Singularity-free Workspace Analysis of General 6-UPS Parallel

best performance, the 2-DOF 3-parallelogram PPM should possess the largest workspace and

excelisms

acteristics and simple structure have been widely used in the industry eld. PMs have been intensively studied since the 1980s and

ted themselves to thisstematically discussedthe kinematic optimalon of performance and

Mechanism and Machine Theory 87 (2015) 117

Contents lists available at ScienceDirect

Mechanism and Machine Theory

j ourna l homepage: www.e lsev ie r .com/ locate /mechmtthe optimumdesign issue of planar 5R symmetrical PPMs. However, all 2-DOF PPMs discussed in these articles have amirror symmet-rical structure. The 2-DOF 3-parallelogram PPM is another type of 2-DOF PPM with a non-symmetrical structure. Although manyRegardless of how simple a 2-DOF PPM is, the optimal design is challenging. Many researchers have devoissue for many years. Gao et al. [10] presented a physical model of the solution space of 2-DOF PPMs and sythe comprehensive classication of this kind of manipulators. Huang et al. [11] presented a hybrid method fordesign of 2-DOF PPMswith a mirror symmetrical geometry. Liu et al. [12] addressed the graphical representatistill attract much attention up to the present [26]. For PMs one of the most important and challenging problems is kinematic opti-mization in which two issues are concerned: performance evaluation and dimensional synthesis [7,8].

The 2-DOF 3-parallelogram PPM, actuated by two coaxial revolute joints, is a typical 2-DOF PPM. It has a non-symmetrical struc-ture composed of three parallelogrammechanisms. This PPM has a prominent function that does not only follow an arbitrary planarcurve preciselywithin theworkspace but also keep the end-effector in a denite posture at all times. Given its outstanding advantagesand simple structure, this PPM has been widely applied in palletizing robots [9].Parallel mechanisms (PMs) exhibitcapacity, compared with serial mechan Corresponding author at: Room 1502, Building 900E-mail address: [email protected] (X.-

http://dx.doi.org/10.1016/j.mechmachtheory.2014.12.010094-114X/ 2014 Elsevier Ltd. All rights reserved.lent characteristics, such as higher rigidity, better positioning accuracy, and higher load[1]. As an important branch of PMs, the 2-DOF PPMs possessing both outstanding char-1. Introductionmost stable transmissibility. This study addresses the performance evaluation and kinematic op-timization of this manipulator. First, the kinematics of the manipulator is analyzed, and perfor-mance indices that consider desirable workspace and transmissibility are proposed. Then, theprocess to determine optimal geometry parameters by using performance atlases is presented. Fi-nally, a 2-DOF 3-parallelogram PPM with desirable workspace and transmissibility is identied.

2014 Elsevier Ltd. All rights reserved.Keywords:Parallel manipulatorOptimal designPerformance atlasWorkspaceTransmissibilityKinematic optimal design of a 2-degree-of-freedom3-parallelogram planar parallel manipulator

Xin-Jun Liu a,b,, Jie Li a, Yanhua Zhou a,c

a State Key Laboratory of Tribology & Institute of Manufacturing Engineering, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, PR Chinab Beijing Key Lab of Precision/Ultra-precision Manufacturing Equipments and Control, Tsinghua University, Beijing 100084, PR Chinac Institute of Aircraft Engineering, Naval Aeronautical and Astronautical University, Yantai, Shandong Province, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 December 2013Received in revised form 18 December 2014Accepted 25 December 2014Available online 10 January 2015

A 2-degree-of-freedom (2-DOF) 3-parallelogramplanar parallelmanipulator (PPM) can follow anarbitrary planar curve and keep the end-effector in a denite posture. Such features are valuablefor spray-painting robots. Considering these advantages, authors proposed a new spray-paintingrobot containing a 2-DOF 3-parallelogram PPM. In order to obtain a spray-painting robot with the3, Tsinghua University, Beijing 100084, PR China. Tel.: +86 10 6278 9211.J. Liu).

4

articles have investigated its application in palletizing robots, including structural optimization [13], optimummotion control [14] andlayout analysis [15], articles about the kinematic optimal design of this manipulator in spray-painting robots are seldom found.

The traditional kinematic optimization method for PMs is the objective function method that involves establishing an objectivefunction and obtaining a result via algorithms [1618]. Three unavoidable difculties are encountered in this method. (a) The highlynon-linear objective function is difcult to establish, and some simplication are inevitably involved in objective function. Thus, theiterative solving process is not only time consuming, but the optimal result based on objective function method is sometimes unreli-able. (b) This method may provide an optimal result, but users cannot know how optimal the result is because the relationship be-tween design parameters and performance indices is unknown. (c) If the design conditions vary, then users have no choice but torestart theirwork from the beginning [19]. Comparedwith the traditional objective functionmethod, the performance atlasesmethodexhibits several obvious advantages. It not only graphically and globally shows the relationship between performance indices and de-sign parameters but also provides all possible solutions to users [20]. Users can obtain the optimal results by comparing all possiblesolutions in the same chart, which is convincing and convenient.

Performance evaluation and dimensional synthesis are two important issues in kinematic optimization. To conduct kinematic op-timal design for a 2-DOF 3-parallelogramPPM, performance indices are required. The local conditioning index (LCI) [21] and the glob-al conditioning index (GCI) [2] have been widely used by numerous researchers. However, a recent study [22] found seriousinconsistencieswhen these indiceswere applied to the kinematic optimal design ofmixed-DOF PMs. Furthermore, the LCI cannot pro-vide amathematical distance between current position and singularity [7], which also limits the application of this index.Wu et al. [7]proposed two simple but useful frame-free indices, namely, the local transmission index (LTI) and the global transmission index (GTI).

the design space of the manipulator, including its length and angular design parameters. The GTW of the manipulator is discussed in

2 X.-J. Liu et al. / Mechanism and Machine Theory 87 (2015) 117Section 4. According to the application requirements in the spray-painting robots, performance indices based on desirable workspaceand transmissibility are proposed in Section 5. Performance atlases based on these performance indices are also illustrated in thissection. Section 6 introduces the use of performance atlases to determine the optimum parameters and provides an optimal manip-ulator. Conclusions are presented in Section 7.

2. Kinematics of a 2-DOF 3-parallelogram PPM

2.1. Architecture

Fig. 1(a) shows a 2-DOF 3-parallelogramPPM. Fig. 1(b) presents the physical model of a spray-painting robot proposed by authorswhich contains a 2-DOF 3-parallelogram PPM. The 2-DOF 3-parallelogram PPM is widely used in palletizing robots, but seldom ap-plied in the spray-painting robots. As shown in Fig. 1(a), the 2-DOF 3-parallelogram manipulator consists of three parallelogrammechanisms, and each parallelogram mechanism can be congured as a parallelogram form or an anti-parallelogram form. Thus,there can be several conguration modes for the end-effector. However, the anti-parallelogram mechanisms are easy to cause

(a)

driving motor spray gun

(b)

Fig. 1. The 2-DOF 3-parallelogram PPM: (a) kinematic structure and (b) application in a spray-painting robot.The LTI can judge the effectiveness of transmissibility in a single pose, whereas the GTI represents the average effectiveness of a seriesof poses in the good transmission workspace (GTW). Although the GTI can provide a PMwith good average transmissibility, how theeffectiveness of transmissibility in each pose varies around the average transmissibility remains unclear. This variation demonstratesthe uctuation of the LTI around the GTI, which is particularly important for the optimal design of driving motors. According to theapplication requirements in the spray-painting robots, new indices that consider desirable workspace and transmissibility are intro-duced to comprehensively solve the kinematic optimization problems of the 2-DOF 3-parallelogram PPM. Then based on these per-formance indices, the kinematic optimization is achieved by using performance atlas method. Finally, a 2-DOF 3-parallelogramPPM with desirable workspace and transmissibility is identied.

This paper is organized as follows. The kinematics of a 2-DOF 3-parallelogram PPM is analyzed in Section 2. Section 3 investigates

geometry interference problemswhich are inappropriate for both palletizing robots and the spray-painting robots. In order to concen-trate the target, only the congurationmode that contains no anti-parallelograms is studied in this article. The end-effector of thema-nipulator is the link IP that is connected to the base through three parallelogrammechanisms, namely,OABC,ODEC, and CFGH. Link CHin parallelogrammechanism CFGH and link CE in parallelogrammechanism ODEC are xed together. In addition, link CF in parallelo-gram mechanism CFGH and link CB in parallelogram mechanism OABC are the two edges of triangle link BFC. Link HI represents thedistance between the center of the end-effector (link IP) and that of joint H. Links IP and HI are xed together. All revolute joints inthis manipulator are perpendicular to plane XOY. Parallelogram mechanisms OABC, ODEC, and CFGH share a revolute center locatedin point C. Parallelogrammechanisms OABC and ODEC share a revolute center located in point O. LinkOA is xed on the serial manip-ulator assembly unit, as shown in Fig. 1(b). Two coaxial actuating revolute joints are located in pointO, as shown in Fig. 2.When linksOC and OD are driven by actuating joints, the end-effector (link IP) can follow an arbitrary planar curve within the workspace andmaintain a denite posture at all times.

Geometric parameters in this manipulator can be described as follows. is the angle between xed link OA and axis OX in coor-dinate system XOY. is the vertex angle between edges CF and CB in triangle link CFB. Link OC is the arm of the manipulator, andits length is R1. Link CH is the forearm, and its length is R3. The length of linkHI is R, which represents the distance between the centerof joint H and that of the end-effector. The end-effector is a cylindrical spray gun assembled on a moving platform, as shown inFig. 1(a). R2 is the distance between the spray nozzle and the moving platform. This distance must satisfy the spray requirementsof the spray gun. Eight kinematic angular parameters are observed in this manipulator. 1 is the angle between link OD and axis OX

tigatedequati

Ac

3X.-J. Liu et al. / Mechanism and Machine Theory 87 (2015) 117above axis OX are meaningful for this spray-painting robot. Furthermore, according to the practical application of this 2-DOF 3-paralelloggrammanipulator, in order to avoid geometric interferencewith cars to be painted and singularity caused by out of control,

Fig. 2. Kinematic scheme of the common working mode for a 2-DOF 3-parallelogram PPM.XP R1 cos 1R3 cos 2Yp R1 sin 1R3 sin 2R2

: 2

cording to Figs. 1 and 2, joint O and link OA are xed on the serial manipulator assembly unit and only the driven anglesXP R1 cos 1R3 cos 2RY 0P R1 sin 1R3 sin 2R2

: 1

In practical applications, R should be as small as possible to obtain a compact structure. In general, the end-effector is a cylindricalspray gun, shown in Fig. 1(a), and R can be regarded as the sum of the radius of jointH and that of the spray gun. Hence, R is a constantvalue that cannot change the position range of the end-effector. Consequently, Eq. (1) can be reasonably simplied as follows:conduct performance evaluation and dimensional synthesis, the kinematics of the 2-DOF 3-parallelogram PPMmust be inves-rst. According to Fig. 2, the position of the end-effector (point P) in coordinate system XOY can be expressed by the followingon:

02.2. Kinematic analysis

To1 DEC 12; 1 ODE 1 2;2 ABC 1; 2 BCO 1 ;

3 FGH 2; 3 CFG 2:that is driven by the actuating joint. 2 is the angle between link OC and axis OX that is driven by another actuating joint. The othersix angular parameters are passive angles expressed by

for thetem XO

gles (that co

2 (paralle

parallemaintalationsposturlength

3.1. Le

Theto innship b

The

4 X.-J. Liu et al. / Mechanism and Machine Theory 87 (2015) 117r1 R1=D; r2 R2=D; and r3 R3=D: 5

Obviously,

r1 r2 r3 3 0 b r1 b 3; 0 b r2 b 3 and 0 b r3 b 3 : 6

Therefore, Eq. (2) can be rewritten in non-dimensional space as follows:

xP r1 cos 1r3 cos 2yp r1 sin 1r3 sin 2r2

7D R1 R2 R3 =3: 4

n length parameters Ri (i= 1, 2, 3) are divided by D to dene three non-dimensional parameters as follows:dimensionalized. Dividing all the length parameters by their average is an efcient method, as used in [23] and [24]. Letlogram mechanism OABC, thus indicating that the posture of triangle link CFB is constant. Similarly, the end-effector (link IP)ins a dened posture via parallelogrammechanisms CFGH and OABC. Thus, the working posture of the end-effector has no re-hip with the length parameters and is determined only by angular parameters and . That is, theworkspace and theworkinge are uncoupled in the design space. Therefore, we can divide the design space into two independent subspaces, namely, thedesign space and the angular design space.

ngth design space

length design space consists of three length parameters R1, R2, and R3. Theoretically, each length parameter can vary from zeroite. Thus, it is impractical to conduct kinematic optimization using these length parameters directly. To illustrate the relation-etween performance indices and length parameters in a performance atlas, these length parameters should be non-where

ui a ib

ci 1;2 ; vi

d ibe

i 1;2 ;a 2R1 R2 Yp

; b

k lm k ln

q;

c R1 Xp 2 R2 Yp 2R32; d 2R3 R2 Yp ;

e XpR3 2 R2 Yp 2R12; k Xp2;

l R2 Yp 2

; m R1R3 2;n R1 R3 2; 1 1; 2 1:

3. Design space

According to Fig. 2, both length parameters (R1, R2 and R3) and angular parameters ( and ) are involved to determine a 2-DOF 3-parallelogramPPM. All these geometric parameters constitute a design space for themanipulator. According to Eq. (2), when the driv-en angles are given, the position of point P is determined by length parameters. Hence, in the samedriving space, theworkspace of thismanipulator is determined by the length parameters, includingR1, R2 and R3. In otherwords, theworkspace of thismanipulator has norelationship with the angular parameters. Meanwhile, as link OA is xed, the angle between link CB and axis OX remains constant via1 2 tan ui i 1;22 2 tan1vi i 1;2

; 3

1

(1

0, 90). Only the solution in Eq. (3) where i = 1 can be used and the corresponding working mode of the 2-DOF 3-logram PPM is shown in Fig. 2.1, 2 to reach the same position (XP, YP) according to Eq. (2). Therefore, there can be theoretically twoworkingmodes for the 2-DOF 3-parallelogram PPM to obtain the same pose. However, the aforementioned driven angles have been dened as (90, 180) and1 and 2) although thedesirableworkspace is given by awork task. To regulate theworking trace, the values of thedriven anglesrrespond with the positions of the end-effector are required. There are two solutions of 1 and 2 shown as Eq. (3) where i=In the practical application of a 2-DOF 3-parallelogram PPM, theworking trace of the end-effector is determined by the driven an-iven angles are always dened as 1 (90, 180) and 2 (0, 90). The ranges of the driven angles constitute a driving spacemanipulator.When the driven angles vary in the driving space, the end-effector can reach different positions in coordinate sys-Y. All possible positions of the end-effector constitute the workspace of the 2-DOF 3-parallelogram PPM.the dr

where (xp, yp) is the non-dimensional position of the end-effector in the workspace correspondingwith (Xp, Yp). According to Eq. (6),the distribution of the non-dimensional parameters (r1, r2, and r3) can be illustrated as triangle UVW in the non-dimensional designspace in Fig. 3.

Although three non-dimensional length parameters are indicated in Eq. (7), only two of them are independent according toEq. (6). A linear map can be made to convert r1, r2, and r3 in the spatial coordinate system to a planar coordinate system dened bys and t, thus allowing performance evaluation and dimensional synthesis based on performance atlases to be conducted in a planarcoordinate system conveniently. Let

t r3 0 b t b 3 ; 8

s r3=3

p 2r2=

3

pt=

3

pb s b 6t =

3

p : 9

Then, Eq. (7) can be rewritten as follows:

xP 33

ps=2t=2

cos 1t cos 2

yp 33

ps=2t=2

sin 1t sin 2 st=

3

p 3

p=2

80.3v AGTW >1.6v

No No2 2

Yes YesR DrLGTW AGTWReduce

Reducev vLGTW/D AC v

LGTW =2.5vStart: andAGTW =4v LGTW =2.5v

should rst nd out the region with the largest GTI and then the subregion with the smallest FTI. This subregion is the optimum an-

Based on the optimum geometric parameters in Fig. 15, the optimum workspace and transmissibility of the optimum 2-DOF 3-

Fig. 14. The intersection of the GTI and the FTI of the 2-DOF 3-parallelogram PPM.

15X.-J. Liu et al. / Mechanism and Machine Theory 87 (2015) 117parallelogram PPM is illustrated in Fig. 16.Furthermore, based on Fig. 14, we can determine that the GTI of the 2-DOF 3-parallelogram PPM is greater than 0.75, and that the

FTI of the manipulator is lower than 0.025.gular parameter region wherein the optimum angular parameters can be selected, as shown in the shaded region in Fig. 14.According to Fig. 14, the center of the shaded region should be selected as the optimal point with the largest GTI and the smallest

FTI, namely, = 90 and = 45.

6.4. Establishing a 2-DOF 3-parallelogram PPM

The optimum length parameters, i.e., R1= 974.14 mm,R2= 300.475 mm, and R3= 528.24 mm, are obtained. The optimuman-gular parameters, i.e., =45 and =90, are also identied by the performance atlas in Fig. 14. All optimum values of the geometricparameters in the design space are determined. However, other geometric parameters not mentioned earlier are still necessary tobuild the 2-DOF 3-parallelogram PPM. These geometric parameters include the lengths of links OA, OD, HI (R), and GH. These param-eters do not affect the workspace and transmissibility of the manipulator and should be determined based on practical circumstanceor the experience of the designer. An optimum 2-DOF 3-parallelogram PPM is established by the authors, as shown in Fig. 15.Fig. 15. An optimum 2-DOF 3-parallelogram PPM.

16 X.-J. Liu et al. / Mechanism and Machine Theory 87 (2015) 1177. Conclusions

Fig. 16.Workspace and transmissibility of the optimum 2-DOF 3-parallelogram PPM.The kinematic optimal design of a 2-DOF 3-parallelogram PPM is addressed in this study by considering desirable workspace andtransmissibility. The kinematics of commonworkingmode for the 2-DOF 3-parallelogramPPM is conducted, and the capable solutionis selected for the working pattern. The design space of the manipulator is analyzed and divided into the uncoupled length designspace and the angular design space. The GTW is dened by considering the transmission angles involved to determine the positionof the end-effector. All possible manipulators are qualitatively investigated by considering their GTWs, and four qualitative optimalregions are obtained. Based on the GTW, the LGTW is proposed to evaluate the painting region of the spray-painting robot. TheAGTW is also proposed to evaluate the working exibility. The CGTW is dened by considering all the transmission angles associatedwith both working position and working posture. Based on the CGTW, the LTI is proposed to investigate the transmissibility of themanipulator in a denite position. Based on the LTI, the GTI is introduced to reect the average transmissibility in the GTW. The FTIis presented to describe the dynamic wave of the LTI around the GTI. Based on the performance atlas of the LGTW and the AGTW,the optimum length parameters are obtained as R1 = 974.14 mm, R2 = 300.475 mm and R3 = 528.24 mm. According to the per-formance atlases of the GTI and the FTI, the optimum angular parameters are obtained as = 45 and = 90. Finally, an optimum2-DOF 3-parallelogram PPM is identied. The optimum workspace and transmissibility of this manipulator are described. Althoughthe performance indices based on the workspace are specic to the 2-DOF 3-parallelogram PPM, the performance indices based onthe transmissibility and the optimal design method used in this study are general and can also be applied to other PPMs.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 51425501, 51375251) and theNational Basic Research Program (973 Program) of China under Grant No. 2013CB035400.

Appendix A. Supplementary data

Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.mechmachtheory.2014.12.014.

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17X.-J. Liu et al. / Mechanism and Machine Theory 87 (2015) 117

Kinematic optimal design of a 2-degree-of-freedom 3-parallelogram planar parallel manipulator1. Introduction2. Kinematics of a 2-DOF 3-parallelogram PPM2.1. Architecture2.2. Kinematic analysis

3. Design space3.1. Length design space3.2. Angular design space

4. Presentation of the GTW5. Performance indices5.1. Performance indices based on desirable workspace5.1.1. Area of the GTW (AGTW)5.1.2. Length of the GTW along axis X (LGTW)

5.2. Performance indices of transmissibility5.2.1. Definition of the comprehensive GTW (CGTW)5.2.2. The LTI5.2.3. The GTI5.2.4. The fluctuant transmission index (FTI)

5.3. Summery of the performance indices

6. Optimal kinematic design based on performance atlases6.1. Requests of kinematic optimal design6.2. Determining the length parameters6.3. Determining the angular parameters6.4. Establishing a 2-DOF 3-parallelogram PPM

7. ConclusionsAcknowledgmentsAppendix A. Supplementary dataReferences