Int. J. Mech. Eng. & Rob. Res. 2013 Pundru Srinivasa Rao ... · Lung-Wen (2003) and Yuefa and Lung-Wen (2004). Song and Zhang (1995) and Joshi and Tsai (2002) determined two singular

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Int. J. Mech. Eng. & Rob. Res. 2013 Pundru Srinivasa Rao and Nalluri Mohan Rao, 2013

POSITION ANALYSIS OF SPATIAL 3-RPSPARALLEL MANIPULATOR

Pundru Srinivasa Rao1* and Nalluri Mohan Rao2

*Corresponding Author: Pundru Srinivasa Rao, [email protected]

Parallel manipulators increase quality because of their large payload carrying capacity, highstiffness, high accuracy and good precision in positioning. Several parallel manipulators with6-degree of freedom were developed, but 6-dof manipulators have complicated analysis andmoreover distinct applications require less than 6-dof such as surgical manipulators, 3-dofsensors and flight simulations, etc. Therefore manipulators with less than 6-dof have beendeveloped. They are called low-dof parallel manipulators like 3-dof, 4-dof or may be 5-dof. Aintend position analysis of spatial 3-RPS parallel manipulator consists of three identical limbs,Base platform, mobile platform, revolute joints, prismatic joints and spherical joints. In the presenttask, it is intend to carryout, the position analysis of a spatial 3-RPS parallel manipulator. Theposition analysis of forward kinematic equations and inverse kinematic equations of 3-RPSspatial parallel manipulator have to be analyzed and simplified by using Sylvester dialyticelimination technique and vector analysis. All the possible solutions of this initial analysis arecalculated by changing the limb lengths through linear actuators.

Keywords: 3-RPS spatial parallel manipulator, Position analysis, Identical limbs, 3-dof, Sylvesterdialytic elimination technique, Vector analysis

INTRODUCTIONSpatial multi-degree of freedom mechanismsare known as parallel mechanisms. The motioncharacteristics of parallel mechanisms may beplanar, Spherical [or] spatial type parallelmanipulators. The kinematic structure ofparallel manipulator is made up of a closedloop

ISSN 2278 – 0149 www.ijmerr.comVol. 2, No. 2, April 2013

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Int. J. Mech. Eng. & Rob. Res. 2013

1 Department of Mechanical Engineering, Mahatma Gandhi Institute of Technology, C.B.I.T (Post), Gandipet, Hyderabad 500075,AP, India.

2 Department of Mechanical Engineering, University College of Engineering, JNT University Kakinada, Kakinada, AP, India.

mechanism. In planar parallel mechanisms,type of each joint must be revolute [R] orprismatic[p] and mobility of each joint must beone. In planar parallel mechanisms, revolutejoint[R] axes are perpendicular to the plane ofmotion and prismatic joint[P]axes lie on theplane of motion. In spherical parallel

Research Paper

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mechanisms, only revolute joints[R] are to bepermissible and all the revolute joint axes mustbe inter-sect at a common point. In spatialparallel mechanisms, the motion of the parallelmanipulator cannot be characterized as planarmotion or spherical motion, means the positionand orientation of mobile platform freely in 3-dimensional space. The parallel mechanismsare sometimes called platform manipulatorsbecause the external load can be shared bylinear actuators or proportional actuators. Theproperties of parallel mechanisms are highstiffness, high accuracy, high speed , lowinertia, large payload capacity and goodprecision in position. The applications ofparallel mechanisms are such as airplanesimulators, mining machines, walkingmachines adjustable articulated trusses,pointing devices, robot manipulator, micromanipulator, Tunnel borer, barbette of warship,satellite surveillance platform, parallel machinetools surgical manipulators, three degree offreedom sensors, etc. For general purposeRobot the parallel manipulators with 6-dof weredeveloped for different applications. But 6-dofparallel manipulators such as Stewart-Goughplatform type and Stewart universal type testmachines (Gough and Whitehall, 1962; andStewart, 1965) are suffer the difficult inkinematic analysis and complex in mechanicaldesign. Moreover, several applications requireless than 6-dof.Therefore, the parallelmechanisms with less than 6-dof (Lee andShah, 1987; Han and Lung-Wen, 2003;Sameer and Lung-Wen, 2003; Liu andShengchia, 2004; Yuefa and Lung-Wen, 2004;Yangmini and Qingsong, 2005; Mohan Raoand Mallikarjuna Rao, 2006; Yangmin andQingsong, 2006; Yi et al., 2008; and Yi et al.,2009) have been developed. They are called

low-dof parallel mechanisms. The low degreeof freedom parallel mechanisms may be 5-dof,4-dof [or]3-Dof.

There are 3-DoF spatial manipulators,called rotational parallel manipulators thatallow the moving platform to rotate about a fixedpoint. Some manipulators, called translationalparallel manipulators provide the movingplatform with pure translational motion and maybe more useful in the fields of automatedassembly and machine tools as alternativesto serial manipulator systems. Some othermanipulators allow the platform to rotate andtranslate. Lee and Shah (1988) developed alow-cost driving simulator using the 3-RPSmanipulator. Kinematic and dynamiccharacteristics of the 3-RPS manipulator havebeen studied by Lee and Shah (1988), Songand Zhang (1995), Yang et al. (1996), Han andLung-Wen (2003) and Yuefa and Lung-Wen(2004). Song and Zhang (1995) and Joshi andTsai (2002) determined two singular positionsof the 3-RPS manipulator from the Jacobian,determined by making use of the theory ofreciprocal screws, while the other directsingular positions of the manipulator weredetermined by Joshi and Tsai (2002) and Liuand Cheng (2004). Kinematics of a highprecision three degree of freedom manipulatorwere determined by Mohan Rao andMallikarjuna Rao (2009).

The design of spatial parallel mechanismshave to be developed for different applicationsthat require both rotational and translationalmotions of the mobile platform. Thesymmetrical 3-dof spatial parallel manipulatorwith 3-identical limbs are to be selected, andeach limb connects fixed base to movingplatform by revolute [R] joint, prismatic [P] joint

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Figure 2: Geometry of Fixed BasePlatform

and spherical [S] joint in sequence. The movingplatform has three degree of freedom that istwo degrees of orientation freedom and onedegree of translation freedom.

In the present work, the position equationsof direct and inverse kinematic analysis of3-PS spatial parallel manipulator have to beanalyzed and simplified by using Sylvesterdialytic elimination technique. The motion ofthe mobile platform is to be constrained byactuating the 3-actuators separately usingelectric drive technology. The motion of the3-actuators must be controlled by microprocessors and micro controllers.

GEOMETRY DESCRIPTIONOF A SPATIAL 3-RPSPARALLEL MANIPULATORThe Spatial 3-RPS parallel manipulatorconsists of a mobile platform [S], i.e., Figure 1connected to a fixed base [B], i.e., Figure 2 bythree identical supporting limbs withsymmetrical kinematic structure. That meansin the selected closed loop structure thenumber of limbs, number of joints, type of joints,number of actuated joints all are identical andequal to the mobility of the mobile platform.Each limb connects by means of a fixedbase[B] to the mobile platform [S] by a revolutejoint [R], a prismatic joint [P], a spherical joint[S] in sequence as in Figure 3, where theprismatic joint is activated by linear actuator.Thus the mobile platform connects to the baseplatform by means of three identical revolute,prismatic and spherical linkages (Figure 4).

In the fixed base platform of a selectedspatial 3-RPS parallel manipulator theCartesian co-ordinate reference frame O{x, y,z} is considered at the centroid of the fixed

Figure 1: 3-DoF Spatial ParallelManipulator

triangular base platform B1, B

2, B

3. Similarly

the mobile co-ordinate reference frame P{u,v, w} is considered at the centroid of thetriangular moving platform S

1, S

2, S

3. For

simplicity of geometry, let us consider X-axis

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Figure 4: Geometry of Moving Platform

as in the direction of 1OB and the u-axis as in

the direction of 1PS . The distance between

corresponding vertices of the fixed platformand mobile platform is denoted by ij LL , thatmeans after activation of each prismatic joints

the distance between corresponding verticesof fixed and moving platforms are ij LL |

j =1, 2, 3

and i = 1, 2, 3, 4, etc., as in Fin(2).

MOBILITY OF A 3-RPSSPATIAL PARALLELMANIPULATORThe mobility of a spatial 3-RPS parallelmanipulator are the number of independenttranslations and rotations of the mobileplatform or the number of input parametersrequired to specify the configuration of aspatial parallel manipulator completely.

The mobility equation depends on numberof links, number of joints and type of jointsincorporated in the selected spatial parallelmanipulator. The spatial parallel manipulatorconsists of 8-links, 3-revolute joints,3-prismaticjoints and 3-spherical joints. The mobility ofspatial parallel manipulator can be calculatedby using Kutz-bach Criterion.

i

ifjnF 1

F: mobility of a spatial parallel manipulator

: Degrees of freedom of space in which amechanism is intended to function.

= 6 for spatial mechanisms [degree offreedom of space with in which themanipulatoroperates]

n: Number of links in a mechanism,including fixed link

j: Number of joints in a mechanism

fi: Degrees of relative motion permitted by

joint i.

Therefore F = 6 (8 – 9 – 1) + (3 + 3 + 3 3)= –12 + 15 = 3

Figure 3: Position Geometry of SpatialParallel Manipulator

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POSITION ANALYSIS OFSPATIAL 3-RPS PARALLELMANIPULATORThe position vector of vertices B

j | j = 1, 2, 3

and Sj

| j = 1, 2, 3

with respect to fixed base frame centroidO{x, y, z} and mobile frame centroid P{u, v, w}can be expressed as follows:

TggOB 0011

T

gggOB

0

2

3

2

122 ...(1)

ThhOS 0011

T

hhhOS

0

2

3

2

122

T

hhhOS

0

2

3

2

133 ...(2)

Let wvu

,, be represents three unit vectors

along u, v and w-axis of the mobile platformreference coordinate frame P{u, v, w}. Thenthe rotation matrix can be expressed in terms

of direction cosines of wvu

,, are

zzz

yyy

xxx

po

wvu

wvu

wvu

wzvzuz

wyvyuy

wxvxux

R

...

...

...

The Position vector of mobile frame verticesS

j |

j = 1, 2, 3 with respect to fixed base frame

centroid O{x, y, z} can be expressed as:

jj PSOPOS

That means iOP |

i = 1, 2, 3, 4, etc., and jiSP |

j = 1,

2, 3 and i = 1, 2, 3, 4, etc.

Represent position vector and rotationmatrix of a reference point P

i |

i = 1, 2, 3, 4, etc., on a

mobile platform respectively at the ith specifiedposition and orientation with respect to thefixed reference frame O{ x, y, z}.

i.e., jiiji SPOPOS

uvwjpi

oTziyixiji hRPPPq ...(3)

where iOP is the transformation from the

moving frame to the fixed frame can bedescribed by position vector

Tziyixii PPPOP and rotation matrix

T

zizizi

yiyiyi

xixixi

pio

wvu

wvu

wvu

R

...(4)

substituting Equation (2) and Equation (4) intoEquation (3) then we can obtain positionkinematic equations of mobile frame verticeswith respect to fixed base frame centroid O{x,y, z}, i.e.,

zizi

yiyi

xixi

i

huP

huP

huP

q1

Similarly

zizizi

yiyiyi

xixixi

i

hvhuP

hvhuP

hvhuP

q

2

3

2

12

3

2

12

3

2

1

2

zizizi

yiyiyi

xixixi

i

hvhuP

hvhuP

hvhuP

q

2

3

2

12

3

2

12

3

2

1

3

...(5)

85



REVOLUTE JOINTCONSTRAINTSDue to revolute joints at B

j |

j = 1, 2, 3 the

mechanical constraints are imposed by therevolute joints, so the spherical joints at S

j |

j = 1,

2, 3 can only move in the planes defined by the

ith linear actuations along the plane of verticesB

jS

j, i.e., OB

jS

j. Therefore the constraints

equations are:

01 yiq

xqq iyi 22 3

Similarly

xqq iyi 33 3 ...(6)

That means

0 yiyi huP ...(7.1)

yiyiyi hvhuP2

3

2

1

xixixi hvhuP

2

3

2

13 ...(7.2)

yiyiyi hvhuP2

3

2

1

xixixi hvhuP

2

3

2

13 ...(7.3)

adding Equation (7.2) and Equation (7.3) thenwe have

xiyi hvP 33 ...(7.4)

adding Equation (7.1) and Equation (7.4) thenwe have

xiyi hvP ...(7.5)

Substituting Equation (7.5) into Equation(7.1) then we have

0 yixi huhv

0 hvu xiyi ...(7.6)

Subtracting Equation (7.3) from Equation(7.2) then we have

xixiyi huPhv 233

xixiyi Puvh 2

yixixi vuhP 2

1...(7.7)

REVOLUTE JOINTCONSTRAINTS FOLLOWEDBY EULER ANGLEREPRESENTATIONThe position analysis of a mobile platform withrespect to fixed base platform centroid O{x, y,z} can be described by direction cosinerepresentation followed Euler anglerepresentation.

Let us consider 3-Euler angles that are

Øi = Rotating about Z-axis of fixed reference

frame

i = Rotating about Y-axis of fixed reference

frame

i = Rotating about X-axis of fixed

reference frame

The 3-Euler angles rotating about Z-X-Yaxis of the fixed reference frame in sequencethen

oRp = Euler (Ø

i,

i,

i)

oRp = Rot(y,

i). Rot(x,

i). Rot(z, Ø

i)

ii

ii

ii

ii

po

cs

sc

cs

sc

R

0

0

001

0

010

0

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100

0

0

ii

ii

cs

sc

iiiii

ii

iiiii

po

scscs

sc

ssscc

R

ii

i

ii

iiiii

ii

iiiii

po

cc

s

cs

ccsss

cc

csssc

R

...(7.8)

where s and c stands for sine and cosinerespectively. Then revolute joint constraints interms of Euler angle representation fromEquation (7.1) are:

iiyi shcP ...(7.9)

From Equation (7.6) are:

ii

iii cc

ssTan

...(7.10)

From Equation (7.7) and Equation (7.9) are

i

iiixi cxccchP

1

22

1 ...(7.11)

Hence 3-constraints on the motion of themoving platform are:

ii

iii cc

ssTan

1

i

iiixi cxccchP

1

22

1

iiyi shcP ...(8)

FORWARD KINEMATICANALYSISForward kinematic analysis are obtained by aset of actuated inputs given, then the position

and orientation of the mobile platform areresolved by Forward kinematic analysis.

Let Lji |

j = 1, 2, 3 are the vectors of the three

actuated input joint variables and

Tiiiziyixi PPP

are the vectors of constrained andunconstrained Cartesian variables whichdescribe the position and orientation ofmobileplatform.

Let Øji |

j = 1, 2, 3 and i = 1, 2, 3, 4, etc., be the angle

made by the ith limb BjS

ji |

j = 1, 2, 3 and i = 1, 2, 3, 4, etc.,

with respect to OBj |

j = 1, 2, 3 then the position

vector of spherical joint Sj with respect to fixed

coordinate system can be expressed as:

ii

ii

i

sL

cLg

q

11

11

1 0

ii

ii

ii

i

sL

cLg

cLg

q

22

22

22

2 2

32

1

ii

ii

ii

i

sL

cLg

cLg

q

33

33

33

3 2

32

1

...(9)

The position and orientation of the limbs canbe determined by the geometric distancebetween two spherical joints S

j |

j = 1, 2, 3 is

constant, i.e., 21SS or 32SS or hSS 313

h

qq

qq

qq

ii

ii

ii

3

13

32

21

87



L1i

L2i

cØ1i

cØ2i – 2L

1iL

2isØ

1isØ

2i – 3g L

1i

cØ1i – 3g L

2i cØ

2i + (L

1i2 + L

2i2 + 3g2 – 3h2) = 0

...(10)

Similarly

L2i

L3i

cØ2i cØ

3i – 2L

2iL

3i sØ

2isØ

3i – 3g L

2i

cØ2i – 3g L

3i cØ

3i + (L

2i2 + L

3i2 + 3g2 – 3h2) = 0

...(11)

L1i

L3i cØ

1icØ

3i – 2L

1i L

3i sØ

1i sØ

3i – 3g L

1i

cØ1i – 3g L

3icØ

3i + (L

1i2 + L

3i2 + 3g2 – 3h2) = 0

...(12)

The above equations are non linearequations and contain 3 unknowns {Ø

1i, Ø

2i,

Ø3i}

The equations can be solved by Sylvesterdialytic elimination technique.

The Syivester dialytic elimination techniqueis applied to reduce the system of non-linearEquation (10) to polynomial in one variable thatis

A1CØ

1iCØ

2i+ B

1SØ

1iSØ

2i+ C

1CØ

1i +

D1CØ

2i + E

1 = 0 ...(13)

Let A1 = L

1iL

2i

B1 = – 2L

1i L

2i

C1 = – 3g L

1i

D1 = – 3g L

2i

E1 = L2

1i + L2

2i + 3g2 – 3h2

Consider

21

1i

i Tant

Then 22

21

11

1

i

ii

t

tC

and 21

11

1

2

i

ii

t

tS

Then

(F1t2

1i+ H

1) t2

2i + (I

1t1i) t

2i + (G

1t2

1i + J

1) = 0

...(14)

where F1 = A

1 – C

1 – D

1 + E

1

G1 = – A

1 – C

1 + D

1 + E

1

H1 = – A

1 + C

1 – D

1 + E

1

I1 = 4B

1

J1 = A

1 + C

1 + D

1 + G

1

Similarly non-linear Equations (11) and (12)can be written as in polynomial variables are

(F2t2

3i+ H

2) t2

2i+ (I

2t3i) t

2i + (G

2t2

3i + J

2) = 0

...(15)

and

(F3t2

1i+ H

3) t2

3i + (I

3t1i) t

3i + (G

3t2

1i+ J

3) = 0

...(16)

The above three Equations (14), (15) and(16) represent in three unknowns t

1i, t

2i and t

3i.

Elimination of t2i then consider

K1

t22i + L

1t2i + M

1 = 0 ...(17)

K2

t22i + L

2t2i + M

2 = 0 ...(18)

where,

K1 = F

1t2

1i + H

1

L1 = I

1t1i

M1 = G

1t2

1i + J

1

K2 = F

2t2

3i + H

2

L2 = I

2t3i

M2 = G

2t2

3i + J

2

(17) x K2 – (18) x K

1 then we have equation

(L1K

2 – L

2K

1) t

2i + (M

1K

2 – M

2K

1) = 0

...(19)

Similarly (17) x M2 – (18) x M

1 then we have

equation

(K1M

2 – K

2M

1) t

2i + (L

1M

2 – L

2M

1) = 0

...(20)

Equations (19) and (20) are two linearequations and one unknown vanishing theirelimination then

(L1K

2 – L

2K

1) (L

1M

2 – L

2M

1) + (M

1K

2 –

M2K

1)2 = 0

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Once t1i is found, a unique value of t

3i can

be calculated and then unique value of t2i can

be computed. That means the unknowns {Ø1i,

Ø2i, Ø

3i} can be calculated.

By substituting Ø1i, Ø

2i, Ø

3i values into

iii qqq 321 ,, can be calculated that means

Tiiiziyixi PPP can be determined.

INVERSE KINEMATICANALYSISInverse kinematic analysis are obtained by alocation of mobile platform given, Then the limblengths L

ji |

j = 1, 2, 3 and i = 1, 2, 3, 4, etc., are resolved by

inverse kinematic analysis. Let us consider theunconstrained variables

i,

i, P

zi then the

constrained variables Pxi, P

yi,

i can be

determined from equations.

ii

iii cc

ssTan

1

i

iiixi cxccchP

1

22

1

iiyi shcP

Then the limb lengths can be determinedby geometry

etc.,4,3,2,1and3,2,1| ijjiji Lgq

iiiiixiziyixii sssccPhPPPd 222221

iiiiiziiiyi csscsPhscPh 22

2222 ghsssccghPg iiiiixi

...(21)

Similarly


iiiiiziiiyi csscsPhscPh

iiyiiiiiixi ccPhsccssPh 33

xiiiiiizi PgccsssPh 3

iiiiiyi sssccghPg 2

13

ii scgh 2

3 iiiii sccssgh

2

3

22

2

3ghccgh ii ...(22)


iiiiiziiiyi csscsPhscPh

iiyiiiiiixi ccPhcssscPh 33

xiiiiiizi PgccsssPh 3

iiiiiyi sssccghPg 2

13

ii scgh 2

3 iiiii cssscgh

2

3

22

2

3ghccgh ii ...(23)

By using above equations the relative limblengths of the manipulator can be determined.

CONCLUSIONIn this work the position analysis of spatial3-RPS parallel manipulator are investigatedin this paper. First ly the mobil i ty ofmanipulator is analyzed by Kutz-bachcriterion. The physical constraints areintroduced along the plane of spherical jointsand actuated link lengths are considered.The forward kinematics problem is solvedthrough Sylvester dialytic eliminationtechnique. The inverse kinematics solutionsare derived in closed form. All the possiblesolutions of this initial analysis are calculatedby changing the limb lengths through linearactuators, and also the posit ion and

89



orientation equations of the mobile platformwith respect to fixed base platform areanalyzed in terms of three limb lengths ofspatial 3-RPS parallel manipulator.

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Documents

Int. J. Mech. Eng. & Rob. Res. 2013 Pundru Srinivasa Rao ... · Lung-Wen (2003) and Yuefa and Lung-Wen (2004). Song and Zhang (1995) and Joshi and Tsai (2002) determined two singular