrdquo Journal of Biomechanics vol35 no 5 pp 649ndash655 2002

[10] A Sato and M Buehler ldquoA planar hopping robot with oneactuator design simulation and experimental resultsrdquo inProceedings of the 2004 IEEERSJ International Conferenceon Intelligent Robots and Systems (IROS rsquo04) pp 3540ndash3545Sendai Japan October 2004

[11] A Seyfarth M Gunther and R Blickhan ldquoStable operation ofan elastic three-segment legrdquo Biological Cybernetics vol 84 no5 pp 365ndash382 2001

[12] A H Hansen D S Childress S C Miff S A Gard and KP Mesplay ldquoThe human ankle during walking implications fordesign of biomimetic ankle prosthesesrdquo Journal of Biomechan-ics vol 37 no 10 pp 1467ndash1474 2004

[13] J Rummel F Iida J A Smith and A Seyfarth ldquoEnlargingregions of stable runningwith segmented legsrdquo inProceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo08) pp 367ndash372 Barcelona Spain May 2008

[14] J D Karssen and M Wisse ldquoRunning with improved distur-bance rejection by using non-linear leg springsrdquo The Interna-tional Journal of Robotics Research vol 30 no 13 pp 1585ndash15952011

[15] K Sreenath H-W Park I Poulakakis and J W GrizzleldquoEmbedding active force control within the compliant hybridzero dynamics to achieve stable fast running on MABELrdquo TheInternational Journal of Robotics Research vol 32 no 3 pp 324ndash345 2013

[16] A Sprowitz A Tuleu M Vespignani M Ajallooeian E Badriand A J Ijspeert ldquoTowards dynamic trot gait locomotiondesign control and experiments with Cheetah-cub a com-pliant quadruped robotrdquo The International Journal of RoboticsResearch vol 32 no 8 pp 932ndash950 2013

[17] K Sreenath H-W Park and J W Grizzle ldquoDesign and exper-imental implementation of a compliant hybrid zero dynamicscontroller with active force control for running on MABELrdquoin Proceedings of the IEEE International Conference on Roboticsand Automation (ICRA rsquo12) pp 51ndash56 St Paul Minn USAMay2012

[18] C V Jutte and S Kota ldquoDesign of nonlinear springs forprescribed load-displacement functionsrdquo Journal of MechanicalDesign Transactions of the ASME vol 130 no 8 Article ID081403 2008

[19] R M Alexander ldquoElastic energy stores in running vertebratesrdquoAmerican Zoologist vol 24 no 1 pp 85ndash94 1984

[20] H Zhao Y-N Wu M Hwang et al ldquoChanges of calf muscle-tendon biomechanical properties induced by passive-stretchingand active-movement training in children with cerebral palsyrdquoJournal of Applied Physiology vol 111 no 2 pp 435ndash442 2011

[21] S Fukashiro M Rob Y Ichinose Y Kawakami and T Fuku-naga ldquoUltrasonography gives directly but noninvasively elasticcharacteristic of human tendon in vivordquo European Journal ofApplied Physiology and Occupational Physiology vol 71 no 6pp 555ndash557 1995

[22] D A Winter A E Patla S Rietdyk and M G Ishac ldquoAnklemuscle stiffness in the control of balance during quiet standingrdquoJournal of Neurophysiology vol 85 no 6 pp 2630ndash2633 2001

[23] S Vogel Catsrsquo Paws and Catapults W W Norton New YorkNY USA 1998

[24] R Blickhan and R J Full ldquoSimilarity in multilegged loco-motion bouncing like a monopoderdquo Journal of ComparativePhysiology A vol 173 no 5 pp 509ndash517 1993

[25] C T Farley andO Gonzalez ldquoLeg stiffness and stride frequencyin human runningrdquo Journal of Biomechanics vol 29 no 2 pp181ndash186 1996

[26] H Geyer A Seyfarth and R Blickhan ldquoSpring-mass runningsimple approximate solution and application to gait stabilityrdquoJournal of Theoretical Biology vol 232 no 3 pp 315ndash328 2005

[27] N C C Sharp ldquoTimed running speed of a cheetah (Acinonyxjubatus)rdquo Journal of Zoology vol 241 no 3 pp 493ndash494 1997

[28] H Hobara Y Kobayashi E Yoshida and M Mochimaru ldquoLegstiffness of older and younger individuals over a range of hop-ping frequenciesrdquo Journal of Electromyography and Kinesiologyvol 25 no 2 pp 305ndash309 2015

[29] A Seyfarth and H Geyer ldquoNatural control of spring-likerunning optimized selfstabilizationrdquo in Proceedings of the5th International Conference on Climbing and Walking Robots(CLAWAR rsquo02) pp 81ndash85 September 2002

[30] P Nanua andK JWaldron ldquoInstability and chaos in quadrupedgalloprdquo Journal of Mechanical Design Transactions Of theASME vol 116 no 4 pp 1096ndash1101 1994

[31] J Rummel F Iida and A Seyfarth ldquoOne-legged locomotionwith a compliant passive jointrdquo in Proceedings of the 9thInternational Conference on Intelligent Autonomous Systems(IAS rsquo06) pp 566ndash573 Tokyo Japan March 2006

[32] J P Schmiedeler and K J Waldron ldquoLeg stiffness and articu-lated leg design for dynamic locomotionrdquo in Proceedings of theASME International Design Engineering Technical Conferencespp 1105ndash1112 September 2002

[33] L R Palmer III and D E Orin ldquoIntelligent control of high-speed turning in a quadrupedrdquo Journal of Intelligent and RoboticSystems Theory and Applications vol 58 no 1 pp 47ndash68 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



m

k

l0

(a) Model parameters



Forward

yap

exi

y

x

mg

0

Fleg

1205720

Step (i)

yap

exi

+1

(b) Locomotion phases and transition conditions

Figure 1 SLIP model during a running cycle

speed) compared with the SLIP model At present the two-segment leg model with constant spring stiffness is widelyapplied to the design of compliant legged robots such as theintersegmental joint configuration of ldquoJenaHopperrdquo designedby Rummel et al [13]

It is well known that the SLIPmodel and the two-segmentleg model with constant spring stiffness can be seen as aneffective tool to research on bouncing gaits for legged robotsIn addition to these two models Karssen and Wisse [14] alsopropose the running model with nonlinear leg spring andstudy the effect of the nonlinear leg springs on disturbancerejection behavior Some results of this optimization revealthat the push (push forward and backward) disturbancesrejection with the optimal nonlinear leg spring is much betterthan with the optimal linear spring What is more it can beseen from the size of the basin of attraction that the rangeof running speeds of the optimal nonliner spring is largelyincreased compared with the optimal linear spring But thisrunningmodel is analyzed in the low-speed case At present anumber of prominent high-speed spring-legged robots suchas bipedal robot MABEL [15] and Cheetah-cub quadrupedrobot [16] are developed Here it is interesting to note thata speed record of 306m sminus1 is implemented on MABEL byadjusting its effective leg stiffness [17]

Although there are many running robots and runningmodels at present we focus only on the two-segment legmodel in this study This is because it is the most reducedleg configuration in spite of the low complexity of this two-segment leg model it is still suitable to solve the questionof how leg segmentation and joint stiffness influence thestability of running at different speeds [6] On the otherhand so far the potential effects of joint stiffness on runningspeeds during high-speed locomotion remain an unresolvedissue Therefore in the present paper we propose the two-segment leg model with ldquoJrdquo-curve spring stiffness Here ldquoJrdquo-curve spring stiffness is inspired by biological materials [18]Subsequently we employ this proposed model to investigate

the potential role of ldquoJrdquo-curve spring stiffness on runningspeeds during high-speed locomotion Here in this paperrunning models focus only on the SLIP model the two-segment leg model with constant spring stiffness and withldquoJrdquo-curve spring stiffness This is to develop a deeper under-standing of the benefits and drawbacks of the two-segmentleg model with ldquoJrdquo-curve spring stiffness compared with theother two models regarding high-speed running capacity

In this study we not only hope that the two-segment legmodel with ldquoJrdquo-curve spring stiffness will show the largestrange of running speed for self-stable high-speed running inall threemodels but also expect that results of ourworkwill beregarded as a promising concept for the design of bioinspiredhigh-speed robots

2 Methods

21 SLIP Model As shown in Figure 1(a) the SLIP model ismodeled as a point mass 119898 attached to a massless spring legwith linear stiffness of 119896 and rest length of 1198970 This model forrunning can be represented as the flight and stance phasesalternatively which is shown in Figure 1(b) During the flightphase the system dynamics is determined by the point-massgravity which results in a ballistic trajectory of the pointmassand then the equation of motion can be expressed as

Fflight = 119898g (1)

where Fflight is the total force vector during flight and g =[0 minus119892]T denotes the gravitational acceleration vector Inaddition considering that the total energy 119864total is assumedto be conserved during flight 119864total is

119864total =12119898V02 + 119898119892119910apex119894 (2)

where V0 is the constant horizontal speed of the point massand 119910apex119894 denotes the apex height (the point mass has themaximum vertical height at the beginning of the flight phase


Hip joint

Upper leg


Cable pulley

Lower leg

Spring properties

m

R

c1205730

120573



119897 (119905) = 1198970 (3)



(4)




119888 (Δ120573) =119865spring119877Δ120573

Δ120573 = 1205730 minus 120573(5)






11989711198972119888 (120573 minus 1205730)sin120573

(6)





Forward

y

x

mg

0

F leg

1205720

Step (i)

yap

exi

yap

exi

+1





119865119904 (119897119904) = 1198864 (119897119904)4 + 1198863 (119897119904)3 + 1198862 (119897119904)2 + 1198861119897119904 + 1198860 (7)


1198864 = minus16119865max

25 (119897max)4

1198863 =48119865max

25 (119897max)3

1198862 = minus11119865max

25 (119897max)2

1198861 =4119865max25119897max

1198860 = 0

(8)






(10)


120591 = 119865119904 (119897119904) 119877 (11)


Δ120573 = (180∘119897119904)(119877120587) (12)






119877 = 180∘119897max120587Δ12057330 (15)




119904 = 119889119910apex119894+1119889119910apex119894

100381610038161003816100381610038161003816100381610038161003816119910lowast (16)



11989610 =1198651011989710 (17)


= 1198961198970119898119892 (18)


10 =119896101198970119898119892 (19)


0 5 10 15 20 25 30

SLIP model

87 88 89 9 91 92 9308509

0951


0123456789

Nor

mal

ized

virt

ual l

eg fo

rce


115 deg140 deg

165 deg

Fle

gF10


3 Results













4 Discussion




The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]

The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]


10

20

30

40

50

10

20

30

40

50

15

25

35

45

5 10 15 20 25 30 35 40

SLIP model


SLIP model


(a)












The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

0

5

10

15

20

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]

0

5

10

15

20

25

30

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]



(b)

SLIP model


SLIP model


The n

umbe

r of t

he eq

uidi

stan

t grid

The n

umbe

r of t

he eq

uidi

stan

t grid



0

50

100

150

200

250

300

[indi

catin

g th

e sta

bilit

y re

gion

]

0

100

200

300

400

500

[indi

catin

g th

e sta

bilit

y re

gion

]

(c)



0

10

20

30

40

50



30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10



SLIP model












1205730

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

0

10

20

30

40

50

0

10

20

30

40

50

0

10

20

30

40

50




08

1

12

14

16

18Ap

ex h

eigh

t at s

tepi+1

(yap

exi

+1) (

m)

115 deg75 deg95 deg


08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)


Fixed point

31ms27ms29ms






08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)

1205720 = 49∘

1205720 = 50∘1205720 = 51∘

1205720 = 52∘1205720 = 53∘







0

50

100

150

200

250

0 01006 008004 012 014 016002

Virt

ual l

eg fo

rce


Fle

g(N

)



5 Conclusion



Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Hip joint

Upper leg


Cable pulley

Lower leg

Spring properties

m

R

c1205730

120573



119897 (119905) = 1198970 (3)



(4)




119888 (Δ120573) =119865spring119877Δ120573

Δ120573 = 1205730 minus 120573(5)






11989711198972119888 (120573 minus 1205730)sin120573

(6)





Forward

y

x

mg

0

F leg

1205720

Step (i)

yap

exi

yap

exi

+1





119865119904 (119897119904) = 1198864 (119897119904)4 + 1198863 (119897119904)3 + 1198862 (119897119904)2 + 1198861119897119904 + 1198860 (7)


1198864 = minus16119865max

25 (119897max)4

1198863 =48119865max

25 (119897max)3

1198862 = minus11119865max

25 (119897max)2

1198861 =4119865max25119897max

1198860 = 0

(8)






(10)


120591 = 119865119904 (119897119904) 119877 (11)


Δ120573 = (180∘119897119904)(119877120587) (12)






119877 = 180∘119897max120587Δ12057330 (15)




119904 = 119889119910apex119894+1119889119910apex119894

100381610038161003816100381610038161003816100381610038161003816119910lowast (16)



11989610 =1198651011989710 (17)


= 1198961198970119898119892 (18)


10 =119896101198970119898119892 (19)


0 5 10 15 20 25 30

SLIP model

87 88 89 9 91 92 9308509

0951


0123456789

Nor

mal

ized

virt

ual l

eg fo

rce


115 deg140 deg

165 deg

Fle

gF10


3 Results













4 Discussion




The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]

The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]


10

20

30

40

50

10

20

30

40

50

15

25

35

45

5 10 15 20 25 30 35 40

SLIP model


SLIP model


(a)












The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

0

5

10

15

20

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]

0

5

10

15

20

25

30

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]



(b)

SLIP model


SLIP model


The n

umbe

r of t

he eq

uidi

stan

t grid

The n

umbe

r of t

he eq

uidi

stan

t grid



0

50

100

150

200

250

300

[indi

catin

g th

e sta

bilit

y re

gion

]

0

100

200

300

400

500

[indi

catin

g th

e sta

bilit

y re

gion

]

(c)



0

10

20

30

40

50



30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10



SLIP model












1205730

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

0

10

20

30

40

50

0

10

20

30

40

50

0

10

20

30

40

50




08

1

12

14

16

18Ap

ex h

eigh

t at s

tepi+1

(yap

exi

+1) (

m)

115 deg75 deg95 deg


08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)


Fixed point

31ms27ms29ms






08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)

1205720 = 49∘

1205720 = 50∘1205720 = 51∘

1205720 = 52∘1205720 = 53∘







0

50

100

150

200

250

0 01006 008004 012 014 016002

Virt

ual l

eg fo

rce


Fle

g(N

)



5 Conclusion



Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Forward

y

x

mg

0

F leg

1205720

Step (i)

yap

exi

yap

exi

+1





119865119904 (119897119904) = 1198864 (119897119904)4 + 1198863 (119897119904)3 + 1198862 (119897119904)2 + 1198861119897119904 + 1198860 (7)


1198864 = minus16119865max

25 (119897max)4

1198863 =48119865max

25 (119897max)3

1198862 = minus11119865max

25 (119897max)2

1198861 =4119865max25119897max

1198860 = 0

(8)






(10)


120591 = 119865119904 (119897119904) 119877 (11)


Δ120573 = (180∘119897119904)(119877120587) (12)






119877 = 180∘119897max120587Δ12057330 (15)




119904 = 119889119910apex119894+1119889119910apex119894

100381610038161003816100381610038161003816100381610038161003816119910lowast (16)



11989610 =1198651011989710 (17)


= 1198961198970119898119892 (18)


10 =119896101198970119898119892 (19)


0 5 10 15 20 25 30

SLIP model

87 88 89 9 91 92 9308509

0951


0123456789

Nor

mal

ized

virt

ual l

eg fo

rce


115 deg140 deg

165 deg

Fle

gF10


3 Results













4 Discussion




The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]

The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]


10

20

30

40

50

10

20

30

40

50

15

25

35

45

5 10 15 20 25 30 35 40

SLIP model


SLIP model


(a)












The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

0

5

10

15

20

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]

0

5

10

15

20

25

30

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]



(b)

SLIP model


SLIP model


The n

umbe

r of t

he eq

uidi

stan

t grid

The n

umbe

r of t

he eq

uidi

stan

t grid



0

50

100

150

200

250

300

[indi

catin

g th

e sta

bilit

y re

gion

]

0

100

200

300

400

500

[indi

catin

g th

e sta

bilit

y re

gion

]

(c)



0

10

20

30

40

50



30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10



SLIP model












1205730

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

0

10

20

30

40

50

0

10

20

30

40

50

0

10

20

30

40

50




08

1

12

14

16

18Ap

ex h

eigh

t at s

tepi+1

(yap

exi

+1) (

m)

115 deg75 deg95 deg


08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)


Fixed point

31ms27ms29ms






08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)

1205720 = 49∘

1205720 = 50∘1205720 = 51∘

1205720 = 52∘1205720 = 53∘







0

50

100

150

200

250

0 01006 008004 012 014 016002

Virt

ual l

eg fo

rce


Fle

g(N

)



5 Conclusion



Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119904 = 119889119910apex119894+1119889119910apex119894

100381610038161003816100381610038161003816100381610038161003816119910lowast (16)



11989610 =1198651011989710 (17)


= 1198961198970119898119892 (18)


10 =119896101198970119898119892 (19)


0 5 10 15 20 25 30

SLIP model

87 88 89 9 91 92 9308509

0951


0123456789

Nor

mal

ized

virt

ual l

eg fo

rce


115 deg140 deg

165 deg

Fle

gF10


3 Results













4 Discussion




The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]

The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]


10

20

30

40

50

10

20

30

40

50

15

25

35

45

5 10 15 20 25 30 35 40

SLIP model


SLIP model


(a)












The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

0

5

10

15

20

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]

0

5

10

15

20

25

30

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]



(b)

SLIP model


SLIP model


The n

umbe

r of t

he eq

uidi

stan

t grid

The n

umbe

r of t

he eq

uidi

stan

t grid



0

50

100

150

200

250

300

[indi

catin

g th

e sta

bilit

y re

gion

]

0

100

200

300

400

500

[indi

catin

g th

e sta

bilit

y re

gion

]

(c)



0

10

20

30

40

50



30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10



SLIP model












1205730

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

0

10

20

30

40

50

0

10

20

30

40

50

0

10

20

30

40

50




08

1

12

14

16

18Ap

ex h

eigh

t at s

tepi+1

(yap

exi

+1) (

m)

115 deg75 deg95 deg


08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)


Fixed point

31ms27ms29ms






08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)

1205720 = 49∘

1205720 = 50∘1205720 = 51∘

1205720 = 52∘1205720 = 53∘







0

50

100

150

200

250

0 01006 008004 012 014 016002

Virt

ual l

eg fo

rce


Fle

g(N

)



5 Conclusion



Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















4 Discussion




The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]

The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]


10

20

30

40

50

10

20

30

40

50

15

25

35

45

5 10 15 20 25 30 35 40

SLIP model


SLIP model


(a)












The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

0

5

10

15

20

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]

0

5

10

15

20

25

30

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]



(b)

SLIP model


SLIP model


The n

umbe

r of t

he eq

uidi

stan

t grid

The n

umbe

r of t

he eq

uidi

stan

t grid



0

50

100

150

200

250

300

[indi

catin

g th

e sta

bilit

y re

gion

]

0

100

200

300

400

500

[indi

catin

g th

e sta

bilit

y re

gion

]

(c)



0

10

20

30

40

50



30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10



SLIP model












1205730

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

0

10

20

30

40

50

0

10

20

30

40

50

0

10

20

30

40

50




08

1

12

14

16

18Ap

ex h

eigh

t at s

tepi+1

(yap

exi

+1) (

m)

115 deg75 deg95 deg


08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)


Fixed point

31ms27ms29ms






08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)

1205720 = 49∘

1205720 = 50∘1205720 = 51∘

1205720 = 52∘1205720 = 53∘







0

50

100

150

200

250

0 01006 008004 012 014 016002

Virt

ual l

eg fo

rce


Fle

g(N

)



5 Conclusion



Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]

The t

oler

ated

rang

e of t

he d

imen

sionl

ess

refe

renc

e stiff

ness

[sho

win

g th

e spe

edra

nge o

f sta

ble r

unni

ng]


10

20

30

40

50

10

20

30

40

50

15

25

35

45

5 10 15 20 25 30 35 40

SLIP model


SLIP model


(a)












The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

The t

oler

ated

rang

e of120572

1an

d1205722

(deg

)

0

5

10

15

20

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]

0

5

10

15

20

25

30

[indi

catin

g la

ndin

g an

gle s

ensit

ivity

]



(b)

SLIP model


SLIP model


The n

umbe

r of t

he eq

uidi

stan

t grid

The n

umbe

r of t

he eq

uidi

stan

t grid



0

50

100

150

200

250

300

[indi

catin

g th

e sta

bilit

y re

gion

]

0

100

200

300

400

500

[indi

catin

g th

e sta

bilit

y re

gion

]

(c)



0

10

20

30

40

50



30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10



SLIP model












1205730

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

0

10

20

30

40

50

0

10

20

30

40

50

0

10

20

30

40

50




08

1

12

14

16

18Ap

ex h

eigh

t at s

tepi+1

(yap

exi

+1) (

m)

115 deg75 deg95 deg


08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)


Fixed point

31ms27ms29ms






08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)

1205720 = 49∘

1205720 = 50∘1205720 = 51∘

1205720 = 52∘1205720 = 53∘







0

50

100

150

200

250

0 01006 008004 012 014 016002

Virt

ual l

eg fo

rce


Fle

g(N

)



5 Conclusion



Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

10

20

30

40

50



30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10



SLIP model












1205730

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

30 40 50 60 70 80 900

10

20

30

40

50

Dim

ensio

nles

s ref

eren

ce st

iffne

ssk10

0

10

20

30

40

50

0

10

20

30

40

50

0

10

20

30

40

50




08

1

12

14

16

18Ap

ex h

eigh

t at s

tepi+1

(yap

exi

+1) (

m)

115 deg75 deg95 deg


08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)


Fixed point

31ms27ms29ms






08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)

1205720 = 49∘

1205720 = 50∘1205720 = 51∘

1205720 = 52∘1205720 = 53∘







0

50

100

150

200

250

0 01006 008004 012 014 016002

Virt

ual l

eg fo

rce


Fle

g(N

)



5 Conclusion



Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











08

1

12

14

16

18Ap

ex h

eigh

t at s

tepi+1

(yap

exi

+1) (

m)

115 deg75 deg95 deg


08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)


Fixed point

31ms27ms29ms






08

1

12

14

16

18

Apex

hei

ght a

t ste

pi+1

(yap

exi

+1) (

m)

1205720 = 49∘

1205720 = 50∘1205720 = 51∘

1205720 = 52∘1205720 = 53∘







0

50

100

150

200

250

0 01006 008004 012 014 016002

Virt

ual l

eg fo

rce


Fle

g(N

)



5 Conclusion



Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













0

50

100

150

200

250

0 01006 008004 012 014 016002

Virt

ual l

eg fo

rce


Fle

g(N

)



5 Conclusion



Competing Interests


Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Influence of -Curve Spring Stiffness on Running...

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