proceedings of the institution of mechanical engineers, part c- journal of mechanical engineering...

http://pic.sagepub.com/Engineering Science

Engineers, Part C: Journal of Mechanical Proceedings of the Institution of Mechanical

http://pic.sagepub.com/content/early/2014/02/20/0954406214525366The online version of this article can be found at:

DOI: 10.1177/0954406214525366online 24 February 2014

publishedProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering ScienceHaibo Qu, Yuefa Fang, Sheng Guo and Wei Ye

A novel 4-UPU translational parallel mechanism with fault-tolerant configurations

Published by:

http://www.sagepublications.com

On behalf of:

Institution of Mechanical Engineers

can be found at:ScienceProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical EngineeringAdditional services and information for

http://pic.sagepub.com/cgi/alertsEmail Alerts:

http://pic.sagepub.com/subscriptionsSubscriptions:

http://www.sagepub.com/journalsReprints.navReprints:

http://www.sagepub.com/journalsPermissions.navPermissions:

What is This?

- Feb 24, 2014OnlineFirst Version of Record >>

at Beijing Jiaotong University on May 21, 2014pic.sagepub.comDownloaded from at Beijing Jiaotong University on May 21, 2014pic.sagepub.comDownloaded from

XML Template (2014) [17.2.20144:53pm] [113]//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PICJ/Vol00000/140036/APPFile/SG-PICJ140036.3d (PIC) [PREPRINTER stage]

Original Article

A novel 4-UPU translational parallelmechanism with fault-tolerantconfigurations

Haibo Qu, Yuefa Fang, Sheng Guo and Wei Ye

Abstract

This paper aims at designing a pure translational parallel mechanism constructed by UPU (universal-prismatic-universal

joint) kinematic limbs. First, the typical problem of unexpected rotations is pointed out from analyzing the typical 3-UPU

parallel mechanism, and the reason for unexpected rotations of parallel mechanism constructed by UPU kinematic limbs

is analyzed. Then, in order to design a pure translational parallel mechanism constructed by UPU without the unexpected

rotations, the 2-UPU single loop is chosen as the basic structure to construct the 4-UPU translational parallel mech-

anism. Each 2-UPU single loop can be used to constrain a rotation about an axis of the linear complexes, which defined

the unexpected rotations. Therefore, a novel type of 4-UPU pure translational parallel mechanism with redundant

actuations is proposed. Since the existence of redundantly actuated kinematic limb, this proposed parallel mechanism

possesses analytical forward kinematics, and its singularity can be avoided completely. Finally, the workspace and fault-

tolerant performance are analyzed. When the proposed 4-UPU parallel mechanism is located in a fault-tolerant config-

uration, the moving platform can still possess movable ability to realize the given task even if one kinematic limb is in

locked-joint failure mode, and the fault-tolerant workspace is obtained.

Keywords

Parallel mechanism, redundant actuations, unexpected rotations, fault-tolerant

Date received: 11 October 2013; accepted: 31 January 2014

Introduction

Since some potential industrial applications, such aspick-and-place, machining operations and so on, thepure translational parallel mechanisms with threedegrees of freedom (DOF) have attracted much atten-tion. Many researchers have studied the 3-DOF transla-tional parallel mechanisms using dierent theory, suchas Clavel,1 Herve and Sparacino,2 Tsai,3,4 Carricato andParenti-Castelli,5,6Kong andGosselin,7Gogu,8,9Huangand Li,10 Jin and Yang,11 Liu et al.,12 and Yu et al.13,14

Among these mechanisms, the most popular transla-tional parallel mechanism is the Delta robot designedby Clavel1 based on the parallelogram structure.

In addition, one translational mechanism that hasattracted a lot of academic interest is the 3-UPU (uni-versal-prismatic-universal joint) parallel mechanismproposed by Tsai.3 After that, Joshi and Tsai1517

carried out an extensive study of this robot, andGregorio and Parenti-Castelli18 performed the mobil-ity analysis of the 3-UPU parallel manipulatorassembled for obtaining a pure translationmotion, respectively, and discussed the singularity

congurations. However, the unexpected rotationswere observed in a manufactured prototype of3-UPU parallel mechanism at Seoul NationalUniversity (SNU). There occurred unexpected largerotations of the moving platform, even the three pris-matic joints are locked and the mechanism seemed tocollapse under its weight.

Many researchers tried to explain the unexpectedrotations. Han et al.19 proved that this SNU 3-UPUparallel mechanism is very sensitive to manufacturingtolerance clearance. Zlatanov et al.20,21 pointed thatthe singularities of this SUN parallel mechanismbelong to constraint singularity, i.e. the screwsystem formed by the constraint wrenches in all legsloses rank. Wolf and Shoham22,23 investigated the

School of Mechanical, Electronic and Control Engineering, Robotics

Institute, Beijing Jiaotong University, Beijing, China

Corresponding author:

Haibo Qu, School of Mechanical, Electronic and Control Engineering,

Robotics Institute, Beijing Jiaotong University, No.3 Shang Yuan Cun,

Hai Dian District, Beijing 100044, China.

Email: [email protected]

Proc IMechE Part C:

J Mechanical Engineering Science

0(0) 113

! IMechE 2014Reprints and permissions:

sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/0954406214525366

pic.sagepub.com

at Beijing Jiaotong University on May 21, 2014pic.sagepub.comDownloaded from


singularities and self-motion of 3-UPU parallel mech-anism by using the linear complex approximation.Meanwhile, Conconi and Carricato24 have made anassessment on the singularities of general parallelkinematic chains with dierent hierarchical levels,and discussed the constraint singularity of 3-UPUparallel mechanism. Gogu25 addressed the constraintsingularities in connection with the structural param-eters of parallel mechanism. Chebbi and Parenti-Castelli26 collected the most relevant results onsingularities and workspace of the 3-UPU parallelmechanism proposed by Tsai, and investigated theinuence of some geometric parameters especiallythe orientation of revolute axes and the locations ofits legs on the singularity loci and workspace size.Chebbi et al.27 studied the singularity based on thekinetostatic of the 3-UPU parallel mechanism. Ganet al.28 designed a new metamorphic parallel mechan-ism by changing the universal joint to the recongur-able joint in the general 3-UPU parallel mechanism.Qu et al.29 performed the analysis of the unexpectedrotation evaluation and avoidance of the 3-UPU par-allel mechanism. Merlet30 also pointed out that the3-UPU mechanism will exhibit only translationmotion if it satises exactly geometrical constraints,which in practice cannot be satised. Also, someresearchers have studied the performance of parallelmechanisms with UPU kinematic limbs. Lu and Hu31

addressed a family of 2-UPU-X parallel mechanismsand obtained an asymmetric 3-UPU parallel mechan-ism. Zhao et al.32,33 analyzed the mobility and singu-larity of a 4-UPU parallel mechanism with Schoniesmotion.

A parallel mechanism with redundant actuationsrefers to the use of more actuators than minimallyrequired for doing the prescribed task. The actuatorredundancy in parallel mechanism can be used toreducing some singular congurations,34 enlargingthe workspace,35 and improving the stiness,36 andso on. Therefore, adding redundant actuations forthe parallel mechanisms is considered as an eectiveapproach to improve the performance.

In this paper, a novel type of 4-UPU pure transla-tional parallel mechanism with redundant actuationsis proposed. The paper is organized as follows. InSection Unexpected rotations, the reason for unex-pected rotations of parallel mechanism constructed byUPU kinematic limbs is analyzed. In Section A novelpure translational parallel mechanism, a novel typeof 4-UPU pure translational parallel mechanism withredundant actuations is proposed by choosing the2-UPU single loop as basic constructing unit. InSection Kinematic and singularity analysis, thekinematic analysis of the improved 4-UPU pure trans-lational parallel mechanism is performed. In SectionWorkspace analysis and fault-tolerant perform-ance, the fault-tolerant performance is analyzedand the fault-tolerant workspace is obtained.Finally, some conclusions involving the design and

fault-tolerant considerations are given in SectionConclusion.

Unexpected rotations

A translational parallel mechanism constructed by theUPU kinematic limbs is easy to exhibit the unex-pected rotations in practice.30 We take the typical3-UPU parallel mechanism proposed by Tsai3 as theexample to illustrate this phenomenon of unexpectedrotations. The typical 3-UPU parallel mechanismconsists of a moving platform, a xed base, andthree limbs of identical kinematic structures. Themoving platform and the xed base possess the similardimensions, and they are connected by a universaljoint, a prismatic joint, and another universal joint.The three prismatic joints are chosen as the actuatedjoints.

Although many researchers15,16,22,26,27 have donesome related analyses on the 3-UPU parallel mechan-ism, it is based on the following assumption:

For each limb, the rst revolute joint axis is parallelto the last revolute joint axis, and the two intermedi-ate revolute joint axes are parallel to one another.

This assumption is the necessary and sucient con-dition for the 3-UPU parallel mechanism to possessthree pure translational motions. However, due to theexistence of manufacturing clearances or assemblyerrors, this condition cannot be satised over thewhole workspace of this parallel mechanism. Forexample, one rotational conguration of this parallelmechanism can be seen in Figure 1. In this case, thereexists an angle 2 between the joint axes s21 and s25, ofthe revolute joints attached to the xed base and tothe moving platform, respectively.

Figure 1. Rotational configuration of 3-UPU parallel

mechanism.

2 Proc IMechE Part C: J Mechanical Engineering Science 0(0)



Therefore, a more reasonable hypothesis for theanalyses of the 3-UPU parallel mechanism is toassume that there may be an angle i between thejoint axis of the revolute joint attached to the xedbase and the revolute joint attached to the movingplatform. Under this assumption, it may be shownthat the UPU limb is an instantaneous structure,which can provide a constraint couple or a constraintforce according to the value of i, as shown inFigure 2.

Take one UPU limb as an example, as shown inFigure 2, all the feasible twists of a limb form a ve-system with established coordinate system Ai xyz.

$i1 1 0 0; 0 0 0 T

$i2 0 sini cosi; 0 0 0 T

$i3 0 0 0; sini cosi cosi cosi sini T

$i40 sini cosi; li cosi

li cosi sini li sini sini

" #T

$i5cosi cosi sini sini sini;0 li cosiisini li cosi cosii

" #T

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

where i is the angle from the limb direction to thenormal vector of universal joint (Ai) plane of the ithlimb, i denotes the angle from the normal vector ofuniversal joint (Ai) plane of the ith limb to the xedbase plane, and i represents the angle between thejoint axes of revolute joints attached to the movingplatform and xed base, respectively.

The reciprocal screw of the kinematic UPU chaincan be obtained by using the algebra operation ofreciprocal product37

$r 0

seci i sin ili

coti seci i sin ili

0 coti 1

T1

If i 0, then the reciprocal screw $r can be sim-plied to $rC, which denotes a pure constraint couple

$rC 0 0 0; 0 coti 1 T 2

If i possesses a nite value, i 6 0, then the recip-rocal screw $r can be rewritten as $rF, which representsa pure constraint force

$rF 0

seci i sinili

coti seci i sinili

0 coti 1

T3

where the subscript F and C denote the constraintforce and constraint couple, respectively.

In these two cases, we can obtain the physicalexpressions of a constraint force and a constraintcouple with an established coordinate system asshown in Figure 2.

Figure 2. The possible wrench type of UPU kinematic limb. (a) The constraint couple provided by single UPU limb and (b) the

constraint force provided by single UPU limb.

Qu et al. 3



When the UPU limb provides a constraint force, itcan be called F-limb, as shown in Figure 2(a). Thenecessary and sucient condition for an F-limb isthat all the revolute joints of an F-limb must be inter-secting or parallel to the constraint force and all theprismatic joint axes of an F-limb are perpendicular tothe constraint force. When the UPU limb provides aconstraint couple, it can be called C-limb, as shown inFigure 2(b). Now the necessary and sucient condi-tion for a C-limb is that all the revolute joint axes of aC-limb must be perpendicular to the constraint coupleand the prismatic joint axes, if any, of a C-limb thatcan be oriented arbitrarily as long as they are linearlyindependent.

When compared with these two cases, the mainreason for the change from an F-limb to a C-limbor a C-limb to an F-limb is the possible existingangle i. From equations (2) and (3), we know thatwhen the angle i is changed from nite value to zero,the wrench applied on the moving platform is alsochanged from a constraint force $rF at point Mi to aconstraint couple $rC. We can also believe that case 2,as shown in Figure 2(b), is a special form of case 1, asshown in Figure 2(a). When the UPU limb provides aconstraint force, as shown in equation (3), the paral-lelism between the xed base and the moving platformcannot be guaranteed, namely there occurs unex-pected rotations. Within the workspace of 3-UPUparallel mechanism, the angle i cannot be guaranteedto be zero even in ideal conditions.

Therefore, when there are unexpected rotations,the constraint system of the moving platform willnot be constituted by only constraint couples, andthe constraint system can be classied in three cases:

Case 1: The constraint system is constituted by two

constraint couples and one constraint force.

However, considered the congurations in practice,

this case cannot be realized.

The two existed constraint couples denote that thereare two intersected axes of revolute joints attached tothe moving platform are parallel to the xed base.

While the existed constraint force represents thatthere appears an angle i from the joint axis of therevolute joint attached to the xed base and the revo-lute joint attached to the moving platform, as shownin Figure 3.

When the two intersected axes, s15 and s25, of revo-lute joints attached to the moving platform are paral-lel to the xed base simultaneously, the movingplatform and xed base are proved to be parallel toeach other. Therefore, the angle 3 from the joint axis,s35, of the revolute joint attached to the xed base willbe equal to zero, which is conict to the existence ofconstraint force.

Case 2: The constraint system is constituted by one

constraint couple and two constraint forces, as shown

in Figure 4.

Since a constraint force constrains the possibletranslation along the constraint force and a constraintcouple constrains the possible rotation around theaxis, which is parallel to the constraint couple, themovement of moving platform should be two rota-tions and one translation.

Figure 3. The possible configurations of case 1.

Figure 4. The possible configuration of case 2.

Figure 5. The possible configurations of case 3.




Case 3: The constraint system is constituted by three

constraint forces, as shown in Figure 5.

As shown in Figure 5, the constraint force providedby each kinematic limb is perpendicular to a certainplane, which is determined by two intersecting axes ofrevolute joints attached to the moving platform andxed base, respectively.

While each two of the three determined planes areintersected with an angle, the three constraint forcesare neither intersected nor parallel with each other.Therefore, the three lines along the three constraintforces are the generators of the regulus. The regulus isdened that there is a set of lines that intersect thesethree lines, and this set of lines builds a quadric sur-face, i.e. a regulus.29

Without loss of generality, the general expressionof line generator of this regulus can be written as

x

y

z

264

375 R z,

x0

y0

z0

264

375

a cos bt sin bt cos a sin

ct

264

375 4

where Rz, denotes the rotation matrix about z axis.x0 a, y0 bt, and z0 ct are the parameter equa-

tions of one generator on the regulus. a, b, and c arethe parameters of semi axis of the regulus.

From equation (4), we can determine the positionof the constraint force by a vector r, and the directionof the constraint force by a vector s. The Pluckercoordinates of the constraint force can be expressed as

$r s; s0 T b sin b cos c; ac sin ac cos ab T

5

where r a cos , a sin , 0T, s b sin b cos c T.Therefore, the possible basic motion of the moving

platform can be obtained by performing the algebraoperation of reciprocal product35 on the three con-straint forces

$1 1 0 0;ac

b0 0

h iT$2 0 1 0; 0

ac

b0

h iT$3 0 0 1; 0 0 ab

c

T

8>>>>>>>>>:

6

Equation (6) indicates that the moving platformpossesses three screw motions.

Therefore, the 3-UPU parallel mechanism is aninstantaneous parallel mechanism, which cannotkeep the moving platform from changing its orienta-tion over the whole workspace. Although the initialdesign objective is to obtain a parallel mechanismwith pure translations, this 3-UPU parallel

mechanism cannot guarantee the parallelism betweenthe joint axes of revolute joints attached to themoving platform and xed base, and then the paral-lelism between the moving platform and xed basecannot be guaranteed.

A novel pure translational parallelmechanism

For the parallel mechanism constructed by UPU kine-matic limbs, the sucient geometric condition to keepthe moving platform from changing its orientation isto eliminate the possibility of the angle between thejoint axes, si1 and si5, of the revolute joints attached tothe xed base and the moving platform, i.e. i 0.Considered the conguration requirements of case 1,as shown in Figure 3, when the two intersected axes,s15 and s25, of revolute joints attached to the movingplatform are parallel to the xed base simultaneously,the moving platform and xed base are proved to beparallel to each other.

Therefore, the aim of designing a translationalparallel mechanism constructed by UPU kinematiclimbs is transformed to guarantee that there are atleast two parallel structures. And each parallelstructure can be depicted as the parallelism betweenthe axes of revolute joints attached to the movingplatform and the xed base in a UPU kinematiclimb.

In order to eliminate the possible rotations asshown in Figures 4 and 5, namely to eliminate thepossible angle i, we propose to use a 2-UPU unitstructure, as shown in Figure 7, to guarantee a kindof parallelism between the axes of revolute joints

Figure 6. The 4-UPU parallel mechanism with redundant

actuations.

Qu et al. 5



attached to the moving platform and the xed base ineach UPU kinematic limb, and the proofwill be stated in Section The proof of expectedparallelism.

Here, we propose a 4-UPU parallel mechanism bytwo 2-UPU unit structure assembled with cross struc-ture, as shown in Figure 6. Each 2-UPU unit structureensures the parallelism between the axes of revolutejoints attached to the moving platform and the xedbase in each UPU kinematic limb. The cross assemblystructure in this proposed 4-UPU parallel mechanismhas guaranteed the basic translational requirementsfor parallel mechanism constructed by UPU kine-matic limbs, namely there are at least two parallelismsbetween the axes of revolute joints attached to themoving platform and the xed base in a UPU kine-matic limb. Therefore, this structure leads to puretranslational motions of the moving platform inthree-dimensional space.

In this 4-UPU parallel mechanism, the four pris-matic joints are chosen as the actuated joints. And thethree translational motions of the moving platformare controlled by varying the lengths of the four pris-matic joints. As a result, the proposed 4-UPU parallelmechanism is a pure translational mechanism withredundant actuations.

The proof of expected parallelism

Since the proposed 4-UPU parallel mechanism is con-structed by the basic 2-UPU unit structure and thesucient geometric condition to keep the movingplatform from changing its orientation is i 0, thecriteria to determine the pure translations of movingplatform of 4-UPU parallel mechanism can be

transformed to prove that there does not exist thejoint angle i in the basic 2-UPU structure.

One basic 2-UPU unit structure is shown inFigure 7, and its congurations are as follows:

1. The two joint axes of revolute joints attached tothe xed base are parallel with each other, i.e.s11==s31.

2. The two joint axes of revolute joints attached tothe moving platform are parallel with each other,i.e. s15==s35.

3. The xed base and moving platform possess dif-ferent dimensions, and therefore the two planes,1 and 2, are not parallel with each other, whichare determined by the joint axes s11 and s15, s31and s35, respectively. In order to facilitate the ana-lysis, an assisted plane 3 is added, as shown inFigure 7. The assisted plane 3 is parallel withplane 1.

4. The plane 3 and the base plane 4 intersect at theintersecting line s011, and there exist the geometricconguration, s11==s

011.

This proof can be performed by the method ofproof by conict. The proof by conict is a form ofproof that establishes the truth of a proposition byshowing that the propositions being falsewould imply a contradiction. The proposition canbe stated as,

Proposition. The joint angle between the joint axes,si1 and si5, of the revolute joints attached to the xedbase and the moving platform is equal to zero, i.e.i 0.

Proof. (proof by contradiction). Suppose this propos-ition is false, i.e. i 6 0.

This conditional statement being false means thejoint angle between the the joint axes, si1 and si5, ofthe revolute joints attached to the xed base and themoving platform is not equal to zero, i.e. hsi1, si5i i.Based on the geometric conditions s11==s31 ands15==s35, we get hs11, s15i hs31, s35i 1.

With the added geometric condition, s11==s011, it

results that s011==s31, and then results hs011, s35i 1.From the conditions hs31, s35i 1, hs011, s35i 1,

we get that the joint axis s35 intersectswith joint axes s31 and s

011 simultaneously.

Since s011==s31, the above condition indicates that thejoint axes s35, s31, and s

011 are located in the same

plane.However, the non-coincidence of plane 2

and 3 reveals that the axes s35, s31, and s011 are not

located in the same plane. Thus, there exists acontradiction.

In other words, the proposition is true, which indi-cates that the moving platform of this proposed4-UPU parallel mechanism possesses no rotations.The proof is completed.

Figure 7. The 2-UPU unit structure.




DOF analysis

In this section, we will study the DOF of the movingplatform as shown in Figure 4 by considering the rela-tionship between the geometric constraint andDOFs.38 Since the moving platform surely shows nounexpected rotations during the movement accordingto Section Unexpected rotations, each kinematiclimb provides a constraint couple to the movingplatform

$rA1B1 0 0 0; 0 cot1 1 T

$rA2B2 0 0 0; cot2 0 1 T

$rA3B3 0 0 0; 0 cot3 1 T

$rA4B4 0 0 0; cot 2 0 1 T

8>>>>>>>:

7

where i denotes the angle from the normal vector ofuniversal joint (Ai) plane of the ith limb to the xedbase plane.

The constraint couples in equation (7) form theterminal constraint system (TCS) applied on themoving platform

$rTCS $rA1B1 $rA2B2 $rA3B3 $rA4B4h iT

8

where the subscript TCS represents the terminal con-straint system.

At ordinary position, the rank of the TCS, $rTCS, is

Rank $rTCS 3 constrain three rotational DOFs

9So according to Zhao et al.,39 the DOF of this

4-UPU parallel mechanism at ordinary position is

F 6 Rank $rTCS 6 3 3 10

Namely, this parallel mechanism possesses threetranslational DOFs.

Kinematic and singularity analysis

Due to the existence of redundantly actuated kine-matic limb, the forward kinematics of the proposed4-UPU parallel mechanism is simplied and the sin-gular congurations are reduced. The performanceanalysis will be carried out in the following section.For the purpose of analysis, a coordinate systemo xyz is attached to the xed base at point o, andanother coordinate system p uvw is attached to themoving platform at point p. The axes x, y, z are par-allel with the axes u, v, w, respectively. The four jointcenter points A1,A2,A3,A4 of the universal jointsmounted to the xed base form a square with 2Lside length, and the four joint center pointsB1,B2,B3,B4 of the universal joints mounted to themoving platform form a square with 2 h side length.

The geometric center points of the two squares arecoincident with point o and p, respectively.

The position vectors of point Ai expressed in thecoordinate system o xyz and Bi expressed in thecoordinate system p uvw are given as

A1 0 L 0 T

A2 L 0 0 T

A3 0 L 0 T

A4 L 0 0 T

8>>>>>>>:

,

pB1 0 h 0 T

pB2 h 0 0 T

pB3 0 h 0 T

pB4 h 0 0 T

8>>>>>>>:

11where the left superscript p denotes that the positionvector of Biis expressed in the coordinate systemp uvw.

Then, the position vector of point Bi expressed inthe coordinate system o xyz is

Bi1

E33 p

0 1

pBi1

12

where E33 is a unit matrix and p is the position vectorof point p expressed in the coordinate system o xyz,p xp yp zpT.

Therefore, the position vector of point Bi isobtained

B1 xp yp h zp T

B2 xp h yp zp T

B3 xp yp h zp T

B4 xp h yp zp T

8>>>>>>>:

13

Kinematic analysis

1. Inverse kinematics

The inverse kinematic problem is to determine thelength li of AiBi, being given the coordinates of themoving platform center point p.

The position vectors of point Ai and Bi in coord-inate system o xyz can refer to equations (11) and(13), the limb length, li, can be solved as follows

l21 A1B1j j2 x2p yp L h2 z2p 14

l22 A2B2j j2 xp h L2 y2p z2p 15

l23 A3B3j j2 x2p y0 h L2 z2p 16

l24 A4B4j j2 xp L h2 y2p z2p 17

Since the four limbs are located at the same side ofthe base, the lengths of the four input limbs are sim-ultaneously positive or negative, here we assume thelimb lengths are all positive.

Qu et al. 7



2. Forward kinematics

For forward kinematics, the lengths of thefour input limbs are given, and the problem is tond the position vectors of center point p ofthe moving platform. This can be accomplishedthrough solving the equation system from equations(14) to (17).

Subtracting equation (14) from equation (16) yields

l21 l23 4L h yp 18

Subtracting equation (15) from equation (17) yields

l22 l24 4h Lxp 19

The coordinate of xp and yp can be deduced fromequations (18) and (19), respectively

xp l22 l24

4h L , yp l21 l234L h 20

Substituting xp and yp into any one of equations(14) to (17), a quadratic equation of zp can beobtained. By ignoring the negative value, the analyticexpression of zp is obtained

z01 14

l41 l22 l242 2l21l23 4h L2

l23 4h2 8hL 4L2h L2

vuuuut 21

z02 14

l42 l21 l232 2l22l24 4h L2

l24 4h2 8hL 4L2h L2

vuuuut 22

The above analysis shows that the forward kine-matics of this improved 4-UPU translational parallelmechanism with redundant actuations can be doneexplicitly.

Jacobian and singularity analysis

The Jacobian matrix is a multidimensional formof the derivative, which transforms the joint rates inthe actuated joint space to the velocity space of themoving platform. For the parallel mechanism,Gosselin and Angeles40 suggested a separation ofthe Jacobian matrix into two matrices: one associatedwith the forward kinematics and another with theinverse kinematics.

Dierentiating equations (14) to (17) with respectto time and writing in matrix form, the dierential

relations between the actuated joint space and thevelocity space of the moving platform can bededuced as

Jp _p Jq _q 23

where

Jp 2xp 2yp L h 2zp2xp h L 2yp 2zp2xp 2 yp h L 2zp2xp L h 2yp 2zp

2664

3775

Jq 2l1 0 0 00 2l2 0 00 0 2l3 00 0 0 2l4

2664

3775

and where _p _xp _yp _zpT represents the velocityof point p of the moving platform and_q _l1 _l2 _l3 _l4T denotes the actuated jointvelocities.

Two Jacobian matrices are obtained as above, Jp isnamed as the forward Jacobian matrix and Jq isnamed as the inverse Jacobian matrix. The parallelmechanism is said to be at a singular congurationwhen either Jp or Jq or both are singular, which resultsin three dierent types of singularities.

Since the length of the input limb li cannot be zero,the determinant of inverse Jacobian matrix will not bezero, only the determinant of forward Jacobianmatrix should be considered. While Jp is a 4 3 rect-angular matrix, we cannot solve the determinant dir-ectly. Here we select an arbitrary 3 3 sub-matrix J0pfrom Jp and solve its determinant

detJ0p 16h L2zp 24

where det represents the determinant of a matrix,zp is the coordinate of point p of moving platform indirection z and will not be equal to zero.

The only singularity of this 4-UPU parallel mech-anism is obtained by setting the determinant of J0p tobe zero

h L 25

The singular conguration is occurred when thegeometric dimensions of the xed base and themoving platform are identical. At this singular geo-metric condition, the concrete singular forms can bedescribed as:

1. When the lengths of the four input limbs are equalto each other, the moving platform has innitesi-mal motion with all the actuate joints that arelocked as shown in Figure 8.

2. The moving platform can perform a rotationalmotion, as shown in Figure 9. However, the




position and orientation of the moving platformstill can be controlled because of the existence ofthe redundant limb.

Workspace analysis and fault-tolerantperformance

Workspace analysis

In order to evaluate the designed mechanism, it isnecessary to determine the boundaries of the work-space.41,42 For the parallel mechanism with puretranslational motions, the workspace is dened asthe set of referred point on moving platform thatcan be reached. In this section, the workspace ofthis 4-UPU parallel mechanism with redundant actu-ations is analyzed. Since the moving platform pos-sesses pure translational motions, to calculate thereachable workspace of the moving platform, it canbe transformed to get the reachable position set ofpoint p. The reachable workspace of this 4-UPU par-allel mechanism can be expressed as

W xp, yp, zp 2 R3jdi,min4di4di,max , i 1, 2, 3, 4

26where W is the reachable position set of point p,xp, yp, zp denotes the position vector of point p,and di,min and di, max represent the minimum and max-imum length values of the actuated limb.

Based on the consideration of the length value ofactuated limb, the workspace boundary can beobtained when the actuated limbs are stretched tothe maximum length value. The calculation is pre-sented in the following three steps:

1. Dene the structural parameters of the 4-UPUparallel mechanism, including L, h, di, min, anddi, max.

2. With the discretization of the length value of theactuated limb, the discrete point cloud of point pcan be depicted based on the forward kinematics.The point cloud represents the discrete reachable

Figure 8. The first form of singular configuration. (a) The

initial position and (b) the position after moving.

Figure 9. The second form of singular configuration.

Table 1. The structural parameters of this 4-UPU parallel

mechanism with redundant actuations (mm).

Variable di, min di, max h L

Value 30 60 30 50

Table 2. The estimated position range of point p (mm).

Variable minxp maxxp minyp maxyp minzp maxzp

Value 33.75 33.75 33.75 33.75 28.80 94.94

Qu et al. 9



workspace of point p, and the estimated positionrange of point p can be obtained.

3. Check the point p in the estimated position rangewhether it satises the length constraint of theactuated limb. If the position coordinates ofpoint p meet the length constraint, then the pos-ition coordinates of point p should be recordedand projected to the three-dimensional positionspace, and the reachable workspace of point pcan be obtained. If not, then the point p shouldbe discarded.

Next, the reachable workspace of this 4-UPU parallelmechanism is performed based the above analysisprocedure. The structural parameters are listed inTable 1. Since the four limbs possess identical struc-tures, we assume that the length ranges of each actu-ated limb are the same.

Through the forward kinematics of this parallelmechanism, the estimated position range of point pis obtained as listed in Table 2.

Finally, the point p in the estimated position rangeis checked to ensure that satises the length constraintof the actuated limb. After projecting the satisedpoint p to the three-dimensional position space, the

reachable workspace of point p is obtained, as shownin Figures 10(a), 11(a), and 12(a).

Workspace of fault-tolerant configuration

The fault-tolerant performance is a critical indexwhen the parallel mechanism is applied to some spe-cial elds, such as space, clear-up of hazardous waste,and so on. Therefore, the used parallel mechanismshould be fault-tolerant to ensure the working reliabil-ity. Even in the failure mode, the fault-tolerant paral-lel mechanism can still perform the whole or part ofthe task operations. The failure modes of a mechan-ism43 include locked-joint, free-swinging joint, andfollowing-motion joint, where the locked-joint is oneof the most common modes.

In this section, the fault-tolerant workspace isdened as the reachable workspace of moving plat-form when the parallel mechanism is in locked-jointfailure mode. And the fault-tolerant workspace isused to evaluate the fault-tolerant performance ofthe proposed 4-UPU parallel mechanism. The existingfault-tolerant workspace indicates that the parallelmechanism can still perform some continuousmotions even there occurs the locked-joint failure.

Figure 10. Comparison of the continuous workspace and fault-tolerant workspace.

Figure 11. Comparison of the maximum XZ section of workspace and fault-tolerant workspace.




We assume that one actuated joint of this proposed4-UPU parallel mechanism is broken and lockedwhen the length of rst actuated limb is equal to45 mm, and the other structural parameters arelisted in Table 1.

Based on the obtained forward kinematics and thediscretization of the length value of the other threeactuated limbs, the discrete point cloud of point pcan be depicted as shown in Figure 13, and the esti-mated position range of point p in locked-joint failuremode is obtained as listed in Table 3.

After that, the point p in the estimated positionrange is checked whether it satises the length con-straint of the other three actuated limbs. The point p,which meets the length constraint of the other threeactuated limbs, is recorded and projected to the three-dimensional position space. And then, the continuous

fault-tolerant reachable workspace of point p isobtained when one failure actuated limb is locked at45mm, as shown in Figures 10(b), 11(b), and 12(b).

From the above fault-tolerant workspace analysis,we know that the proposed 4-UPU 3-DOF transla-tional parallel mechanism can still perform the fault-tolerant operations even in the locked-joint failuremode.

Conclusion

This paper addresses a novel type of 4-UPU 3-DOFtranslational parallel mechanism with redundantactuations. The main work and conclusions of thispaper are drawn as follows:

1. The reason for unexpected motions of parallelmechanism constructed by UPU kinematic limbsis analyzed.

2. A novel type of 4-UPU 3-DOF translational par-allel mechanism with redundant actuations is pro-posed. This redundantly actuated parallelmechanism is constructed by basic 2-UPU struc-ture, and the possible unexpected rotations areeliminated. Since the existence of redundantlyactuated kinematic limb, this proposed parallelmechanism possesses analytical forward kine-matics, and its singularity can be avoidedcompletely.

3. This proposed parallel mechanism provides cer-tain fault-tolerant performance, and its fault-tol-erant workspace is obtained. The 4-UPU parallelmechanism will still perform some spatial transla-tions when one kinematic limb is in locked-jointfailure mode.

Funding

This work was supported by the National Natural Science

Foundation of China [grant numbers 51175029, 51075025];the Beijing Natural Science Foundation [grant number

Figure 12. Comparison of the maximum XY section of workspace and fault-tolerant workspace.

Figure 13. The discrete point cloud of point p in locked-joint

failure.

Table 3. The estimated position range of point p (mm).

Variables minxp maxxp minyp maxyp minzp maxzp

Value 19.69 14.06 33.75 33.75 28.80 44.99

Qu et al. 11



3132019]; and the Program for New Century Excellent

Talents in University [grant number NCET-12-0769].

Conflict of interest

None declared.

Acknowledgements

This paper has been partly accomplished in INRIA

Sophia Antipolis where Dr Haibo Qu is a visiting scholarin project COPRIN, and that part is revised by Dr J-PMerlet.

References

1. Clavel R. DELTA, a fast robot with parallel geometry.In: Proceedings of the 18th international symposiumon industrial robots, Lausanne, Switzerland, 1988,

pp.91100. Berlin: Springer.2. Herve JM and Sparacino F. Structural synthesis of par-

allel robots generating spatial translation. In:

Proceedings of the 5th international conference onadvanced robotics, Pisa, Italy, 1991, pp.808813.New York: IEEE.

3. Tsai LW. Kinematics of a three-DOF platform with

three extensible limbs. In: Proceedings of 5th recentadvances in robot kinematics, Ljubljana, Slovenia,1996, pp.401410. Berlin: Springer.

4. Tsai LW. Systematic enumeration of parallel manipu-lator. In: Proceedings of the 1st European-AmericanForum on parallel kinematic machines, Milan, Italy,

1998, pp.3349. Berlin: Springer.5. Carricato M and Parenti-Castelli V. A family of 3-DOF

translational parallel manipulators. J Mech Des 2003;

125: 302307.6. Carricato M and Parenti-Castelli V. Singularity-free

fully-isotropic translational parallel mechanisms. Int JRobot Res 2002; 21: 161174.

7. Kong XW and Gosselin CM. Type synthesis of 3-DOFtranslational parallel manipulators based on screwtheory. J Mech Des 2004; 126: 8392.

8. Gogu G. Structural synthesis of parallel robots, part 2:translational topologies with two and three degrees offreedom. Netherlands: Springer, 2009.

9. Gogu G. Structural synthesis of fully-isotropictranslational parallel robots via theory of lineartransformations. Eur J Mech A/Solids 2004; 23:

10211039.10. Huang Z and Li QC. Type synthesis of symmetrical

lower-mobility parallel mechanisms using the con-straint-synthesis method. Int J Robot Res 2003; 22:

5979.11. Jin Q and Yang TL. Theory for topology synthesis of

parallel manipulators and its application to three-

dimension-translation parallel manipulators. J MechDes 2004; 126: 625639.

12. Liu XJ, Jeong JI and Kim J. A three translational

DOFs parallel cube-manipulator. Robotica 2003; 21:645653.

13. Yu JJ, Zhao TS, Bi SS, et al. Type synthesis of parallelmechanisms with three translational degrees of free-

dom. Prog Nat Sci 2003; 13: 536545.14. Yu JJ, Dai JS, Bi SS, et al. Type synthesis of a class of

spatial lower-mobility parallel mechanisms with

orthogonal arrangement based on Lie group enumer-ation. Sci China Tech Sci 2010; 53: 388404.

15. Joshi SA and Tsai LW. Jacobian analysis of limited-

DOF parallel manipulator. J Mech Des 2002; 124:254258.

16. Tsai LW and Joshi SA. Kinematics and optimization of

a spatial 3-UPU parallel manipulator. J Mech Des 2000;122: 439446.

17. Joshi SA and Tsai LW. A comparison study of two 3-

DOF parallel manipulators one with three and the otherwith four supporting legs. IEEE Trans Robot Autom2003; 19: 200209.

18. Gregorio RD and Parenti-Castelli V. Mobility analysis

of the 3-UPU parallel mechanism assembled for a puretranslational motion. J Mech Des 2002; 124: 259264.

19. Han C, Kim J, Kim J, et al. Kinematic sensitivity ana-

lysis of the 3-UPU parallel mechanism. Mech MachTheory 2002; 37: 787798.

20. Zlatanov D, Bobev IA and Gosselin CM. Constraint

singularity of parallel mechanism. In: Proceedings ofthe IEEE international conference on robotics and auto-mation, Washington DC, USA, 2002, pp.469502.

New York: IEEE.21. Zlatanov D, Bonev IA and Gosselin CM. Constraint

singularities as c-space singularities. In: Proceedings of8th international symposium on advances in robot kine-

matics, Caldes de Malavella, Spain, 2002, pp.183192.Berlin: Springer.

22. Wolf A and Shoham M. Investigation of parallel

manipulators using linear complex approximation.J Mech Des 2003; 125: 564572.

23. Wolf A, Shoham M and Park FC. Investigation of sin-

gularities and self-motions of the 3-UPU robot. In:Proceedings of the 8th international symposium onadvances in robot kinematics, Caldes de Malavella,Spain, 2002, pp.155164. Berlin: Springer.

24. Conconi M and Carricato M. A new assessment ofsingularities of parallel kinematic chains. IEEE TransRobot Autom 2009; 25: 757770.

25. Gogu G. Constraint singularities and the structural par-ameters of parallel robots. In: Proceedings of the 11thinternational symposium on advances in robot kinematics,

Batz-sur-Mer, France, 2008, pp.2128. Berlin: Springer.26. Chebbi AH and Parenti-Castelli V. Geometric and

manufacturing issues of the 3-UPU pure translational

manipulator. In: Proceedings of the 3rd European con-ference on mechanism science, Cluj-Napoca, Romania,2010, pp.595604. Berlin: Springer.

27. Chebbi AH, Affi Z and Romdhane L. Kinetostatic and

singularity analyses of the 3-UPU translational parallelrobot. In: Proceedings of 5th international workspace oncomputational kinematics, Duisburg, Germany, 2009,

pp.6168. Berlin: Springer.28. Gan DM, Dai JS and Liao QZ. Constraint analysis on

mobility change in the metamorphic parallel mechan-

ism. Mech Mach Theory 2010; 45: 18641876.29. Qu HB, Fang YF and Guo S. Parasitic rotation evalu-

ation and avoidance of 3-UPU parallel mechanism.Front Mech Eng 2012; 7: 210218.

30. Merlet JP. Parallel robots. Netherlands: Springer, 2006.31. Lu Y and Hu B. Analysis of kinematics and solution of

active/constrained forces of asymmetric 2UPUX par-allel manipulators. Proc IMechE, Part C: J MechanicalEngineering Science 2006; 220: 18191830.




32. Zhao JS, Fu YZ, Zhou K, et al. Mobility properties of aSchoenflies-type parallel manipulator. Robot Comput-Integr Manuf 2006; 22: 124133.

33. Zhao JS, Chu FL and Feng ZJ. Singularities within theworkspace of spatial parallel mechanisms with symmet-ric structure. Proc IMechE, Part C: J Mechanical

Engineering Science 2010; 224: 459472.34. Ropponen T and Nakamura Y. Singularity-free param-

eterization and performance analysis of actuation

redundancy. In: Proceedings of IEEE international con-ference on robotics and automation, Cincinnati, USA,1990, pp.806811. New York: IEEE.

35. Zanganeh KE and Angeles J. Mobility and position

analyses of a novel redundant parallel manipulator In:Proceedings of IEEE international conference on roboticsand automation, San Diego, USA, 1994, pp.30493054.

New York: IEEE.36. Chakarov D. Study of the antagonistic stiffness of par-

allel manipulators with actuation redundancy. Mech

Mach Theory 2004; 39: 583601.37. Tsai LW. Robot analysis, the mechanics of serial and

parallel manipulators. New York: Wiley, 1999.

38. Zhang KT, Dai JS and Fang YF. Geometric con-straint and mobility variation of two 3SvPSv meta-morphic parallel mechanisms. J Mech Des 2013; 135:

011001.39. Zhao JS, Zhou K and Feng ZJ. A theory of degrees of

freedom for mechanisms. Mech Mach Theory 2004; 39:

621643.40. Gosselin CM and Angeles J. Singularity analysis of

closed-loop kinematic chains. IEEE Trans Robot

Autom 1990; 6: 281290.41. Dibakar S and Mruthyunjaya TS. A computational

geometry approach for determination of boundary ofworkspaces of planar mechanisms with arbitrary top-

ology. Mech Mach Theory 1999; 34: 149169.42. Zhao JS, Chu FL and Feng ZJ. Symmetrical character-

istics of the workspace for spatial parallel mechanisms

with symmetric structure. Mech Mach Theory 2008; 43:427444.

43. English JD and Maciejewski AA. Failure tolerance

through active braking: a kinematic approach. Int JRobot Res 2001; 20: 287299.

Qu et al. 13


proceedings of the institution of mechanical engineers, part c- journal of mechanical engineering...

Documents

upu parallel mechanism

institution of mechanical

upu kinematic limbsis

upu single loop

keywordsparallel mechanism

upuparallel mechanism

upu universalprismatic

faulttolerant workspace