Structural synthesis of a class of two-loop generalized parallel mechanisms

Mechanism and Machine Theory 128 (2018) 429–443

Contents lists available at ScienceDirect

Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

Structural synthesis of a class of two-loop generalized parallel mechanisms Chunxu Tian a, Yuefa Fang a,∗, Q.J. Ge b a b

School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, PR China Department of Mechanical Engineering, Stony Brook University, Stony Brook, New York 11794-2300. United States

a r t i c l e

i n f o

Article history: Received 2 March 2018 Revised 18 April 2018 Accepted 16 June 2018

Keywords: Parallel mechanism Coupling sub-chain Structural synthesis Kinematic chains Screw theory

a b s t r a c t The conventional parallel mechanism (CPM) is characterized by connecting serial kinematic limbs or chains to the moving platform or linkage. The mobility and motion pattern of the CPM may be further constrained by adding a coupling sub-chain to serial kinematic chains. That will result in a new class of mechanisms called generalized parallel mechanisms or GPMs for short. While the systematic mobility analysis and structural synthesis of GPMs are more challenging and have rarely been studied, the application of the coupling subchain possesses the potential for enhanced functionality and performance. This paper proposes a novel approach for synthesizing two-loop mechanisms with coupling sub-chains that form the basic building blocks for GPMs. The screw theory based constraint synthesis method is advocated. Examples are enumerated to demonstrate the feasibility of the proposed approach. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction A parallel mechanism is a multi-loop mechanism with several independent serial kinematic limbs connecting from the moving platform to the fixed base [1–3]. To reduce the total amount of floating weights, the actuated joints in the connecting chains are often placed at or near the base. Such conventional parallel mechanisms (CPMs) have been focused on in the research during the past decades. In recent years, however, significant research attention has been drawn to a new class of parallel mechanisms that are characterized by coupling sub-chains [4–6] or configurable platform [7,8]. They are called generalized parallel mechanisms (GPMs) and possess the potential for more tailored functionality or improved performance [9,10]. As the structural synthesis of GPM is much more complex than that of CPM, the current research focus has been on the two-loop mechanism, which is the basic building block of GPM [4,11]. It should be mentioned that the two-loop mechanisms proposed in this paper are two longitudinal loop mechanisms. This paper aims at developing a systematic approach for the structural synthesis of 2-DOF and two-loop mechanisms. In general, to design a novel mechanism, structural synthesis and dimensional synthesis are two significant processes. In the creative design process of parallel manipulators, the structural synthesis is the essential step and received much attention in the past decades [12–15]. As for CPMs, Gogu [16] investigated the linear transformation to synthesize parallel manipulators. Huang and Li [17] explored parallel mechanisms with identical limbs using screw theory method. Fang and Tsai [18] proposed 4-DOF (degrees of freedom) and 5-DOF serial kinematic limbs to synthesize symmetrical parallel mechanisms. Kong and Gosselin [19] extended virtual chain approach to constructing 3T1R parallel manipulators. Hervé [20] and ∗

Corresponding author. E-mail address: [email protected] (Y. Fang).

https://doi.org/10.1016/j.mechmachtheory.2018.06.008 0094-114X/© 2018 Elsevier Ltd. All rights reserved.

430

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

Fig. 1. Generating GPMs by assembling appropriate SOCs.

Fig. 2. The planar two-loop mechanism.

Li et al. [21] proposed several parallel mechanisms by displacement group. Yang et al. [22] presented a kind of characteristic set and equations to create parallel manipulators. In the process of synthesizing parallel mechanisms, Gao et al. [23] put forward the novel concept of Gf set. By using the motion synthesis and kinematic mapping approach, Zhao et al. [24] designed planar mechanisms. In recent years, the structural synthesis of GPMs has received much attention and efforts. Parallel mechanisms with coupling sub-chains or configurable moving platform have been synthesized for specified output motion [8,25] and possess the potential for higher accuracy and stiffness of the overall parallel mechanism. Based on chain groups, Campos, et al. [26] investigated the use of Assur groups for constructing generalized kinematic limbs and manipulators. Ding et al. [4,11] explored the mobility analysis method of two-layer and two-loop (TLTL) mechanisms, and synthesized novel hybrid parallel mechanisms with various DOFs. By using the embedment methodology, Shen et al. [27] presented several 6-DOF hybrid manipulators. Zeng and Fang [6] developed the displacement group theory for synthesizing spatial multi-loop manipulators and proposed the novel manipulators with deployability and kinematotropic property. For the sake of obtaining agile underwater stereo vision, Zoppi and Molfino [25,28] designed and analyzed a 3-DOF flexible parallel-hybrid mechanism. The complete set of mechanism solutions was presented by Bałchanowski [29] to synthesize spatial mechanisms. This paper proposes a systematic structural synthesis approach for the 2-DOF two-loop mechanisms. This paper is organized as follows. Section 2 details the method for designing GPMs by adding appropriate serial kinematic chains. Section 3 gives the screw theory of closed-loop mechanisms and identifies mutual constraints between different loops. It then describes in detail the procedure for synthesizing two-loop mechanisms. Section 4 presents the constraint synthesis and virtual work based method for various kinematic legs and single-loop mechanisms. This method is then used to deduce various combinations of kinematic limbs and general single-loop mechanisms. Finally, some representative 2-DOF and two-loop mechanisms are constructed to demonstrate the feasibility of the proposed method. 2. A generative approach to the synthesis of GPMs Based on screw theory, structural synthesis of GPMs can be realized by synthesizing different single-loop mechanisms from the end moving platform to the base [11]. According to the displacement subgroup method, the spatial multi-loop detachment method and the configuration cards method for displacement subsets [6] are developed for designing and analyzing coupling sub-chains. In addition, the GPMs derived by constructing parallel mechanisms with configurable platforms offer the higher accuracy than CPMs [8]. A GPM can be generated by starting with a single-loop mechanism and then adding several open kinematic chains on the top of one another [30]. Thus, a V independent closed-loop GPM is decomposed into a V-1 closed-loop mechanism with an additional serial open chain (SOC) on the top of it, as shown in Fig. 1. In this scheme, a two-loop mechanism, which will be used as the basic building block of GPMs, can be obtained by assembling one SOC to a single-loop mechanism. Taking the planar two-loop mechanism as an example, the process to construct GPMs can be described in detail. The mechanism in Fig. 2(a) is a planar five-bar closed-loop mechanism with 2-DOF, in which, IMP and SIMP respectively represent the first and second intermediate moving platforms of GPMs, and EMP is the end moving platform. After arranging one appropriate SOC on the five-bar closed-loop mechanism, a two-loop mechanism with 2-DOF can be derived, as shown in Fig. 2(c). In this way, the coupling sub-chain can be effectively determined. Similarly, more mutual constraints and less passive joints

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

431

Fig. 3. The serial kinematic chains of two-loop mechanism.

between two different closed loops can be derived [4,6,11]. The construction process of spatial two-loop mechanisms with 2-DOF will be developed in the subsequent sections. 3. The analysis of mutual constraints between two loops When synthesizing the first single-loop mechanism, the expected mobility and motion pattern of GPMs cannot be determined. Consequently, the DOF of novel GPMs cannot be derived as synthesizing the first single-loop mechanism with initial DOF [31]. Since the GPMs are closed-loop mechanisms with different independent loops [14], the loop-closure condition should be satisfied. Assuming that the limb with n kinematic joints is composed of u revolute joints and (n - u) prismatic joints. The summation of twists in the kinematic pairs of a single-loop kinematic chain is equal to zero [32], i.e., u 

ξi s i = 0

(1)

i=1 u  i=1

ξi ( r i × s i ) +

n 

ξi s i = 0

(2)

i=u+1

where ξ i represents the angular rate (ωi ) in the revolute pair and the linear rate (ν i ) in the prismatic pair; si denotes the unit direction vector collinear with the direction of the screw axis; ri is the position vector of a random point on the screw axis. For convenience, the revolute joints are arranged at the first u kinematic joints.

ω1 (r1 × s1 ) + · · · + ωu (ru × su ) + νu+1 su+1 + · · · + νn sn = 0

(3)

It follows that, at the origin of a coordinate frame, the sum of linear velocities generated by prismatic joints and revolute joints in a closed-loop mechanism should be zero. The first single-loop mechanism, which is near to the base, is a parallel mechanism with two legs, as shown in Fig. 3. Note that the first and second moving platforms in the first single-loop mechanisms are represented by FMP and SMP, respectively. When a basic link [33] connected with two 1-DOF points is treated as the fixed base, the first h (h ≤ u) revolute joints and first g (g ≤ n − u) prismatic joints are assumed to be distributed in one leg. The twist systems of two serial kinematic chains of the closed-loop mechanisms are obtained as follows

    ω j1 r j1 × s j1 + · · · + ω jh r jh × s jh + ν j (u+1) s j (u+1) + · · · + ν j (u+g) s j (u+g)     = −ω j (h+1) r j (h+1) × s j (h+1) − · · · − ω ju r ju × s ju − ν j (u+g+1) s j (u+g+1) − · · · − ν jn s jn

(4)

in which j (j = 1, 2) represents the jth single-loop mechanism. The intermediate moving platform of the GPM is considered as the original basic link of the second single-loop mechanism. IMP is the end-effector of the first single-loop mechanism. Therefore, the twist system of the second single-loop mechanism can be obtained similarly. It should be noted that the twist system of the coupling sub-chains is developed in the opposite direction. Without loss of generality, assuming that the coupling sub-chain consists of m (m ≤ u − h) revolute joints and k (k ≤ n − u − g) prismatic joints, then the twist system of the coupling sub-chain is as follows.

ω1 (r1 × s1 ) + · · · + ωm (rm × sm ) + νm+1 sm+1 + · · · + νm+k sm+k   = −ω1(u−m+1) r1(u−m+1) × s1(u−m+1) − · · · − ω1u (r1u × s1u ) − ν1(n−k+1) s1(n−k+1) − · · · − ν1n s1n     = ω2(h+1) r2(h+1) × s2(h+1) + · · · + ω2(h+m) r2(h+m) × s2(h+m) + ν2(u+g+1) s2(u+g+1) + · · · + ν2(u+g+k) s2(u+g+k)

(5)

According to the general methodology for mobility analysis [11], the dimension of the solution space will be changed when the coefficient matrix is changed in the continual motion. In other words, as long as there is a dependent parameter in the coupling sub-chain, the mobility of the equivalent sub-parallel mechanism is different from that of the second singleloop mechanism. Consequently, the DOF of a two-loop mechanism is less than or equal to the sum of DOFs of the two dependent single-loop parallel mechanisms.

432

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

Fig. 4. Two constraints are exerted on a rigid body instantaneously.

Without loss of generality, one of the two ternary links is regarded as the end effector of the first single-loop parallel mechanism. The relation between twist systems can be established as



$IMP = $S1 ∩ $S2 + $S3



or



$IMP = $S2 ∩ $S1 + $S3



(6)

where $S1 is the twist system of the limb without including coupling sub-chains in the first single-loop mechanism; $S2 and $S3 are combined into the twist system of the other limb; considering Fig. 3, “∩” signifies the intersection operation of twist systems. In the second single-loop mechanisms, the IMP serves as the fixed base. Therefore, the following relation between twist systems can be derived.



$EMP = $S4 ∩ $S5 − $S3





or $EMP = $S5 ∩ $S4 − $S3



(7)

where $S4 is the twist system of the limb without including coupling sub-chains in the second single-loop mechanism; the twist system of the other is contributed by −$S3 and $S5 . According to screw theory [3,14], new composite translations or rotations can be derived after performing the linear combination of some translations and rotations as follows: a new translation is obtained from the combination of two dependent translations with different directions. After combining two rotations, whose axes are intersecting at a fixed point, a new rotation can be derived. The axis of the derived rotation coincides with the existing point. By combining the rotation and two perpendicular translations or two parallel rotations, a new rotation with the axis parallel to the given rotation can be generated. Three 3-DOF planar motions can be used to generate any rotations, which are parallel to the rotation of the kinematic chain. Finally, the motion pattern of the overall two-loop mechanism is derived by the linear combinations of two interconnected single-loop mechanisms. The instantaneous mobility of the end moving platform in the mechanism is expressed as follows.

− → − → − → M EMP = M IMP + M REMP

(8)

− → − → − → where M EMP is the mobility of the EMP in the overall two-loop mechanism; M IMP and M REMP respectively denote the mobility of the moving platform in the first and second single-loop mechanisms.

4. Structural synthesis of various kinematic limbs The planar two-loop mechanisms have been systematically constructed in reference [34–36]. When synthesizing spatial two-loop mechanisms, SOCs and single-loop mechanisms of GPMs need to be constructed. Therefore, various serial kinematic chains should be analyzed first. Both screw theory and displacement group can be the basic theory to construct two-loop mechanisms. There are some differences between the above two methods. When constructing GPMs, the simpler operation may be involved in the displacement group method [6,20,21]. Although the screw theory [12,17,18] is complex, the relation among coupling sub-chains, IMP and EMP can be revealed in detail. As a consequence, the screw theory and virtual work based constraint synthesis method are performed in the following sections. This method is used to construct various legs and single-loop mechanisms.

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

433

Table 1 L02CF limbs structures. Kinematic joints

No.

Four-link chains

No.

Three-link chains

P-3R

1 2 3 4 5 6 7 8

PRRR RPRR PPRR P RP R PRRP RPPR PPPR PRPP

9 10 11 12 – 13 14 15

CRR RCR PCR PRC – CP R PPC PCP

2P-2R

3P-R

The underline indicates the same direction of rotations.

4.1. Structural synthesis of L02CF limb, L20CF limb and L11CF limb By reference [2,14,32], when a rigid body undergoes the infinitesimal twist with constraints exerted on it as shown in Fig. 4, the virtual work can be developed by the constraint as follows

δW = θ ·

t 

[s · (rrc × src + hrc src ) + src · (r × s + hs )]

(9)

c=1

After carrying out the corresponding operations, the specific form can be expressed as

δW = θ ·

t 

[(h + hrc )(s · src ) + src · (r × s ) + s · (rrc × src )]

(10)

c=1

where δ W is the infinitesimal work of the rigid body; θ is the product of wrench intensity and joints rates; t (t = 1, 2, 3, 4, 5) is the number of constraints in the wrench system, and h represents the pitch of the screw. The subscript r in sr , rr and hr represents the corresponding vectors and pitch in the wrench system. According to the geometry of the two screws, the following relations are derived as

s · sr = cos α

(11)

sr · (r × s ) + s · (rr × sr ) = −a · (s × sr ) = −a sin α

(12)

in which, α represents the twist angle between s and sr ; a signifies the common perpendicular vector of s and sr , measured from the twist to the wrench. It satisfies the right-hand rule. After substituting Eqs. (11) and (12) into (10), the following equation can be obtained.

δW = θ ·

t 

[(h + hrc ) cos αc − ac sin αc ]

(13)

c=1

When t = 1, there is only one constraint in the limb. As it has been known, L01CF limbs and L10CF limbs [18] are two special kinds of limbs that are capable of providing a constraint couple and a constraint force, respectively. There may exist a LPF QC limb (P = 0, 1, 2, 3; Q = 0, 1, 2, 3) that exerts P constraint forces and Q constraint couples on the end-effector. With the rank of the wrench system at two, i.e., t = 2, two constraint forces or two constraint couples, or one constraint force and one constraint couple are simultaneously exerted on an end-effector. It can be L02CF limb, L20CF limb or L11CF limb [31]

δW = θ · [s · (rr1 × sr1 + hr1 sr1 ) + sr1 · (r × s + hs) + s · (rr2 × sr2 + hr2 sr2 ) + sr2 · (r × s + hs)]

(14)

δW = θ · [(h + hr1 ) cos α1 − a1 sin α1 + (h + hr2 ) cos α2 − a2 sin α2 ]

(15)

In general, θ is not equal to zero. The reciprocal condition can be satisfied by adjusting the twist angle α and vector a. When two constraint couples of infinite pitches act as the wrench system, the L02CF limbs are formed. It follows from Eq. (15) that angles α 1 and α 2 must be 90° simultaneously. It means that all the axes of revolute joints must be parallel to the common normal, which is defined by the two constraint couples. However, the linearly independent prismatic joints are capable of being arranged arbitrarily. Since the common normal is uniquely defined by two constraint couples, only one type of revolute joint exists in the limb. Note that the cylindrical joints (C), universal joint (U) and spherical joint (S) can be contributed by the combination of a P joint and an R joint or a series of R joints within a kinematic limb. Besides, the wrench system of the limb will not be influenced by the composite joints instantaneously, as shown in Table 1. After taking the adopted geometric criteria into consideration, the following cases can arise: The revolute joints (RR or RRR) which are parallel to each other are indicated with the underline. The combination of the revolute joint and the universal joint (RU) represents that the first axis of the U joint is parallel to the axis of the R joint. This combination results from three revolute joints, and two of them are parallel.

434

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443 Table 2 L20CF limbs structures. Kinematic joints

No.

Four-link chains

No.

Three/Two-link chains

4R

1 2

RF RF RR RF RRR

3 4 5 6 7

R’RRR R’R’RR PRF RR RF PRR PRRR

8 9 10 11 12 13 14 15 16

RF UR RF S URR R’S R’UR PUR RF CR CRR PS

P-3R

The italic R and U joints intersect at a point. Table 3 L11CF limbs structures. Kinematic joints

No.

Four-link chains

No.

Three-link chains

4R P-3R

1 2 3 4 5 6 7 8

RRF RF RF RPRF RF RRF PRF RRF RF P RPPRF RPRF P RRF PP RRPP

9 10 11 12 13 14 15 16 17

URF RF C R F RF UPRF URF P CPRF C RF P UPP(1) UPP(2) RCP

2P-2R

The combination of the revolute joint and the cylindrical joint (RC) indicates that the axis of the C joint is parallel to the axis of the R joint. Resulting from the sequence RRP, the axes of revolute joints are parallel to the direction of the prismatic joint P. The cylindrical joint and revolute joints (CR or CRR), which are written in italics, denote that the axis of the C joint is intersecting with the revolute joints at a common point. Resulting from the sequence P RR or P RRR, the direction of the P joint is parallel to the axis of the revolute joint in the sequence RR or RRR. Also, the axes of the revolute joints are intersecting at a common point. The universal joint and revolute joints (UR or URR), which are resulted from the sequence RRR or RRRR, signify that the axes of R joints are intersecting with the first axis of the U joint at a comment point. In the L20CF limb [9], both pitches of the constraints are zero, i.e., hr 1 = 0 and hr 2 = 0. For a revolute joint in theL20CF limbs, h = 0. Eq. (15) implies that the magnitude of the vectors a1 and a2 need to be zero, or the angles α 1 and α 2 are equal to zero. It follows that all the revolute joints are intersecting with or parallel to the two constraint forces. Moreover, Eq. (15) requires that the axis of the prismatic joint is perpendicular to the two constraint forces simultaneously. In other words, the prismatic joint is the common normal of the two given constraint forces. Therefore, more than two revolute joints exist in the kinematic chain. Since the unique existence of the plane or the common normal determined by the two given constraint forces, there is at most one independent prismatic joint in the limb. For simplicity, if the twist angle α between s and sr is equal to zero, the corresponding revolute joint is indicated as RF . Besides, when the magnitude of the common perpendicular vector a of s and sr is equal to zero, the corresponding revolute joint is denoted as R (in italics). To realize the finite motion in a parallel mechanism, RF revolute joints need to be connected in serials with several intermediate prismatic joints, and R joints could only be assembled in series. Furthermore, to distinguish the existing intersecting point and R joints, the revolute joint intersecting with another point is indicated as R’, as shown in Table 2. The wrench system of the L11CF limb [31,37,38] is the combination of the wrenches of a L10CF limb and a L01CF limb. The 1 F L0C limb and the L01CF limb correspond to a constraint force and a constraint couple respectively. The second constraint is assumed to be a constraint force, i.e. hr 2 = 0. To satisfy the reciprocal conditions, the Equations given by (11) and (12) need to function together. It follows that cos α1 = 0 and a2 sin α2 = 0 should be simultaneously satisfied in one limb. That implies the angle α 1 is 90°, while the magnitude of the vector a2 or the angle α 2 need to be equal to zero. Therefore, all the revolute joints are intersecting with or parallel to the constraint force, and perpendicular to the constraint couple. The direction of the prismatic joint is perpendicular to the constraint force. To avoid redundant joints, there exist at least two R joints in the L11CF limb. Since the P joint coinciding with the RF joint is impossible, the C joint composed of an RF joint and a P joint is excluded, as shown in Table 3. 4.2. Structural synthesis of other limbs With the rank of the wrench system at three, i.e., t = 3, the joint screw of the limb is reciprocal to three constraints:

δW = θ · [(h + hr1 ) cos α1 − a1 sin α1 + (h + hr2 ) cos α2 − a2 sin α2 + (h + hr3 ) cos α3 − a3 sin α3 ]

(16)

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

435

Table 4 L12CF limbs structures. Kinematic joints

No.

Three-link chains

No.

Two-link chains

3R P-2R

1 2 3 4 5 6 7

R F RF RF PRF RF RF P RF RF PP PRF P RPP PRP

– – – – – 8

– – – – – CP

Kinematic joints

No.

Three-link chains

No.

Two-link chains

3R P-2R

1 2 3 4 5

R F RF R PRF R RF P R RF R P PRR

6 – 7

RF U – RF C

8 9

PU CR

2P-R

Table 5 L21CF limbs structures.

Table 6 L22CF limbs structures. Kinematic joints

No.

Two-link chains

No.

One-link chains

2R 1R-P

1 2 3

R F RF P RF RP

– – 4

– – C

When three independent constraint couples are exerted on the end-effector, the L03CF limb can be derived. According to Eq. (11), it is geometrically impossible for a revolute joint to realize the equation α 1 = α 2 = α 3 = 90°. It follows that there is no revolute joint that is perpendicular to three constraint couples. The L03CF limb is uniquely composed of three linearly independent prismatic joints. For the L30CF limb, all the pitches of the constraints are equal to zero, i.e., hr 1 = hr 2 = hr 3 = 0. Since three constraint forces intersect at one point, we have a1 = a2 = a3 . Therefore, three independent revolute joints constitute the L30CF limb. As for L12CF limbs [39] and L21CF limbs, the constraints exerted on the end-effector consist of one (two) infinitesimal pitch and two (one) infinite pitches. To satisfy the reciprocal condition, some vector a and angle α need to be equal to zero. Meanwhile, the rest of the angles are 90°. The geometrical requirements of the revolute joints can be deduced according to L11CF limbs. In the L12CF limbs, there is at least one revolute joint in the limb. As a result, it can be constructed by 3R, 1P2R or 2P1R. Since two constraint couples are included in the L12CF limbs, RF joint and R joint cannot exist simultaneously. Also, the U joint cannot be used as the composite joint to create L12CF limb, as shown in Table 4. For the L21CF limb, it can only be composed of 3R or 1P2R, as shown in Table 5. It should be noted that there is a special kind of limb, named L3H limbs, where H denotes the wrench of the limb is a constraint helix. This kind of limb is composed of three revolute joints. The axes of the first joint and the third joint of the limb are intersecting at a common moving point. After selecting an appropriate reference coordinate, the joint screws of the limb are reciprocal to three helical constraints. The classic Bennett single loop mechanism is constructed by the L3H limb [40,41]. The closed-loop mechanism which is constructed by L3H limbs can be used to synthesize the GPMs with a configurable platform [8]. When t = 4, the joint screws of the limb are reciprocal to three constraint couples and one constraint force, or three constraint forces and one constraint couple, or two constraint couples and two constraint forces. They can be L13CF limbs, L31CF limbs or L22CF limbs [39]. The L13CF limb uniquely consists of two linearly independent prismatic joints, and only two independent revolute joints can be used to construct the L31CF limb. The L22CF limb can be constructed by 2R or 1P1R. According to the analysis in the L12CF limb, the U joint cannot be used as a composite joint to construct the L22CF limb, as shown in Table 6. As to the analysis above, the L32CF limb and the L23CF limb respectively signify the single joint R and P. When P = Q = 0, the significant no constraint limb represented as LD can be derived. This kind of limb can be derived by adding an appropriate single joint on the top of the L10CF limb or the L01CF limb. Besides, the RR, US, RS, PS, and SS dyads derived in this section have been widely used to construct general single-loop mechanisms. According to Tables 2 and 6, the parallelogram mechanism with two identical RR limbs, RS limbs, and SS limbs are constructed from two identical L22CF limbs, L20CF limbs and L10CF limbs respectively. Regarding the constraint, the mechanism with U (U = 0, 1, 2, 3) translational motion and V (V = 0, 1, 2, 3) rota−U )F tional motion can be represented by the M((33−V mechanism. The RRR and RRC serial limbs which are used to construct M31CF )C −U )F and M30CF mechanisms can be derived from the L12CF limbs and L02CF limbs respectively. The wrench systems of these M((33−V )C

436

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

Table 7 General single-loop mechanisms and the equivalent SOCs.

PF where the symbol “+” denotes that the mechanism is constructed by two different LPF QC limbs and signifies the incorporation of two different LQC limbs in 

PF LPF QC + LQ  C -SOC.

mechanisms are obtained from the constraint combination of the LPF QC limbs. By the equivalent condition of the wrench sys-

−U )F )F tem, when synthesizing GPMs, the sub-parallel M((33−V mechanisms can be treated as the equivalent serial L((33−U limbs. )C −V )C Taking some general single-loop mechanisms [42] which are usually used as sub-parallel mechanisms in the GPMs as the example, their equivalent serial limbs, together with the types of mechanism and equivalent SOCs are listed in the following Table 7.

5. Construction of single-loop mechanisms and two-loop mechanisms With the above various kinematic limbs, a general procedure for synthesizing two-loop mechanisms is urgently desired. After applying the various kinematic limbs into the following general procedure, the systematic approach for the structural synthesis of two-loop mechanisms can be derived. In this section, the results derived from the preceding sections will be applied for the construction of single-loop mechanisms and 2-DOF two-loop mechanisms. When synthesizing two-loop mechanisms, the single-loop mechanisms are broken up at the fixed base link and viewed as the incorporation of two LPF QC 

limbs, i.e., LPF + LPQ FC -SOC. QC 5.1. The procedure for synthesizing two-loop mechanisms According to the above analysis, the procedure for synthesizing two-loop mechanisms is divided into a few steps as follows. Meanwhile, the FMP and SMP respectively denote the IMP and SIMP in the overall two-loop mechanisms. 1) Construct the first single-loop mechanism. The motion pattern of the FMP, which is also the IMP of the two-loop mechanism, should be analyzed first. Since the motion pattern of the EMP is generated by the linear combination of two connected single-loop mechanisms, the selection of the FMP is subject to the motion pattern of the EMP. 2) According to the motion pattern of the EMP and the corresponding coupling sub-chains, the other ternary basic link SMP in the first single-loop mechanism is determined. Thus, an initial coupling sub-chain (ICS-C) can be obtained. 3) While fixing the FMP, the mobility of the SMP and the relative motion of the second single-loop mechanism can be derived. Since the existence of mutual constraint between two loops, the SMP might not fully perform the motion as its kinematic chain shows. In other words, the relative motion between the FMP and the SMP of the first single-loop mechanism may be constrained by the closed loop. If that happens, the ICS-C should be equivalent to the corresponding coupling sub-chain (CCS-C) from the point of the relative motion between the FMP and the SMP. 4) Construct the second single-loop mechanism that includes the CCS-C between FMP and SMP. It should be noted that the passive joints may be included in the CCS-C when synthesizing the second single-loop mechanism. When the FMP of the second single-loop mechanism is broken up, the corresponding SOC can be derived. 5) Assemble the corresponding SOC on the first single-loop mechanism and verify the relative motion between the IMP and the SIMP in the overall two-loop mechanism. When the IMP is fixed, the motion pattern of the SIMP should

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

437

Fig. 5. The procedure to synthesize two-loop mechanisms.

be analyzed, and verify whether it is the same as the motion pattern of the CCS-C. Otherwise, the ICS-C should be equivalent to the final coupling sub-chain (FCS-C) according to the relative motion between the IMP and the SIMP. Alternatively, return to step (4) and synthesize a new second single-loop mechanism. 6) Identify the actuation scheme. With actuated joints locked, the combination of actuation wrenches of actuated joints and basis wrenches of the wrench system should constitute the basis wrenches of the 6-system. Consequently, the derived two-loop mechanism is rational. Otherwise, return to step (4) (or step (2)), and synthesize a new second (or first) single-loop mechanism, as shown in Fig. 5. During the process of synthesizing two-loop mechanisms, a special case should be taken into consideration. When all the kinematic joints in the ICS-C of the first single-loop mechanism act as passive joints in the second single-loop mechanism, the two-connected single-loop mechanisms are independent. In this case, there is no mutual constraint between two single-loop mechanisms. The motion pattern of the EMP is the union operation of the two single-loop mechanisms. When

438

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

Table 8 1-DOF mechanism SOCs with 2-DOF output motion patterns. DOF

Motion type

Mechanisms type

Seven-link chains

L23CF + LD 1-DOF Single Loop Mechanisms (7 ≥ n2 ≥ 4)

1T

M32CF

L22CF + L01CF L21CF L20CF L13CF L12CF L23CF

1R

M23CF

+ L02CF + L03CF + L10CF + L11CF + LD

Six-link chains L23CF L23CF L22CF L22CF L21CF L21CF L20CF L13CF L12CF

+ L10CF + L01CF + L11CF + L02CF + L12CF + L03CF + L13CF + L11CF + L12CF

Five-link chains

Four-link chains

L23CF + L20CF

L23CF + L21CF

L23CF + L11CF

L23CF + L12CF

+ L02CF + L21CF + L12CF + L21CF + L12CF

L23CF + L03CF

L23CF L13CF L13CF L22CF L22CF L22CF

L13CF + L13CF L22CF + L22CF L22CF + L13CF L23CF + L21CF

+ 3C

L32CF + L10CF

L32CF + L20CF

L32CF + LD

L32CF + L01CF

L32CF + L11CF

L32CF + L30CF

L31CF + L01CF

L31CF + L11CF

L32CF + L02CF

L32CF + L21CF

L30CF + L02CF

L31CF + L02CF

L31CF + L21CF

L32CF + L12CF

L22CF + L10CF

L30CF + L12CF

L31CF + L12CF

L31CF + L31CF

L21CF + L11CF

L22CF + L20CF

L30CF + L22CF

L31CF + L22CF

+ L21CF + L12CF

L22CF + L22CF

L20CF

+ L12CF

L22CF L21CF L21CF

+ L11CF + L21CF + L12CF

L22CF L22CF

where n2 is the number of binary links.

synthesizing two-loop mechanisms, for the sake of simplification, the passive joints in the ICS-C can be separately calculated in the second single-loop mechanism. 5.2. 2-DOF two-loop mechanisms: 2T, 1T1R and 2R In a single-loop mechanism with basic links, the number of basic links is equal to the number of kinematic joints. By reference [14,32], the number of basic links or kinematic joints in the 2-DOF single-loop parallel mechanisms is no more than eight. Based on the constraint synthesis and the above analysis in Section 4, all the possible SOCs, which are the combination of various limbs, can be found to construct two-loop mechanisms, as shown in Table 8. Besides, since different constraint forces of different limbs are simultaneously provided to the end-effector, new constraint couples can be generated in the process of assembling different limbs. Therefore, the following geometrical requirements need to be met. One constraint force and one constraint couple, whose axis is perpendicular to the constraint force, can be generated by two parallel constraint forces of two L10CF limbs; One constraint force and two constraint couples, whose axes are perpendicular to the constraint force, can be derived by three spatially parallel constraint forces of three L10CF limbs. Two coplanar constraint forces and one constraint couple (perpendicular to one of the constraint forces) applied to the endeffector can be obtained from three constraint forces of three L10CF limbs. Therefore, when the overconstraints are contributed by the constraint forces, they should be coincident with each other. For instance, in the M32CF mechanism constructed by L23CF limb and L10CF limb, the constraint force in the L10CF limb should be perpendicular to the prismatic joint in the L23CF limb. As a result, some L10CF limbs cannot fit this condition well. For example, the axes of R joints in different groups of the RRRR’R’ limb are respectively intersecting at different common points, and then this limb is excluded. First, 2-DOF two-loop mechanisms with the end-effector producing two translations (2T) are considered. The 2T twoloop mechanisms can be obtained from two independent single-loop mechanisms with one pure translational motion, represented by 1T^1T. Similarly, 2T and 1T single-loop mechanisms can be used to construct a 2T two-loop mechanism, denoted as 2T♦1T or 1T♦2T. Note that symbol “^” and symbol “♦” indicate that the output motions of the two connected single-loop mechanisms are independent and dependent, respectively. For the sake of simplification, the GPMs possessing two or more mutual constraints, such as 2T♦2T, will be neglected in this paper. The L22CF + L01CF -SOC is used to synthesize the first single-loop mechanism with one pure translation. After connecting the L22CF + L01CF -SOC end to end, the M32CF mechanism can be derived, as shown in Fig. 6(a). In the M32CF mechanism, the first serial kinematic chain (FSKC) acted as one limb is a L22CF limb, and involves one virtual prismatic pair. The other limb serves as a L01CF limb. This limb is composed of the second serial kinematic chain (SSKC) and ICS-C and involves three independent virtual prismatic pairs. As shown in Fig. 6(b), to analyze the relative motion between the FMP and the SMP, the FMP in the first single-loop mechanism is fixed. Then a new parallel mechanism is obtained. After analyzing the motion pattern of the SMP in the new parallel mechanism, the CCS-C can be derived, as drawn in Fig. 6(c). As described before, the motion pattern of the first single-loop mechanism and CCS-C are determined. The 2T second single-loop mechanism constructed by the third serial kinematic chain (TSKC), the fourth serial kinematic chain (TFSKC) and ICS-C need to be synthesized. As shown in Fig. 6(c) and (d), the CCS-C is included in one of the kinematic limbs

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

439

Fig. 6. The 2T two-loop mechanism constructed by 1T♦2T: L22CF + L01CF -SOC and L12CF + L02CF -SOC.

in the second single-loop mechanism. When attaching the fixed base of the second single-loop mechanism at the FMP, the structure constructed by the L12CF limb and the L02CF limb is derived. After breaking up the fixed base of the structure, the L12CF + L02CF -SOC can be obtained. At last, the redundant common joints in the TFSKC and CCS-C are removed, and the L12CF + L02CF -SOC is connected to the first single-loop mechanism. Then a 2T two-loop mechanism is obtained, as shown in Fig. 6(e) and Fig. 6(f). In this mechanism, the EMP of the overall two-loop mechanism can output two translational DOFs through mobility analysis. The second single-loop mechanism is constructed by the ICS-C and a L12CF + L02CF -SOC. Similarly, more synthesized results can be obtained, as shown in Tables 8 and 9. Note that some two-loop mechanisms can also be constructed by a rigid structure and a non-rigid loop mechanism, as shown in Fig. 6(g). Because of the mutual constraint, the system of twist equations [4] may have zero solution. In other words, the second single-loop mechanism is a rigid substructure, i.e., the top substructure is a rigid structure. Besides, there exists initial DOF (2T) in the first single-loop mechanism. With consideration of the mutual constraints between loops, the loops composed of TSKC, TFSKC and ICS-C are constrained to rigid loops. Therefore, the initial DOF of the first single-loop mechanism degenerates into a M32CF mechanism with 1T output motion. This situation will not be discussed in this paper. Also, the two-loop mechanisms which can be obtained by combining a rigid structure and a non-rigid loop mechanism are viewed as unavailable.

440

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

Table 9 2-DOF mechanism SOCs with 2-DOF output motion patterns. DOF 2-DOF Single Loop Mechanisms (8 ≥ n2 ≥ 5)

Motion type

Mechanisms type

2T

M31CF

Eight-link chains

L13CF + LD L12CF + L01CF L11CF L10CF 1T1R

+ L02CF + L03CF

M22CF L22CF + LD L21CF + L01CF L20CF + L02CF L12CF + L10CF L11CF + L11CF

2R

M13CF

L31CF + LD L30CF + L01CF L20CF + L11CF L21CF

+ L10CF

Seven-link chains L13CF L13CF L12CF L11CF L11CF L22CF L22CF L21CF L21CF L12CF L12CF L31CF L31CF L30CF L21CF L21CF

+ L10CF + L01CF + L02CF + L12CF + L03CF + L10CF + L01CF + L11CF + L02CF + L20CF + L11CF + L10CF + L01CF + L11CF + L20CF + L11CF

Six-link chains

Five-link chains

L13CF + L11CF

L13CF + L12CF

L13CF + L02CF

L13CF + L03CF

L12CF L13CF L22CF L22CF L22CF L21CF L21CF L12CF L31CF L31CF L21CF

+ L12CF + L11CF + L20CF + L11CF + L02CF + L21CF + L12CF + L12CF + L20CF + L11CF + L21CF

L22CF + L21CF L22CF + L12CF

L31CF + L30CF L31CF + L21CF

Fig. 7. The 1T1R two-loop mechanism constructed by 1T♦1T1R: L22CF + L01CF -SOC and L22CF + L12CF -SOC.

The 1T1R two-loop mechanisms can be obtained from the following structures: 1T^1R, 1T1R♦1R and 1T1R♦1T. The 1T♦1T1R two-loop mechanisms are constructed by a 1T mechanism and a 1T1R mechanism. As shown in Fig. 7(a), the first single-loop mechanism is synthesized by the L22CF + L01CF -SOC. The binary link with one translational motion is chosen as the moving platform in the first single-loop mechanism, as shown in Fig. 7(b). The relative motion between two intermediate moving platforms is a 2-DOF translational motion. Besides, the CCS-C must be included in the second single-loop mechanism, which is constructed by the ICS-C and a L22CF + L12CF -SOC, as shown in Fig. 7(c) and (d). Then the 1T1R two-loop mechanisms shown in Fig. 7(e) and Fig. 7(f) are obtained. The 2R two-loop mechanisms can be obtained from the structures of 1R^1R and 1R♦2R. The 1R♦2R two-loop mechanisms are constructed by the single-loop mechanism with two rotations and the single-loop mechanism with one rotation. As shown in Fig. 8(a), the first single-loop mechanism is synthesized by the L31CF + L30CF -SOC. The binary link which is next to the

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

441

Fig. 8. 2R two-loop mechanism constructed by 1R♦2R: L31CF + L30CF -SOC and L32CF + L30CF -SOC.

fixed base is viewed as the moving platform in the first single-loop mechanism, as shown in Fig. 8(b). The relative motion between two intermediate moving platforms is two rotations. Besides, the CCS-C must be included in the second single-loop mechanism, which is constructed by the ICS-C and a L32CF + L30CF -SOC. Then the 2R two-loop mechanism shown in Fig. 8(e) is obtained. 5.3. Actuation scheme and potential advantages In the GPMs, all the actuators are located on the fixed base and distributed on different limbs to reduce the inertia and avoid floating weights. When arranging the actuated joints, the conditions of rational actuated joints need to be satisfied [19]. In the 2T two-loop mechanism synthesized in Fig. 6(e), the actuators are mounted in the kinematic joints which are connected to the fixed base. After locking the actuated joints, the L22CF + L01CF -SOC degenerates into L32CF + L01CF -SOC. Then the first loop mechanism degenerates into a rigid loop, and the CCS-C is a rigid structure. The actuation wrenches of the actuated joints together with the wrench system of the two-loop mechanism constitute the basis wrenches. As a result, the input scheme for the mechanism is feasible. Similarly, the proposed 1T1R mechanism in Fig. 7(e) and 2R mechanism in Fig. 8(e) are rational. As it has been known, the loops in the mechanisms constructed by 1T^1T, 1T^1R and 1R^1R are independent. When the actuated joints are locked, the first single-loop mechanism may degenerate into a rigid loop. However, the second single-loop mechanisms are still movable, as shown in Fig. 8(f). Therefore, only some of the mechanisms constructed by 1T♦2T, 1T♦1T1R, 1R♦1T1R and 1R♦2R are rational 2-DOF two-loop mechanisms. Due to the special topological structures, there are some advantages and potential applications in this kind of mechanisms. With the existence of the mutual constraints, the number of overconstraint is increased and then the accuracy of the mechanisms is significantly improved. Besides, since the two single-loop mechanisms are dependent, the singularity of each single-loop mechanism may be avoided. Moreover, two-loop generalized parallel mechanisms are capable of fulfilling some specific tasks [43,44]. It also casts light in the direction of designing reconfigurable mechanisms by assembling kinematotropic linkages as coupling sub-chains. Associated with distinct phases of the coupling sub-chain, the two-loop generalized parallel mechanism can perform different output motions. Furthermore, according to the scheme proposed in this paper, generalized parallel mechanisms with more than two independent loops can be derived. On the contrary, those mechanisms can also be simplified into basic blocks, i.e., two-loop generalized parallel mechanisms. 6. Conclusion This paper presented a systematic approach for the structural synthesis of two-loop mechanisms, which are the basic building blocks for GPMs. The 2-DOF two-loop mechanisms are obtained by adding a SOC on the top of the existing

442

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

closed-loop mechanism. Based on screw theory, the mutual constraints generated by the coupling sub-chain between two connected loops have been studied. Then, taking advantage of the virtual work of kinematic joints, the screw theory based constraint synthesis method for kinematic limbs and single-loop mechanisms has been developed. Moreover, the procedure for synthesizing two-loop mechanisms with various kinematic legs has been analyzed. Furthermore, the feasible incorporation of different limbs and single-loop mechanisms which are used to construct two-loop mechanisms has been presented. At last, several typical 2-DOF two-loop mechanisms have been constructed to demonstrate the feasibility of the proposed approach. The approach presented in this paper is also available for synthesizing other GPMs. Acknowledgement This work was supported by the National Natural Science Foundation of China [grant number 51675037]; the NSF grant to Stony Brook University [grant number CMMI-1563413]; and the China Scholarship Council [grant number 201607090032]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]

J.-P. Merlet, Parallel Robots, Springer Science & Business Media, 2012. L.-W. Tsai, Robot analysis: the Mechanics of Serial and Parallel Manipulators, John Wiley & Sons, 1999. Z. Huang, Q. Li, H. Ding, Theory of Parallel Mechanisms, Springer Science & Business Media, 2012. H. Ding, W.-a. Cao, Z. Chen, A. Kecskeméthy, Structural synthesis of two-layer and two-loop spatial mechanisms with coupling chains, Mechan. Mach. Theor. 92 (2015) 289–313. X. Ding, Y. Yang, J.S. Dai, Design and kinematic analysis of a novel prism deployable mechanism, Mechan. Mach. Theor. 63 (2013) 35–49. Q. Zeng, Y. Fang, Structural synthesis and analysis of serial–parallel hybrid mechanisms with spatial multi-loop kinematic chains, Mechan. Mach. Theor. 49 (2012) 198–215. B.-J. Yi, H.Y. Na, J.H. Lee, Y.-S. Hong, S.-R. Oh, I.H. Suh, W.K. Kim, Design of a parallel-type gripper mechanism, Intern. J. Robot. Res. 21 (2002) 661–676. Q. Zeng, K.F. Ehmann, Design of parallel hybrid-loop manipulators with kinematotropic property and deployability, Mechan. Mach. Theor. 71 (2014) 1–26. C. Tian, Y. Fang, Q.J. Ge, Structural synthesis of parallel manipulators with coupling sub-chains, Mechan. Mach. Theor. 118 (2017) 84–99. M.G. Mohamed, C.M. Gosselin, Design and analysis of kinematically redundant parallel manipulators with configurable platforms, IEEE Trans. Robot. 21 (2005) 277–287. W.-a. Cao, H. Ding, Z. Chen, S. Zhao, Mobility analysis and structural synthesis of a class of spatial mechanisms with coupling chains, Robotica (2016) 1–19. G. Gogu, Structural Synthesis of Parallel Robots, Springer, 2008. C.M. Gosselin, Type synthesis of 3-DOF translational parallel manipulators based on screw theory, (2004). K.H. Hunt, Kinematic Geometry of Mechanisms, Press Oxford, Clarendon, 1978. Q. Li, L. Xu, Q. Chen, W. Ye, New family of RPR-equivalent parallel mechanisms: design and application, Chinese J. Mechan. Eng. 30 (2017) 217–221. G. Gogu, Structural synthesis of fully-isotropic parallel robots with Schönflies motions via theory of linear transformations and evolutionary morphology, Eur. J. Mechan. A/Solids 26 (2007) 242–269. Z. Huang, Q. Li, Type synthesis of symmetrical lower-mobility parallel mechanisms using the constraint-synthesis method, Intern. J. Robot. Res. 22 (2003) 59–79. Y. Fang, L.-W. Tsai, Structure synthesis of a class of 4-DoF and 5-DoF parallel manipulators with identical limb structures, Intern. J. Robot. Res. 21 (2002) 799–810. X. Kong, C.M. Gosselin, Type synthesis of 3T1R 4-DOF parallel manipulators based on screw theory, IEEE Trans. Robot. Automat. 20 (2004) 181–190. S. Refaat, J.M. Hervé, S. Nahavandi, H. Trinh, Asymmetrical three-DOFs rotational-translational parallel-kinematics mechanisms based on Lie group theory, Eur. J. Mechan.A/Solids 25 (2006) 550–558. Q. Li, Z. Huang, J.M. Hervé, Type synthesis of 3R2T 5-DOF parallel mechanisms using the Lie group of displacements, IEEE Trans. Robot. Automat. 20 (2004) 173–180. T.-L. Yang, A.-X. Liu, Q. Jin, Y.-F. Luo, H.-P. Shen, L.-B. Hang, Position and orientation characteristic equation for topological design of robot mechanisms, J. Mechan. Design 131 (2009) 021001. F. Gao, J. Yang, Q.J. Ge, Type synthesis of parallel mechanisms having the second class GF sets and two dimensional rotations, J. Mechan. Robot. 3 (2011) 011003. P. Zhao, X. Li, L. Zhu, B. Zi, Q.J. Ge, A novel motion synthesis approach with expandable solution space for planar linkages based on kinematic-mapping, Mechan. Mach. Theor. 105 (2016) 164–175. M. Zoppi, R. Molfino, ArmillEye: flexible platform for underwater stereo vision, J. Mechan. Design 129 (2007) 808–815. A. Campos, C. Budde, J. Hesselbach, A type synthesis method for hybrid robot structures, Mechan. Mach. Theor. 43 (2008) 984–995. H. Shen, T. Yang, L.-z. Ma, Synthesis and structure analysis of kinematic structures of 6-dof parallel robotic mechanisms, Mechan. Mach. Theor. 40 (2005) 1164–1180. M. Zoppi, R.M. Molfino, Forward kinematics equations of a 3-dof hybrid PM for underwater camera active support, robot motion and control, 2005. RoMoCo’05, in: Proceedings of the Fifth International Workshop on, IEEE, 2005, pp. 231–236. J. Bałchanowski, General method of structural synthesis of parallel mechanisms, Archiv. Civil Mechan. Eng. 16 (2016) 256–268. T. Yang, A. Liu, H. Shen, Y. Luo, L. Hang, Q. Jin, Composition principle based on SOC unit and coupling degree of BKC for general spatial mechanisms, in: Proc. 2015 IFToMM World Congress, 2015, pp. 25–30. C. Tian, Y. Fang, S. Guo, Structural synthesis of a class of 2R2T hybrid mechanisms, Chinese J. Mechan. Eng. 29 (2016) 703–709. Y. Fang, L.-W. Tsai, Enumeration of a class of overconstrained mechanisms using the theory of reciprocal screws, Mechan. Mach. Theor. 39 (2004) 1175–1187. R. Johnson, Mechanical Design Synthesis-Creative Design and Optimize, Huntington, New York, 1987. H.-J. Lee, Y.-S. Yoon, Automatic method for enumeration of complete set of kinematic chains, JSME international journal. Ser. C, Dynamics, control, robotics, design and manufacturing 37 (1994) 812–818. M.A. Pucheta, N.E. Ulrich, A. Cardona, Automated sketching of non-fractionated kinematic chains, Mechan. Mach. Theor. 68 (2013) 67–82. J. Titus, Further applications of group theory to the enumeration and structural analysis of basic kinematic chains, (1989). W. Ye, Y. Fang, K. Zhang, S. Guo, A new family of reconfigurable parallel mechanisms with diamond kinematotropic chain, Mechan. Mach. Theor. 74 (2014) 1–9. C. Tian, Y. Fang, S. Guo, H. Qu, A class of reconfigurable parallel mechanisms with five-bar metamorphic linkage, in: Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 231, 2017, pp. 2089–2099. C. Tian, Y. Fang, S. Guo, H. Qu, Structure synthesis of reconfigurable parallel mechanisms with closed-loop metamorphic linkages, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2017.

C. Tian et al. / Mechanism and Machine Theory 128 (2018) 429–443

443

[40] Z. Huang, P. Xia, The mobility analyses of some classical mechanism and recent parallel robots, in: ASME 2006 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers, 2006, pp. 977–983. [41] X. Kong, C. Huang, Type synthesis of single-DOF single-loop mechanisms with two operation modes, reconfigurable mechanisms and robots, in: 2009. ReMAR 2009. ASME/IFToMM International Conference on, IEEE, 2009, pp. 136–141. [42] Q. Jin, T.-L. Yang, Theory for topology synthesis of parallel manipulators and its application to three-dimension-translation parallel manipulators, J. Mechan. Design 126 (2004) 625. [43] G.S. Soh, J.M. McCarthy, The synthesis of six-bar linkages as constrained planar 3R chains, Mechan. Mach. Theor. 43 (2008) 160–170. [44] B. Parrish, J. McCarthy, in: Identification of a Usable Six-Bar Linkage For Dimensional synthesis, New Trends in Mechanism and Machine Science, Springer, 2013, pp. 255–262.