formulationofunified jacobianforserial-parallelmanipulators

Formulation of unied Jacobian for serial-parallel manipulators

Bo Hu a,b,c

a Parallel Robot and Mechatronic System Laboratory of Hebei Province, Yanshan University, Qinhuangdao, Hebei 066004, Chinab Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministry of National Education, Yanshan University, Qinhuangdao,Hebei 066004, Chinac State Key Laboratory of Robotics and System (HIT)

a r t i c l e i n f o

Article history:Received 9 May 2013Received in revised form5 March 2014Accepted 5 March 2014Available online 2 April 2014

Keywords:Serial-parallel manipulatorJacobianKinematicsGeometrical constraint

a b s t r a c t

This paper presents a general approach for Jacobian analysis of serial-parallel manipulators (S-PMs)formed by two lower mobility parallel manipulators (PMs) connected in serials. Based on the kinematicrelation, coupling and constraint properties of each PM, the unied forward and inverse Jacobianmatrices for S-PMs are derived in explicit form. It is shown that the Jacobian matrices of S-PMs haveunied forms, which include the complete information of each PM. A (3-RPS)(3-SPS/UP) S-PM is usedas an example to demonstrate the proposed approach. The established model is applicable for S-PMswith various architectures.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Serial manipulators (SMs) and PMs are two important branchesin the eld of robotics [1,2]. Intensive efforts over the years haveled to the development of fundamental theories related to SMsand PMs in their current and mature form. In recent years, S-PMshave attracted continuous interests in the eld of robotics [312].S-PMs have the advantages of both SMs and PMs which overcomethe issues related to rigidity and workspace limitations. Due totheir advantages, S-PMs have some potential applications for robotarms, machine tools, sensors, surgical manipulators, tunnel borers,satellite surveillance platforms [35].

The research of S-PMs mainly focused on the 6-degree offreedom (DOF) manipulators due to their compact topology,non-actuation redundancy, large workspace and relatively simplecontrol [612]. In this aspect, Romdhane [6] proposed a stewart-like hybrid S-PM and studied its displacement kinematics. Zheng[7] designed an S-PM by adding a pure rotational 3-UPU PM onto apure translational 3-UPU PM. Lu and Hu [810] studied theforward kinematics, statics, stiffness and workspace for someS-PMs [10]. Alvarado [11,12] studied the kinematics and dynamicsfor S-PMs using screw theory.

Compared with SMs and PMs, the research of S-PMs progressedat a slow pace. There are still some important unresolved pro-blems for this class of manipulators. An important reason for theslowness of the research progress for S-PMs is that the integral

Jacobian of this class of manipulators has not yet been established[512]. It is well known that Jacobian matrix plays an importantrole in the design and analysis for manipulators. The singularity,accuracy, kinematic performance evaluation and optimizationcan't be decoupled with this mapping matrix. For SMs, theJacobian can be easily derived [1]. For PMs, the Jacobian hasvarious forms due to their special properties and has been widelystudied by several approaches. The general Jacobian matrix forPMs is the nn form Jacobian, which relates the n actuators'velocities to the n components of the end effector velocity vector.This form of Jabobian [13,14] does not contain constraint informa-tion in the manipulators. Another form is the dimensionallyhomogeneous Jacobian [1517] which is established to overcomethe unit inconsistency of the Jacobian in optimization design. Thewidely used Jacobian is the 66 form Jacobian which containskinematics and constraint information [1820] for PMs.

Since each of S-PMs has its special kinematic characteristics,establishing the unied Jacobian form, which is applicable forS-PMs with various architectures, is obviously signicant. Due tothe complicated structure, high coupling and various constraintrelations for the S-PMs, solving the unied Jacobian for S-PMs is achallenging work. Up to now, there has been no effort to establishthe integral Jacobian of S-PMs [512]. For this reason, this paperfocuses on establishing the unied Jacobian for S-PMs.

The remainder of this paper is organized as follows. In Section2, the unied forward and inverse Jacobian matrices for S-PMsare established based on the kinematic relation, coupling andconstraint properties for each PM of the S-PMs. In Section 3,after a brief description of a novel (3-RPS)(3-SPSUP) S-PM in

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Robotics and Computer-Integrated Manufacturing

http://dx.doi.org/10.1016/j.rcim.2014.03.0010736-5845/& 2014 Elsevier Ltd. All rights reserved.

E-mail address: [email protected] (Bo Hu).

Robotics and Computer-Integrated Manufacturing 30 (2014) 460467

Section 3.1, the inverse kinematics of this S-PM is derived in closedform in Section 3.2, and the forward and inverse Jacobian matricesof this S-PM are derived based on the established Jacobian modelin Section 3.3. In Section 4, a numerical example concerned withthe inverse kinematics and Jacobian of the (3-RPS)(3-SPSUP)S-PM is provided. Finally, some concluding remarks are given inSection 5.

2. General Jacobian of S-PMs

In this paper, we only consider 6-DOF non-redundant S-PMs.Suppose that the S-PM is formed by one lower PM with k1 DOF andone upper PM with k2 DOF connected in series. Here, k1k26.Let the two PMs be PM 1 and PM 2 in sequence from bottom to up.Let ni0 and ni1 (i1, 2) be the lower platform and upper platformof PM i, respectively. Then n10 and n21 denote the base andterminal platform of the whole S-PM, respectively. In structure,n11 and n20 are xed with their centers kept coincidence (seeFig. 1).

Let oi (i1, 2) be the center of the upper platform of PM i. Forconvenience of analysis, we establish coordinate frames {ni0} and{ni1}(i1,2) at the center of the lower platform ni0 and the upperplatform ni1 for PM i, respectively. Then, {n10} can be seen as thebase coordinate frame and {n21} can be seen as the terminalcoordinate frame of the S-PM, respectively.

Let ncdnabR andncdnab (a1,2; b0,1; c1,2; d0,1) denote the

rotational matrix and angular velocity of {nab} relative to {ncd},respectively. Let ncdoi and

ncdvoi denote the position and linearvelocity vectors of oi relative to {ncd}, respectively. Then ni0oi,

ni0ni1R ,

ni0voi andni0ni1 (i1,2) denote the position, rotational matrix, linear

velocity and angular velocity of ni1 relative to ni0 for PM i, n10o2 ,n10n21R,

n10vo2 andn10n21 denote the position, rotational matrix, linear

velocity and angular velocity of n20 relative to n10 for the wholeS-PM.

2.1. The velocity decoupling and constraint matricesfor lower-mobility PMs

The integral forward and inverse Jacobian matrices of S-PMscontain the information about each PM of the S-PMs including thekinematic relation, coupling and constraint properties. Thus, thevelocity decoupling and velocity constraint matrices for each PMof S-PMs must be analyzed rst.

Due to the architecture constraints in lower mobility PMs, thekinematic parameters of the terminal platform are coupled. Ingeneral case, one PM with ki DOF has 6-ki independent poseconstraint equations. The pose constraint equations can be derivedbased on the geometrical constraints in PM i as follows:

f 1ni0Xoi; ni0Yoi; ni0Zoi; ni0i ; ni0i ; ni0i 0f 2ni0Xoi; ni0Yoi; ni0Zoi; ni0i; ni0i; ni0i 0

f 6ki

ni0Xoi;ni0Yoi;

ni0Zoi;ni0i;

ni0i;ni0i 0

1

where ni0Xoi;ni0Yoi;

ni0Zoi are three components of the positionvector ni0oi,

ni0i;ni0i and

ni0i are three Euler angles, which areused for representing the orientation of ni1 relative to nio for PM i.

From Eq. (1), the 6-dimensional pose parametersni0Xoi;

ni0Yoi;ni0Zoi;

ni0i;ni0i;

ni0i in {nio} can be expressed by kiindependent parameters ni0 si1;

ni0 si2;;ni0 siki for i-th PM as follows:

ni0Xoi Xoini0si1; ni0 si2;; ni0 siki ;ni0Yoi Yoini0si1; ni0si2;; ni0siki ;

ni0Zoi Zoini0si1; ni0si2;; ni0siki ;ni0i ini0 si1; ni0 si2;; ni0siki ;

ni0i ini0 si1; ni0 si2;; ni0 siki ;ni0i ini0si1; ni0si2;; ni0siki 2

where, ni0si1;ni0si2;;

ni0siki are ki independent generalized coordi-nates of ni1 for PM i. Eq. (2) is the pose decoupling equation of PMi.

By differentiating both sides of Eq. (2) with respect to time, itleads to

ni0 _Xoi ki

j 1

ni0Xoini0sij

Uni0 _sij ;ni0 _Yoi

ki

j 1

ni0Yoini0 sij

Uni0 _sij ;

ni0 _Zoi ki

j 1

ni0Zoini0 sij

Uni0 _sij ;ni0 _i

ki

j 1

ni0ini0sij

Uni0 _sij ;

ni0 _i ki

j 1

ni0ini0 sij

Uni0 _sij ;ni0 _i

ki

j 1

ni0ini0sij

Uni0 _sij 3

Let Ri ;Ri and Ri be three axes of i, i and i, respectively. Theangular velocity of ni1 relative to nio for PM i can be expressed asfollowing:

ni0ni1 ni0Ri ni0Ri

ni0Ri

ni0 _ini0 _ini0 _i

2664

3775 4

where ni0Ri ,ni0Ri and

ni0Ri are the unit vectors along Ri ;Ri andRi , respectively.

From Eqs. (3) and (4), it leads to

ni0voi Jvini0vsi ; Jvi

ni0Xoini0 si1

ni0Xoini0 siki

ni0 Yoini0 si1

ni0 Yoini0 siki

ni0 Zoini0 si1

ni0 Zoini0 siki

2666664

3777775;

ni0vsi ni0 _si1

ni0 _siki

264

375 5a

ni0ni1 Jini0vsi ; Ji

ni0Rini0Ri

ni0Ri

ni0ini0 si1

ni0ini0 s

iki

ni0ini0 si1

ni0ini0 s

iki

ni0 ini0 si1

ni0 ini0 s

iki

2666664

3777775

5b

n10

n21

n11

n20

o2

o1

{n21}

{n20}

{n10}

{n11}

Fig. 1. A general S-PM.

B. Hu / Robotics and Computer-Integrated Manufacturing 30 (2014) 460467 461

From Eqs. (5a) and (5b), it leads to

ni0voini0ni1

Joini0vsi ; Joi

JviJi

" #6

where Joi(i1, 2) is a 6 ki form matrix which is dened as thevelocity decoupling matrix, vsi is a vector formed by ki indepen-dent velocity parameters of ni1 for PM i.

The actuation velocities for PM i can be expressed based on theresults of the single lower mobility PM as following [2,20]:

vri Jini0voini0ni1

" #7

where Ji is a ki6 form matrix which is dened as the actuationJacobian, vri is a vector formed by the actuation velocities of PM i.

Unlike 6-DOF PM, the constrained wrenches (forces/torques)exist in lower mobility PMs. The constrained wrenches limit themotion of the terminal platform. In the general case, one PM withki DOFs has 6ki independent constrained wrenches. The con-strained wrenches can be determined by the geometricalapproach, which is shown as follows [20]:

1. In each sub-chain of lower mobility PMs, the constrained forcesmust be perpendicular to all prismatic joints and must becoplanar with all revolute joints respectively.

2. In each sub-chain of lower mobility PMs, the constrainedtorques must be perpendicular to all revolute joints.

Support that there are pi constrained forces and qi constrainedtorques existed in PM i. Let ni0 f ij(i1,2; j1, 2, , pi) be the unitvector of constrained force, ni0 tij be the corresponding vector fromthe center of terminal platform to ni0 f ij for PM i. Let

ni0 im(m1, 2,, qi) be the unit vector of constrained torque for PM i. Since theconstrained forces/torques of PM i do no work to ni1, it leads to[20]ni0 f ij U

ni0voini0 tij ni0 f ijUni0ni1 0 j 1;; pi 8a

ni0 im Uni0ni1 0 m 1;; qi 8b


Ji

n10voini0ni1

" # 06ki1; Ji

ni0 fTi1 ni0ti1

ni0 f i1T

ni0 fTipi

ni0 tipi ni0 f ipi

T

0 ni0 Ti1 0 ni0 Tiqi

266666666664

377777777775; k2 6k1

9where Ji is a (6ki)6 form matrix which is dened as thevelocity constraint matrix. Eq. (9) is dened as the velocityconstraint equation for PM i. This equation can also be obtainedby other approaches [18,19,21,22].

2.2. Derivation of general Jacobian for S-PMs

Let g[gx gy gz]T, h[hx hy hz]T be two arbitrary vectors, S(g) bea skew-symmetric matrix. There must be

Sg 0 gz gygz 0 gxgy gx 0

264

375; Sg SgT ; g h Sgh 10

The center of the terminal platform o2 in the base coordinatesystem {n10} can be expressed as following:n10o2 n10o1n10n20Rn20o2 ;

n10n20R

n10n11R

n11n20R 11

A composite rotational matrix n10n21R from {n21} to {n10} can beexpressed as following:n10n21R

n10n11R

n11n20R

n20n21R 12

n11 and n20 are xed connected, it leads ton10n11 n10n20 13a

The angular velocity of the terminal platform relative to base ofS-PM can be expressed as following:n10n21 n10n20

n10n20R

n20n21 n10n11

n10n20R

n20n21 13b

By differentiating both sides of Eq. (11) with respect to timeand combining with Eqs. (10) and (13a), it leads ton10vo2 n10vo1n10n20Rn20vo2n10n20

n10n20Rn20o2

n10vo1n10n20Rn20vo2Sn10n20R

n20o2 n10n11 14

From Eqs. (13b) and (14),it leads ton10vo2n10n21

" # T1

n10vo1n10n11

" #T2

n20vo2n20n21

" #;

T1 E33 Sn10n20Rn20o2 033 E33

" #; T2

n10n20R 033033

n10n20R

" #15

where E33 is an 33 form unit matrix.From Eqs. (6) and (7), it leads to

ni0voini0ni1

" # JoiJi Joi1vri 16

From Eqs. (15) and (16), it leads ton10vo2n10n21

" # JF

vr1vr2

" #; JF T1Jo1J1Jo11 T2Jo2J2 Jo21 17

here, JF is the forward Jacobian for S-PMs.From Eq. (15), it can be known that jT1jjT2j1. Thus, the

invertible matrix of Ti (i1, 2) exists. Multiplying both sides ofEq. (15) by T11 , it leads to

T11

n10vo2n10n21

" #

n10vo1n10n11

" #T11 T2

n20vo2n20n21

" #18

Multiplying both sides of Eq. (18) by J1 and combining with Eq.(9), it leads to

J1T11

n10vo2n10n21

" # J1T11 T2

n20vo2n20n21

" #19


J1T11

n10vo2n10n21

" # J1T11 T2Jo2n20vs2 20

Since k1k26, J1 is a k26 form matrix and J1T11 T2Jo2 is ak2 k2 form matrix. From Eq. (20), it leads to

n20vs2 J1T11 T2 Jo21J1T11

n10vo2n10n21

" #21

Multiplying both sides of Eq. (21) by Jo2 and combining withEq. (6), it leads to

n20vo2n20n21

" # Jo2J1T11 T2 Jo21J1T11

n10vo2n10n21

" #22

Multiplying both sides of Eq. (22) by J2 and combining withEq. (7), it leads to

vr2 J2 Jo2J1T11 T2 Jo21J1T11n10vo2n10n21

" #23

B. Hu / Robotics and Computer-Integrated Manufacturing 30 (2014) 460467462

Using the same approach, vr1 can be derived from Eqs. (6), (7),(9) and (15) as following:

vr1 J1 Jo1J2T12 T1Jo11J2T12n10vo2n10n21

" #24

where J2T12 T1Jo1 is a k1 k1 form matrix.


vr1vr2

" # J

n10vo2n10n21

" #; J

J1Jo1J2T12 T1Jo11J2T12J2Jo2J1T11 T2 Jo21J1T11

24

35 25

where J is the unied inverse Jacobian for S-PMs.Let Fo and To be the external force and torque applied on the

terminal platform of S-PMs, Fri(i1, 2) be the vector formed by theactuation forces/torques of PM i.

From the principle of virtual work, it leads to

FTr1 FTr2 vr1vr2

" #

n0Fon0To

" #T n10vo2n10n21

" #26


Fr1Fr2

" # JT1

n0Fon0To

" # JTF

n0Fon0To

" #27

From Eq. (27), the statics of S-PM can be solved.

3. Example

In order to demonstrate the approach developed in this paper,we perform here a generalized Jacobian analysis for a novel (3-RPS)(3-SPSUP) S-PM.

3.1. Description of the (3-RPS)(3-SPSUP) S-PM

The (3-RPS)(3-SPSUP) S-PM is formed by one lower 3-RPSPM and one upper 3-SPSUP PM connected in series (See Fig. 2).The lower 3-RPS PM includes a lower platform n10, an upper

platform n11 and three RPS-type driving legs r1k (k1, 2, 3). n10and n11 are two equilateral triangles. The k-th RPS(revolute joint-active prismatic joint-spherical joint)-type leg connects n10 withn11 by using a revolute joint R1k at A10k on n10, a prismatic joint Palong r1k, and a spherical joint S at A11k on n11. The upper3-SPSUP PM includes a lower platform n20, an upper platformn21, three SPS (spherical joint-active prismatic joint-sphericaljoint)-type active legs r2k (k13) and one UP (universal joint-active prismatic joint)-type constrained leg r20. Each of the SPS-type active legs connects n20 to n21 by a spherical joint S at A20k onn20, an active leg with a prismatic joint P, and a spherical joint S atA21k on n21. The UP-type constrained leg r20 connects n20 with n21by a universal joint U on n20 at o1, a prismatic joint P along r20 andperpendicularly xed to n21 at o2. The U joint is composed of twocrossed revolute joints R21 and R22. n11 and n20 have an angle of601 between them. n10 and n21 can be seen as the base andterminal platform of the (3-RPS)(3-SPSUP) S-PM, respectively.

In the (3-RPS)(3-SPSUP) S-PM, there are 1 terminal plat-form n21 with a piston-rod, 1 composite platform n11/n20, 1 basen10, 7 cylinders, and 6 piston-rods, thus the number of links isc016. There are 7 prismatic joints, 3 revolute joints, 9 sphericaljoints, and 1 universal joint in the S-PM, thus the number of jointsis c20. The DOF of this S-PM is calculated as below [19]:

M 6c0c1 c

i 1mim0 6 16201

7 13 19 31 23 6 28

here, m11 is the DOF of revolute joint, m21 is the DOF ofprismatic joint, m32 is the DOF of universal joint, m43 is theDOF of spherical joint. The redundancy DOF is mo3 for 3 SPS-type active legs rotating about their own axes. mo has no inuenceon the kinematic characteristics.

Let ? be a perpendicular constraint and || be a parallelconstraint. Establish coordinate frames {nij} (i1, 2; j0, 1) atthe center of nij with Xij, Yij and Zij(i1, 2; j0, 1) are threeorthogonal coordinate axes and some constraints (Xij||Aij1Aij3,Yij?Aij1Aij3, Zij?nij) are satised. In the 3-RPS PM, the geometrical

Fig. 2. A (3-RPS)(3-SPSUP) S-PM: (a) sketch of (3-RPS)(3-SPSUP) S-PM and (b) CAD model of (3-RPS)(3-SPSUP) S-PM.


constraints can be expressed as follows:

R11jjA102A103;R12jjA101A103;R13jjA101A102;R1k ? r1k k 13 29aIn the 3-SPSUP PM, the geometrical constraints can be

expressed as follows:

R21jjA201A203;R21 ? R22;R22 ? r20;R22jjY21;r20 ? n21 29b

3.2. Inverse position analysis of the (3-RPS)(3-SPSUP) S-PM

The position vectors Aijk (i1, 2; j0, 1; k1, 2, 3) of threevertices Aijk in {nij} can be expressed as follows:

nijAij1 eij2

q10

264

375; nijAij2

0eij0

264

375; nijAij3 eij2

q10

264

375; q 3p

30awhere eij denotes the distance from the center of nij to Aijk.

The position vectors Ai1k (i1, 2; k1, 2, 3) of three vertices ofni1 in {ni0} for each PM can be expressed as following:

ni0Ai1k ni0ni1Rni1Ai1kni0oi; ni0oi

ni0Xoini0Yoini0Zoi

264

375 30b

Let ni0ni1R be formed by XYX Euler rotations with i, i and i arethree Euler angles, it leads to

ni0ni1R

ni0xli ni0ylini0zli

ni0xmi ni0ymini0zmi

ni0xni ni0ynini0zni

264

375

ci si si si ci

si si ci ci si ci si ci si si ci cici si si ci ci ci si si si ci ci ci

264

375 31

where ni0xli ni0xmi ni0xni ni0yli ni0ymi ni0yni ni0zli ni0zmi ni0zniare nine orientation parameters of ni0ni1R .

n11 and n20 have an angle of 601 between them, thus therotational matrix for n20 relative to n11 can be expressed asfollowing:

n11n20R

12

1 q 0q 1 00 0 1

264

375 32a

A composite rotational matrix n10n21R from n21 relative to n10 canbe expressed as following:

n10n21R

n10n11R

n11n20R

n20n21R

n20xl2 n20yl2n20zl2

n20xm2 n20ym2n20zm2

n20xn2 n20yn2n20zn2

264

375 32b

3.2.1. The pose decoupling equations for the 3-RPS PMFor the 3-RPS PM, the unit vectors R1k of R1k (k1, 2, 3) in {n10}

can be expressed as follows:

n10R11 12

1q

0

264

375; n10R12

100

264

375; n10R13 12

1q

0

264

375 33

From the geometrical constraints of the 3-RPS PM, it leads ton10R1k Un10A11kn10A10k 0 34

From Eqs. (30a), (30b) and (34)), it leads ton10Xo1 e11n10yl1 35a

n10Yo1 e11n10ym1n10xl1=2 35b

n10xm1 n10yl1; n10xn1 n10zl1; n10yn1 n10zm1 35c

From Eqs. (31) and (35c), it leads to

1 1 36a

From Eqs. (31) and (36a), n10n11R for the 3-RPS PM can besimplied as following:

n10n11R

c1 s1s1 s1c1s1s1 c

21 s21c1 c1 s1 s1c1c1

c1 s1 s1c1 s1c1c1 s21 c21c1

2664

3775 36b

From Eqs. (35a), (35b) and (36b), it leads to

n10Xo1 e11s1s1 37a

n10Yo1 e11c21 s21c1 c1 =2 37b

3.2.2. The pose decoupling equations for the 3-SPSUP PMFor the 3-SPSUP PM, the unit vectors R2k of R2k (k1, 2) in

{n20} can be expressed as following:

n20R21 100

264

375; n20R22 n20y21

n20yl2n20ym2n20yn2

264

375 38

From Eq. (29b), it leads to

n20R21 Un20R22 0 39

From Eqs. (29b) and (39), it leads to

n20yl2 0; n20Xo2 r20zl2; n20Yo2 r20zm2; n20Zo2 r20zn2 40

From Eqs. (31) and (40), n20n21R for the 3-SPSUP PM can besimplied as following:

n20n21R

c2 0 s2s2s2 c2 s2c2c2 s2 s2 c2c2

264

375 41


n20Xo2 r20s2 ;n20Yo2 r20s2c2 ;

n20Zo2 r20c2c2 42

3.2.3. Inverse position modeling of (3-RPS)(3-SPSUP) S-PMThe inverse position analysis of the (3-RPS)(3-SPSUP) S-PM

is to determine the required actuated variables from a given poseof n21 relative to n10.

From the structure property of this S-PM, it leads to

n10o1 n10o2r2o Un10z2

n10Xo2r20 Un10zl2n10Yo2r20 Un10zm2n10Zo2r20 Un10zn2

0B@

1CA 43

where n10o2 n10Xo2 n10Yo2 n10Zo2 TFrom Eqs. (37a), (37b) and (43), it leads to

r20 n10Xo2e11s1s1

n10zl2

n10Yo2e11c21 s21c1 c1 =2n10zm2

44

From Eq. (44), it leads to

n10zm2e11s1 s1 n10zl2e11c21 s21c1 c1 =2 n10zl2n10Yo2n10zm2n10Xo2

45


R21 and R22 can be expressed in {n10} as following:

n10R21 n10n11Rn11n20R

n20R21 12

c1 qs1s1s1s1 qc21 s21c1

c1s1 qs1c1 s1c1c1

2664

3775;

n10R21 n10yl2n10ym2n10yn2

264

375 46

Since R21?R22, it leads ton10yl2c1 qs1 s1

n10ym2s1 s1 qc21 s21c1

n10yn2c1s1 qs1c1 s1c1c1 0 47

Let b1 tg1=2; b2 tg1=2;there must be

s1 2b11b21

; c1 1b211b21

; s1 2b21b22

; c1 1b221b22

48

Substituting Eq. (48) into Eq. (45), it leads to

c6b41b

22c5b21b22c4b22c3b1b2c2b41c1b21c0 0 49a

where

c6 n10zl2n10Yo2n10zm2n10Xo2n10zl2e11;c5 2n10zl2n10Yo2n10zm2n10Xo2n10zl2e11;c4 n10zl2n10Yo2n10zm2n10Xo22n10zl2e11;c3 4n10zm2e11; c2 n10zl2n10Yo2n10zm2n10Xo2;c1 2n10zl2n10Yo2n10zm2n10Xo22n10zl2e11;c0 n10zl2n10Yo2n10zm2n10Xo2

Substituting Eq. (48) into Eq. (47), it leads to

d16b1b42d15b21b32d14b32d13b41b22d12b31b22d11b21b22

d10b1b22d9b22d8b31b2d7b21b2d6b1b2d5b2d4b41d3b31d2b21d1b1d0 0 49bwhere

d16 2qn10yn2; d15 2n10yn2;d14 2n10yn2; d13 qn10ym2n10yl2;d12 2qn10yn2; d11 2qn10ym22n10yl2;d10 2qn10yn2;d9 n10yl2qn10ym2; d8 4qn10yl2n10ym2;d7 2n10yn2; d6 4qn10yl2n10ym2;d5 2n10yn2; d4 n10yl2qn10ym2; d3 2qn10yn2;d2 2n10yl26qn10ym2; d1 4qn10yn2; d0 n10yl2qn10ym2;

From Eq. (49a) and (49b), it leads to

g12b22g11b2g10 0 50a

g24b42g23b32g22b22g21b2g20 0 50bwhere

g12 c6b41c5b21c4; g11 c3b1; g10 c2b41c1b21c0;g24 d16b1; g23 d15b21d14; g22 d13b41d12b31d11b21d10b1d9;

g21 d8b31d7b21d6b1d5; g20 d4b41d3b31d2b21d1b1d0Multiplying both sides of Eq. (50a) by b2, b

22, b

32 respectively, it

leads to

g12b32g11b22g10b2 0

g12b42g11b32g10b22 0

g12b52g11b42g10b32 0 51aMultiplying both sides of Eq. (50b) by b2, it leads to

g24b52g23b42g22b32g21b22g20b2 0 51b

From Eqs. (50a), (50b), (51a) and (51b), it leads to

G

b52b42b32b22b21

266666666664

377777777775 0; G

0 0 0 g12 g11 g100 0 g12 g11 g10 00 g12 g11 g10 0 0g12 g11 g10 0 0 00 g24 g23 g22 g21 g20g24 g23 g22 g21 g20 0

26666666664

37777777775

52

The necessary condition for Eq. (52) to have nontrivial solutionsis

jGj 0 53

Eq. (53) is a nonlinear equation with regard to b1. When thepose of n21 relative to n10 is given, b1 can be solved from Eq. (53).

After b1 is solved, b2 can be solved from Eq. (50a) as following:

b2 g117

g2114g12g10

q2g12

54

After b1 and b2 are solved, 1 and 1 can be solved from Eq. (48)and r20 can be solved from Eq. (44), then

n10n11Rcan be solved from

Eq. (36b) and n10o1 can be solved from Eq. (43) subsequently. FromEq. (12), it leads to

n10n21R

n10n11R

n11n20R

1n10n21R 55a

Combining with Eqs. (41) and (55a), 2 and 2 can be obtained,then n20o2 can be solved from Eq. (42), subsequently.

From Eqs. (30a) and (30b), ni0Ai1k can be solved, then rik (i1,2;k13) can be solved as following:

rik jni0Ai1kni0Ai0kj 55b

3.3. Jacobian establishment of the (3-RPS)(3-SPSUP) S-PM

From Eqs. (37a) and (37b), the linear velocity of the upperplatform relative to the lower platform of the 3-RPS PM can beexpressed as following:

n10vo1 Jv1n10vs1 ; n10vs1 _1_1_Zo1

264

375;

Jv1 e11c1 s1 e11s1c1 0

e11c1s1 1c1 e11s1 s21 1=2 00 0 1

264

375 56a

From Eqs. (5b) and (36b), the angular velocity of the upperplatform relative to the lower platform of the 3-RPS PM can beexpressed as following:

n10n11

n10R1 _1n10R1 _1n10R1

_1 J1n10vs1 ;

J1 1c1 0 0s1s1 c1 0c1 s1 s1 0

264

375 56b

here,

n10R1 100

264

375; n10R1

0c1s1

264

375; n10R1

c1s1s1c1 s1

264

375



Jo1

e11c1s1 e11s1c1 0e11c1s1 1c1 e11s1 s21 1=2 0

0 0 11c1 0 0s1s1 c1 0c1s1 s1 0

26666666664

37777777775

56c

Here, Jo1 is the 63 form velocity decoupling matrix of the 3-RPS PM.

From the geometrical approach for determining constrainedwrenches, one constrained force, which is parallel with R1i andpasses through S joint, can be determined in each RPS leg. Thus,the velocity constraint equation for the 3-RPS PM can be deter-mined as following:

J1n10vo1n10n11

031;

J1

n10 f T11 n10 t11 n10 f 11 Tn10 f T12 n10 t12 n10 f 12 Tn10 f T13 n10 t13 n10 f 13 T

2664

3775;

n10 f 1k n10R1kn10 t1k n10o1n10A11k

57

Here, J1 is the 36 form velocity constraint matrix for the 3-RPS PM.

From Eq. (42), the linear velocity of the upper platform relativeto the lower platform of the 3-SPSUP PM can be expressed asfollowing:

n20vo2 Jv2n20vs2 ; Jv2 0 r20c2 s2

r20c2c2 r20s2s2 s2c2r20s2c2 r20c2 s2 c2c2

264

375;

n20vs2 _2_2_r20

264

375 58a

The angular velocity of the upper platform relative to the lowerplatform of the 3-SPSUP PM can be expressed as following:

n20n21

n20R2 _2n20R2 _2 J2n20vs2 ; J2

1 0 00 c2 00 s2 0

264

375;

n20R2 100

264

375; n20R2

0c2s2

264

375 58b


Jo2

0 r20c2 s2r20c2c2 r20s2 s2 s2c2r20s2c2 r20c2s2 c2c2

1 0 00 c2 00 s2 0

26666666664

37777777775

58c

Here, Jo2 is the 63 form velocity decoupling matrix of the3-SPSUP PM.

Based on the geometrical approach for determining the con-strained wrenches, one constrained torque, which is perpendicularwith R21 and R22, can be determined in UP type leg. In addition,two constrained forces, which are parallel with x21 and y21 andpass through the center of U joint, can be determined in UPleg respectively. Thus, the velocity constraint equation for the

3-SPSUP can be determined as following:

J2n20vo2n20n21

031; J2

n20 f T21 n20 t21 n20 f 21 Tn20 f T22 n20 t22 n20 f 22 T0 n20T21

2664

3775;

n20 f 21 n20x21 ; n20 f 22 n20y21n20 t2i n20o2n20o1

n20T21 n20R21 n20R2259

Here, J2 is the 36 form velocity constraint matrix for the3-SPSUP PM.

The 3-RPS PM and the 3-SPSUP PM have linear active legs,the actuation velocities can be expressed as following [20]

vri Jini0voini0ni1

; Ji

ni0Ti1 ni0ei1 ni0i1 Tni0Ti2 ni0ei2 ni0i2 Tni0Ti3 ni0ei3 ni0i3 T

26664

37775;

ni0ik ni0Ai1kni0Ai0kni0Ai1kni0Ai0k ; ni0eik ni0oini0Ai1k i 1;2; k 1;2;3

60

Substituting Jo1 derived from Eq. (56c), Jo2 derived from Eq.(58c), J1 derived from Eq. (57), J2 derived from Eq. (59) and Ji(i1,2) derived from Eq. (60) into Eqs. (17) and (25), the forwardand inverse Jacobian of the (3-RPS)(3-SPSUP) S-PM can bedetermined.

4. Analytically solved example

In this section, the computation of the inverse position andJacobian for the (3-RPS)(3-SPSUP) S-PM is performed applyingthe established kinematics and Jacobian models at a given pose.Set the dimension parameters as e10120 mm, e11e2080 mm,e2160 mm. The position of the terminal platform n21 relative tobase n10 is given as n10Xo2 61:5; n10Yo2 20; n10Zo2 229Tmm.The orientation of n21 relative to n10 is given by XYX-type Eulerrotations with (64, 65.5, -56)1 are three correspondingEuler angles. Then b1 can be solved from Eq. (53) as following (seeTable 1):

It is known from Table 1 that b1 has 16 solutions. In order todetermine the acceptable analytic solution from multi-solutions,the simulation mechanism of the (3-RPS)(3-SPSUP) S-PM iscreated using CAD variation geometry approach in CAD software[23]. When given the dimension parameters for the simulationmechanism according to the parameters used in analytic model,the value of 1 can be obtained. By comparing the analytic andsimulation results, it is known that the simulation solution is inexcellent agreement with the 13th solution obtained from theanalytic model. After 1 is determined, other pose parameters canbe solved based on the established analytic model as following:

1 28:981; 2 3:93051; 2 3:52171n10o1 2:9191 2:1554 131:5129 Tmm;n20o2 7:0717 7:8763 114:6379 Tmm

Table 1The 16 solutions of b1.

14 01.0000i 01.0000i 01.0000i 01.0000i58 01.0000i 01.0000i 01.0000i 01.0000i912 0.2614 4.8685 8.7318 15.26771316 0.0655 0.1145 0.2054 3.8254


Then the inverse kinematics of the (3-RPS)(3-SPSUP) S-PMcan be solved as following:

r11 120:2790 mm; r12 122:3843 mm; r13 159:9683 mm;r21 116:5750 mm; r22 118:8374 mm; r23 111:6689 mm:The inverse Jacobian of the (3-RPS)(3-SPSUP) S-PM at this

pose can be determined as following:

J

1:0757 1:8237 0:9802 230:6453 148:6923 33:21411:7630 0:4169 0:9831 52:1565 240:6251 11:16804:3408 8:9113 0:9772 632:9442 396:5137 935:48370:6485 4:2911 0 369:5345 130:5829 351:28454:1771 7:2913 0 572:5189 323:9157 667:48923:3511 4:5831 0 296:1434 250:2898 644:5454

2666666664

3777777775

Using the derived Jacobian, when the linear and angularvelocities of n21 relative to n10 are given as n10vo2 1 2 3 mm=s, n10n21 3 4 5 rad=s, the inverse velocity ofthe (3-RPS)(3-SPSUP) S-PM can be solved as follows:vr1 0:5798 20:5752 4:4000 T mm=s;vr2 11:1923 4:7466 17:4525 T mm=s

The results are veried by CAD variation geometry approach[23] in CAD software.

5. Conclusion

The main contribution of this paper lies in the derivation of theunied Jacobian model for S-PMs with 6-DOF. The forward andinverse Jacobian matrices are derived based on the kinematicsmodel, coupling and constraint properties of each PM of the S-PMs. The established Jacobian matrices contain all information ofeach PM, including the velocity coupling relations which arecontained in velocity decoupling matrix Joi(i1,2), the velocityconstraint relations which are contained in velocity constraintmatrix Ji, and the velocity transmission relations which arecontained in matrix Ji. The existence and uniqueness of theintegral Jacobian for S-PMs are shown from the explicit form.

The novel (3-RPS)(3-SPSUP) S-PM is analyzed to illustratethe proposed approach. The inverse displacement of the (3-RPS)(3-SPSUP) S-PM is derived in closed form. The computing resultshows that the terminal platform can reach at most 16 differentposes with respect to base. The inverse Jacobian is computed at areasonable pose. The unied model provides a general andeffective approach for establishing the Jacobian of various S-PMs.

Acknowledgments

The authors are grateful to the project (No. 51305382) sup-ported by National Natural Science Foundation of China, ExcellentYouth Foundation of Science and Technology of Higher Education

of Hebei Province (YQ2013011) and the nancial support of StateKey Laboratory of Robotics and System (HIT) (SKLRS-2012-MS-01).

References

[1] Angeles J. Fundamentals of robotic mechanical systems. Springer-Verlag;2003.

[2] Merlet JP. Parallel robots. London: Kluwer Academic Publishers; 2000.[3] Hanan MW, Walker IA. Kinematics and the implementation of an elephant's

trunk manipulator and other continuum style robots. J Robot Syst 2003;20(2):4563.

[4] Moosavian S, Pourreza A. Heavy object manipulation by a hybrid serial-parallel mobile robot. Int J Robot Autom 2010;25(2):10920.

[5] Liang C, Ceccarelli M. Design and simulation of a waisttrunk system for ahumanoid robot. Mech Mach Theory 2012;53(7):5065.

[6] Romdhane L. Design and analysis of a hybrid serial-parallel manipulator. MechMach Theory 1999;34(7):103755.

[7] Zheng XZ, Bin HZ, Luo YG. Kinematic analysis of a hybrid serial-parallelmanipulator. Int J Adv Manuf Technol 2004;23(1112):92530.

[8] Lu Y, Hu B. Analyses of kinematics/statics and workspace of a 2(SPSPRSPU) serial-parallel manipulator. Multibody Syst Dyn 2009;21(4):36170.

[9] Hu B, Lu Y, et al. Analysis of stiffness and elastic deformation of a 2(SPSPRSPU) serial-parallel manipulator. Robot Comput Integr Manuf2011;27(2):41825.

[10] Hu B, Lu Y, et al. Statics and stiffness model of serial-parallel manipulatorformed by k parallel manipulators connected in series. ASME J Mech Robot2012;4(2):021012.

[11] Gallardo-Alvarado J, Aguilar-Njera CR, et al. Kinematics and dynamics of2(3-RPS) manipulators by means of screw theory and the principle of virtualwork. Mech Mach Theory 2008;43(10):128194.

[12] Gallardo-Alvarado J, Aguilar-Njera CR, et al. Solving the kinematics anddynamics of a modular spatial hyper-redundant manipulator by means ofscrew theory. Multibody Syst Dyn 2008;20(4):30725.

[13] Zhu SJ, Huang Z, Guo. XJ. Forward/reverse velocity and acceleration analysisfor a class of lower-mobility parallel mechanisms. ASME J Mech Des 2007;129(4):3906.

[14] Lu Y, Hu. B. A unied approach to solving driving forces in spatial parallelmanipulators with less than 6-DOF. Trans ASME J Mech Des 2007;129(11):115360.

[15] Gosselin CM. The optimum design of robotic manipulators using dexterityindices. J. Robot Auton Syst 1992;9(4):21326.

[16] Kim SG, Ryu J. New dimensionally homogeneous Jacobian matrix formulationby three end-effector points for optimal design of parallel manipulators. IEEETrans Robot Autom 2003;19(4):7316.

[17] Pond G, Carretero J. Formulating Jacobian matrices for the dexterity analysis ofparallel manipulators. Mech Mach Theory 2006;41(12):150519.

[18] Joshi S, Tsai LW. Jacobian analysis of limited-DOF parallel manipulators. ASMEJ Mech Des 2002;124(2):2548.

[19] Huang Z, Zhao YS, Zhao TS. Advanced spatial mechanism. Beijing, China:Higher Education Press; 2006.

[20] Lu Y, Hu B. Unication and simplication of velocity/acceleration of limited-dof parallel manipulators with linear active legs. Mech Mach Theory 2008;43(9):111228.

[21] Cheng G, Yu J, Gu W. Kinematic analysis of 3SPS1PS bionic parallel testplatform for hip joint simulator based on unit quaternion. Robot Comput-Integr Manuf 2012;28(2):25764.

[22] Hong MB, Choi YJ. Kinestatic analysis of nonsingular lower mobility manip-ulators. IEEE Trans Robot 2009;25(4):93842.

[23] Lu Y. Using cad variation geometry for solving velocity and acceleration ofparallel manipulators with 35 linear driving limbs. ASME J Mech Des2006;128(4):73846.


Formulation of unified Jacobian for serial-parallel manipulatorsIntroductionGeneral Jacobian of S-PMsThe velocity decoupling and constraint matrices for lower-mobility PMsDerivation of general Jacobian for S-PMs

ExampleDescription of the (3-RPS)+(3-SPS+UP) S-PMInverse position analysis of the (3-RPS)+(3-SPS+UP) S-PMThe pose decoupling equations for the 3-RPS PMThe pose decoupling equations for the 3-SPS+UP PMInverse position modeling of (3-RPS)+(3-SPS+UP) S-PM

Jacobian establishment of the (3-RPS)+(3-SPS+UP) S-PM

Analytically solved exampleConclusionAcknowledgmentsReferences

formulationofunified jacobianforserial-parallelmanipulators

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