kinematics, dynamics, and optimal control of pneumatic ...the hexapod robot is driven by electric...

Research ArticleKinematics Dynamics and Optimal Control ofPneumatic Hexapod Robot

Long Bai1 Lu-hanMa1 Zhifeng Dong1 and Xinsheng Ge2

1School of Mechanical Electronic amp Information Engineering China University of Mining and Technology (Beijing)Beijing 100083 China2Mechanical amp Electrical Engineering School Beijing Information Science amp Technology University Beijing 100192 China

Correspondence should be addressed to Long Bai bailong0316jn126com

Received 8 August 2016 Revised 16 January 2017 Accepted 12 February 2017 Published 9 March 2017

Academic Editor Francisco Valero

Copyright copy 2017 Long Bai et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Pneumatic hexapod robot is driven by inert gas carried by itself which has board application prospect in rescue operation of disasterconditions containing flammable gas Cruising ability is main constraint for practical engineering application which is influencedby kinematics and dynamics character The matrix operators and pseudospectral method are used to solve dynamics modelingand numerical calculation problem of robot under straight line walking Kinematics model is numerically solved and relationshipof body joints and drive cylinders is obtained With dynamics model and kinematics boundary conditions the optimal input gaspressure of leg swing and bodymoving in one step is obtained by pseudospectralmethod According to action character ofmagneticvalve calculation results of control inputs satisfy engineering design requirements and cruising ability under finite gas is obtained

1 Introduction

In recent years more and more robots are used in industrialaccidentrsquos detect and rescue operation The commonly usedrobots include motor and hydraulic drive types but they arenot suited for some close space accident environment whichfills with flammable gas such as gas explosion accident of coalmine because electric devices of themmay lead to secondaryexplosion The pneumatic robot is driven by inert gas and isconvenient to be controlled which iswidely used in industrialandmedical domains Verrelst et al [1] designed a pneumaticbiped robot which verifies feasibility of using pneumaticsystem as power source Lavoie and Desbiens [2] designed acockroach type pneumatic hexapod robotMorimoto et al [3]designed a rehabilitation used soft touch manipulator by softcylinder and obtained a high working accuracy Qiu et al [4]designed a pipe inspection robot by soft cylinders Diez et al[5] designed a neural rehabilitation pneumatic robot Lowet al [6] explored a soft pneumatic massager used in jointsauxiliary motion and Ramsauer et al [7] explored an errordetection using pneumatic Stewart platform

With these backgrounds a natural antiexplosion pneu-matic hexapod robot (PHR) which is driven by inert gas isdesigned in this exploration However cruising ability is a biginfluence in robotrsquos engineer application for the carried gasrsquosvolume is limited by self-weight of robot The cruising abilityof PHR is measured by straight line walking distance limitedby product of volume and pressure of carried gas Duringstraight line walking the same characters of each gait decidethey have same gas consumption so cruising ability problemchanges to be calculation of distance and gas consumption ofone step Gas consumption of one step is defined as productof cylinderrsquos volume and drive pressure The cylinder volumeis known and pressure is influenced by dynamics characterof robot The optimal control method is used to calculateminimum drive pressure

In the last few years there are many explorations on opti-mal control problems of hexapod robots Sliva and Machado[8] reviewed optimization method used in legged robotsenergypower optimal control objective functions are listedout Sanz-Merodio et al [9] explored energy consumptionof mammal and insect type robots and concluded that leg

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 6841972 16 pageshttpsdoiorg10115520176841972

2 Mathematical Problems in Engineering

dynamics accounts for most energy consumption Chen et al[10] designed an insect type hexapod robot and leg has a seriesmechanism type the optimal control of leg swing is solved bypseudospectral method Roy et al [11ndash14] explored kinematicdynamics and optimal control problems of hexapod robotthe hexapod robot is driven by electric motor so it has aseries mechanism type Luneckas et al [15] analyzed hexapodrobotrsquos energy consumption by motion of body and stepheight Deng et al [16] explored energy reducing problemof hexapod robot by kinematics analysis Gonzalez de Santoset al [17] explored minimization of hexapod robot in irreg-ular terrain the optimal analysis is based on statically stablegait Jin et al [18] explored hexapod walking robotrsquos powerconsumption optimization problem by torque distributionalgorithm and parameters include duty factor stride lengthbogy height and foot trajectory lateral offset Zhu et al [19]explored optimal design of hexapod robot with kinematicmodel

Fundamentally PHR is a parallel mechanism most ofoptimal control explorations of it are static or simplify itas serial mechanism so the real dynamics character cannotfaithfully represent itThe first reason is that complex dynam-ics character of parallel mechanism makes it difficult to usetriangle functions to calculate it and complex triangle andantitriangle transformations will lead to unsolvable modelSecondly complex nonlinear characters need optimal controlalgorithm that has high calculation accuracy and stability butclassic algorithms such as Runge-Kutta method do not satisfythese two characters

According to references of dynamics modeling by Liegroup [20 21] and optimal control with pseudospectralmethod [22 23] the matrix and vector operators can avoidtriangle and antitriangle transformations which makes dyna-mics modeling easier Pseudospectral method is a globalnumericalmethodwhich has high stability and is widely usedin many domains many engineering problems are solvedsuccessfully [24 25] So in this explorationmatrix and vectoroperators are used as units for dynamic modeling and opti-mal control problem is solved by pseudospectralmethodThecontrol inputs curves which satisfy pneumatic control char-acters are obtained and then cruising ability calculationmethod is built at last which offers a reference for theimprovement of robot

2 The Mechanism and Gait ofPneumatic Hexapod Robot

PHR is a biorobot so it has two design schemes insect typein Figure 1 and mammal type in Figure 2 Many people likeinsect type but it has some problems Firstly it walks along119883 direction by triangle gait of hip joints bear great yawingforces which will lead to jointsrsquo rapid abrasion and to notbeing suited for engineering application Secondly it has a bigwidth so crosswise passing ability is restricted Thirdly therealization of straight linewalking needs combinationmotionof three joints which is more difficult to realize by pneumaticsystem and has high gas consumption The mammal typePHR does not have these problems The nitrogen gas bottleis in trunk the maximum pressure can reach 15MPa The

1

3

5

6

ThighHip

Body

2

4

6 Shank

z

xo

y

Z

XY O

Figure 1 The insect type hexapod robot

O

Gas bottle

35

6

PRV

Body

Shank

Thigh

Magnetic valves

1 24

DP

z

xoy

Z

XY

Figure 2 The mammal type hexapod robot

high pressure gas is decompressed to 1MPa by PRV (pressurereducing valve) and then gas can be decomposed to differentlow pressures from 015MPa to 08MPa by DP (duplexpieces) All the magnetic valves and control devices canbe packaged in box which is convenient for antiexplosiondesign Each leg is composed of shank and thigh which aredriven by cylinder

According to comparison between Figures 1 and 2 eachleg ofmammal type hexapod robot has two jointswhich is lessthan the insect type Based on this mechanism the PHR canrealize straight line walking as gait in Figure 3 The gait canbe divided into four actions Firstly the shank drive cylindershrinks to make foot tip separate from ground secondlythe thigh drive cylinder stretches out to make the leg stepforward thirdly the shank drive cylinder stretches out tomake foot contact with ground again fourthly the thigh drivecylinder shrinks to make body move forward According tothe above gait analysis the two joints work at different timeand have no coincides So the swing of thigh and shankcan be treated as a parallel pendulum When body movesforward by support of legs PHR and ground form a closeloop According to straight line walking gait the movementof body is only decided by motion of thigh So the movement

Mathematical Problems in Engineering 3

igh of 1 4 5

igh of 2 3 6Shank of 1 4 5

Shank of 2 3 605TT

Figure 3 The straight line walking gait of PHR

12

3

45

Figure 4 The legrsquos motion process under straight walking gait

process can be expressed only by one parameter The motionprocess of straight line walking can be expressed as inFigure 4

3 Kinematics Modeling of PHR

In this part the pneumatic hexapod robotrsquos whole straightline walking kinematic model is built with matrix andvector operators The parallel pendulum kinematic modelthat corresponds to leg swing process is built at first Thenthe kinematics model of whole machine under straight linewalking process is derived as follows

31 The Kinematic Model of Leg Swing The mechanism andparameters of thigh and shank are given in Figure 5 Accord-ing to Figure 5 thigh and shank have samemechanismwhichconsists of rocker (11987411198743) and push rod 11987421198743 So thigh andshank can be expressed by same kinematic model Supposerotate angle of rocker along 1198741 is 1205791 rotate angle of push rodalong 1198742 is 1205792 and length of push rod is 119897 rotation matrix ofrocker and push rod can be written as

R119894 = [cos 120579119894 minus sin 120579119894sin 120579119894 cos 120579119894 ] 119894 = 1 2 (1)

The rotation matrix satisfies RR119879 = I2times2 detR = 1Suppose position vector of point1198743 on rocker is r1 = (119909119886 119910119886)

so the coordinate of 1198743 in inertial frame is R1r1 Supposeposition vector of 1198742 in inertial frame is r2 = (119909119887 119910119887)According to translate process as 1198741 rarr 1198742 rarr 1198743 thecoordinate of 1198743 in inertial frame is r2 + 119897R2e1 According tocoordinates of 1198742 1198743 and length between them the relationcan be expressed as

1003817100381710038171003817R1r1 minus r210038171003817100381710038172 = 1198972 (2)

Equation (2) can be unfolded as

r1198791 r1 minus 2r1198792R1r1 + r1198792 r2 = 1198972 (3)

Equation (1) can be decomposed by R119894 = 119901119894I + 119902119894119878(1)Bring it into (3) and 119901119894 119902119894 can be solved by nonlinearequations as (4) 1199012119894 + 1199022119894 = 1 is the constraint between 119901119894and 119902119894

119902119894119860 + 119861119901119894 = 1198621199012119894 + 1199022119894 = 1 (4)

In (4) 119860 = r1198792 119878(1)r1 119861 = r1198792 r1 119862 = (r1198791 r1 + r1198792 r2 minus 1198972)2According to (4) R1 can be calculated The relation of pushrod and rockerrsquos attitudes and length of push rod is

119897R2e1 = R1r1 minus r2 (5)

The attitude matrix satisfies the differential relation as R =R119878(120596) then differentiating (3) the velocity relation can bederived as

1205961r1198791 119878 (1)R1198791 r2 = 119897V (6)

In (6) 119897 = V After transposition of terms the relationbetween angular velocity of rocker and velocity of push rod iswritten as

1205961 = 119897r1198791 119878 (1)R1198791 r2 V (7)

Differentiating (3) the relation of pose attitude velocityand angular velocity is written as

1205962119897R2119878 (1) e1 = 1205961R1119878 (1) r1 minus VR2e1 (8)

Systemizing (8) the relation between angular velocity ofpush rod and rocker is written as

1205962 = 1205961 e1198792R1198792R1119878 (1) r1119897 (9)

Differentiating (6) and then systemizing it the angularacceleration of rocker is written as

1 = V2 + 119897119886 + 12059621r1198791R1r2r1198791 119878 (1)R1198791 r2 (10)

In (10) 119886 = V Differentiating (8) the angular accelerationof push rod is written as

2 = 12059621 e1198791 119878 (1)R1198792R1r1119897 minus 21205962V119897

minus 1 e1198791 119878 (1)R1198792R1119878 (1) r1119897

(11)


Y1

Y1

O1

O1

O3y1

y1

1205791

1205791

1205792

1205792

xb

xa

xax1x1

ya

l

l

yb

xb

X1

X1

O3

x2x2

O2

O2

Y2

Y2y2

y2

yaX2

X2

Figure 5 The parallel pendulum diagram of thigh and shank

O0

O1

Y0

Y1

X0

X1

x0

y0

y1x1

O2

YI

rI2

pB

XBOB

OIXI

rB0

YB

Figure 6 The mechanisms schematic diagram of pneumatic hexapod robot under straight line walking

32 KinematicModel of BodyMoving Themovement of bodyis expressed by pose of body framersquos origin 119874119861 relative toinertial frame 119874119868119883119868119884119868 as in Figure 6 Define p119861 and r1198682 asposition vectors of body and foot tip in inertial frame119874119868119883119868119884119868respectively r1198610 is position vector of thigh joint 1198740 in frame119874119861119883119861119884119861 r01 and r12 are position vectors of shank joint 1198741in thighrsquos body frame 119874011990901199100 and foot tip 1198742 in shankrsquos bodyframe 119874111990911199101 respectively Supposing that rotation matrixesof thigh and shank are R0 and R1 respectively so the closeloop relation of body thigh shank and ground is derived as

r1198682 minus r1198610 minus R0 (r01 + R1r12) = p119861 (12)

According to (12) and straight line walking gait theposition of body is decided by rotation of thigh for shank jointkeeps still during bodymovesThe velocity relation of leg andbody is derived out by differential calculation of (12) and theresult is

minusR0 (r01 + R1r12) minus R0R1r12 = p119861 (13)

Equation (13) can be written as an expansion type as

minus1205960R0S1 (r01 + R1r12) minus 1205961R0R1S1r12 = k119861 (14)

The acceleration relation of leg and body is derived bydifferentiating (14) and the result is

minus 0R0S1 (r01 + R1r12) minus 1205960R0S1 (r01 + R1r12)minus 1205960R0S1R1r12 minus 1R0R1S1r12 minus 1205961R0R1S1r12minus 1205961R0R1S1r12 = a119861

(15)

The neat type of (15) is

minus 1205720R0S1 (r01 + R1r12) minus 1205721R0R1S1r12+ 12059620R0 (r01 + R1r12) + (212059601205961 + 12059621)R0R1r12

= a119861(16)

4 Dynamics Modeling of PHR

41 Dynamics Model of Leg Swing With kinematics modelthe dynamics model of parallel single pendulum can be builtby Lagrange theory For the control objective is rocker sorotation angle 1205791 of rocker is chosen as generalized coordi-nate of dynamics system The complete expression form of


rotation matrix of push rod R2 can be derived by multiplyinge1198791 and e1198792 on both sides of (5) respectively It shows in

R2 = 1198991I + 1198992119878 (1)119897 (17)

In (17) 1198991 = e1198791R1r1 minus e1198791 r2 and 1198992 = e1198792R1r1 minus e1198792 r2 thenbring (17) into (9) which is the expression of 1205962 so it has anew type as (13)

1205962 = 1198993119897 1205961 (18)

In (18) 1198993 = 1198991e1198792R1119878(1)r1 minus 1198992e1198791R1119878(1)r1 Based on (6)V can be written as

V = 1205961r1198791 119878 (1)R1198791 r2119897 (19)

According to systemrsquos pose-attitude relation the kine-matic energy of system can be written as

1198791 = 12119869112059621 + 12 (1198692 + 1198693) 12059622 + 121198983V2 (20)

The potential energy of system can be written as

1198811 = 1198981119892e1198792R1r1198981 + (1198982 + 1198983) 119892e1198792 r2+ (11989821198921198971198982 + 1198983119892 (119897 minus 1198971198983))119897 1198992

(21)

Bringing (18) and (19) into (20) and (21) energy formulabased on R1 and 1205961 can be obtained According to Lagrangetheory dynamics equation of conservation system can bederived out by

119889119889119905 ( 1205971198711205971205961) minus (120597119871120597R1) = 0 (22)

After expanding dynamics equation the complete equa-tion of dynamics system can be expressed as

1 = minus1198991612059621 + 1198691198861205962119899119895 (11989912 minus 11989910) minus 1198983120596211198991411989991198972119899119895

minus 1198991511989913 minus 119898312059611198994119886119897119899119895 (1198995 minus 11989911) minus 1198983119892119899119895 1198994119899119898minus 1198981119892119899119895 11989917

(23)

In (23) the expressions of parameters are shown asfollows

119897119886 = r1198791 r1 + r1198792 r2119897119887 = r1198792R1r11198991119886 = e1198791R1r1119897119887 = r1198792R1r11198991119886 = e1198791R1r11198992119886 = e1198792R1r11198993119886 = e1198792R1119878 (1) r11198993119887 = e1198791R1119878 (1) r11198994119886 = r1198792R1119878 (1) r11198995119886 = r1198791R

1198791 r2

11989917 = e1198792R1119878 (1) r1198981 119897 = (119897119886 minus 2119897119887)12 1198994 = minus1198994119886119897 1198991 = 1198991119886 minus e1198791 r21198992 = 1198992119886 minus e1198792 r21198993 = 11989911198993119886 minus 11989921198993119887119899119898 = 1198992119897 1198995 = 1205961 1198995119886119897 + 119899411989941198861198972 119869119886 = 1198692 + 11986931198999 = minus1198994119886119897 1198998 = 11989931198871198993119886 minus 11989911198992119886 minus 11989931198861198993119887 + 11989921198991119886119899119895 = (1198691 + 119869119886 119899

231198972 ) + 1198983 (1198994119886119897 )

2 11989910 = 12059611198972 (1198998119897 minus 11989941198993)11989911 = 12059611198972 (1198995119886119897 + 11989941198994119886) 11989912 = 1205961 1198998119897 minus 119899411989931198972 11989913 = (1198993119886119897 minus 11989921198994)1198972 11989914 = 1198995119886119897 + 1198994119899411988611989915 = (11989821198921198971198982 + 1198983119892 (119897 minus 1198971198983))119899119895

11989916 = 1198691198861198993 (1198998119897 minus 11989941198993)1198973119899119895

(24)


Ta0

120579a0

Fa1

Fay

Fax

Fa0

Ta1

120579a1

Tb0

120579b0

Fb1

Fb0

Fby

Fbx

120579b1Tb1

Fc0

YB

OB XB

120579c0

Tc0

120579c1

Tc1

Fc1

Ot

Yt

rfra

Fcyy

Y

XOFcx

x

yt

Xt

M

ℎM

Yd

Od

Xd

ydxd

xt

120579t120579d

120579s

120579

Figure 7 The mechanisms of PHRrsquos whole machine and the equivalent type under straight line gait

42 Dynamics Model of Body Moving The straight line walk-ing of PHR is realized by rotation of thigh joints and shankjoints keep still duringmotion process According to Figure 7PHR is supported by three legs contact forces are on feet tipswhich are F119886 = [119865119886119909 119865119886119910] F119887 = [119865119887119909 119865119887119910] F119888 = [119865119888119909 119865119888119910]respectively Thigh joints are driven by 1198791198860 1198791198870 1198791198880 shankjoints are driven by 1198791198861 1198791198871 1198791198881 The torque corresponds todrive forces of cylinders When shank joints keep still shankthigh and shank drive cylinder can be handled as one unit Sodrive forces of shank have no influence on the walking Thesix legs of PHRhave samemechanismcharacter and bodyhasno rotation during straight line walking so rotation anglesof thighs are equal which means 1205791198860 = 1205791198870 = 1205791198880 1205791198861 =1205791198871 = 1205791198881 = 119862 in Figure 7 Supposing that feet tips have nomotion relative to ground so the mechanism of the wholemachine can be equivalent to themechanism type as the rightpart in Figure 7 In this mechanism displacement of bodyis equal to displacement of thigh joint in inertial frame anddynamics character of whole machine can be expressed byone parameter

Define 119898119887 as mass of body mass center is defined at119872For body has no rotation during straight line walking sopotential energy is only related to vertical distance betweenmass center and hinge joint1198981 is gathermass of shank thighand drive cylinder of shank 1198982 1198983 are masses of cylindertube and rod respectively Define 1198691 as inertia moment of legunit which rotates along foot tip119874 1198692 1198693 are inertia momentsof drive cylinderrsquos tube and rod along119874119889 and119874119886 respectivelySuppose r119891 is position vector of point 119874119905 in body frame 119874119909119910of leg unit and position vector of 119874119886 in frame 119874119909119910 is r119886 Sokinetic energy and potential energy of leg unit are written as

1198791 = 121198691 12057921198811 = 1198981119892e1198792R (120579) r1198981

(25)

In formula (25) 120579 = 120587 minus 120579119905 minus 120579119904 so 120579 = minus 120579119905 = minus120596119905 120596119905 isangular velocity of thigh and r1198981 is position vector of masscenter The position of 119874119905 in frame 119874119883119884 is written as

r119905 = R (120579) r119891k119905 = minus120596119905R (120579) s1r119891r119886 = R (120579) r119886k119886 = minus120596119905R (120579) s1r119886

(26)

So kinetic energy and potential energy of body is writtenas

1198792 = 12119898119887k2119905 = 121198981198871205962119905 (R (120579) s1r119891)119879 (R (120579) s1r119891)

= 121198981198871205962119905 10038171003817100381710038171003817r119891100381710038171003817100381710038172

1198812 = 119898119887119892e1198792 (R (120579) r119891 + ℎ119872e2) (27)

The motion of drive cylinder is analyzed as follows Themotion of drive cylinder consists of motion of cylinder tubeand rod As in Figure 7 the tube connects with thigh and rodconnects with body so the motion of tube is combinationof displacement of point 119874119886 and rotation along 119874119886 Themotion of rod is combination of displacement of point119874119889 androtation along119874119889 So kinetic and potential energy of cylindertube are written as (28) and the rodrsquos are written as (29)

1198793 = 121198982k2119886 + 1211986921205962119889 = 12 (1198982 1003817100381710038171003817r11988610038171003817100381710038172 1205962119905 + 11986921205962119889)1198813 = 1198982119892e1198792 (R (120579) r119886 + 1198971198982R (120579119889) e1)

(28)


1198794 = 121198983k2119905 + 1211986931205962119889 = 12 (1198982 10038171003817100381710038171003817r119891100381710038171003817100381710038172 1205962119905 + 11986921205962119889)

1198814 = 1198983119892e1198792 (R (120579) r119891 + r119889 + 1198971198983R (120587 minus 120579119889) e1) (29)

For straight line walking is realized by three legsrsquo motion soLagrange function of whole system is written as

119871 = 119879 minus 119881 = (31198791 + 1198792 + 31198793 + 31198794) minus (31198811 + 1198812+ 31198813 + 31198814) = 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817

2 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172+ 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)1205962119905 + 32 (1198692 + 1198693) 1205962119889minus [31198981119892e1198792R (120579) r1198981 + 119898119887119892e1198792R (120579) r119891+ 31198983119892e1198792R (120579) r119891 + 31198982119892e1198792R (120579) r119886]+ 311989831198921198971198983e1198792R119879 (120579119889) e1 minus 311989821198921198971198982e1198792R (120579119889) e1minus 119898119887119892ℎ119872 minus 31198983119892e1198792 r119889

(30)

According to kinematic character of parallel pendulumR(120579119889) can be expressed by R(120579119905) and 120596119889 can be expressed by120596119905 and R(120579119905) The concrete expressions are in

R119889 = 1198991198891I + 1198991198892119878 (1)119897120596119889 = 11989911988931198972 1205961199051198991198891 = minuse1198791R119905r1 + e1198791 r2

1198991198892 = e1198792R119905r1 minus e1198792 r21198991198893= (minuse1198791R119905r1e2 + e1198791 r2e2 minus e1198792R119905r1e1 + e1198792 r2e1)119879R119905s1r1r1198791 r1 minus 2r1198792R119905r1 + r1198792 r2 = 1198972

(31)

In (31)R119905 = R(120579119905) R119889 = R(120579119889) Lagrange function can bewritten as (32) with (31)

119871 = 119871 (120596119905R119905)= 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817

2 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)1205962119905+ 32 (1198692 + 1198693) (11989911988931198972 )

2 1205962119905 + (R119905119904e2)119879sdot [31198981119892r1198981 + 119898119887119892r119891 + 31198983119892r119891 + 31198982119892r119886]minus (311989831198921198971198983 + 311989821198921198971198982) 1198991198892119897 minus 119898119887119892ℎ119872minus 31198983119892e1198792 r119889

(32)

Ta0

Fy1205790

1205793

1205792

Fx

F

Figure 8 The force of the single joint

In (32) R119905119904 = R(120579119905 + 120579119904) According to Lagrange theorythe dynamics equation of system is obtained as

119889119889119905 120597119871120597120596119905 minus120597119871120597120579119905 = 1198961119905 + 1198962120596

2119905 + 1198963

1198961 = [(31198691 + 119898119887 10038171003817100381710038171003817r119891100381710038171003817100381710038172 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)+ 3 (1198692 + 1198693) (11989911988931198972 )

2]1198962 = [6 (1198692 + 1198693) (11989911988931198972 )(1205971198991198893120597120579119905

11198972 +2r1198792R119905s1r111989911988931198974 )]

minus 3 (1198692 + 1198693) (11989911988931198972 )(120597119899119889312059712057911990511198972 +

2r1198792R119905s1r111989911988931198974 )

1198963 = (311989831198921198971198983 + 311989821198921198971198982) (12059711989911988921205971205791199051119897 +

r1198792R119905s1r11198973 )minus (R119905119904s1e2)119879 [31198981119892r1198981 + 119898119887119892r119891 + 31198983119892r119891+ 31198982119892r119886]

(33)

In (33) 1205971198991198893120597120579119905 = (minuse1198791R119905s1r1e2 minus e1198792R119905s1r1e1)119879R119905s1r1 minus(e1198792 r2e1 + e1198791 r2e2 minus e1198791R119905r1e2 minus e1198792R119905r1e1)119879R119905r1 1205971198991198892120597120579119905 =e1198792R119905s1r1minuse1198792 r2 r1198791 r1minus2r1198792R119905r1+r1198792 r2 = 1198972 So (33) is dynamicsequation of PHR under straight line walking

43 The Force Analysis The dynamics models of leg swingand straight line walking are built in Sections 31 and 32respectively According to dynamics models rotation anglesof joints are chosen as generalized coordinates Actually thejoints are driven by cylinders so the relation between jointdrive torques and cylinder drive forces should be constructed

In order to construct the relation between joint torqueand cylinder drive force the parallel pendulum structure ofthigh joint is used as example in Figure 8 Supposing thecylinderrsquos push force is 119865 the angle between push rod andthigh is 1205793 According to triangle character 1205793 = 120587minus1205790minus1205792 two


components of cylinder drive force on orthogonal directionsof thigh are

119865119909 = sin (1205790 + 1205792) 119865 = (sin 1205790 cos 1205792 + cos 1205790 sin 1205792) 119865119865119910 = minus cos (1205790 + 1205792) 119865= minus (cos 1205790 cos 1205792 minus sin 1205790 sin 1205792) 119865

(34)

So the torque on joint is

119879 = 119865119909119903119909 + 119865119910119903119910 = 119865 [(sin 1205790 cos 1205792 + cos 1205790 sin 1205792) 119903119909minus (cos 1205790 cos 1205792 minus sin 1205790 sin 1205792) 119903119910] (35)

According to rotation relation suppose thatR0 andR2 are

R0 = [119888 (minus1205790) minus119904 (minus1205790)119904 (minus1205790) 119888 (minus1205790) ] = [1198881205790 1199041205790minus1199041205790 1198881205790]

R2 = [119888 [minus (120587 minus 1205792)] minus119904 [minus (120587 minus 1205792)]119904 [minus (120587 minus 1205792)] 119888 [minus (120587 minus 1205792)] ]

= [minus1198881205792 1199041205792minus1199041205792 minus1198881205792]

(36)

So the projections of hinge joint on thigh and drive forcevector are as in

R0r1 = [ 1198881205790 1199041205790minus1199041205790 1198881205790][

119903119909119903119910] = [

1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]

119865R2e1 = 119897 [minus1198881205792 1199041205792minus1199041205792 minus1198881205792][

10] = 119865[

minus1198881205792minus1199041205792]

(37)

The cross product of (37) is

119879 = 119865[ 1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]119879

[0 minus11 0 ] [

minus1198881205792minus1199041205792]

= 119865 [119903119909 (11988812057901199041205792 + 11990412057901198881205792) minus 119903119910 (11988812057901198881205792 minus 11990412057901199041205792)] (38)

According to analysis on geometry the joint torqueis cross product of legrsquos hinge pointrsquos position vector andcylinder direction vector which can be expressed as matrixtype as in

120591 = (R0r1)119879 s1 (119865R2e1) (39)

According to the relation between R2 and R0 (39) can bewritten as

120591 = 119865119897 (R0r1)119879 s1 (1198991e1 + 1198992s1e1)= 119865119897 (R0r1)119879 (1198991e2 minus 1198992e1)

(40)

Furthermore the drive force is generated by gas and therelation is 119865 = 12058711988921199014 119889 is cylinder bore (mm) and 119901is gas pressure (MPa) When cylinder is confirmed the gasconsumption is only influenced by pressure 119901

5 The Optimal Control withPseudospectral Method

Themain character of pseudospectral method is that the stateand control variables of ordinary differential equations arediscrete on Legendre-Gauss points The discrete points areused as nodes to construct Lagrange interpolating polyno-mial which is used to approximate state variables and controlvariables The derivatives of state variables are approximatedby differentiating the overall interpolating polynomial sothat differential equation constraints are changed to bealgebra constraints The integral part of performance indexis calculated by Gauss integral From above transformationsthe optimal control problem is translated to be a nonlinearprogramming problem with a series of algebra constraints

51 The Problem Description According to (23) and (40)state equation of parallel pendulum which represents legswing is written as (41) According to (23) and (33) stateequation of robot during straight line walking is as in (42)

1205791 = 12059611 = minus1198991612059621 + 1198691198861205962119899119895 (11989912 minus 11989910) minus 1198983

1198991411989991198972119899119895 12059621

minus 1198991511989913 minus 119898312059611198994119886119897119899119895 (1198995 minus 11989911) minus 1198983119892119899119895 1198994119899119898minus 1198981119892119899119895 11989917 +

12058711988921199014119897119899119895 (R1r1)119879 (1198991e2 minus 1198992e1)

(41)

120579119905 = 120596119905119905 = 1198651198961119897 (R119905r1)

119879 (1198991198891e2 minus 1198991198892e1) minus 119896211989611205962119905 minus 11989631198961

(42)

The common optimal control problem can be describedas searching the control variable u(119905) which satisfies mini-mum objective function In (41) and (42) state variables arex(119905) = [1205791 1205961] and x(119905) = [120579119905 120596119905] respectively control inputis gas pressure 119901 so u(119905) = [0 119901] The minimum objectivefunction is

119869 = Φ (x (1199050) 1199050 x (119905119891) 119905119891) + int1199051198911199050

119892 (x (119905) u (119905)) 119889119905 (43)

In (43) state variable x(119905) initial time 1199050 and end time 119905119891satisfy dynamics equation as (44) which represents ordinarydifferential equations as (41) and (42)

x (119905) = f [x (119905) u (119905) 119905] (44)

The boundary conditions are 120601(x(1199050) 1199050 x(119905119891) 119905119891) = 0In this exploration boundary conditions include initialterminal and boundary values of rotation angle 120579 and angularvelocities 120596 The control constraint is written as C(x(119905)u(119905) 119905) le 0 In this exploration the constraint is variationboundaries of input gas pressure


52 The Time Domain Transformation Before using Gauss-pseudospectral method the time interval of optimal controlshould be transformed from 119905 isin [1199050 119905119891] to 120591 isin [minus1 1] firstThe process is shown as

120591 = 2119905119905119891 minus 1199050 minus119905119891 + 1199050119905119891 minus 1199050 (45)

The transformation process for minimum performanceindex is written as

119869 = Φ (x (minus1) 1199050 x (1) 119905119891)+ 119905119891 minus 11990502 int1

minus1119892 (x (120591) u (120591) 120591) 119889120591 (46)

The dynamics differential equation constraints can betransformed to be

x (120591) = 119905119891 minus 11990502 f [x (120591) u (120591) 120591] 120591 isin [minus1 1] (47)

The boundary condition 120601(x(minus1) 1199050 x(1) 119905119891) = 0The path constraints C(x(120591) u(120591) 120591) le 0

53 The State and Control Variables Approximated by theOverall Interpolating Polynomial Gauss-pseudospectralmethod uses 119899 Legendre-Gauss points and 1205910 = minus1 asnodes which forms 119899+1 Lagrange interpolating polynomials119871 119894(120591) 119894 = 0 119899 as primary function to approximate thestate variables as in

x (120591) asymp 119883 (120591) = 119899sum119894=0

119871 119894 (120591) x (120591119894) (48)

In (48) base function of Lagrange interpolating polyno-mials can be expressed as (49) which makes approximatestate on nodes equal to virtual conditions as x(120591119894) asymp119883(120591119894) 119894 = 0 119899

119871 119894 (120591) = 119899prod119895=0119895 =119894

120591 minus 120591119895120591119894 minus 120591119895 (49)

The Lagrange interpolating polynomials are used as basisfunction for approximate control variables as

u (120591) asymp U (120591) = 119899sum119894=1

119871 119894 (120591)U (120591119894) (50)

In above equations 120591119894 119894 = 1 119899 are Legendre-Gausspoints

54 The Transformation of Differential Constraints to AlgebraConstraints Differentiating state variable dynamics differ-ential equation constraints can be transformed to be algebraconstraints as

x (120591119896) asymp X (120591119896) = 119899sum119894=0

119894 (120591119896) x (120591119894) = 119899sum119894=0

D119896119894 (120591119896) x (120591119894) (51)

The expression of differential matrix is written as

D119896119894 = 119894 (120591119896)

=

(1 + 120591119896) 119899 (120591119896) + 119875119899 (120591119896)(120591119896 minus 120591119894) [(1 + 120591119894) 119899 (120591119894) + 119875119899 (120591119894)] 119894 = 119896(1 + 120591119894) 119873 (120591119894) + 2119873 (120591119894)2 [(1 + 120591119894) 119873 (120591119894) + 119875119873 (120591119894)] 119894 = 119896

(52)

In (52) 119896 = 1 119899 119894 = 0 119899 From the above trans-formations dynamics differential constraints are translated tobe algebra constraints

119899sum119894=0

D119896119894 (120591119896)X (120591119894)

minus 119905119891 minus 11990502 f (X (120591119896) U (120591119896) 120591119896 1199050 119905119891) = 0(53)

55 The Terminal State Constraints under Discrete ConditionFor nodes of Gauss-pseudospectral method excludes endpoint 120591119891 = 1 so the terminal state X119891 is not definite indynamics differential equation constraintsThe terminal stateshould satisfy dynamics constraints as

x (120591119891) = x (1205910) + int1minus1f (x (120591) u (120591) 120591) 119889120591 (54)

The termianl constraints are discreted and approximatedby the Gauss integral method which can be written as (55)terminal constraint is written as

X (120591119891) = X (1205910)+ 119905119891 minus 11990502

119899sum119896minus1

119908119896f (X (120591119896) U (120591119896) 120591 1199050 119905119891) (55)

In (55) 119908119896 = int1minus1119871 119894(120591)119889120591 is Gauss weight 120591119896 is Legendre-

Gauss points

56 The Performance Index under Discrete Condition Inte-gral parts of performance index are approximated by Gaussintegral and performance index with pseudospectral typecan be obtained as

119869 = Φ (X0 1199050X119891 119905119891)+ 119905119891 minus 11990502

119899sum119896=1

119908119896119892 (X119896U119896 120591119896 1199050 119905119891) (56)

Therefore the continuous optimal control problem istransformed to be a nonlinear programming problem withdiscrete work of pseudospectral method Then discrete con-trol and state variables can obtain a minimum performanceindex which satisfies state constraints terminal constraintsboundary conditions 120601(X0 1199050X119891 119905119891) = 0 and path con-straints C(X119896U119896 120591119896 1199050 119905119891) le 0


57 The Optimal Control of PHR The gas consumption opti-mal control of PHR under straight line walking has two partsThe first part is gas consumption optimal control of leg swingwhen foot has no contact with ground which correspondsto state equation (41) The second part is gas consumptionoptimal control of body moving by support of legs in whichfeet contact with ground This part corresponds to stateequation (42) The sum of these optimal control results isgas consumption of one step For the goal of optimal controlis minimum of gas consumption so objective function is119869 = int1199051198911199050119901119889119905 The optimal control of PHR can be expressed as

searching control input 119901 to make system move from initialcondition x(1199050) = x0 to terminal condition x(119905119891) = x119891under minimum energy consumption and satisfy a certainof constraints in a given time interval The process can bewritten as follows

The Functional Extreme Value Problem of Optimal Control

The performance index 119869 = int11990511989111990501199012119889119905

The constraints of initial value x(1199050) = x0The state equation x = f[x u 119905]The constraints of control umin le u le umaxThe constraints of states xmin le x le xmaxThe boundary conditions 119905 le 119905119891 119909(119905119891) le x119891

6 The Kinematics Analysis

The kinematic process of PHRrsquos straight line walking isanalyzed in this part The structure parameters of PHR areas follows The parameters are obtained from the 3D modelof PHR as Figure 2 In order to verify the correctness ofmathematicalmodel the 3Dmodel is kinematic simulated byADAMS and the simulation results are used as criterions forthe correctness of numerical results of mathematical modelThe numerical solution path is designed as follows

The Numerical Solution Path

The initialization of the variables 119897 = 1198970 V = V0 119886 =1198860For loop

solve the following formulas as sequence (3)(4) (6) (8) (9) and (10)calculate the following parameters R1R2 12059611205962 1 2the initial value update is as follows 119897 = 119897(119905) V =V(119905) 119886 = 119886(119905)

End

The structure parameters of thigh are r1 = [11155825] r2 = [2725 3175] the length of thigh is 250mmThe structure parameters of shank are r1 = [1555 minus5825]r2 = [minus1975 minus5825] and r119891 = [463 2165] the length ofshank is 338mm The initial length of thigh drive cylinder

minus600

minus500

minus400

minus300

minus200

minus100

0

y(m

m)

minus300 minus200 minus100minus400 100 2000x (mm)

Figure 9 The pose-attitude variation of leg

minus90

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

120579 s(∘)

120579 t(∘)

Figure 10 The jointsrsquo rotation angle variation

is 198mm and shankrsquos is 248mm The cylinderrsquos stroke is50mm For the pneumatic experiment has not proceeded theacceleration of cylinder motion is supposed as 10000mms2in simulation For magnetic valversquos minimum action time is01 s so the action time of cylinder is supposed to be 01 sAccording to the above parameters and motion relationsthe variation curves of legrsquos kinematics parameters are as inFigures 9ndash15

The track of foot tip and shank joint is expressed inFigure 9The black blue and red curves are the tracks of footswing process The coordinates of initial and terminal pointsare [minus65 506] and [minus2558 minus4455] respectively These twocoordinates indicate that the forward distance of one stepis 190mm and the difference of coordinates on 119910 directionindicates that body has 119910 direction motion during foottransformation which is 60mm


minus35

minus30

minus25

31

32

33

34

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

120579 s2(∘)

120579 t2(∘)

Figure 11 Rotation angle of drive cylinder

minus10

minus5

0

5

120596t

(rad

s)

005 01 015 02 025 030Time (sec)

minus10

0

10

120596s

(rad

s)

005 01 015 02 025 030Time (sec)

Figure 12 Angular velocity of thigh and shank joints

minus05

0

05

1

120596t2

(rad

s)

005 01 015 02 025 030Time (sec)

minus2

0

2

4

120596s2

(rad

s)

005 01 015 02 025 030Time (sec)

Figure 13 Angular velocity of drive cylinder

minus100

minus50

0

120572t

(rad

s2)

005 01 015 02 025 030Time (sec)

minus100

0

100

120572s

(rad

s2)

005 01 015 02 025 030Time (sec)

Figure 14 Angular acceleration of joints

minus50

0

50120572t2

(rad

s2)

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

minus50

0

50

120572s2

(rad

s2)

Figure 15 Angular acceleration of drive cylinders

The rotation angles variation curves of thigh and shankare expressed in Figure 10 According to Figure 10 theinitial and terminal angles of thigh joint are minus622∘ andminus8556∘ the variation range of thigh joint is 2336∘ andthe three key values of shank joint are minus60∘ minus7964∘ and1964∘ respectivelyThe curves are smooth and have parabolacharacter which means that the rotation of thigh and shankjoints have stable acceleration which is identical to themotion character of drive cylinders

The rotation angles variation curves of drive cylindersof thigh and shank are expressed in Figure 11 According toFigure 11 the initial and terminal rotation angles of thighjointrsquos drive cylinder are 3142∘ and 3392∘ the variation rangeof it is 25∘ and the three key values of shank jointrsquos drivecylinder are minus2515∘ minus3203∘ and 687∘ respectively Thevariation ranges are far smaller than main joints

The angular velocities variation curves of thigh and shankjoints are expressed in Figure 12 The variation range of


thigh jointrsquos angular velocity is 7552 rads and shank jointrsquosvariation ranges are 6898 rads and 6192 rads on differentrotation directions The curves are smooth and variationtendencies are similar to a straight line which means thegradients of angular velocity are constant

The angular velocities variation curves of drive cylindersare expressed in Figure 13 According to Figure 13 thevariation curve of thigh drive cylinderrsquos angular velocity hasa parabola character and the maximum value is 06729 radsthe variation ranges of shank drive cylinderrsquos angular veloci-ties are 1958 and 2063 radsThe values and variation rangesof drive cylinders angular velocities are far smaller than jointsexpressed in Figure 12

The angular acceleration variation curves of thigh andshank joints are expressed in Figure 14 According to Fig-ure 14 the variation range of thigh jointrsquos angular accel-eration is between minus8699 rads2 and minus8026 rads2 whichhas a small variation range The variation ranges of shankjoint drive cylinderrsquos angular acceleration on two differentmotion directions which are [minus7317 minus5433] rads2 and[6488 9257] rads2 respectively The variation curves aresmooth and continuous which means the motion of thighand shank joints is second-order continuous

The angular acceleration of drive cylinders is shown inFigure 15 The variation range of thigh joint drive cylin-derrsquos angular acceleration is [2245 minus2938] rads2 and shankjointrsquos is [minus2779 0] rads2 and [2022 4147] rads2 on twodirections respectively This means that although rotationangels and angular velocities of the drive cylinders are smallthe variation processes are rapid

The displacement velocities and acceleration variationcurves of body which moves by support of legs are expressedin Figures 16 17 and 18 respectively According to Figure 16the initial and terminal positions of thigh joint relative to foottip are [2585 4409]mm and [524 5084]mm respectivelyThe variation curve is a smooth arc According to two coordi-nates the forward displacement of one step is 2061mm andthe displacement on vertical direction is 675mm Accordingto Figure 17 the velocityrsquos maximum value on 119909 direction is448ms and on 119910 direction is minus0461ms the velocity on 119909direction is 10 times the velocity on 119910 direction Accordingto Figure 18 the maximum values of acceleration on 119909 and119910 directions are 7298ms2 and 3237ms2 respectively andmaximum values are occurring at the end of motion process

According to kinematics analysis the conclusions can besummarized as follows

(1) The variation curves of angles angular velocities andangular acceleration are all smooth which meansthe straight line walking process has second-ordersmooth character

(2) The rotation angles velocities and acceleration valuesof thigh and shank joints are much larger than that ofdrive cylindersThismeans that themechanism of legcan be improved to be the type where drive cylindershave no rotation during motion process in the future

(3) The maximum values of angular velocities and angu-lar acceleration appear at the end of motion so in

0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)

Figure 16 The track of thigh joint with body moving

0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)

Figure 17 The velocity of mass center

times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

Figure 18 The acceleration of mass center


the actual engineering design the accelerate motionat the end of cylinders stroke should be avoided

(4) The straight line walking distance of one step is 02mand the process needs 4 cylinder strokesThe gas con-sumption volumes of each stroke are product of cylin-der inner area and stroke distanceThe front 3 strokesonly need to drive a small mass so they only need alow pressure the 4th stroke needs to drive the massof the whole machine so it needs a high pressureThepressure values of each stroke are calculated in thefollowing part with optimal control method

7 The Optimal Control Analysis

In order to analyze dynamics and control of PHR the pseudo-spectral optimal control method is used to solve dynamicsequations of straight line walking as (36) and (37) With thiscalculation the variation curves of motion parameters asrotation angles angular velocities and angular accelerationof straight line will be obtained and control input pressurescurves of each action will also be obtained With thesecurves the dynamics and control of PHR can be syntheticallyanalyzed

According to straight line gait the first step is upliftprocess of shank The mass and inertia of shank are 119898119904 =08154 kg and 119869119904 = 00492 kgsdotm2 and the mass centerrsquos posi-tion vector is r119898119904 = [0223 0]m According to the kinematicsanalysis results the initial length of cylinder is 248mmso initial rotation angle of shank joint is 60∘ and terminallength of cylinder is 198mm so terminal angle of shankjoint is 7964∘ initial and terminal values of (36) are 120579intial1 =1205873 120596intial

1 = 0 120579end1 = 044120587 120596end1 = 0 variation ranges

of 1205791 and 1205961 are 1205873 le 1205791 le 044120587 and minus10 le 1205961 le 10respectively The control input pressure range is 015MPa le119901 le 08MPa for minimum and maximum valid pressures ofmagnetic valve are 015MPa and 08MPa

The second step is rotation of thigh joint In this stepshank joint keeps still so thigh and shank can seem as awhole The mass and inertia of this whole part are 119898119905119904 =1576 kg and 119869119905119904 = 01276 kgsdotm2 and the mass centerrsquos posi-tion vector is r119898119904 = [02 0133]m The initial length ofcylinder is 198mm so initial rotation angle of shank joint is622∘ and terminal length of cylinder is 248mm so terminalangle of shank joint is 8556∘ the initial and terminal values of(36) are 120579intial1 = 0346120587 120596intial

1 = 0 120579end1 = 0475120587 120596end1 = 0

the variation ranges of 1205791 and 1205961 are 034120587 le 1205791 le 048120587 andminus10 le 1205961 le 10 respectively Control input pressure range isas the first step

The third step is rotation of shank joint The parametersof this step are identical to the first step the initial andterminal values of (36) are 120579intial1 = 044120587 120596intial

1 = 0 120579end1 =1205873 120596end1 = 0

The fourth step is body moving with rotation of thighjoint In this step foot tip contacts ground and shank keepsstill The mass of body is 119898119887 = 30 kg the mass of leg is1198981 = 1576 kg the moment of inertia along the foot tip is

1198691 = 01247 kgsdotm2 the mass center position is r1198981 = [02270059]m the initial and terminal values of (37) are 120579intial119905 =0475120587 120596intial119905 = 0 120579end119905 = 0346120587 120596end

119905 = 0 and variationranges of 120579119905 and120596119905 are 034120587 le 120579119905 le 048120587 andminus10 le 120596119905 le 10respectively

The drive cylinder two partsrsquo mass moment of inertiaand distance of mass center to hinge joint are as follows1198981 = 0257 kg 1198691 = 53 times 10minus4 kgm2 1198971198981 = 0069m 1198982 =0043 kg 1198692 = 215 times 10minus4 kgm2 1198971198982 = 0058m

The 4 steps of one straight line walking gait are solved bypseudospectral method and simulation results are expressedin Figures 19ndash22 According to Figure 19 the uplift process ofshank only needs a low pressure as 015MPa but the controltime only needs 008 s as the left graph of Figure 9 If thecontrol time is 01 s the control input pressure only needs01MPa On point of engineering the magnetic valve will notact if the pressure is lower than 015MPa so the first optimalcontrol result as in Figure 19 meets the need of engineeringapplication However the lowest action time of magneticvalve is 01 s so impact may occur during the experimentAccording to Figure 20 thigh joint has a stable motionprocess when input pressure is 015MPa and control time is01 s

The optimal control results of third step are shown inFigure 21 which are similar to results as in Figure 19 andcontrol input pressure is also 015MPa The optimal controlresults of fourth step are shown in Figure 22 When the inputpressure is 06MPa and control time is 01 s variation ofrotation angle is not smooth andwhen control input pressureis 05MPa and control time is 03 s the variation of rotationangle and angular velocity are smooth so control input is05MPa with control time 03 s being the best choice

According to the above optimal control calculationresults the conclusions can be summarized as follows

(1) The leg swing only needs a low gas pressure as015MPa and body moves by support of legs needpressure of 05MPa with mass of whole machinebeing 30 kg

(2) The optimal control results indicate that it needs atleast two gas pressure stages to be designed in practi-cal engineering design

(3) The gas consumption of one gait is 015MPa times (1205874)1198892times3times3+05MPatimes(1205874)1198892 = 185MPatimes(1205874)1198892 =004MPasdotL with mass of whole machine being 30 kgand inner diameter of cylinder being 32mm In thisexploration the volume and gas pressure of highpressure bottle are 8 L times 15MPa so the carried highpressure gas can support the PHR to walk 3000 stepsstraightly According to kinematic analysis the for-ward distance of one step is 02m so the cruisingability of PHR is 600m

8 Conclusion

In this paper kinematic dynamics and optimal controlproblemof PHR are explored and cruising ability of designed


0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)

Figure 19 The optimal control results of shank joint on the first step

minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)

Figure 20 The optimal control results of thigh joint on the second step

PHR is analyzed According to this exploration the conclu-sions can be summarized as follows

(1) Matrix and vector operator are a good modelingmethod that can replace triangle method whichmakes kinematic and dynamic modeling of complex

parallel mechanism easier In this exploration kine-matic and dynamic model of PHR are built by matrixand vector operators which are successfully solved

(2) Pseudospectralmethod is convenient to solve optimalcontrol problems of nonlinear dynamics systems


0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)

Figure 21 The optimal control results of shank joint on the third step

minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)

Figure 22 The optimal control result of thigh joint on the fourth step


for its high accuracy In this exploration dynamicsequation of PHR is successfully solved and optimalinput gas pressures of different actions in one gaitare obtained which offers guidance for engineeringdesign

(3) Cruising ability of PHR is influenced by kinematicand dynamics characters synthetically Forward dis-tance of one step is obtained by kinematics analysisand gas pressures are obtained by dynamics analysisThe results indicate that cruising ability of designedPHR satisfies engineering needs

Competing Interests

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

Acknowledgments

The exploration is supported by the Natural Science Founda-tion of China (11472058)

References

[1] BVerrelst B Vanderborght J Vermeulen RVHam JNaudetand D Lefeber ldquoControl architecture for the pneumaticallyactuated dynamic walking biped lsquolucyrsquordquo Mechatronics vol 15no 6 pp 703ndash729 2005

[2] M Lavoie and A L Desbiens ldquoDesign of a cockroach-like run-ning robot for the 2004 SAE walking machine challengerdquo inClimbing and Walking Robots pp 311ndash318 Springer 2006

[3] T Morimoto M Aliff T Akagi and S Dohta ldquoDevelopmentof flexible haptic robot arm using flexible pneumatic cylinderswith backdrivability for bilateral controlrdquo in Proceedings ofthe 3rd International Conference on Intelligent Technologies andEngineering Systems (ICITES rsquo14) vol 345 of Lecture Notes inElectrical Engineering pp 231ndash237 Springer 2016

[4] H Qiu S Dohta T Akagi S Shimooka and S Fujimoto ldquoAna-lytical model of pipe inspection robot using flexible pneumaticcylinderrdquo in Proceedings of the 3rd International Conference onIntelligent Technologies and Engineering Systems (ICITES rsquo14)vol 345 of Lecture Notes in Electrical Engineering pp 325ndash334Springer International Publishing Cham 2016

[5] J A Diez F J Badesa L D Lledo et al ldquoDesign and develop-ment of a pneumatic robot for neurorehabilitation therapiesrdquoin Robot 2015 Second Iberian Robotics Conference vol 418 ofAdvances in Intelligent Systems and Computing pp 315ndash326Springer 2016

[6] F-Z Low H H Tan J H Lim and C-H Yeow ldquoDevelopmentof a soft pneumatic sock for robot-assisted ankle exerciserdquo Jour-nal of Medical Device vol 10 no 1 Article ID 014503 2016

[7] M Ramsauer M Kastner P Ferrara R Naderer and HGattringer ldquoA pneumatically driven stewart platform used asfault detection devicerdquo Applied Mechanics and Materials vol186 pp 227ndash233 2012

[8] M F Sliva and J A T Machado ldquoA literature review on theoptimization of legged robotsrdquo Journal of Vibration and Controlvol 18 no 12 pp 1753ndash1761 2011

[9] D Sanz-Merodio E Garcia and P Gonzalez-De-Santos ldquoAna-lyzing energy-efficient configurations in hexapod robots for

demining applicationsrdquo Industrial Robot vol 39 no 4 pp 357ndash364 2012

[10] J Chen Y Liu J Zhao H Zhang and H Jin ldquoBiomimeticdesign and optimal swing of a hexapod robot legrdquo Journal ofBionic Engineering vol 11 no 1 pp 26ndash35 2014

[11] S S Roy and D K Pratihar ldquoDynamic modeling of energy effi-cient crab walking of hexapod robotrdquo Applied Mechanics andMaterials vol 110-116 pp 2730ndash2739 2012

[12] S S Roy P S Choudhury andD K Pratihar ldquoDynamicmodel-ing of energy efficient hexapod robotrsquos locomotion over gradientterrainsrdquo in Trends in Intelligent Robotics vol 103 pp 138ndash145Springer 2010

[13] S S Roy and D K Pratihar ldquoEffects of turning gait parameterson energy consumption and stability of a six-legged walkingrobotrdquo Robotics and Autonomous Systems vol 60 no 1 pp 72ndash82 2012

[14] S S Roy and D K Pratihar ldquoKinematics dynamics and powerconsumption analyses for turningmotion of a six-legged robotrdquoJournal of Intelligent amp Robotic Systems vol 74 no 3-4 pp 663ndash688 2014

[15] M Luneckas T Luneckas D Udris and N M F FerreiraldquoHexapod robot energy consumption dependence on bodyelevation and step heightrdquoElektronika ir Elektrotechnika vol 20no 7 pp 7ndash10 2014

[16] Z Deng Y Liu L Ding H Gao H Yu and Z Liu ldquoMotionplanning and simulation verification of a hydraulic hexapodrobot based on reducing energyflow consumptionrdquo Journal ofMechanical Science and Technology vol 29 no 10 pp 4427ndash4436 2015

[17] P Gonzalez de Santos E Garcia R Ponticelli and M Arm-ada ldquoMinimizing energy consumption in hexapod robotsrdquoAdvanced Robotics vol 23 no 6 pp 681ndash704 2009

[18] B Jin C Chen and W Li ldquoPower consumption optimizationfor a hexapod walking robotrdquo Journal of Intelligent and RoboticSystems Theory and Applications vol 71 no 2 pp 195ndash2092013

[19] Y Zhu B Jin W Li and S Li ldquoOptimal design of hexapodwalking robot leg structure based on energy consumption andworkspacerdquo Transactions of Canadian Society for MechanicalEngineering vol 38 no 3 pp 305ndash317 2014

[20] T LeeComputational GeometricMechanics andControl of RigidBodies University of Michigan Ann Arbor Mich USA 2008

[21] Z Terze A Muller and D Zlatar ldquoLie-group integrationmethod for constrained multibody systems in state spacerdquoMultibody System Dynamics vol 34 no 3 pp 275ndash305 2015

[22] S-B Xu S-B Li and B Cheng ldquoTheory and application ofLegendre pseudo-spectral method for solving optimal controlproblemrdquo Control and Decision vol 29 no 12 pp 2113ndash21202014

[23] K Tong J Zhou and L He ldquoLegendre-gauss pseudospectralmethod for solving optimal control problemrdquoActa Aeronauticaet Astronautica Sinica vol 29 no 6 pp 1531ndash1537 2008

[24] Y Sun M R Zhang and X L Liang ldquoImproved Gauss pseudo-spectral method for solving a nonlinear optimal control prob-lem with complex constraintsrdquo Acta Automatica Sinica vol 39no 5 pp 672ndash678 2013

[25] Y Liu Y Zhao J Xu and W Liu ldquoVehicle handling inversedynamics based on Gauss pseudo-spectral method while enc-ountering emergency collision avoidancerdquo Journal of Mechani-cal Engineering vol 48 no 22 pp 127ndash132 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


dynamics accounts for most energy consumption Chen et al[10] designed an insect type hexapod robot and leg has a seriesmechanism type the optimal control of leg swing is solved bypseudospectral method Roy et al [11ndash14] explored kinematicdynamics and optimal control problems of hexapod robotthe hexapod robot is driven by electric motor so it has aseries mechanism type Luneckas et al [15] analyzed hexapodrobotrsquos energy consumption by motion of body and stepheight Deng et al [16] explored energy reducing problemof hexapod robot by kinematics analysis Gonzalez de Santoset al [17] explored minimization of hexapod robot in irreg-ular terrain the optimal analysis is based on statically stablegait Jin et al [18] explored hexapod walking robotrsquos powerconsumption optimization problem by torque distributionalgorithm and parameters include duty factor stride lengthbogy height and foot trajectory lateral offset Zhu et al [19]explored optimal design of hexapod robot with kinematicmodel

Fundamentally PHR is a parallel mechanism most ofoptimal control explorations of it are static or simplify itas serial mechanism so the real dynamics character cannotfaithfully represent itThe first reason is that complex dynam-ics character of parallel mechanism makes it difficult to usetriangle functions to calculate it and complex triangle andantitriangle transformations will lead to unsolvable modelSecondly complex nonlinear characters need optimal controlalgorithm that has high calculation accuracy and stability butclassic algorithms such as Runge-Kutta method do not satisfythese two characters

According to references of dynamics modeling by Liegroup [20 21] and optimal control with pseudospectralmethod [22 23] the matrix and vector operators can avoidtriangle and antitriangle transformations which makes dyna-mics modeling easier Pseudospectral method is a globalnumericalmethodwhich has high stability and is widely usedin many domains many engineering problems are solvedsuccessfully [24 25] So in this explorationmatrix and vectoroperators are used as units for dynamic modeling and opti-mal control problem is solved by pseudospectralmethodThecontrol inputs curves which satisfy pneumatic control char-acters are obtained and then cruising ability calculationmethod is built at last which offers a reference for theimprovement of robot

2 The Mechanism and Gait ofPneumatic Hexapod Robot

PHR is a biorobot so it has two design schemes insect typein Figure 1 and mammal type in Figure 2 Many people likeinsect type but it has some problems Firstly it walks along119883 direction by triangle gait of hip joints bear great yawingforces which will lead to jointsrsquo rapid abrasion and to notbeing suited for engineering application Secondly it has a bigwidth so crosswise passing ability is restricted Thirdly therealization of straight linewalking needs combinationmotionof three joints which is more difficult to realize by pneumaticsystem and has high gas consumption The mammal typePHR does not have these problems The nitrogen gas bottleis in trunk the maximum pressure can reach 15MPa The

1

3

5

6

ThighHip

Body

2

4

6 Shank

z

xo

y

Z

XY O

Figure 1 The insect type hexapod robot

O

Gas bottle

35

6

PRV

Body

Shank

Thigh

Magnetic valves

1 24

DP

z

xoy

Z

XY

Figure 2 The mammal type hexapod robot

high pressure gas is decompressed to 1MPa by PRV (pressurereducing valve) and then gas can be decomposed to differentlow pressures from 015MPa to 08MPa by DP (duplexpieces) All the magnetic valves and control devices canbe packaged in box which is convenient for antiexplosiondesign Each leg is composed of shank and thigh which aredriven by cylinder

According to comparison between Figures 1 and 2 eachleg ofmammal type hexapod robot has two jointswhich is lessthan the insect type Based on this mechanism the PHR canrealize straight line walking as gait in Figure 3 The gait canbe divided into four actions Firstly the shank drive cylindershrinks to make foot tip separate from ground secondlythe thigh drive cylinder stretches out to make the leg stepforward thirdly the shank drive cylinder stretches out tomake foot contact with ground again fourthly the thigh drivecylinder shrinks to make body move forward According tothe above gait analysis the two joints work at different timeand have no coincides So the swing of thigh and shankcan be treated as a parallel pendulum When body movesforward by support of legs PHR and ground form a closeloop According to straight line walking gait the movementof body is only decided by motion of thigh So the movement


igh of 1 4 5


Shank of 2 3 605TT


12

3

45









1003817100381710038171003817R1r1 minus r210038171003817100381710038172 = 1198972 (2)


r1198791 r1 minus 2r1198792R1r1 + r1198792 r2 = 1198972 (3)


119902119894119860 + 119861119901119894 = 1198621199012119894 + 1199022119894 = 1 (4)


119897R2e1 = R1r1 minus r2 (5)


1205961r1198791 119878 (1)R1198791 r2 = 119897V (6)


1205961 = 119897r1198791 119878 (1)R1198791 r2 V (7)


1205962119897R2119878 (1) e1 = 1205961R1119878 (1) r1 minus VR2e1 (8)


1205962 = 1205961 e1198792R1198792R1119878 (1) r1119897 (9)


1 = V2 + 119897119886 + 12059621r1198791R1r2r1198791 119878 (1)R1198791 r2 (10)


2 = 12059621 e1198791 119878 (1)R1198792R1r1119897 minus 21205962V119897

minus 1 e1198791 119878 (1)R1198792R1119878 (1) r1119897

(11)


Y1

Y1

O1

O1

O3y1

y1

1205791

1205791

1205792

1205792

xb

xa

xax1x1

ya

l

l

yb

xb

X1

X1

O3

x2x2

O2

O2

Y2

Y2y2

y2

yaX2

X2


O0

O1

Y0

Y1

X0

X1

x0

y0

y1x1

O2

YI

rI2

pB

XBOB

OIXI

rB0

YB










(15)



= a119861(16)





R2 = 1198991I + 1198992119878 (1)119897 (17)


1205962 = 1198993119897 1205961 (18)


V = 1205961r1198791 119878 (1)R1198791 r2119897 (19)


1198791 = 12119869112059621 + 12 (1198692 + 1198693) 12059622 + 121198983V2 (20)


1198811 = 1198981119892e1198792R1r1198981 + (1198982 + 1198983) 119892e1198792 r2+ (11989821198921198971198982 + 1198983119892 (119897 minus 1198971198983))119897 1198992

(21)


119889119889119905 ( 1205971198711205971205961) minus (120597119871120597R1) = 0 (22)




(23)



1198791 r2


231198972 ) + 1198983 (1198994119886119897 )


11989916 = 1198691198861198993 (1198998119897 minus 11989941198993)1198973119899119895

(24)


Ta0

120579a0

Fa1

Fay

Fax

Fa0

Ta1

120579a1

Tb0

120579b0

Fb1

Fb0

Fby

Fbx

120579b1Tb1

Fc0

YB

OB XB

120579c0

Tc0

120579c1

Tc1

Fc1

Ot

Yt

rfra

Fcyy

Y

XOFcx

x

yt

Xt

M

ℎM

Yd

Od

Xd

ydxd

xt

120579t120579d

120579s

120579




1198791 = 121198691 12057921198811 = 1198981119892e1198792R (120579) r1198981

(25)



(26)


1198792 = 12119898119887k2119905 = 121198981198871205962119905 (R (120579) s1r119891)119879 (R (120579) s1r119891)

= 121198981198871205962119905 10038171003817100381710038171003817r119891100381710038171003817100381710038172

1198812 = 119898119887119892e1198792 (R (120579) r119891 + ℎ119872e2) (27)


1198793 = 121198982k2119886 + 1211986921205962119889 = 12 (1198982 1003817100381710038171003817r11988610038171003817100381710038172 1205962119905 + 11986921205962119889)1198813 = 1198982119892e1198792 (R (120579) r119886 + 1198971198982R (120579119889) e1)

(28)


1198794 = 121198983k2119905 + 1211986931205962119889 = 12 (1198982 10038171003817100381710038171003817r119891100381710038171003817100381710038172 1205962119905 + 11986921205962119889)

1198814 = 1198983119892e1198792 (R (120579) r119891 + r119889 + 1198971198983R (120587 minus 120579119889) e1) (29)


119871 = 119879 minus 119881 = (31198791 + 1198792 + 31198793 + 31198794) minus (31198811 + 1198812+ 31198813 + 31198814) = 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817


(30)


R119889 = 1198991198891I + 1198991198892119878 (1)119897120596119889 = 11989911988931198972 1205961199051198991198891 = minuse1198791R119905r1 + e1198791 r2


(31)


119871 = 119871 (120596119905R119905)= 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817

2 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)1205962119905+ 32 (1198692 + 1198693) (11989911988931198972 )


(32)

Ta0

Fy1205790

1205793

1205792

Fx

F



119889119889119905 120597119871120597120596119905 minus120597119871120597120579119905 = 1198961119905 + 1198962120596

2119905 + 1198963

1198961 = [(31198691 + 119898119887 10038171003817100381710038171003817r119891100381710038171003817100381710038172 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)+ 3 (1198692 + 1198693) (11989911988931198972 )

2]1198962 = [6 (1198692 + 1198693) (11989911988931198972 )(1205971198991198893120597120579119905

11198972 +2r1198792R119905s1r111989911988931198974 )]

minus 3 (1198692 + 1198693) (11989911988931198972 )(120597119899119889312059712057911990511198972 +

2r1198792R119905s1r111989911988931198974 )

1198963 = (311989831198921198971198983 + 311989821198921198971198982) (12059711989911988921205971205791199051119897 +


(33)







(34)







(36)


R0r1 = [ 1198881205790 1199041205790minus1199041205790 1198881205790][

119903119909119903119910] = [

1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]


10] = 119865[

minus1198881205792minus1199041205792]

(37)


119879 = 119865[ 1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]119879

[0 minus11 0 ] [

minus1198881205792minus1199041205792]

= 119865 [119903119909 (11988812057901199041205792 + 11990412057901198881205792) minus 119903119910 (11988812057901198881205792 minus 11990412057901199041205792)] (38)


120591 = (R0r1)119879 s1 (119865R2e1) (39)



(40)






1198991411989991198972119899119895 12059621


12058711988921199014119897119899119895 (R1r1)119879 (1198991e2 minus 1198992e1)

(41)

120579119905 = 120596119905119905 = 1198651198961119897 (R119905r1)


(42)


119869 = Φ (x (1199050) 1199050 x (119905119891) 119905119891) + int1199051198911199050

119892 (x (119905) u (119905)) 119889119905 (43)


x (119905) = f [x (119905) u (119905) 119905] (44)







minus1119892 (x (120591) u (120591) 120591) 119889120591 (46)





x (120591) asymp 119883 (120591) = 119899sum119894=0

119871 119894 (120591) x (120591119894) (48)


119871 119894 (120591) = 119899prod119895=0119895 =119894

120591 minus 120591119895120591119894 minus 120591119895 (49)


u (120591) asymp U (120591) = 119899sum119894=1

119871 119894 (120591)U (120591119894) (50)



x (120591119896) asymp X (120591119896) = 119899sum119894=0

119894 (120591119896) x (120591119894) = 119899sum119894=0

D119896119894 (120591119896) x (120591119894) (51)


D119896119894 = 119894 (120591119896)

=

(1 + 120591119896) 119899 (120591119896) + 119875119899 (120591119896)(120591119896 minus 120591119894) [(1 + 120591119894) 119899 (120591119894) + 119875119899 (120591119894)] 119894 = 119896(1 + 120591119894) 119873 (120591119894) + 2119873 (120591119894)2 [(1 + 120591119894) 119873 (120591119894) + 119875119873 (120591119894)] 119894 = 119896

(52)


119899sum119894=0

D119896119894 (120591119896)X (120591119894)

minus 119905119891 minus 11990502 f (X (120591119896) U (120591119896) 120591119896 1199050 119905119891) = 0(53)


x (120591119891) = x (1205910) + int1minus1f (x (120591) u (120591) 120591) 119889120591 (54)


X (120591119891) = X (1205910)+ 119905119891 minus 11990502

119899sum119896minus1

119908119896f (X (120591119896) U (120591119896) 120591 1199050 119905119891) (55)


Gauss points


119869 = Φ (X0 1199050X119891 119905119891)+ 119905119891 minus 11990502

119899sum119896=1

119908119896119892 (X119896U119896 120591119896 1199050 119905119891) (56)













End


minus600

minus500

minus400

minus300

minus200

minus100

0

y(m

m)



minus90

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

120579 s(∘)

120579 t(∘)





minus35

minus30

minus25

31

32

33

34

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

120579 s2(∘)

120579 t2(∘)


minus10

minus5

0

5

120596t

(rad

s)

005 01 015 02 025 030Time (sec)

minus10

0

10

120596s

(rad

s)

005 01 015 02 025 030Time (sec)


minus05

0

05

1

120596t2

(rad

s)

005 01 015 02 025 030Time (sec)

minus2

0

2

4

120596s2

(rad

s)

005 01 015 02 025 030Time (sec)


minus100

minus50

0

120572t

(rad

s2)

005 01 015 02 025 030Time (sec)

minus100

0

100

120572s

(rad

s2)

005 01 015 02 025 030Time (sec)


minus50

0

50120572t2

(rad

s2)

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

minus50

0

50

120572s2

(rad

s2)















0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)


0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)


times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)











1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










igh of 1 4 5


Shank of 2 3 605TT


12

3

45









1003817100381710038171003817R1r1 minus r210038171003817100381710038172 = 1198972 (2)


r1198791 r1 minus 2r1198792R1r1 + r1198792 r2 = 1198972 (3)


119902119894119860 + 119861119901119894 = 1198621199012119894 + 1199022119894 = 1 (4)


119897R2e1 = R1r1 minus r2 (5)


1205961r1198791 119878 (1)R1198791 r2 = 119897V (6)


1205961 = 119897r1198791 119878 (1)R1198791 r2 V (7)


1205962119897R2119878 (1) e1 = 1205961R1119878 (1) r1 minus VR2e1 (8)


1205962 = 1205961 e1198792R1198792R1119878 (1) r1119897 (9)


1 = V2 + 119897119886 + 12059621r1198791R1r2r1198791 119878 (1)R1198791 r2 (10)


2 = 12059621 e1198791 119878 (1)R1198792R1r1119897 minus 21205962V119897

minus 1 e1198791 119878 (1)R1198792R1119878 (1) r1119897

(11)


Y1

Y1

O1

O1

O3y1

y1

1205791

1205791

1205792

1205792

xb

xa

xax1x1

ya

l

l

yb

xb

X1

X1

O3

x2x2

O2

O2

Y2

Y2y2

y2

yaX2

X2


O0

O1

Y0

Y1

X0

X1

x0

y0

y1x1

O2

YI

rI2

pB

XBOB

OIXI

rB0

YB










(15)



= a119861(16)





R2 = 1198991I + 1198992119878 (1)119897 (17)


1205962 = 1198993119897 1205961 (18)


V = 1205961r1198791 119878 (1)R1198791 r2119897 (19)


1198791 = 12119869112059621 + 12 (1198692 + 1198693) 12059622 + 121198983V2 (20)


1198811 = 1198981119892e1198792R1r1198981 + (1198982 + 1198983) 119892e1198792 r2+ (11989821198921198971198982 + 1198983119892 (119897 minus 1198971198983))119897 1198992

(21)


119889119889119905 ( 1205971198711205971205961) minus (120597119871120597R1) = 0 (22)




(23)



1198791 r2


231198972 ) + 1198983 (1198994119886119897 )


11989916 = 1198691198861198993 (1198998119897 minus 11989941198993)1198973119899119895

(24)


Ta0

120579a0

Fa1

Fay

Fax

Fa0

Ta1

120579a1

Tb0

120579b0

Fb1

Fb0

Fby

Fbx

120579b1Tb1

Fc0

YB

OB XB

120579c0

Tc0

120579c1

Tc1

Fc1

Ot

Yt

rfra

Fcyy

Y

XOFcx

x

yt

Xt

M

ℎM

Yd

Od

Xd

ydxd

xt

120579t120579d

120579s

120579




1198791 = 121198691 12057921198811 = 1198981119892e1198792R (120579) r1198981

(25)



(26)


1198792 = 12119898119887k2119905 = 121198981198871205962119905 (R (120579) s1r119891)119879 (R (120579) s1r119891)

= 121198981198871205962119905 10038171003817100381710038171003817r119891100381710038171003817100381710038172

1198812 = 119898119887119892e1198792 (R (120579) r119891 + ℎ119872e2) (27)


1198793 = 121198982k2119886 + 1211986921205962119889 = 12 (1198982 1003817100381710038171003817r11988610038171003817100381710038172 1205962119905 + 11986921205962119889)1198813 = 1198982119892e1198792 (R (120579) r119886 + 1198971198982R (120579119889) e1)

(28)


1198794 = 121198983k2119905 + 1211986931205962119889 = 12 (1198982 10038171003817100381710038171003817r119891100381710038171003817100381710038172 1205962119905 + 11986921205962119889)

1198814 = 1198983119892e1198792 (R (120579) r119891 + r119889 + 1198971198983R (120587 minus 120579119889) e1) (29)


119871 = 119879 minus 119881 = (31198791 + 1198792 + 31198793 + 31198794) minus (31198811 + 1198812+ 31198813 + 31198814) = 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817


(30)


R119889 = 1198991198891I + 1198991198892119878 (1)119897120596119889 = 11989911988931198972 1205961199051198991198891 = minuse1198791R119905r1 + e1198791 r2


(31)


119871 = 119871 (120596119905R119905)= 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817

2 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)1205962119905+ 32 (1198692 + 1198693) (11989911988931198972 )


(32)

Ta0

Fy1205790

1205793

1205792

Fx

F



119889119889119905 120597119871120597120596119905 minus120597119871120597120579119905 = 1198961119905 + 1198962120596

2119905 + 1198963

1198961 = [(31198691 + 119898119887 10038171003817100381710038171003817r119891100381710038171003817100381710038172 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)+ 3 (1198692 + 1198693) (11989911988931198972 )

2]1198962 = [6 (1198692 + 1198693) (11989911988931198972 )(1205971198991198893120597120579119905

11198972 +2r1198792R119905s1r111989911988931198974 )]

minus 3 (1198692 + 1198693) (11989911988931198972 )(120597119899119889312059712057911990511198972 +

2r1198792R119905s1r111989911988931198974 )

1198963 = (311989831198921198971198983 + 311989821198921198971198982) (12059711989911988921205971205791199051119897 +


(33)







(34)







(36)


R0r1 = [ 1198881205790 1199041205790minus1199041205790 1198881205790][

119903119909119903119910] = [

1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]


10] = 119865[

minus1198881205792minus1199041205792]

(37)


119879 = 119865[ 1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]119879

[0 minus11 0 ] [

minus1198881205792minus1199041205792]

= 119865 [119903119909 (11988812057901199041205792 + 11990412057901198881205792) minus 119903119910 (11988812057901198881205792 minus 11990412057901199041205792)] (38)


120591 = (R0r1)119879 s1 (119865R2e1) (39)



(40)






1198991411989991198972119899119895 12059621


12058711988921199014119897119899119895 (R1r1)119879 (1198991e2 minus 1198992e1)

(41)

120579119905 = 120596119905119905 = 1198651198961119897 (R119905r1)


(42)


119869 = Φ (x (1199050) 1199050 x (119905119891) 119905119891) + int1199051198911199050

119892 (x (119905) u (119905)) 119889119905 (43)


x (119905) = f [x (119905) u (119905) 119905] (44)







minus1119892 (x (120591) u (120591) 120591) 119889120591 (46)





x (120591) asymp 119883 (120591) = 119899sum119894=0

119871 119894 (120591) x (120591119894) (48)


119871 119894 (120591) = 119899prod119895=0119895 =119894

120591 minus 120591119895120591119894 minus 120591119895 (49)


u (120591) asymp U (120591) = 119899sum119894=1

119871 119894 (120591)U (120591119894) (50)



x (120591119896) asymp X (120591119896) = 119899sum119894=0

119894 (120591119896) x (120591119894) = 119899sum119894=0

D119896119894 (120591119896) x (120591119894) (51)


D119896119894 = 119894 (120591119896)

=

(1 + 120591119896) 119899 (120591119896) + 119875119899 (120591119896)(120591119896 minus 120591119894) [(1 + 120591119894) 119899 (120591119894) + 119875119899 (120591119894)] 119894 = 119896(1 + 120591119894) 119873 (120591119894) + 2119873 (120591119894)2 [(1 + 120591119894) 119873 (120591119894) + 119875119873 (120591119894)] 119894 = 119896

(52)


119899sum119894=0

D119896119894 (120591119896)X (120591119894)

minus 119905119891 minus 11990502 f (X (120591119896) U (120591119896) 120591119896 1199050 119905119891) = 0(53)


x (120591119891) = x (1205910) + int1minus1f (x (120591) u (120591) 120591) 119889120591 (54)


X (120591119891) = X (1205910)+ 119905119891 minus 11990502

119899sum119896minus1

119908119896f (X (120591119896) U (120591119896) 120591 1199050 119905119891) (55)


Gauss points


119869 = Φ (X0 1199050X119891 119905119891)+ 119905119891 minus 11990502

119899sum119896=1

119908119896119892 (X119896U119896 120591119896 1199050 119905119891) (56)













End


minus600

minus500

minus400

minus300

minus200

minus100

0

y(m

m)



minus90

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

120579 s(∘)

120579 t(∘)





minus35

minus30

minus25

31

32

33

34

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

120579 s2(∘)

120579 t2(∘)


minus10

minus5

0

5

120596t

(rad

s)

005 01 015 02 025 030Time (sec)

minus10

0

10

120596s

(rad

s)

005 01 015 02 025 030Time (sec)


minus05

0

05

1

120596t2

(rad

s)

005 01 015 02 025 030Time (sec)

minus2

0

2

4

120596s2

(rad

s)

005 01 015 02 025 030Time (sec)


minus100

minus50

0

120572t

(rad

s2)

005 01 015 02 025 030Time (sec)

minus100

0

100

120572s

(rad

s2)

005 01 015 02 025 030Time (sec)


minus50

0

50120572t2

(rad

s2)

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

minus50

0

50

120572s2

(rad

s2)















0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)


0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)


times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)











1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Y1

Y1

O1

O1

O3y1

y1

1205791

1205791

1205792

1205792

xb

xa

xax1x1

ya

l

l

yb

xb

X1

X1

O3

x2x2

O2

O2

Y2

Y2y2

y2

yaX2

X2


O0

O1

Y0

Y1

X0

X1

x0

y0

y1x1

O2

YI

rI2

pB

XBOB

OIXI

rB0

YB










(15)



= a119861(16)





R2 = 1198991I + 1198992119878 (1)119897 (17)


1205962 = 1198993119897 1205961 (18)


V = 1205961r1198791 119878 (1)R1198791 r2119897 (19)


1198791 = 12119869112059621 + 12 (1198692 + 1198693) 12059622 + 121198983V2 (20)


1198811 = 1198981119892e1198792R1r1198981 + (1198982 + 1198983) 119892e1198792 r2+ (11989821198921198971198982 + 1198983119892 (119897 minus 1198971198983))119897 1198992

(21)


119889119889119905 ( 1205971198711205971205961) minus (120597119871120597R1) = 0 (22)




(23)



1198791 r2


231198972 ) + 1198983 (1198994119886119897 )


11989916 = 1198691198861198993 (1198998119897 minus 11989941198993)1198973119899119895

(24)


Ta0

120579a0

Fa1

Fay

Fax

Fa0

Ta1

120579a1

Tb0

120579b0

Fb1

Fb0

Fby

Fbx

120579b1Tb1

Fc0

YB

OB XB

120579c0

Tc0

120579c1

Tc1

Fc1

Ot

Yt

rfra

Fcyy

Y

XOFcx

x

yt

Xt

M

ℎM

Yd

Od

Xd

ydxd

xt

120579t120579d

120579s

120579




1198791 = 121198691 12057921198811 = 1198981119892e1198792R (120579) r1198981

(25)



(26)


1198792 = 12119898119887k2119905 = 121198981198871205962119905 (R (120579) s1r119891)119879 (R (120579) s1r119891)

= 121198981198871205962119905 10038171003817100381710038171003817r119891100381710038171003817100381710038172

1198812 = 119898119887119892e1198792 (R (120579) r119891 + ℎ119872e2) (27)


1198793 = 121198982k2119886 + 1211986921205962119889 = 12 (1198982 1003817100381710038171003817r11988610038171003817100381710038172 1205962119905 + 11986921205962119889)1198813 = 1198982119892e1198792 (R (120579) r119886 + 1198971198982R (120579119889) e1)

(28)


1198794 = 121198983k2119905 + 1211986931205962119889 = 12 (1198982 10038171003817100381710038171003817r119891100381710038171003817100381710038172 1205962119905 + 11986921205962119889)

1198814 = 1198983119892e1198792 (R (120579) r119891 + r119889 + 1198971198983R (120587 minus 120579119889) e1) (29)


119871 = 119879 minus 119881 = (31198791 + 1198792 + 31198793 + 31198794) minus (31198811 + 1198812+ 31198813 + 31198814) = 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817


(30)


R119889 = 1198991198891I + 1198991198892119878 (1)119897120596119889 = 11989911988931198972 1205961199051198991198891 = minuse1198791R119905r1 + e1198791 r2


(31)


119871 = 119871 (120596119905R119905)= 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817

2 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)1205962119905+ 32 (1198692 + 1198693) (11989911988931198972 )


(32)

Ta0

Fy1205790

1205793

1205792

Fx

F



119889119889119905 120597119871120597120596119905 minus120597119871120597120579119905 = 1198961119905 + 1198962120596

2119905 + 1198963

1198961 = [(31198691 + 119898119887 10038171003817100381710038171003817r119891100381710038171003817100381710038172 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)+ 3 (1198692 + 1198693) (11989911988931198972 )

2]1198962 = [6 (1198692 + 1198693) (11989911988931198972 )(1205971198991198893120597120579119905

11198972 +2r1198792R119905s1r111989911988931198974 )]

minus 3 (1198692 + 1198693) (11989911988931198972 )(120597119899119889312059712057911990511198972 +

2r1198792R119905s1r111989911988931198974 )

1198963 = (311989831198921198971198983 + 311989821198921198971198982) (12059711989911988921205971205791199051119897 +


(33)







(34)







(36)


R0r1 = [ 1198881205790 1199041205790minus1199041205790 1198881205790][

119903119909119903119910] = [

1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]


10] = 119865[

minus1198881205792minus1199041205792]

(37)


119879 = 119865[ 1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]119879

[0 minus11 0 ] [

minus1198881205792minus1199041205792]

= 119865 [119903119909 (11988812057901199041205792 + 11990412057901198881205792) minus 119903119910 (11988812057901198881205792 minus 11990412057901199041205792)] (38)


120591 = (R0r1)119879 s1 (119865R2e1) (39)



(40)






1198991411989991198972119899119895 12059621


12058711988921199014119897119899119895 (R1r1)119879 (1198991e2 minus 1198992e1)

(41)

120579119905 = 120596119905119905 = 1198651198961119897 (R119905r1)


(42)


119869 = Φ (x (1199050) 1199050 x (119905119891) 119905119891) + int1199051198911199050

119892 (x (119905) u (119905)) 119889119905 (43)


x (119905) = f [x (119905) u (119905) 119905] (44)







minus1119892 (x (120591) u (120591) 120591) 119889120591 (46)





x (120591) asymp 119883 (120591) = 119899sum119894=0

119871 119894 (120591) x (120591119894) (48)


119871 119894 (120591) = 119899prod119895=0119895 =119894

120591 minus 120591119895120591119894 minus 120591119895 (49)


u (120591) asymp U (120591) = 119899sum119894=1

119871 119894 (120591)U (120591119894) (50)



x (120591119896) asymp X (120591119896) = 119899sum119894=0

119894 (120591119896) x (120591119894) = 119899sum119894=0

D119896119894 (120591119896) x (120591119894) (51)


D119896119894 = 119894 (120591119896)

=

(1 + 120591119896) 119899 (120591119896) + 119875119899 (120591119896)(120591119896 minus 120591119894) [(1 + 120591119894) 119899 (120591119894) + 119875119899 (120591119894)] 119894 = 119896(1 + 120591119894) 119873 (120591119894) + 2119873 (120591119894)2 [(1 + 120591119894) 119873 (120591119894) + 119875119873 (120591119894)] 119894 = 119896

(52)


119899sum119894=0

D119896119894 (120591119896)X (120591119894)

minus 119905119891 minus 11990502 f (X (120591119896) U (120591119896) 120591119896 1199050 119905119891) = 0(53)


x (120591119891) = x (1205910) + int1minus1f (x (120591) u (120591) 120591) 119889120591 (54)


X (120591119891) = X (1205910)+ 119905119891 minus 11990502

119899sum119896minus1

119908119896f (X (120591119896) U (120591119896) 120591 1199050 119905119891) (55)


Gauss points


119869 = Φ (X0 1199050X119891 119905119891)+ 119905119891 minus 11990502

119899sum119896=1

119908119896119892 (X119896U119896 120591119896 1199050 119905119891) (56)













End


minus600

minus500

minus400

minus300

minus200

minus100

0

y(m

m)



minus90

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

120579 s(∘)

120579 t(∘)





minus35

minus30

minus25

31

32

33

34

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

120579 s2(∘)

120579 t2(∘)


minus10

minus5

0

5

120596t

(rad

s)

005 01 015 02 025 030Time (sec)

minus10

0

10

120596s

(rad

s)

005 01 015 02 025 030Time (sec)


minus05

0

05

1

120596t2

(rad

s)

005 01 015 02 025 030Time (sec)

minus2

0

2

4

120596s2

(rad

s)

005 01 015 02 025 030Time (sec)


minus100

minus50

0

120572t

(rad

s2)

005 01 015 02 025 030Time (sec)

minus100

0

100

120572s

(rad

s2)

005 01 015 02 025 030Time (sec)


minus50

0

50120572t2

(rad

s2)

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

minus50

0

50

120572s2

(rad

s2)















0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)


0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)


times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)











1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











R2 = 1198991I + 1198992119878 (1)119897 (17)


1205962 = 1198993119897 1205961 (18)


V = 1205961r1198791 119878 (1)R1198791 r2119897 (19)


1198791 = 12119869112059621 + 12 (1198692 + 1198693) 12059622 + 121198983V2 (20)


1198811 = 1198981119892e1198792R1r1198981 + (1198982 + 1198983) 119892e1198792 r2+ (11989821198921198971198982 + 1198983119892 (119897 minus 1198971198983))119897 1198992

(21)


119889119889119905 ( 1205971198711205971205961) minus (120597119871120597R1) = 0 (22)




(23)



1198791 r2


231198972 ) + 1198983 (1198994119886119897 )


11989916 = 1198691198861198993 (1198998119897 minus 11989941198993)1198973119899119895

(24)


Ta0

120579a0

Fa1

Fay

Fax

Fa0

Ta1

120579a1

Tb0

120579b0

Fb1

Fb0

Fby

Fbx

120579b1Tb1

Fc0

YB

OB XB

120579c0

Tc0

120579c1

Tc1

Fc1

Ot

Yt

rfra

Fcyy

Y

XOFcx

x

yt

Xt

M

ℎM

Yd

Od

Xd

ydxd

xt

120579t120579d

120579s

120579




1198791 = 121198691 12057921198811 = 1198981119892e1198792R (120579) r1198981

(25)



(26)


1198792 = 12119898119887k2119905 = 121198981198871205962119905 (R (120579) s1r119891)119879 (R (120579) s1r119891)

= 121198981198871205962119905 10038171003817100381710038171003817r119891100381710038171003817100381710038172

1198812 = 119898119887119892e1198792 (R (120579) r119891 + ℎ119872e2) (27)


1198793 = 121198982k2119886 + 1211986921205962119889 = 12 (1198982 1003817100381710038171003817r11988610038171003817100381710038172 1205962119905 + 11986921205962119889)1198813 = 1198982119892e1198792 (R (120579) r119886 + 1198971198982R (120579119889) e1)

(28)


1198794 = 121198983k2119905 + 1211986931205962119889 = 12 (1198982 10038171003817100381710038171003817r119891100381710038171003817100381710038172 1205962119905 + 11986921205962119889)

1198814 = 1198983119892e1198792 (R (120579) r119891 + r119889 + 1198971198983R (120587 minus 120579119889) e1) (29)


119871 = 119879 minus 119881 = (31198791 + 1198792 + 31198793 + 31198794) minus (31198811 + 1198812+ 31198813 + 31198814) = 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817


(30)


R119889 = 1198991198891I + 1198991198892119878 (1)119897120596119889 = 11989911988931198972 1205961199051198991198891 = minuse1198791R119905r1 + e1198791 r2


(31)


119871 = 119871 (120596119905R119905)= 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817

2 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)1205962119905+ 32 (1198692 + 1198693) (11989911988931198972 )


(32)

Ta0

Fy1205790

1205793

1205792

Fx

F



119889119889119905 120597119871120597120596119905 minus120597119871120597120579119905 = 1198961119905 + 1198962120596

2119905 + 1198963

1198961 = [(31198691 + 119898119887 10038171003817100381710038171003817r119891100381710038171003817100381710038172 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)+ 3 (1198692 + 1198693) (11989911988931198972 )

2]1198962 = [6 (1198692 + 1198693) (11989911988931198972 )(1205971198991198893120597120579119905

11198972 +2r1198792R119905s1r111989911988931198974 )]

minus 3 (1198692 + 1198693) (11989911988931198972 )(120597119899119889312059712057911990511198972 +

2r1198792R119905s1r111989911988931198974 )

1198963 = (311989831198921198971198983 + 311989821198921198971198982) (12059711989911988921205971205791199051119897 +


(33)







(34)







(36)


R0r1 = [ 1198881205790 1199041205790minus1199041205790 1198881205790][

119903119909119903119910] = [

1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]


10] = 119865[

minus1198881205792minus1199041205792]

(37)


119879 = 119865[ 1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]119879

[0 minus11 0 ] [

minus1198881205792minus1199041205792]

= 119865 [119903119909 (11988812057901199041205792 + 11990412057901198881205792) minus 119903119910 (11988812057901198881205792 minus 11990412057901199041205792)] (38)


120591 = (R0r1)119879 s1 (119865R2e1) (39)



(40)






1198991411989991198972119899119895 12059621


12058711988921199014119897119899119895 (R1r1)119879 (1198991e2 minus 1198992e1)

(41)

120579119905 = 120596119905119905 = 1198651198961119897 (R119905r1)


(42)


119869 = Φ (x (1199050) 1199050 x (119905119891) 119905119891) + int1199051198911199050

119892 (x (119905) u (119905)) 119889119905 (43)


x (119905) = f [x (119905) u (119905) 119905] (44)







minus1119892 (x (120591) u (120591) 120591) 119889120591 (46)





x (120591) asymp 119883 (120591) = 119899sum119894=0

119871 119894 (120591) x (120591119894) (48)


119871 119894 (120591) = 119899prod119895=0119895 =119894

120591 minus 120591119895120591119894 minus 120591119895 (49)


u (120591) asymp U (120591) = 119899sum119894=1

119871 119894 (120591)U (120591119894) (50)



x (120591119896) asymp X (120591119896) = 119899sum119894=0

119894 (120591119896) x (120591119894) = 119899sum119894=0

D119896119894 (120591119896) x (120591119894) (51)


D119896119894 = 119894 (120591119896)

=

(1 + 120591119896) 119899 (120591119896) + 119875119899 (120591119896)(120591119896 minus 120591119894) [(1 + 120591119894) 119899 (120591119894) + 119875119899 (120591119894)] 119894 = 119896(1 + 120591119894) 119873 (120591119894) + 2119873 (120591119894)2 [(1 + 120591119894) 119873 (120591119894) + 119875119873 (120591119894)] 119894 = 119896

(52)


119899sum119894=0

D119896119894 (120591119896)X (120591119894)

minus 119905119891 minus 11990502 f (X (120591119896) U (120591119896) 120591119896 1199050 119905119891) = 0(53)


x (120591119891) = x (1205910) + int1minus1f (x (120591) u (120591) 120591) 119889120591 (54)


X (120591119891) = X (1205910)+ 119905119891 minus 11990502

119899sum119896minus1

119908119896f (X (120591119896) U (120591119896) 120591 1199050 119905119891) (55)


Gauss points


119869 = Φ (X0 1199050X119891 119905119891)+ 119905119891 minus 11990502

119899sum119896=1

119908119896119892 (X119896U119896 120591119896 1199050 119905119891) (56)













End


minus600

minus500

minus400

minus300

minus200

minus100

0

y(m

m)



minus90

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

120579 s(∘)

120579 t(∘)





minus35

minus30

minus25

31

32

33

34

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

120579 s2(∘)

120579 t2(∘)


minus10

minus5

0

5

120596t

(rad

s)

005 01 015 02 025 030Time (sec)

minus10

0

10

120596s

(rad

s)

005 01 015 02 025 030Time (sec)


minus05

0

05

1

120596t2

(rad

s)

005 01 015 02 025 030Time (sec)

minus2

0

2

4

120596s2

(rad

s)

005 01 015 02 025 030Time (sec)


minus100

minus50

0

120572t

(rad

s2)

005 01 015 02 025 030Time (sec)

minus100

0

100

120572s

(rad

s2)

005 01 015 02 025 030Time (sec)


minus50

0

50120572t2

(rad

s2)

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

minus50

0

50

120572s2

(rad

s2)















0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)


0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)


times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)











1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ta0

120579a0

Fa1

Fay

Fax

Fa0

Ta1

120579a1

Tb0

120579b0

Fb1

Fb0

Fby

Fbx

120579b1Tb1

Fc0

YB

OB XB

120579c0

Tc0

120579c1

Tc1

Fc1

Ot

Yt

rfra

Fcyy

Y

XOFcx

x

yt

Xt

M

ℎM

Yd

Od

Xd

ydxd

xt

120579t120579d

120579s

120579




1198791 = 121198691 12057921198811 = 1198981119892e1198792R (120579) r1198981

(25)



(26)


1198792 = 12119898119887k2119905 = 121198981198871205962119905 (R (120579) s1r119891)119879 (R (120579) s1r119891)

= 121198981198871205962119905 10038171003817100381710038171003817r119891100381710038171003817100381710038172

1198812 = 119898119887119892e1198792 (R (120579) r119891 + ℎ119872e2) (27)


1198793 = 121198982k2119886 + 1211986921205962119889 = 12 (1198982 1003817100381710038171003817r11988610038171003817100381710038172 1205962119905 + 11986921205962119889)1198813 = 1198982119892e1198792 (R (120579) r119886 + 1198971198982R (120579119889) e1)

(28)


1198794 = 121198983k2119905 + 1211986931205962119889 = 12 (1198982 10038171003817100381710038171003817r119891100381710038171003817100381710038172 1205962119905 + 11986921205962119889)

1198814 = 1198983119892e1198792 (R (120579) r119891 + r119889 + 1198971198983R (120587 minus 120579119889) e1) (29)


119871 = 119879 minus 119881 = (31198791 + 1198792 + 31198793 + 31198794) minus (31198811 + 1198812+ 31198813 + 31198814) = 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817


(30)


R119889 = 1198991198891I + 1198991198892119878 (1)119897120596119889 = 11989911988931198972 1205961199051198991198891 = minuse1198791R119905r1 + e1198791 r2


(31)


119871 = 119871 (120596119905R119905)= 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817

2 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)1205962119905+ 32 (1198692 + 1198693) (11989911988931198972 )


(32)

Ta0

Fy1205790

1205793

1205792

Fx

F



119889119889119905 120597119871120597120596119905 minus120597119871120597120579119905 = 1198961119905 + 1198962120596

2119905 + 1198963

1198961 = [(31198691 + 119898119887 10038171003817100381710038171003817r119891100381710038171003817100381710038172 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)+ 3 (1198692 + 1198693) (11989911988931198972 )

2]1198962 = [6 (1198692 + 1198693) (11989911988931198972 )(1205971198991198893120597120579119905

11198972 +2r1198792R119905s1r111989911988931198974 )]

minus 3 (1198692 + 1198693) (11989911988931198972 )(120597119899119889312059712057911990511198972 +

2r1198792R119905s1r111989911988931198974 )

1198963 = (311989831198921198971198983 + 311989821198921198971198982) (12059711989911988921205971205791199051119897 +


(33)







(34)







(36)


R0r1 = [ 1198881205790 1199041205790minus1199041205790 1198881205790][

119903119909119903119910] = [

1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]


10] = 119865[

minus1198881205792minus1199041205792]

(37)


119879 = 119865[ 1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]119879

[0 minus11 0 ] [

minus1198881205792minus1199041205792]

= 119865 [119903119909 (11988812057901199041205792 + 11990412057901198881205792) minus 119903119910 (11988812057901198881205792 minus 11990412057901199041205792)] (38)


120591 = (R0r1)119879 s1 (119865R2e1) (39)



(40)






1198991411989991198972119899119895 12059621


12058711988921199014119897119899119895 (R1r1)119879 (1198991e2 minus 1198992e1)

(41)

120579119905 = 120596119905119905 = 1198651198961119897 (R119905r1)


(42)


119869 = Φ (x (1199050) 1199050 x (119905119891) 119905119891) + int1199051198911199050

119892 (x (119905) u (119905)) 119889119905 (43)


x (119905) = f [x (119905) u (119905) 119905] (44)







minus1119892 (x (120591) u (120591) 120591) 119889120591 (46)





x (120591) asymp 119883 (120591) = 119899sum119894=0

119871 119894 (120591) x (120591119894) (48)


119871 119894 (120591) = 119899prod119895=0119895 =119894

120591 minus 120591119895120591119894 minus 120591119895 (49)


u (120591) asymp U (120591) = 119899sum119894=1

119871 119894 (120591)U (120591119894) (50)



x (120591119896) asymp X (120591119896) = 119899sum119894=0

119894 (120591119896) x (120591119894) = 119899sum119894=0

D119896119894 (120591119896) x (120591119894) (51)


D119896119894 = 119894 (120591119896)

=

(1 + 120591119896) 119899 (120591119896) + 119875119899 (120591119896)(120591119896 minus 120591119894) [(1 + 120591119894) 119899 (120591119894) + 119875119899 (120591119894)] 119894 = 119896(1 + 120591119894) 119873 (120591119894) + 2119873 (120591119894)2 [(1 + 120591119894) 119873 (120591119894) + 119875119873 (120591119894)] 119894 = 119896

(52)


119899sum119894=0

D119896119894 (120591119896)X (120591119894)

minus 119905119891 minus 11990502 f (X (120591119896) U (120591119896) 120591119896 1199050 119905119891) = 0(53)


x (120591119891) = x (1205910) + int1minus1f (x (120591) u (120591) 120591) 119889120591 (54)


X (120591119891) = X (1205910)+ 119905119891 minus 11990502

119899sum119896minus1

119908119896f (X (120591119896) U (120591119896) 120591 1199050 119905119891) (55)


Gauss points


119869 = Φ (X0 1199050X119891 119905119891)+ 119905119891 minus 11990502

119899sum119896=1

119908119896119892 (X119896U119896 120591119896 1199050 119905119891) (56)













End


minus600

minus500

minus400

minus300

minus200

minus100

0

y(m

m)



minus90

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

120579 s(∘)

120579 t(∘)





minus35

minus30

minus25

31

32

33

34

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

120579 s2(∘)

120579 t2(∘)


minus10

minus5

0

5

120596t

(rad

s)

005 01 015 02 025 030Time (sec)

minus10

0

10

120596s

(rad

s)

005 01 015 02 025 030Time (sec)


minus05

0

05

1

120596t2

(rad

s)

005 01 015 02 025 030Time (sec)

minus2

0

2

4

120596s2

(rad

s)

005 01 015 02 025 030Time (sec)


minus100

minus50

0

120572t

(rad

s2)

005 01 015 02 025 030Time (sec)

minus100

0

100

120572s

(rad

s2)

005 01 015 02 025 030Time (sec)


minus50

0

50120572t2

(rad

s2)

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

minus50

0

50

120572s2

(rad

s2)















0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)


0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)


times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)











1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198794 = 121198983k2119905 + 1211986931205962119889 = 12 (1198982 10038171003817100381710038171003817r119891100381710038171003817100381710038172 1205962119905 + 11986921205962119889)

1198814 = 1198983119892e1198792 (R (120579) r119891 + r119889 + 1198971198983R (120587 minus 120579119889) e1) (29)


119871 = 119879 minus 119881 = (31198791 + 1198792 + 31198793 + 31198794) minus (31198811 + 1198812+ 31198813 + 31198814) = 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817


(30)


R119889 = 1198991198891I + 1198991198892119878 (1)119897120596119889 = 11989911988931198972 1205961199051198991198891 = minuse1198791R119905r1 + e1198791 r2


(31)


119871 = 119871 (120596119905R119905)= 12 (31198691 + 119898119887 10038171003817100381710038171003817r11989110038171003817100381710038171003817

2 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)1205962119905+ 32 (1198692 + 1198693) (11989911988931198972 )


(32)

Ta0

Fy1205790

1205793

1205792

Fx

F



119889119889119905 120597119871120597120596119905 minus120597119871120597120579119905 = 1198961119905 + 1198962120596

2119905 + 1198963

1198961 = [(31198691 + 119898119887 10038171003817100381710038171003817r119891100381710038171003817100381710038172 + 31198982 1003817100381710038171003817r11988610038171003817100381710038172 + 31198983 10038171003817100381710038171003817r119891100381710038171003817100381710038172)+ 3 (1198692 + 1198693) (11989911988931198972 )

2]1198962 = [6 (1198692 + 1198693) (11989911988931198972 )(1205971198991198893120597120579119905

11198972 +2r1198792R119905s1r111989911988931198974 )]

minus 3 (1198692 + 1198693) (11989911988931198972 )(120597119899119889312059712057911990511198972 +

2r1198792R119905s1r111989911988931198974 )

1198963 = (311989831198921198971198983 + 311989821198921198971198982) (12059711989911988921205971205791199051119897 +


(33)







(34)







(36)


R0r1 = [ 1198881205790 1199041205790minus1199041205790 1198881205790][

119903119909119903119910] = [

1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]


10] = 119865[

minus1198881205792minus1199041205792]

(37)


119879 = 119865[ 1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]119879

[0 minus11 0 ] [

minus1198881205792minus1199041205792]

= 119865 [119903119909 (11988812057901199041205792 + 11990412057901198881205792) minus 119903119910 (11988812057901198881205792 minus 11990412057901199041205792)] (38)


120591 = (R0r1)119879 s1 (119865R2e1) (39)



(40)






1198991411989991198972119899119895 12059621


12058711988921199014119897119899119895 (R1r1)119879 (1198991e2 minus 1198992e1)

(41)

120579119905 = 120596119905119905 = 1198651198961119897 (R119905r1)


(42)


119869 = Φ (x (1199050) 1199050 x (119905119891) 119905119891) + int1199051198911199050

119892 (x (119905) u (119905)) 119889119905 (43)


x (119905) = f [x (119905) u (119905) 119905] (44)







minus1119892 (x (120591) u (120591) 120591) 119889120591 (46)





x (120591) asymp 119883 (120591) = 119899sum119894=0

119871 119894 (120591) x (120591119894) (48)


119871 119894 (120591) = 119899prod119895=0119895 =119894

120591 minus 120591119895120591119894 minus 120591119895 (49)


u (120591) asymp U (120591) = 119899sum119894=1

119871 119894 (120591)U (120591119894) (50)



x (120591119896) asymp X (120591119896) = 119899sum119894=0

119894 (120591119896) x (120591119894) = 119899sum119894=0

D119896119894 (120591119896) x (120591119894) (51)


D119896119894 = 119894 (120591119896)

=

(1 + 120591119896) 119899 (120591119896) + 119875119899 (120591119896)(120591119896 minus 120591119894) [(1 + 120591119894) 119899 (120591119894) + 119875119899 (120591119894)] 119894 = 119896(1 + 120591119894) 119873 (120591119894) + 2119873 (120591119894)2 [(1 + 120591119894) 119873 (120591119894) + 119875119873 (120591119894)] 119894 = 119896

(52)


119899sum119894=0

D119896119894 (120591119896)X (120591119894)

minus 119905119891 minus 11990502 f (X (120591119896) U (120591119896) 120591119896 1199050 119905119891) = 0(53)


x (120591119891) = x (1205910) + int1minus1f (x (120591) u (120591) 120591) 119889120591 (54)


X (120591119891) = X (1205910)+ 119905119891 minus 11990502

119899sum119896minus1

119908119896f (X (120591119896) U (120591119896) 120591 1199050 119905119891) (55)


Gauss points


119869 = Φ (X0 1199050X119891 119905119891)+ 119905119891 minus 11990502

119899sum119896=1

119908119896119892 (X119896U119896 120591119896 1199050 119905119891) (56)













End


minus600

minus500

minus400

minus300

minus200

minus100

0

y(m

m)



minus90

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

120579 s(∘)

120579 t(∘)





minus35

minus30

minus25

31

32

33

34

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

120579 s2(∘)

120579 t2(∘)


minus10

minus5

0

5

120596t

(rad

s)

005 01 015 02 025 030Time (sec)

minus10

0

10

120596s

(rad

s)

005 01 015 02 025 030Time (sec)


minus05

0

05

1

120596t2

(rad

s)

005 01 015 02 025 030Time (sec)

minus2

0

2

4

120596s2

(rad

s)

005 01 015 02 025 030Time (sec)


minus100

minus50

0

120572t

(rad

s2)

005 01 015 02 025 030Time (sec)

minus100

0

100

120572s

(rad

s2)

005 01 015 02 025 030Time (sec)


minus50

0

50120572t2

(rad

s2)

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

minus50

0

50

120572s2

(rad

s2)















0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)


0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)


times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)











1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(34)







(36)


R0r1 = [ 1198881205790 1199041205790minus1199041205790 1198881205790][

119903119909119903119910] = [

1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]


10] = 119865[

minus1198881205792minus1199041205792]

(37)


119879 = 119865[ 1199031199091198881205790 + 1199041205790119903119910minus1199031199091199041205790 + 1199031199101198881205790]119879

[0 minus11 0 ] [

minus1198881205792minus1199041205792]

= 119865 [119903119909 (11988812057901199041205792 + 11990412057901198881205792) minus 119903119910 (11988812057901198881205792 minus 11990412057901199041205792)] (38)


120591 = (R0r1)119879 s1 (119865R2e1) (39)



(40)






1198991411989991198972119899119895 12059621


12058711988921199014119897119899119895 (R1r1)119879 (1198991e2 minus 1198992e1)

(41)

120579119905 = 120596119905119905 = 1198651198961119897 (R119905r1)


(42)


119869 = Φ (x (1199050) 1199050 x (119905119891) 119905119891) + int1199051198911199050

119892 (x (119905) u (119905)) 119889119905 (43)


x (119905) = f [x (119905) u (119905) 119905] (44)







minus1119892 (x (120591) u (120591) 120591) 119889120591 (46)





x (120591) asymp 119883 (120591) = 119899sum119894=0

119871 119894 (120591) x (120591119894) (48)


119871 119894 (120591) = 119899prod119895=0119895 =119894

120591 minus 120591119895120591119894 minus 120591119895 (49)


u (120591) asymp U (120591) = 119899sum119894=1

119871 119894 (120591)U (120591119894) (50)



x (120591119896) asymp X (120591119896) = 119899sum119894=0

119894 (120591119896) x (120591119894) = 119899sum119894=0

D119896119894 (120591119896) x (120591119894) (51)


D119896119894 = 119894 (120591119896)

=

(1 + 120591119896) 119899 (120591119896) + 119875119899 (120591119896)(120591119896 minus 120591119894) [(1 + 120591119894) 119899 (120591119894) + 119875119899 (120591119894)] 119894 = 119896(1 + 120591119894) 119873 (120591119894) + 2119873 (120591119894)2 [(1 + 120591119894) 119873 (120591119894) + 119875119873 (120591119894)] 119894 = 119896

(52)


119899sum119894=0

D119896119894 (120591119896)X (120591119894)

minus 119905119891 minus 11990502 f (X (120591119896) U (120591119896) 120591119896 1199050 119905119891) = 0(53)


x (120591119891) = x (1205910) + int1minus1f (x (120591) u (120591) 120591) 119889120591 (54)


X (120591119891) = X (1205910)+ 119905119891 minus 11990502

119899sum119896minus1

119908119896f (X (120591119896) U (120591119896) 120591 1199050 119905119891) (55)


Gauss points


119869 = Φ (X0 1199050X119891 119905119891)+ 119905119891 minus 11990502

119899sum119896=1

119908119896119892 (X119896U119896 120591119896 1199050 119905119891) (56)













End


minus600

minus500

minus400

minus300

minus200

minus100

0

y(m

m)



minus90

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

120579 s(∘)

120579 t(∘)





minus35

minus30

minus25

31

32

33

34

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

120579 s2(∘)

120579 t2(∘)


minus10

minus5

0

5

120596t

(rad

s)

005 01 015 02 025 030Time (sec)

minus10

0

10

120596s

(rad

s)

005 01 015 02 025 030Time (sec)


minus05

0

05

1

120596t2

(rad

s)

005 01 015 02 025 030Time (sec)

minus2

0

2

4

120596s2

(rad

s)

005 01 015 02 025 030Time (sec)


minus100

minus50

0

120572t

(rad

s2)

005 01 015 02 025 030Time (sec)

minus100

0

100

120572s

(rad

s2)

005 01 015 02 025 030Time (sec)


minus50

0

50120572t2

(rad

s2)

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

minus50

0

50

120572s2

(rad

s2)















0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)


0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)


times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)











1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














minus1119892 (x (120591) u (120591) 120591) 119889120591 (46)





x (120591) asymp 119883 (120591) = 119899sum119894=0

119871 119894 (120591) x (120591119894) (48)


119871 119894 (120591) = 119899prod119895=0119895 =119894

120591 minus 120591119895120591119894 minus 120591119895 (49)


u (120591) asymp U (120591) = 119899sum119894=1

119871 119894 (120591)U (120591119894) (50)



x (120591119896) asymp X (120591119896) = 119899sum119894=0

119894 (120591119896) x (120591119894) = 119899sum119894=0

D119896119894 (120591119896) x (120591119894) (51)


D119896119894 = 119894 (120591119896)

=

(1 + 120591119896) 119899 (120591119896) + 119875119899 (120591119896)(120591119896 minus 120591119894) [(1 + 120591119894) 119899 (120591119894) + 119875119899 (120591119894)] 119894 = 119896(1 + 120591119894) 119873 (120591119894) + 2119873 (120591119894)2 [(1 + 120591119894) 119873 (120591119894) + 119875119873 (120591119894)] 119894 = 119896

(52)


119899sum119894=0

D119896119894 (120591119896)X (120591119894)

minus 119905119891 minus 11990502 f (X (120591119896) U (120591119896) 120591119896 1199050 119905119891) = 0(53)


x (120591119891) = x (1205910) + int1minus1f (x (120591) u (120591) 120591) 119889120591 (54)


X (120591119891) = X (1205910)+ 119905119891 minus 11990502

119899sum119896minus1

119908119896f (X (120591119896) U (120591119896) 120591 1199050 119905119891) (55)


Gauss points


119869 = Φ (X0 1199050X119891 119905119891)+ 119905119891 minus 11990502

119899sum119896=1

119908119896119892 (X119896U119896 120591119896 1199050 119905119891) (56)













End


minus600

minus500

minus400

minus300

minus200

minus100

0

y(m

m)



minus90

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

120579 s(∘)

120579 t(∘)





minus35

minus30

minus25

31

32

33

34

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

120579 s2(∘)

120579 t2(∘)


minus10

minus5

0

5

120596t

(rad

s)

005 01 015 02 025 030Time (sec)

minus10

0

10

120596s

(rad

s)

005 01 015 02 025 030Time (sec)


minus05

0

05

1

120596t2

(rad

s)

005 01 015 02 025 030Time (sec)

minus2

0

2

4

120596s2

(rad

s)

005 01 015 02 025 030Time (sec)


minus100

minus50

0

120572t

(rad

s2)

005 01 015 02 025 030Time (sec)

minus100

0

100

120572s

(rad

s2)

005 01 015 02 025 030Time (sec)


minus50

0

50120572t2

(rad

s2)

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

minus50

0

50

120572s2

(rad

s2)















0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)


0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)


times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)











1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















End


minus600

minus500

minus400

minus300

minus200

minus100

0

y(m

m)



minus90

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

minus80

minus70

minus60

005 01 015 02 025 030Time (sec)

120579 s(∘)

120579 t(∘)





minus35

minus30

minus25

31

32

33

34

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

120579 s2(∘)

120579 t2(∘)


minus10

minus5

0

5

120596t

(rad

s)

005 01 015 02 025 030Time (sec)

minus10

0

10

120596s

(rad

s)

005 01 015 02 025 030Time (sec)


minus05

0

05

1

120596t2

(rad

s)

005 01 015 02 025 030Time (sec)

minus2

0

2

4

120596s2

(rad

s)

005 01 015 02 025 030Time (sec)


minus100

minus50

0

120572t

(rad

s2)

005 01 015 02 025 030Time (sec)

minus100

0

100

120572s

(rad

s2)

005 01 015 02 025 030Time (sec)


minus50

0

50120572t2

(rad

s2)

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

minus50

0

50

120572s2

(rad

s2)















0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)


0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)


times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)











1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus35

minus30

minus25

31

32

33

34

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

120579 s2(∘)

120579 t2(∘)


minus10

minus5

0

5

120596t

(rad

s)

005 01 015 02 025 030Time (sec)

minus10

0

10

120596s

(rad

s)

005 01 015 02 025 030Time (sec)


minus05

0

05

1

120596t2

(rad

s)

005 01 015 02 025 030Time (sec)

minus2

0

2

4

120596s2

(rad

s)

005 01 015 02 025 030Time (sec)


minus100

minus50

0

120572t

(rad

s2)

005 01 015 02 025 030Time (sec)

minus100

0

100

120572s

(rad

s2)

005 01 015 02 025 030Time (sec)


minus50

0

50120572t2

(rad

s2)

005 01 015 02 025 030Time (sec)

005 01 015 02 025 030Time (sec)

minus50

0

50

120572s2

(rad

s2)















0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)


0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)


times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)











1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















0

100

200

300

400

500

600

y(m

m)

100 200 300 400 500 6000x (mm)


0 002 004 006 008 01Time (sec)

0 002 004 006 008 01Time (sec)

0

2000

4000

6000Vx

(mm

s)

minus1000

minus500

0

Vy

(mm

s)


times104

times104

2

4

6

8

a x(m

ms

2)

minus5

0

5

a y(m

ms

2)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)











1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















1 = 0 120579end1 = 0475120587 120596end1 = 0



1 = 0 120579end1 =1205873 120596end1 = 0











8 Conclusion



0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 002 004 006 008 01Time (sec)

minus02

0

02

p(M

Pa)

002 004 006 0080Time (sec)

Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 0080

minus10

minus5

0

120596(r

ads

)

004002 006 0080Time (sec)

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)


minus02

0

02

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

minus10

minus5

0

120596(r

ads

)

002 004 006 008 010Time (sec)







0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 002 004 006 008Time (sec)

0 002 004 006 008Time (sec)

120579 120579

F F

120596 120596

minus01

0

01

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10

120596(r

ads

)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

002 004 006 008 010Time (sec)

minus02

0

02

p(M

Pa)

minus15

minus1

120579(r

ad)

0

5

10120596

(rad

s)

002 004 006 0080Time (sec)


minus1

0

1

p(M

Pa)

002 004 006 008 010Time (sec)

minus15

minus1

120579(r

ad)

002 004 006 008 010Time (sec)

0

05

1

120596(r

ads

)

002 004 006 008 010Time (sec)

0

1

2

120596(r

ads

)

005 01 015 02 025 030Time (sec)

minus15

minus1

120579(r

ad)

005 01 015 02 025 030Time (sec)

minus05

0

05

p(M

Pa)

005 01 015 02 025 030Time (sec)





Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









kinematics, dynamics, and optimal control of pneumatic ...the hexapod robot is driven by electric...

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