New kinematic structures for 2-, 3-, 4-, and 5-DOFparallel manipulator designs

Feng Gao a,*, Weimin Li a, Xianchao Zhao b, Zhenlin Jin c, Hui Zhao c

a Robotic Research Center, Hebei University of Technology, Tianjin 300130, PR Chinab Robotic Research Center, Yanshan University, Qinhuangdao, Hebei 066004, PR China

c Robotic Research Institute, Beijing University of Aeronautics and Astronautics, Beijing 100083, PR China

Received 11 January 2001; accepted 11 April 2002

Abstract

In this paper, several types of composite pairs and new kinds of sub-chains (limbs or legs) with specicdegrees of freedom are proposed. Based on the special Pluucker coordinates for describing the displacementof the output link of a limb, the principle for design of structures of parallel robotic mechanisms is pre-sented. And several new types of 2-, 3-, 4- and 5-DOF parallel robotic mechanisms are obtained. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Stewart platform; Parallel robots; Mechanisms

1. Introduction

Since Stewart used the 6-DOF parallel mechanism for ight simulator in 1965 [1], the parallelmechanisms have been named Stewart platform which has been used in the robotic area fre-quently, and have been under increasing developments over the last few years from a theoreticalview point as well as for practical applications. The parallel mechanisms have been utilized formany practical applications, where high load carrying capacity (ight simulator, automobilesimulator, tank simulator, earthquake simulator and so on), good kinematic and dynamic per-formance (parallel machine tools, Delta robot for very fast pick-and-place tasks of light loads,force and torque sensors, legs of walking machine) and precise positioning (micro-manipula-tors, manipulator for ophthalmic surgery operation) are of paramount importance. Recently,

*Corresponding author.

0094-114X/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0094-114X(02)00044-7

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Merlet [2], Dasgupta and Mruthyunjaya [3] presented state-of-the-art reviews of the literature onthe Stewart platform manipulators, in detail.The mechanical design of parallel robots requires the application of engineering expertise in

a variety of areas. Important disciplines include mechanism design, structure design, and me-chanical, control and electrical engineering. In the invention and design of parallel robots, thecontriving of the new types of robotic mechanisms is one of the most important activities, becausethe mechanisms determine the performance characteristics of the robots. An interesting anddicult problem is to nd a method to design a mechanical architecture for a parallel manipulatorbeing given its number and type of degree of freedom. Hunt [4] suggested the use of parallel-actuated mechanisms like the ght simulator of Stewart as 6-DOF robot manipulators. Signicantcontributions, which carried the realm of parallel manipulators from its infancy into the status ofa popular research topic, were made by the thought-provoking works of Earl and Rooney [5] andHunt [6]. Earl and Rooney [5] analyzed the kinematic structures for robotic applications and theirinterconnections including both serial and parallel mechanisms and presented methods for syn-thesis of new kinematic structures. Hunt [6] studied the structural kinematics of parallel manip-ulators on the basis of screw theory and enumerated promising kinematic structures. Pernette andHenein [7] presented the method for design of a 3-DOF parallel translating manipulator withUPU joints kinematic chains. The planar 2-DOF parallel manipulators are 5-bar linkages with5 revolute joints [8] or 3 revolute joints and 2 prismatic pairs. The planar 3-DOF parallel ma-nipulators [9,10] are 8-bar linkages with two ternary links (frame and end-eector) connectedthrough three in-parallel legs, each leg consisting of two links. Though a number of mechanismsare possible by dierent combinations of revolute and prismatic joints at the legs, two of themhaving 3-RRR and 3-RPR structures have attracted wide research interest. The spherical 3-DOFparallel manipulator [11,12] consists of two bodies connected through three in-parallel legs(mostly RRR with axes of all revolute joints passing through a point) such that any motion ofthe end-eector is always on a sphere. The Delta robot [13] is with 3 translational degrees offreedom.Although many researchers have paid attention to the design of the parallel robot mechanisms,

and proposed several types of parallel mechanisms, for instance, 2- and 3-DOF planar parallelmechanisms, Delta robots with 3 translational degrees of freedom, 3-DOF spherical robots and

Fig. 1. Traditional types of pairs.

1396 F. Gao et al. / Mechanism and Machine Theory 37 (2002) 13951411

6-DOF parallel mechanisms, there is still lack of the types of parallel mechanisms, especially 2-, 3-,4- and 5-DOF parallel robot mechanisms with desired end-eector motions. The reason for this is

Table 1

Classication of simple limbs

DOF Pairs Types of limbs

6 P, S, S SPS, PSS

U, P, S UPS, PUS

R, S, S RSS, SRS

U, R, S URS, RUS

5 R, R, S RRS, RSR

R, P, S RPS, RSP, PRS, PSR

P, U, U PUU, UPU

4 P, U, R PUR, PRU, UPR, RPU

3 R, R, R RRR

R, P, R RPR, PRR

H, R, P HRP, PRH, RPH

2 R, R

P, R PR, RP

Fig. 2. New types of composite pairs.

F. Gao et al. / Mechanism and Machine Theory 37 (2002) 13951411 1397

Table 2

Classication of composite limbs

DOF Pairs Types of limbs

5 P, U, U PUU, UPU, UPUR, U, U RUU, URU, UURP, ^U, U P^UU, UP^U, ^UPUP, U^, U PU^U, U^PU, UU^PR, ^U, U R^UU, ^URU, ^UURR, U^, U RU^U, U^RU, UU^RR, PU, U RPUU, URPU, PURU

R, UP, U R UPU, UPRU, UUPR

R, U, C CUR, RUC, CRU

C, U^, R CU^R, RU^C, RCU^

4 P, U, R PUR, UPR, RUPR, R, U RUR, URR, RRU

P, ^U, R P^UR, RP^U, R^UPR, ^U, R R^UR, RR^U, ^URRP, UP, R PUPR, RUPP, UPPR

R, UP, R RUPR, RRUP, UPRR

C, U CU, UCC, U^ CU^, U^C

3 P, U PU, UPR, U RU, URP, U^ PU^, U^PR, U^ RU^, U^R

2 PU, UP, ^U, U^, U

Fig. 3. The limbs with traditional structures.

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that there is not the ecient and general theory for type synthesis of parallel mechanisms beinggiven the number and types of degrees of freedom. This paper presents the theory for innovationand invention of new types of 2-, 3-, 4- and 5-DOF parallel mechanisms.

2. Classication of pairs

Parallel robotic mechanisms consist of links and pairs or joints. The traditional types of pairsare prismatic pair (P), revolute joint (R), spherical joint (S), helix pair (H), cylinder pair (C), anduniversal joint (U), as shown in Fig. 1. However, only using the traditional pairs is dicult todesign the parallel robotic mechanisms with specic 2, 3, 4 and 5 DOF. The reason for this is thatit is hard to obtain the limbs with specic 2, 3, 4 and 5 DOF. For example, although both thelimbs RPS and UPU have 5 DOF, it is hard to determine the kinematic characteristics of theirend-eectors because of the coupled motions.In this paper, we propose several new types of composite joints, which are very useful for design

of parallel robotic mechanisms. The joints are pure-translation universal joint (U), translation-and-rotation universal joint (U^ or ^U), and pure-planar-translation universal joint (PU or UP), asshown in Fig. 2.

Fig. 4. The limbs with composite structures.

F. Gao et al. / Mechanism and Machine Theory 37 (2002) 13951411 1399

3. Classication of limbs

For investigation of the type synthesis of the parallel mechanisms, the key issue is to nd thelimbs with known kinematic characteristics, that is to say, with specied degrees of freedom.Therefore, we have to discuss the classication of the limbs for parallel mechanisms. The limbs are

Table 3

2-, 3-, 4- and 5-DOF limbs with specic kinematic characteristics

DOF Pairs of

limbs

Characteristics for structures of

limbs

Kinematic

constraints

Characteristics of kinematic con-

straints of limbs

5 P, U, U No special requirement 1 Constraining rotation along withthe axis perpendicular to the two

axes of universal joint

P, ^U, U Each limb has two parallel axesamong joints R, ^U, U^ and U

R, U, UP, U^, UR, PU, U

R, UP, U

R, ^U, U Each limb has three parallel axesamong joints R, ^U, U^ and U

R, U^, U

R, U, C No special requirement 1 Constraining rotation along withthe axis perpendicular to the two

axes of joints R and C

R, U^, C Each limb has two parallel axesamong joints R, ^U and C

1 Constraining rotation along with

the axis perpendicular to the axes

of joints R, ^U and C

4 P, U, R No special requirement 2 Constraining rotations along withthe tow axes perpendicular to the

axis of joint R

R, R, U The rotation axes of all joints(R, ^U, U^ and C) for eachlimb are parallel each other

2 Constraining rotations along with

the tow axes perpendicular to the

axes of joints R, ^U, U^ and CP, ^U, R

R, ^U, R

P, UP, R

R, UP, R

C, U

C, U^

3 P, U No special requirement 3 Constraining three-axis rotations

P, U^ The translation axis of pair P isperpendicular to the axes of

joint U^ or ^U

3 Constraining the three-axis

rotationsP, ^U

2 PU The translation axis of pair P is

perpendicular to the axes of

4-bar linkage

4 Constraining the three-axis rota-

tions and one-axis translation

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the sub-chains connected between the upper platform and lower platform of parallel roboticmechanisms. Tables 1 and 2 show the classication of the simple and composite limbs for parallelmechanisms, respectively, in which, the rst letter expresses the joint connected with xed frame(lower platform), and the last letter represents the joint connected with moving platform (upperplatform). For instance, the limb UPS means that the limb is connected with xed frame by thejoint U and linked with moving platform by the joint S. By using the joints as shown in Figs. 1 and2, the limbs with traditional structures and composite structures are proposed as shown in Figs. 3and 4, respectively.In Fig. 4, the limb PU^U has 5 DOF, in which U^ is a planar parallelogram; the limb PUU has

5 DOF, in which U represents 3-UU with the same link lengths; and limbs PUR has 4 DOF.Table 3 shows the classication of 2-, 3-, 4- and 5-DOF limbs with specic kinematic charac-teristics, from which we can see that the composite joints U, ^U, U^, PU and UP are very usefulfor design of 2-, 3-, 4- and 5-DOF limbs with specic kinematic characteristics.

4. Principle for type design of parallel robotic mechanisms

Although one has had the theory for calculation of degrees of freedom for planar and spatialmechanisms, the theory cannot be used for analyzing and synthesizing the structural types ofparallel robotic mechanisms, which have less than 6 DOF. The reason for this is that the availabletheory just relates the pairs and links, but the limbs, so that it is hard to calculate the degreesof freedom of some of parallel robotic mechanisms correctly. For example, according to the

Table 4

Special Pluucker coordinates of limbs

DOF Limbs $j

vxj vyj vzj xaj xbj xcj

6 UPS 1 1 1 1 1 1

SPS 1 1 1 1 1 1

PUS 1 1 1 1 1 1

5 R^UU 1 1 1 1 1 0P^UU 1 1 1 1 1 0PUU 1 1 1 1 1 0

4 PUR 1 1 1 1 0 0P^UR 1 1 1 1 0 0R^UR 1 1 1 1 0 0CU^ 1 1 1 1 0 0

3 PU^ 1 1 0 1 0 0PU 1 1 1 0 0 0RPR 1 1 0 0 0 1

RRR 1 1 0 0 0 1

RRR 0 0 0 1 1 1

2 PU 1 1 0 0 0 0

F. Gao et al. / Mechanism and Machine Theory 37 (2002) 13951411 1401

Table 5

Classication of parallel robot mechanisms

DOF Each number

is the DOF of

each limb

Examples Characteristics of

structures

Kinematic performance of upper

platform

Rotations Translations

2 2, 2 2-PU Planar 0 2

3, 2 1-RRR & 1-PU 0 2

1-RPR & 1-PU 0 2

3, 2 3-RRR & 1-RR Planar

1-RPR & 1-RP

6, 2 1-UPS & 1-PU Planar 0 2

3 3, 3, 3 3-RRR Spherical 3 0

3-RRR Planar 1 2

3-RPR 1 2

3-PRR 1 2

3-PU Spatial 0 33-UP 0 3

4, 4, 4 3-PUR Spatial 0 33-P^UR 0 33-CU^ 0 33-^UC 0 33-RPC 0 3

3-RRC 0 3

5, 5, 5 3-PUU Spatial 0 33-P^UU Spatial 0 3Delta Spatial 0 3

3-RPS 3-DOF

3-SPR 3-DOF

6, 6, 6, 3 3-UPS & 1-UP 3-DOF

4, 3, 3 1-P^UR & 2-PU^ 1 24, 4, 3 2-P^UR & 1-PU^ 1 25, 5, 3 2-PUU & 1-PU Spatial 0 36, 6, 3 2-PUS & 1-PU 0 3

4 5, 5, 4, 4 2-PUU & 2-PUR Two axes of pairsR are parallel

each other

1 3

2-PUU & 2-P^UR 1 36, 6, 4, 4 2-PUS & 2-PUR 1 3

2-PUS & 2-P^UR 1 3

6, 6, 6, 4 3-PUS & 1-PUU 2 2

3-PUS & 1-PUR 1 33-PUS & 1-P^UR 1 3

5 6, 6, 6, 6, 5 4-PUS & 1-PUU 2 34-PUS & 1-P^UU 2 34-SPS & 1-PUU 2 3

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available theory for calculation of degrees of freedom, the degrees of freedom for the 3-UPUparallel mechanism could be 3, but actually, the mechanism can have 3 or 4 or 5 DOF, whichdepends on the position of the end-eector.For design of parallel robotic mechanisms with specic kinematic characteristics, it is very

important to discuss the limbs with specic kinematic characteristics. For convenience, we let $ be

Table 5 (continued)

DOF Each number is

the DOF of each

limb

Examples Characteristics of

structures

Kinematic performance of upper

platform

Rotations Translations

6 6, 6, 6, 6, 6, 6 6-PUS 3 3

6-SPS 3 3

6-UPS 3 3

6-RUS 3 3

6-RSS 3 3

6, 6, 6 3-PRPS Each limb has 2

actuators

3 3

Fig. 5. 2-PU.

Fig. 6. 2-PU^ & 1-P^UR.

F. Gao et al. / Mechanism and Machine Theory 37 (2002) 13951411 1403

Fig. 7. 2-CU^ & 1-PU^.

Fig. 8. 2-P^UR & 1-PU^.

Fig. 9. 3-CU^.

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the special Pluucker coordinates for describing the displacement of the output link of a limb for aparallel mechanism, which is

$j vxj vyj vzj ; xaj xbj xcj 1where, vj vxj vyj vzj expresses the translation of the output link of the limb j, andxjxaj xbj xcj denotes the rotation of the output link of the limb j with respect to three

Fig. 10. 3-P^UR.

Fig. 11. 3-PU.

Fig. 12. 3-PUR.

F. Gao et al. / Mechanism and Machine Theory 37 (2002) 13951411 1405

Eulers angles, a, b and c. The special Pluucker coordinates vxj, vyj, vzj, xaj, xbj, xcj can be taken as 1or 0. When taking 1, it means that the limb j has that degree of freedom; when taking 0, it meansthat the limb j has no that degree of freedom. From Eq. (1), we can obtain 2-, 3-, 4-, 5- and 6-DOFlimbs with specic kinematic characteristics, as shown in Table 4, in which the limbs satisfy thekinematic constraints as shown in Table 3.In a parallel mechanism, if the parallel mechanism has specic degrees of freedom ($), the limbs

1; 2; . . . and n by which the upper platform (moving end-eector) is connected with lower platform(xed frame) have to satisfy the following condition:

Fig. 13. 3-PUU.

Fig. 14. 3-RRR (planar).

Fig. 15. 3-RRR (spherical).

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$ $1 \ $2 \ $n 2Eq. (2) expresses that the special Pluucker coordinates of the nal motion generated by theupper platform of a parallel mechanism are equal to the intersection of the special Pluucker co-ordinates of all limbs in the mechanism, which is the principle for type design of parallel robotic

Fig. 16. Delta.

Fig. 17. 3-RPS.

Fig. 18. 2-PUR & 2-PUS.

F. Gao et al. / Mechanism and Machine Theory 37 (2002) 13951411 1407

mechanisms with specic degrees of freedom. Eq. (2) is very useful for design of parallel roboticmechanisms with the specic degrees of freedom.

Fig. 19. 4-PUR.

Fig. 20. 2-P^UU & 2-P^UR.

Fig. 21. 3-UPS & 1-UP.

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5. Structure types of parallel mechanisms

The previous section have considered the procedure for design of parallel robotic mechanisms,especially lower dimensional versions of the parallel mechanisms by the intersection of the specialPluucker coordinates of all limbs in the mechanism. Based on the limbs as shown in Tables 14, theclassication of the parallel mechanisms with specic 2, 3, 4, 5 and 6 DOF can be obtained, as

Fig. 22. 6-RSS.

Fig. 23. 6-PSS (Hexaslider).

Fig. 24. 3-PRPS.

F. Gao et al. / Mechanism and Machine Theory 37 (2002) 13951411 1409

shown in Table 5 and Figs. 525. Fig. 21 shows the 3-DOF parallel robot with 4 limbs (3-UPS &1-UP), in which the 3 DOF depend on the limb UP. Fig. 24 shows the 6-DOF parallel robot with3 limbs (3-PRPS), in which each limb has 2 actuators.

6. Conclusions

Design of structure types of parallel robotic mechanisms is a key issue in robotic area, becauseof lack of the parallel robotic mechanisms in lower dimensional spaces. In this paper, several typesof composite pairs and new kinds of sub-chains (limbs) with specic degrees of freedom areproposed. The principle for design of structures of parallel robotic mechanisms is presented, basedon the special Pluucker coordinates for describing the displacement of the output link of a limb.And several new types of 2-, 3-, 4- and 5-DOF parallel robotic mechanisms are obtained.

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F. Gao et al. / Mechanism and Machine Theory 37 (2002) 13951411 1411

New kinematic structures for 2-, 3-, 4-, and 5-DOF parallel manipulator designsIntroductionClassification of pairsClassification of limbsPrinciple for type design of parallel robotic mechanismsStructure types of parallel mechanismsConclusionsReferences