Type synthesis of parallel mechanisms by utilizing sub-mechanisms and digital topological graphs

Mechanism and Machine Theory 109 (2017) 39–50

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

Type synthesis of parallel mechanisms by utilizing submechanisms and digital topological graphs

MARK

⁎

Yi Lua,b, , Nijia Yea,b a b

College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, PR China Advanced Metal Forming Key Laboratory of Education Ministry & Parallel Robot Key Laboratory of Hebei, PR China

A R T I C L E I N F O

ABSTRACT

Keywords: Type synthesis Digital topology graph Serial limb Closed planar mechanism Parallel mechanism

A type synthesis of various parallel mechanisms by utilizing sub-mechanisms and digital topology graphs is studied. The conditions for type synthesis of the parallel mechanisms with the sub-mechanisms are determined. Based on the determined conditions, the digital topology graphs are derived from different associated linkages, and the digital topology graphs are transformed into revised digital topology graphs for type synthesis of the parallel mechanisms with redundant constraints. The sub-mechanisms are transformed into simple equivalent limbs and their equivalent relations are analyzed. Seventeen novel parallel mechanisms are synthesized by utilizing the derived digital topology graphs, the revised digital topology graphs and the sub-mechanisms or the combinations of sub-mechanisms. The synthesized parallel mechanisms are simplified by replacing the complex sub-mechanisms with their simple equivalent limbs. Finally, the degrees of freedoms of all the synthesized parallel mechanisms are calculated to verify the correctness and effectiveness of the proposed approach.

1. Introduction Type synthesis of mechanisms is a well known method for creating novel mechanisms [1–6]. In this aspect, Gogu 2008 studied the type synthesis of parallel mechanisms (PMs) and presented morphological and evolutionary approaches [1], he 2009 studied the translational topologies of closed mechanisms with two and three degrees of freedom (DoFs) [2], and he 2010 studied the topology synthesis of closed mechanisms with the planar motion of moving platform [3]. Johnson derived some associated linkages for the planar mechanisms using a determining tree and synthesized many planar mechanisms by utilizing the associated linkage [4]. Huang et al. conducted the type synthesis of PMs by utilizing Lie group method and screw-theory [5]. Yang et al. studied the topology structure design of mechanisms [6]. The approach of topology graph has been widely applied to the type synthesis of closed mechanisms [2,3,6,7]. A contracted graph without any binary links is applied to derive the topology graph. In this aspect, Sohn, Vucina and Freudenstein [7,8] and Tsai et al. [9] proposed an approach of topology graph and a contracted graph and applied them to the type synthesis of the closed mechanisms. Yan and Kang [10] studied the configuration synthesis of mechanisms by changing types and/or motion orientations of some joints. Caro et al. [11] presented some synthesis rules to obtain a complete minimum set of serial topologies capable of producing Schönflies-motion motions. Pucheta et al. [12,13] synthesized the planar linkages based on the constrained sub-graph isomorphism detection and the existing mechanisms. Saxena and Ananthasuresh [14] selected the best configuration of some mechanisms based on kinetostatic design specifications. Tarcisio et al. [15] studied the topological synthesis of a PM based on the wrist design requirements. Some (i=3, 4, 5)-DoF PMs were synthesized by Kong and Gosselin by utilizing the

⁎

Corresponding author at: College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, PR China. E-mail address: [email protected] (Y. Lu).

http://dx.doi.org/10.1016/j.mechmachtheory.2016.11.008 Received 10 January 2016; Received in revised form 5 November 2016; Accepted 14 November 2016 Available online 26 November 2016 0094-114X/ © 2016 Elsevier Ltd. All rights reserved.

Mechanism and Machine Theory 109 (2017) 39–50

Y. Lu, N. Ye

n the numbers of joints ns, nu, nc, np, nr the numbers of S, U, C, P, R b, t, q binary, ternary, quaternary links p, h pentagonal, hexagonal links ei the ith edge ni (i=2, …, 6) the numbers of b, t, q, p, h s Lj the sub-serial mechanism with j actuators k Lj k-DoF planar closed mechanism with j actuator, k=0, 1, 2, 3 Lk k-DoF PM with k actuator, k=3, 4, 5 ν the numbers of redundant constraints ζ passive DoF

Nomenclature DoF PM m B AL TG DTG P, R U, C, S J N

degree of freedom parallel mechanism moving platform fixed base associated linkage topology graph digital topological graph prismatic and revolute pairs universal, cylinder, spherical pairs connection joint with one-DoF the numbers of links

screw theory and virtual chain approach [16–18]. Lu et al. derived some unified planar/spatial associated linkages, and several contracted graphs from associated linkage by utilizing topology matrices, they also derived topology graphs from the contracted graph by visual inspection [19]. In addition, Lu et al. discovered the relations between associated linkage, redundant constraint, and DoF of the PMs [20], they also derived valid kinematic limbs of 3-DoF PMs without redundant constraint [21] and derived complicated topology graph using array for the type synthesis of 3-, 4-DoF PMs [22]. The above mentioned studies have their merits and different focuses. Currently, the associated linkage, the topology graph, the contracted graph, arrays and screw theory are mainly used for the type synthesis of the kinematic chains and their structures [4,5,11,14,16–18], the closed planar mechanisms [4,9,12,13] and the PMs without redundant constraint [16–19,21]. When the serial mechanism, the closed planar mechanism and the k-DoF PM as k < i are taken as the sub-mechanisms of the iDoF PMs, the i-DoF PMs may possess the redundant constraints and have some merits, such as enlarged position and orientation workspace, increased the capability of load bearing and the high stiffness, simplified structure and control [13,23]. These merits may enable the PMs to satisfy more requirements in robots, machine tools, automobiles, airplanes, micro mechtronic manipulators, and to standardize the manufacturing and assembly of the sub-mechanisms. However, so far, the relative methods and rules have not been systematically studied for the type synthesis of the i-DoF PMs with the redundant constraints constructed by k-DoF sub-PMs, where k < i, by utilizing the digital topological graph (DTG) method. For this reason, this paper focuses on the type synthesis of the PMs by utilizing the sub-mechanisms and the DTGs. Following problems are solved: (1) construct various DTGs of the PMs from the associated linkages; (2) determine the relationships between the DTGs and the equivalent limbs of the sub-mechanisms; (3) synthesize the i-DoF PMs by utilizing the sub-mechanisms, the DTGs and the revised DTGs. 2. Sub-mechanisms and their merits A sub-mechanism may be a sub-serial limb, or a sub-planar closed mechanism, or a k-DoF PM as k < i or their combination. Let (R, P, U, S) be the (revolute, prismatic, universal, spherical) kinematic pairs, respectively. Let sLj (j=0, 1, 2, 3) be the sub-serial limb with j actuators connected in series. Let kLj (j=0, 1, 2; k=1, 2, 3) be the sub-planar closed mechanism with j actuators and k DoFs. Let Lk (k=3, …, i-1) be a k-DoF sub-PM with k actuators as k < i. Each of them has several merits as follows. When a SP-type or a UP-type sub-serial limb sL0 is taken as a passive constrained limb for connecting the moving platform with the base in the i-DoF PMs, a tiny self motion of the PMs can be removed effectively, and the rotational stiffness of the PMs can be increased [21,24]. When a SPS-type sub-serial limb sL0 is taken as an elastic auxiliary limb for connecting the moving platform with the base in the i-DoF PMs, the workload applied onto the moving platform of the PMs can be born by several SPS-type elastic auxiliary limbs sL0 and the required active force of the active limbs can be reduced largely. When each of the sub-serial limbs sLj (j=2, 3) is taken as a limb for connecting the moving platform with the base in the i-DoF PMs, the total number of the required limbs must be reduced. Thus, the interference between the limbs and the moving platform may be avoided easily, and the workspace of the i-DoF PMs can be enlarged. In addition, since other active limbs of the i-DoF PMs can be transformed into the SPU-type linear active limbs which are not sensitive to the manufacturing errors, not only the tiny self motion of the PMs can be removed effectively, but also their capability of load bearing can be increased. When one of the planar closed mechanisms kLj (j=0, 1, 2; k=1, 2, 3) is taken as a sub-planar limb for connecting the moving platform with the base in the i-DoF PMs, the sub-planar limb may have the following merits [20]: 1. The sub-planar closed mechanism only includes the revolute pair and prismatic pair. The revolute pair has a larger capability of pulling force bearing than that of the spherical pair. 2. The precision of the revolute pair is higher than that of the spherical pair under a large cyclic loading because the backlash of the revolute pair can be eliminated more easily than that of the spherical pair by adding a preload. 3. The workspace of the i-DoF PM can be increased because the rotation angle of the revolute pair is larger than the rotation cone angle of the spherical pair before interference. 4. The whole mechanism can be simplified because the number of the binary links is reduced. 5. It is easy to increase the mechanical advantage, such as to increase the displacement or the velocity of the moving platform, and to 40

Mechanism and Machine Theory 109 (2017) 39–50

Y. Lu, N. Ye Table 1 The acceptable combinations of the sub-serial limbs and the sub-closed planar limbs in i-DoF PMs. PMs

Acceptable combinations of sLj (j=1, 2, 3) and kLj (j=1, 2; k=1, 2, 3) in i-DoF PMs and their numbers

3 DoF

3sL1, 3 1L1, sL1+2 1L1, 2sL1+1L1, sL1+2L2, 1L1+2L2, sL1+sL2, 1L1+sL2, L1+3L2, 1L1+3L2 4sL1, 4 1L1, sL1+3 1L1, 3sL1+1L1, 2sL1+2 1L1, 2sL1+2L2, 2 1L1+2L2, 2 2L2, 2sL1+ s L2, 2sL1+sL2, 2 1L1+sL2, s L1+1L1+sL2, sL1+sL3, 1L1+sL3, 2sL1+3L2, 2 1L1+3L2, 2 3L2, 3L2+2L2 5sL1, 5 1L1, sL1+4 1L1, 4sL1+1L1, 2sL1+3 1L1, 3sL1+2 1L1, 3sL1+2L2, 3 1L1+2L2, s L1+2 2L2, 1L1+2 2L2, sL1+2sL2, 1L1+2sL2, 2sL1+sL3, 2 1L1+sL3, 2L2+sL3, sL2+sL3, sL1+2 3L2, 1L1+2 3L2, 3sL1+3L2, 3 1L1+3L2, 3L2+sL3, sL1+3L2+2L2, 1L1+3L2+2L2 6sL1, 5sL1+1L1, 4sL1+2L2, 4sL1+2 1L1, 3sL1+3 1L1, 3sL1+sL2+1L1, 3sL1+sL3, 2sL1+2sL2, 2sL1+2 2L2, 2sL1+sL2+2L2, 2sL1+sL2+2 1L1, 2sL1+4 1L1, 2sL1+sL3+1L1, sL1+5 1L1, sL1+2L2+sL3, sL1+ s L2+sL3, sL1+sL3+2 1L1, sL1+3 1L1+2L2, s L1+1L1+2 2L2, sL1+2sL2+1L1, 3sL2, 2sL2+2L2, 2sL2+2 1L1, sL3+3 1L1, 2L2+ 4 1L1, 2 2L2+2 1L1, 3 2L2, 6 1L1, 4sL1+3L2, 2sL1+2 3L2, 2sL1+sL2+3L2, sL1+3L2+sL3, sL1+3 1L1+3L2, sL1+1L1+2 3L2, 2sL2+3L2, 3L2+4 1L1, 2 3L2+2 1L1, 3 3L2, s L1+1L1+3L2+2L2, 2sL1+3L2+2L2, 2 1L1+3L2+2L2, 2 3L2+2L2, 2 2L2+3L2,

10

s

4 DoF

5 DoF

6 DoF

18

23

43

increase the stiffness of the active limb. 6. It is easy to construct a compliant mechanism and a hybrid rigid/compliant mechanism (a rigid mechanism with the local compliant limbs) using the planar closed-loop chain. 7. Either the extension or the active force of the active limb may be enlarged. When each of the k-DoF sub-PMs Lk (k=3, …, i-1) is used to construct one limb of the i-DoF PMs, the stiffness of the i-DoF PMs can be increased. Generally, the above mentioned different sub-mechanisms can be built into different standard units with high precision by a special company. Therefore, more novel i-DoF PMs with high precision can be built and assembled easily by utilizing these standard units. The most existing i-DoF PMs are constructed from a moving platform and a base which are connected by i serial limbs with one actuator sL1. When the sub-serial limbs and the sub-planar limbs or their combinations are applied to construct the i-DoF PMs, more novel i-DoF PMs with the above merits can be created. Generally, i is the sum of the actuators in the i-DoF PMs. Thus, the acceptable combinations among the sub-serial limbs sLj (j=1, 2, 3) and the sub-closed planar limbs kLj (j=1, 2; k=1, 2, 3) in the iDoF PMs are obtained, see Table 1. 3. Relation between digital topology graphs (DTGs) and sub-mechanisms 3.1. Associated linkage, topology graph, array, DTGs and their relations The Kutzbach-Grübler formula has been widely used to calculate the DoF of closed mechanisms [4]. By considering the redundant constraints and the passive DoF in the parallel mechanisms (PMs), the Kutzbach-Grübler formula is revised by Huang [5] as below, (1) here, i is the prescribed DoF of the moving platform m (output link); N is the number of the links including the fixed base B; n is the number of kinematic pairs; ν is the number of all the redundant constraints; ζ is the number of passive DoF which does not have Table 2 The number of the basic links nj in 19 i-DoF associated linkages (ALs). AL

n2

n3

n4

n5

n6

AL

n2

n3

n4

n5

n6

AL

n2

n3

n4

n5

n6

AL1 AL2 AL3 AL4 AL5 AL6 AL7

9+i+ζ-ν 14+i+ζ-ν 19+i+ζ-ν 18+i+ζ-ν 12+i+ζ-ν 13+i+ζ-ν 17+i+ζ-ν

2 0 0 2 4 2 2

0 2 0 0 0 1 2

0 0 2 0 0 0 0

0 0 0 1 0 0 0

AL8 AL9 AL10 AL11 AL12 AL13 AL14

16+i+ζ-ν 17+i+ζ-ν 18+i+ζ-ν 15+i+ζ-ν 20+i+ζ-ν 19+i+ζ-ν 20+i+ζ-ν

4 3 1 6 4 6 5

1 0 1 0 2 1 0

0 1 1 0 0 0 1

0 0 0 0 0 0 0

AL15 AL16 AL17 AL18 AL19

18+i+ζ-ν 21+i+ζ-ν 23+i+ζ-ν 18+i+ζ-ν 22+i+ζ-ν

0 4 7 8 2

3 0 0 0 0

0 0 1 0 2

0 1 0 0 0

41

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Y. Lu, N. Ye

b t 4 b e3 e5 b e1 b e2 e b b 4 J b b b b b e6 b t2 b b t3 b t1

t1 e3

t1

t4 1b

4b e2

e1 t2

1b 3b 2b

e6

e4

t4 1

4

4b

4

1 3

e5 t3

t2

2

U, C ↔ S

t3

J

R, P ↔ ↔

J

J J

J

J

Fig. 1. A topology graph with 4t+15b and 21 J (a), a simplified topology graph with 4t+15b and 21 J (b), A DTG with 4t+15b and its array {t1114 t2412 t3234 t4431} (c), Equivalence between kinematic pairs and J (d).

influence on the moving characteristics of m; (ns, nu, nc, np, nr) are the numbers of (S, U, C, P, R) pairs, respectively. When ν=ζ=0, Eq. (1) becomes the Kutzbach-Grübler formula for solving the DoF of the mechanisms without the redundant constraints and the passive DoF. Let (b, t, q, p, h) be the binary link, the ternary link, the quaternary link, the pentagonal link, and the hexagonal link in the topology graph, respectively. Let ni (i=2, …, 6) be the number of (b, t, q, p, h), respectively. Let J be a joint with one-DoF. An associated linkage is an acceptable group of links (b, t, q, p, h, …) with given DoF [5], which is a basic tool for deriving various topology graphs [5,20]. The 19 associated linkages ALj (j=1, …, 19) with i-DoF and different group of (b, t, q, p, h) have been derived [20] and their nj (j=2, …, 6) are solved and listed in Table 2 in order to illustrate the type synthesis of the i-DoF PMs. Generally, it is quite complicated to synthesize the PMs by utilizing the topology graph because the topology graph includes many binary links b, see Fig. 1a. When (t, q, p, h, …) in the topology graph are replaced by the vertices connected by (3, 4, 5, 6, …) edges, respectively, and each of edges in the topology graph is marked by the number of binary links which are connected in series by several one-DoF joints J on this edge, the representation of the topology graph can be simplified, see Fig. 1b. Therefore, a DTG is used to simplify the representation of the topology graph [19], see Fig. 1c. A digital topology graph (DTG) includes several vertices connected by several edges. (t, q, p, h, …) in DTG are represented by vertices which are connected by (3, 4, 5, 6, …) edges, respectively. Each of the edges in DTG is marked by a digit. Each digit represents the number of the binary links b connected in series by several J on this edge. For instance, a topology graph for representing the 3-DoF PMs includes 4t and 15b which are connected by 21 J, see Fig. 1a. The complicated representation of the topology graph can be simplified by DTG, see Fig. 1c. In this DTG, 4 t are represented by 4 vertices which are connected by 6 edges ej (j=1, …., 6) marked by digits (4, 1, 1, 3, 4, 2), respectively. Here, digits (4, 1, 1, 3, 4, 2) represent (4, 1, 1, 3, 4, 2)b connected in series by (5, 2, 2, 4, 6, 3)J in ei (i=1, …., 6), respectively. An array is a basic tool for deriving and representing the DTG [19], and is also used to identify the isometric and invalid DTGs. Thus, the type synthesis of the PMs and their isometric identification can be carried out automatically by utilizing the arrays and the compiled program [22]. The conditions of representing the DTG by utilizing array are explained as follows: 1. Each of the arrays includes several strings, and each string is corresponding to one of (t, q, p, h, …); the number of the digits in the strings is the same as that of the edges of (t, q, p, h, …); the digits of a string are the same as the number of b on the edges of (t, q, p, h, …). For instance, an array {t1114 t2412 t3234 t4431} with 4 strings can be used to represent the DTG in Fig. 1c. In this array, the first string 114 is corresponding to the numbers of b (1, 1, 4) on the 3 edges (e3, e2, e1) of t1, respectively; the second string 412 is corresponding to the numbers of b (4, 1, 2) on the 3 edges (e1, e2, e6) of t2, respectively. Similarly, in the same array, other two strings (234, 431) are corresponding to the number of b on the 3 edges of (t3, t4), respectively. 2. In the same array, the last digit of the jth string is the same as the first digit of the (j+1)th string; the first digit of the left-most string is the same as the last digit of the right-most string. 3. The sum of the digits in every array is 2n2. For instance, the digits sum of {114, 412, 234, 431} is 30 as n2=15. 4. The digits of any 2 edges connected by 2 vertices in a DTG are exchangeable. 3.2. Conditions of using DTGs and sub-mechanisms In order to synthesize more novel i-DoF PMs by utilizing the DTGs and the sub-mechanisms based on the existing design rules in [4,11,19,20], the five conditions for the type synthesis of the i-DoF PMs by utilizing the sub-mechanisms and the DTGs are given as follows: 1. If a (3≤i)-DoF PM has neither redundant constraint nor passive DoF, the number of links in any closed loop chain of the DTG must be 7 or more in order to avoid any local structure. 2. If a (3≤i)-DoF PM has neither redundant constraint nor passive DoF, the number of the binary links in any edge of the DTG should be 6 or less in order to avoid any local structure. 3. If the moving platform is connected with the base by a sub-serial limb, the number of joints J in this sub-serial limb of the DTG must be i or more. 4. If a limb which connects the moving platform with the base includes a k-DoF sub-mechanism as k < i, the number of the joints J in this limb of the DTG must be i-k or more. 5. In order to construct the k-DoF sub-planar closed limb using the DTG, the number of the links in a closed loop chain of the DTG

42

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Y. Lu, N. Ye

must be 6+k in order to avoid local structure. In addition, three existing auxiliary conditions [4,19] (see Fig. 1d) should be considered as follows: 1. It is permissible to replace any one J in the topology graph with revolute pair or prismatic pair in the mechanism. 2. It is permissible to replace any 2J connected in series in the topology graph with universal pair or cylinder pair in the mechanism. 3. It is permissible to replace any 3J connected in series in the topology graph with spherical pair in the mechanism. The theoretical bases of above conditions 1→5 are explained as follows: It is known from [19,20] that the simplest spatial mechanism without both redundant constraint and passive DoF is a one-DoF closed loop mechanism formed by 7 links connected in series by 7J. If the number of links is less than 7, the simplest spatial mechanism becomes a structure. Therefore, the condition 1 must be satisfied. If the number of binary link b on one edge more than 6, DoF of this edge may be redundant, meantime the number of b on other edges of the DTG for the i-DoF PM must be reduced and results in the local structure. For instance, a parallel manipulator may include a limb formed by a moving platform (b) connected with a SPU type kinematic chain. Based on the auxiliary conditions, this limb can be formed from 6 b connected by 7J in series in the DTG. Therefore, the condition 2 must be satisfied. Since J is a joint with one-DoF, the maximum DoF of the moving platform with respect to the base is the number of J in a subserial limb. Hence, when the number of J in this sub-serial limb is less than i, DoF of the moving platform with respect to the base must be less than i DoFs. Therefore, the condition 3 must be satisfied. Since the sub-mechanism has k-DoF as k < i, the maximum DoF of the moving platform with respect to the base is the sum of k and the number of J connected in series in a limb with a sub-mechanism from the moving platform to the base. Hence, when the sum of k and the number of J is less than i, DoF of the moving platform with respect to the base must be less than i DoFs. Hence, the condition 4 must be satisfied. It is known from [19,20] that when a closed loop chain of the DTG is transformed into one of sub-planar limbs kLj (j=0, 1, 2; k=1, 2, 3) with υ=3, the required number of links in the closed loop chain is reduced by υ=3. It is known from [19,20] that if DoF of a spatial closed chain mechanism without both redundant constraints and the passive DoF is (1, 2, 3), the number of the links in this spatial closed chain mechanism must be (7, 8, 9), respectively. It implies that: A closed chain with 7 links in the DTG can be transformed into a 1-DoF planar closed mechanism 1Lj (j=0, 1) with 4 links, υ=3, and 0 actuator or 1 actuator. Therefore, the condition 5 must be satisfied. A closed chain with 8 links in the DTG can be transformed into a 2-DoF planar closed mechanism 2L2 with 5 links, 2 actuators and υ=3. Therefore, the condition 5 must be satisfied. A closed chain with 9 links in the DTG can be transformed into a 3-DoF planar closed mechanism 3L2 with 6 links, 2 actuators and υ=3. Therefore, the condition 5 must be satisfied. 4. Equivalent limbs of sub-mechanisms 4.1. Equivalent limbs of sub-serial limbs and sub-planar limbs Each of the sub-serial limbs sLj (j=0, 1, 2, 3) in the PMs generally includes a lower link ll, an upper link lu and j actuators which are connected in series from the base to the moving platform, see Fig. 2a. The equivalent limb of the sub-serial limbs sLj is represented by a line marked by sLj, see Fig. 2b. sLj in the i-DoF PM can be selected from a edge with i or more b in the DTG for connecting the moving platform with the base. Generally, most existing PMs include the sub-serial limbs sLj (j=0, 1). When different s Lj (j=2, 3) are applied for the type synthesis of the PMs, more novel i-DoF PMs with less limb can be created. Four typical closed planar mechanisms kLj (j=0, 1, 2; k=1, 2, 3) are represented in Fig. 3. In fact, several other different jLk can be constructed and applied for the type synthesis of the PMs. Let ⊥ be the perpendicular constraint. Let (lu, ll, lp, lc) be the upper link, the lower link, the piston link, and the cylinder link, respectively. A one-DoF sub-planar closed mechanism with 0 actuator 1L0 is formed by 4 links (2b, lu, lc) in a planar loop chain connected in series by 4 mutually parallel revolute pairs R, see Fig. 3a1. A one-DoF sub-closed planar mechanism with 1 actuator 1L1 is formed by 4 links (lu, ll, lp, lc) in a planar loop chain which is connected in series by 1 revolute pair R, 1 active prismatic pair P and 2 revolute pairs R. Here 3R pairs are parallel mutually and P⊥R are satisfied, see Fig. 3b1. A 2-DoF sub-planar closed mechanism prismatic pair P, × universal pair U, spherical pair S, cylinder pair C, • revolute pair R, passive P,

weld joint,

translational actuator,

lu

lu

lu

lu

lu

lu

s

s

s

s

s

s

L0 ll

L0 ll

L0 ll

L1 ll

L1 ll

rotational actuator,

lu

L2 ll

s

L2 ll

lu

lu s

L3 ll

lu

s

L3

sub-planar closed mechanism lu s

L3 ll

s

ll

Fig. 2. Ten sub-serial limbssLj (j=0, 1, 2, 3) in i-DoF PM and their equivalent limb.

43

Lj ll

Mechanism and Machine Theory 109 (2017) 39–50

Y. Lu, N. Ye

au

b R ll

R lp

au lu

au

R

R lu

L0

lu

lp

1

b R

P

R

al

al

lc al

al

R

au 1

L1

R lp

P lu

R

au L2

2

P lc R

lp

lc al al

ll

au

au

ll al

al

R lp lc R

au P

lu

R lp

au L2

3

P

al

lc

ll

al R

Fig. 3. Four typical planar closed mechanisms kLj; 1L0 (a1), 1L1 for increasing force and for enlarging displacement (b1), 2L2 (c1), 3L2 (d1) and their simple representations (a2), (b2), (c2), (d2).

with 2 actuators 2L2 is formed by 5 links (lu, ll, 2lp, lc) in a planar loop chain which is connected in series by 1 revolute pair R, 1 active prismatic pair P, 2 revolute pairs R and 1 active prismatic pair P. Here 3R pairs are parallel mutually and P⊥R are satisfied, see Fig. 3c1. A 3-DoF sub-planar closed mechanism with 2 actuators 3L2 is formed from 6 links (lu, ll, 2lp, 2lc) in a planar loop chain which is connected in series by 2 revolute pairs R, 1 active prismatic pair P, 2 revolute pairs R, 1 active prismatic pair P. Here 4R pairs are parallel mutually and P⊥R is satisfied, see Fig. 3d1. Since each of the sub-planar closed mechanisms kLj (j=0, 1, 2; k=1, 2, 3) has no passive DoF, ζ=0 is satisfied. Thus, their redundant constraints υ are solved based on Eq. (1), see Table 3. Since the lower link ll and the upper link lu in the sub-planar closed mechanisms kLj (j=1, 2; k=1, 2, 3) are required to provide three connection points: two for constructing kLj and one for the connection point (al and au) at the two ends of kLj for connecting with the PMs, both ll and lu must be t. In order to simplify the representation of the PMs, the sub-planar closed mechanisms jLk (j=1, 2; k=1, 2, 3) are represented by a line marked by kLj and (al and au) at its two ends, see Fig. 3(a2, b2, c2, d2). 4.2. Equivalent limbs of 3-DoF sub-PM L3 An equivalent limb L3 of the 3-DoF sub-PM can be divided into L3I with 2t, L2II with 2q, L3III with q+2t, and others. The characteristics of (L3I, L3II, L3III) are explained, respectively, as follows: A L3I generally includes 3 active limbs sL1, an upper platform lu formed by t, a lower platform ll formed by t. Since each of ll and lu here is required to provide 4 connection points: 3 for L3I, 1 for connecting with the (3 < i)-DoF PMs, both ll and lu must be q. Therefore, the equivalent limb L3I can be constructed by utilizing a DTG with 2q which are connected by 3 edges in parallel with 12b. For example, the 8 different DTGs for the type synthesis of various 3-DoF PMs can be obtained from [18], see Fig. 4(a1–a8). Their equivalent limb is represented by a line with L3I, see Fig. 4b. If L3I is symmetry in structure, the DTG in Fig. 4a1 can be used to synthesize symmetry L3I. Thus, the 26 different symmetry kinematic chains can be used to construct the 26 different symmetry L3I. When L3I is asymmetry in structure, the DTGs in Fig. 4(a2– a8) can be used to synthesize asymmetry L3I, and more asymmetry kinematic chains can be obtained from [21]. After that, more novel (3 < i)-DoF PMs can be synthesized. A L3II generally includes 3 sL1, 1 sL0, an upper platform lu and a lower platform ll. Since both ll and lu are formed by q and required to provide 5 connection points: 4 for L3II, 1 for connecting with the (3 < i)-DoF PMs, both ll and lu must be p. Therefore, the equivalent limb of L3II can be constructed by utilizing a DTG with 2p which are connected by 4 edges in parallel with 17b. For example, a DTG with 2q+17b can be derived from AL2 in Table 2, see Fig. 4c. After that, three different L3II can be synthesized by utilizing derived DTG with 2q+17b see Fig. 4d1–d3. Their equivalent limb is represented by a line marked by L3II, see Fig. 4e. A L3III generally includes 3sL1, 1sL0, an upper platform lu formed by t, and a lower platform ll formed by q. Since ll and lu are required to provide 4 and 5 connection points, respectively, for L3III and for connecting with the (3 < i)-DoF PMs, ll must be q, and lu must be p. Therefore, the equivalent limb of L3III can be constructed by utilizing a DTG with q and p. For example, a DTG with q+2t +16b can be derived from AL6 in Table 2, see Fig. 4f. Next, two different L3III can be synthesized by utilizing the DTG with q+2t+16b, see Fig. 4(g1, g2). A 3D module of L3III is constructed from Fig. 4g2, see Fig. 4g3. Their equivalent limb is represented by a line marked by L3III, see Fig. 4h. 4.3. Equivalent limb L4 of 4-DoF sub-PM An equivalent limb L4 of the 4-DoF sub-PM can be divided into L4I with 2q, L4II with 2t, L4III with 2p, and others. The characteristics of (L4I, L4II, L4III) are explained, respectively, as follows: A L4I generally includes 4 sL1, an upper platform lu and a lower platform ll. Since both ll and lu are formed by q and required to Table 3 Redundant constraint υ of sub-planar closed mechanisms jLk (j=0, 1, 2 actuators; k=1, 2, 3 DoFs). k

Lj

°L1 1 L1

j 0 1

k 1 1

N 4 4

n 4 4

np 0 1

nr 4 3

ζ 0 0

ν

k

Lj

j

k

N

n

np

nr

ζ

ν

3 3

2

L2 L2

2 2

2 3

5 6

5 6

2 2

3 4

0 0

3 3

3

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Mechanism and Machine Theory 109 (2017) 39–50

Y. Lu, N. Ye

au

au 4 4

4

4 3

5

3 6 3

5 5 2

4 6 2

7

3 2

5 6 1

au L3I

4 1

7

al

al

al au 2

s

L1

s

L1

5 5

5

al DTG with 2q+17b and array {q2555 q5552}

4

L1

L1 s

s

L0

s

L1

s

L1

L1

L0

L1

au

s

s

s

L0

L1

s

L3II al

au

03 4

s

s

t

t

L1 s L0

L1

al DTG with q+2t+16b, array {q4435 t530 t044}

L1

s

s

5

t

t

s

L1

s

L1 s L0

s

L1

R

t R R

s

q

L3III

S

S

S

S

au

P

P

P

q

t

al

q

Fig. 4. The 10 DTGs of L3, 5 different 3-DoF sub-PMs and their equivalent limbs (L3I, L3II, L3III).

provide 5 connection points: 4 for L4I, 1 for connecting with the (4 < i)-DoF PMs, both ll and lu must be p. Therefore, the equivalent limb of L4I can be constructed by utilizing a DTG with 2p which are connected by 4 edges in parallel with 18b. For example, a DTG with 2q+18b with different arrays {4455 5544}, {3555 5553}, {3546 6453} can be derived from AL2 in Table 2, see Fig. 5(a1–a3). After that, the 7 different L4I can be synthesized by utilizing the DTG with 2q+18b, see Fig. 5(b1–b7). Their equivalent limb is represented by a line marked by L4I, see Fig. 5c. A L4II generally includ300. es 2sL1, 1sL2, an upper platform lu and a lower platform ll. Since both ll and lu are formed by t and required to provide 4 connection points: 3 for L4II, 1 for connecting with the (4 < i)-DoF PMs, both ll and lu must be q. Therefore, the equivalent limb of L4II can be constructed by utilizing a DTG with 2q which are connected by 3 edges in parallel with 13b. For example, a DTG with 2t +13b and different arrays {445 544} or {355 553} can be derived from AL1 in Table 2, see Fig. 5(d1, d2). After that, the 3 different L4II can be synthesized by utilizing the DTG with 2t+13b and 2sL1+1sL2, see Fig. 5(e1–e3). Their equivalent limb is represented by a line marked by L4II, see Fig. 5f. A L4III generally includes 4 sL1, 1 sL0, an upper platform lu and a lower platform ll. Since both ll and lu are formed by p and 4 a1

au 4 5

5 al DTG with 2q+18b and array {4455 5544} au 3 5 a2

5

au L4I al c b5 au

au s L1 s 4 d1 L2 5 4 al al DTG with 2t+13b and array {445 544} e1 au

b3

5

al DTG with 2q+18b and array {3555 5553}

5

b2

b1

b4

5

s

L1 s L1

b7

b6

au

s

L2

s

au

au

L1

d2

s

3 5

L2

5

e2 au

au

L1

L1

L4II

al

e3

au

f

au

g 5 3 al DTG with 2p+23b and array {p55355 p55355}

au

s

al

al DTG with 2t+13b and array {355 553}

al

s

L4III

5

h1

al

h2

al

h3

al

al i

Fig. 5. The 5 DTGs of L4, the 13 different 4-DoF sub-PMs and their equivalent limbs (L4I, L4II, L4III).

45

Mechanism and Machine Theory 109 (2017) 39–50

Y. Lu, N. Ye

4

au

5 al DTG with 2p+24b a au 4 4 6 al DTG with 2t+14b d1

L5I al au

b1

au

5

5

b2

au 5

al e1

e2

b4

c au L5II

al e3

al

al e4 au

f au L5III

al DTG with p+q+t+23b g

au

au

au

5

b3 au

5 4

al DTG with 2t+14b d2

al

au

0 4 4

au

5 5 5

al h1

al

h2

al

h3

al

i

Fig. 6. The 4 DTGs of L5, the 5-DoF sub-PM and their equivalent limbs (L5I, L5II, L5III).

required to provide 6 connection points: 5 for L4III, 1 for connecting with the (5≤i)-DoF PMs, both ll and lu must be h. For example, a DTG with 2p+23b with array {55355 55355} can be derived based on Table 2, see Fig. 5g. The 3 different L4III can be synthesized using the DTG with 2p+23b, see Fig. 5(h1–h3). Their equivalent limb is represented by a line with L4II, see Fig. 5i. 4.4. Equivalent limbs L5 of 5-DoF sub-PM An equivalent limb L5 of the 5-DoF sub-PM can be divided into L5I with 2p, L5II with 2t, L5III with t+q+p, and others. The characteristics of (L5I, L5II, L5III) are explained, respectively, as follows: A L5I generally includes 5 sL1, an upper platform lu and a lower platform ll. Since both ll and lu are formed by p and are required to provide 6 connection points: 5 for L5I, 1 for connecting with the (6≤i)-DoF PMs, both ll and lu must be h. For example, a DTG with 2p+24b and its array {p55554 p45555} can be derived based on Table 2, see Fig. 6a. The 4 different L5I can be synthesized using the DTG with 2p+24b, see Fig. 6(b1–b4). Their equivalent limb is represented by a line with L5I, see Fig. 6c. A L5II generally includes 2 sL2, 1 sL1, an upper platform lu formed by b or t, a lower platform ll formed by t. When lu is formed by b or t, lu is required to provide 3 or 4 connection points: 2 or 3 for L5II, 1 for connecting with the (5 < i)-DoF PMs. The lower platform ll is required to provide 4 connection points: 3 for L5II, 1 for connecting with the (5 < i)-DoF PMs. For example, a DTG with 2t+14b and its two arrays {t446 t644} and {t554 t555} can be derived from AL1 in Table 2, see Fig. 6(d1, d2). The 4 different L5II can be synthesized using the DTG with 2t+14b, see Fig. 6(e1–e4). Their equivalent limb is represented by a line with L5II, see Fig. 6f. A L5III generally includes 5 sL1, an upper platform lu formed by q, and a lower platform ll formed by p. Since lu is required to Table 4 The acceptable combinations among sub-mechanisms in i-DoF PM. i DoF

Acceptable combinations among sub-mechanisms in i-DoF PM and their numbers

4 DoF 5 DoF

s L1+L3I, sL1+L3II, sL1+L3III, 1L1+L3I, 1L1+L3II, 1L1+L3III 2sL1+L3I, 2 1L1+L3I, L2+L3I, 2sL1+L3II, 2 1L1+L3II, 2L2+L3II, 2sL1+L3III, 2 1L1+L3III, 2L2+L3III, s L1+L4I, 1 L1+L4I, sL1+L4II, 1L1+L4II, sL1+L4III, 1L1+L4III, sL2+L3I, sL2+L3II, sL2+L3III, 3L2+L3II, 3 L2+L3III, 3sL1+L3I, 3 1L1+L3I, sL1+2L2+L3I, 1L1+2L2+L3I, 3sL1+L3II, 3 1L1+L3II, sL1+2L2+L3II, 1 L1+2L2+L3II, 3sL1+L3III, 3 1L1+L3III, sL1+2L2+L3III, 1L1+2L2+L3III, 2L3I, 2L3II, 2L3III, L3I+L3II, L3I+L3III, L3II+L3III, sL1+L5I, 1L1+ L5I, s L1+L5II, 1L1+L5II, sL1+L5III, 1L1+L5III, 2sL1+L4I, 2 1L1+L4I, 2L2+L4I, 2sL1+L4II, 2 1L1+L4II, 2 L2+L4II, 2sL1+L4III, 2 1L1+L4III, 2L2+L4III, sL2+L4I, sL2+L4II, sL2+L4III, sL3+L3I, sL3+L3II, sL3+L3III, s L1+sL2+L3I, sL1+sL2+L3II, s L1+sL2+L3III, 1L1+sL2+L3I, 1L1+sL2+L3II, 1L1+sL2+L3III, 2sL1+L3I+1L1, 2sL1+ L3II+1L1, 2sL1+L3III+1L1, s L1+L4I+1L1, sL1+L4II+1L1, sL1+L4III+L1, sL1+L3I+2 1L1, sL1+L3II+2 1L1, sL1+L3III+ 2 1L1, sL1+3L2+L3I, 1 L1+3L2+L3I, sL1+3L2+L3II, 1L1+3L2+L3II, sL1+3L2+L3III, 1L1+3L2+L3III, 2L3+L4I, 3L2+ L4II, 3L2+L4III

6 DoF

46

6 20

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Mechanism and Machine Theory 109 (2017) 39–50

Y. Lu, N. Ye

au

s

L1

L2 al 3 L2+sL1 a

3

4-1

4-2

m

4

r-DTG with 2t+(12-3)b b

m

m

B

B

c1

B

c2

c3

Fig. 7. A simplified mechanism with 3L2+sL1 (a), revised DTG (r-DTG) with 2t+(12-3)b (b), three synthesized 3-DoF PMs with 3L2+sL1 (c1-c3).

provide 5 connection points: 4 for L5III, 1 for connecting with the (5 < i)-DoF PMs, lu must be p. Since ll is required to provide 6 connection points: 5 for L5III, 1 for connecting with the (5 < i)-DoF PMs, ll must be h. For example, a DTG with p+q+t+23b and its array {p44555 q5550 t044} can be derived from AL10 in Table 2, see Fig. 6g. Next, the 3 different L5III can be synthesized by utilizing the DTG with p+q+t+23b, see Fig. 6(h1–h3). Their equivalent limb is represented by a line marked by L5III, see Fig. 6i. 5. Type synthesis of i-DoF PMs by utilizing sub-mechanisms When the sub-mechanisms {LkI, LkII, LkIII (k=3, …, i-1); sLj, kLj (j=1, 2; k=1, 2, 3)} are used to synthesize the i-DoF PMs, more novel i-DoF PMs can be created. Generally, i is the sum of the actuators in the i-DoF PM. Thus, the acceptable combinations among {LkI, LkII, LkIII (k=3, …, i-1); s Lj, kLj (j=1, 2, 3; k=1, 2, 3)} in the i-DoF PM can be obtained are listed in Table 4 based on Table 1. Several examples among acceptable combinations in Table 4 for type synthesis of the i-DoF PMs are illustrated as follows: Based on the conditions 3 and 4 in Section 3.2, the moving platform m and the base B in the i-DoF PMs are determined and marked by m and B in follows examples, respectively. Example 1. In order to synthesize the 3-DoF PMs, 1 3L2 and 1 sL1 can be applied to construct a simple mechanism based on Table 1, see Fig. 7a. Next, based on the condition 5 in Section 3.2, a DTG with 2t+12b can be constructed from AL1 in Table 2 and can be transformed into a revised DTG with 2t+(12-3)b, see Fig. 7b. Finally, three 3-DoF PMs with 3L2+sL1 can be synthesized using the revised DTG with 2t+(12-3)b, see Fig. 7(c1–c3). Example 2. In order to synthesize the 4-DoF PMs, 1 L3I and sL1 can be applied to construct a simple mechanism based on Table 4, see Fig. 8a. Next, a DTG with 2q+18b and its array {4446 6444} can be constructed from AL2 in Table 2 (see Fig. 8b) because it includes a sub-DTG with 2t which are connected by 3 edges in parallel with 12b according to the characteristics of L3I in Section 4.2. Finally, three novel 4-DoF PMs with L3I+sL1 can be synthesized, see Fig. 8(c1–c3). Example 3. In order to synthesize the 4-DoF PMs, 1 L3I and 1 1L1 can be applied to construct a simple mechanism, see Fig. 9a. Next, a DTG with 2q+2t+21b and its array {q1444 q4443 t332 t231} can be constructed from AL7 in Table 2 since it includes a sub-DTG with 2t connected by 3 parallel edges. After that, this DTG with 2q+2t+21b is transformed into a revised DTG with 2q+2t +(21-3)b and its array {q1444 q4443 t312 t211} based on condition 5 in Section 3.2, see Fig. 9b1. Thus, two novel 4-DoF PMs with L3I+1L1 can be synthesized using the revised DTG with 2q+2t+(21-3)b, see Fig. 9(c1, c2). Similarly, a DTG with 2q+2t+21b and its array {q2444 q4442 t232 t232} can be constructed and transformed into a new revised DTG with 2q+2t+(21-3)b and its array {q2444 q4442 t212 t212}, see Fig. 9b2. Thus, the third novel 4-DoF PM with L3I+1L1 can be synthesized, see Fig. 9c3. Example 4. In order to synthesize 5-DoF PMs, 1 L3I and 1 sL2 can be combined into a simply mechanism based on Table 3, see Fig. 10a1. a DTG with 2q+19b and its array {q4447 q7444} can be constructed from AL2 in Table 2 since it includes 2q for connecting L3I with sL2, see Fig. 10b1. Thus, the 2 novel 5-DoF PMs can be synthesized using the DTG with 2q+19b, see Fig. 10(c1, c2). Similarly, 1 L3I and 1 3L2 can be combined into a simply mechanism based on Table 3, see Fig. 10a2. A DTG with 2q+2t+22b and its array {q1444 q4442 t243 t341} can be constructed from AL7 in Table 2 since it includes 2q and 2t for connecting L3I with 3L2. After that, this DTG can be transformed into a revised DTG with 2q+2t+(22-3)b and its array {q1444 q4442 t222 t221} based on

au

4 s

L1 L3I al L3I+sL1

au

au

au

m

4 4

6 al DTG with 2q+18b and array {4446 6444}

al

B

m m

B

al

al

au

b

Fig. 8. A simplified mechanism with L3I+sL1 (a), DTG with 2q+18b (b), three synthesized 4-DoF PMs with L3I+sL1 (c1-c3).

47

Mechanism and Machine Theory 109 (2017) 39–50

Y. Lu, N. Ye

au

1

L3I al L3I+1L1 a

4 3

L1

4

m

1 4 3-1 4 4

m

au

au

3 2-2 4 r-DTG with 2q+2t+(21-3)b array {1444 4443 312 211}

m

au

2 4 3-1

4

4 4 2 2-2 4 r-DTG with 2q+2t+(21-3)b array {2444 4442 212 212} b2

b1

al

B

al

c1

B c2

al

B c3

Fig. 9. A simplified mechanism with L3I+1L1 (a), 2q+2t+(21-3)b (b1, b2), three 4-DoF PMs with L3I+1L1 (c1-c3).

au

au s

L2

L3I al L3I+sL1

4 al

4 4

m

m

au

au

7

DTG with 2q+19b and array {4447 7444}

B

al

al

B

au

3

L2 4

L3I al L3I+3L2

2 44 1

au 4-2

m

au

m

3-1

r-DTG with 2q+2t+(22-3)b

B

al

al

B

Fig. 10. A simplified mechanism with L3I+sL2 (a1), A DTG with 2q+19b (b1), two synthesized 5-DoF PMs with L3I+sL2 (c1, c2), A simplified mechanism with L3I+3L2 (a2), revised DTG with 2q+2t+(21-3)b (b2), two synthesized 5-DoF PMs with L3I+3L2 (c3, c4).

condition 5 in Section 3.2. Thus, two novel 5-DoF PMs with L3I+3L2 can be synthesized, see Fig. 10(c3, c4).

Example 5. In order to synthesize the 6-DoF PMs, 1 L4I and 2 sL1 can be combined into a simply mechanism based on Table 4, see Fig. 11a. A DTG with 2p+2t+29b can be constructed from AL19 with (ζ=1, n2=22+i+ζ=22+6+1=29, n3=2, n5=2) in Table 2 since it includes a sub-DTG with 2p for connecting 1 L4I with 2 sL1, see Fig. 11b1. Thus, the 2 novel 6-DoF PMs can be synthesized, see Fig. 11(c1, c2). Here, the moving platform m and the base B are different link in synthesized 2 novel 6-DoF PMs. Similarly, 1 L4I and 1 3L2 can be combined into a simply mechanism based on Table 3, see Fig. 11a2. A DTG with 2p+2t+29b can be constructed from AL19 in Table 2 since it includes sub-DTG with 2p for connecting 1 L4I with 3L2, see Fig. 11b2. After that, this DTG can be transformed into a revised DTG with 2p+2t+(29-3)b based on the condition 5 in Section 3.2. Thus, two novel 6-DoF PMs with L4I+3L2 can be synthesized, see Fig. 11(c3, c4). Here, the moving platform and the base are different link in 2 synthesized novel 6-DoF PMs. The DoFs of above 17 synthesized i-DoF PMs in Fig. 7c1→Fig. 11c4 are calculated and verified based on revised KutzbachGrübler formula of Eq. (1), see Table 5. In the light of applications of the i-DoF PMs, several specifications should be considered for determining actuators, kinematic pairs, link and their positions as follows: 1. 2. 3. 4. 5. 6. 7.

Should Should Should Should Should Should Should

attach the kinematic pairs with more locate DoFs onto the base to increase the workspace. use the linear actuators to bear heavy loads at lower speed. use rotational actuators to increase speed under light loads. use the sub-closed planar mechanism to increase the stiffness of the PM. attribute more actuators onto or close to the base to reduce influence of the vibration on precision. use the link with more connection joints as the base to increase stability. not take one of link of the sub-closed planar mechanisms as the base to prevent revolute pair from a large bending load.

Based on above rules, other more different i-DoF PMs can be synthesized using the DTGs and sub-mechanisms. In fact, only 32 DTGs and few sub-mechanisms are applied for synthesizing the PMs, and many other DTGs and sub-mechanisms have not been considered here. Therefore, more novel PMs can be synthesized from many other DTGs and other sub-mechanisms. In addition, when the order of kinematic pairs and the orientations of kinematic pairs in the kinematic chains are varied [18], more novel i-DoF PMs can be synthesized.

Fig. 11. A simplified mechanism with L4I+2sL1 (a), A DTG with 2p+2t+29b (b), two synthesized 6-DoF PMs with L4I+2sL1 (c1, c2); A simplified mechanism with L4I+3L2 (a2), revised DTG with 2p+2t+(29-3)b (b2), two synthesized 6-DoF PMs with L4I+3L2 (c3, c4).

48

Mechanism and Machine Theory 109 (2017) 39–50

Y. Lu, N. Ye Table 5 DoF verifications of the 17 novel i-DoF PMs in Fig. 7c1→Fig. 11c4. No.

Fig.

n2

n3

n4

n5

n6

N

n

ζ

ν

ns

nu

nc

np

nr

i

s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

7c1 7c2 7c3 8c1 8c2 8c3 9c1 9c2 9c3 10c1 10c1 10c3 10c4 11c1 11c2 11c3 11c4

8 8 8 18 18 18 18 18 18 19 19 19 19 29 29 29 29

2 2 2 2 2 2 2 2 2 0 0 2 2 2 2 2 2

0 0 0 0 0 0 2 2 2 2 2 2 2 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 9 9 11 11 11 13 13 13 11 11 15 15 16 16 18 18

10 10 10 13 13 13 16 16 16 13 13 18 18 20 20 22 22

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1

3 3 3 0 0 0 3 3 3 0 0 3 3 0 0 3 3

1 1 0 4 1 4 3 4 4 4 4 3 3 6 6 4 4

0 0 2 1 7 1 3 1 1 2 2 2 2 5 5 4 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 3 3 4 4 4 4 4 4 4 4 5 5 6 6 6 6

6 6 5 4 1 4 6 7 7 3 3 8 8 3 3 8 8

3 3 3 4 4 4 4 4 4 5 5 5 5 6 6 6 6

s

Lj, kLj, Lk

L1+3L2 L1+3L2 s L1+3L2 s L1+L3I s L1+L3I s L1+L3I 1 L1+L3I 1 L1+L3I 1 L1+L3I s L2+L3I s L2+L3I 3 L2+L3I 3 L2+L3I 2sL1+L4I 2sL1+L4I 3 L2+L4I 3 L2+L4I s

i=6(N–n–1)+3ns+2nu+2nc+np+nr+ν–ζ in Eq. (1); i–DoF; N – the numbers of links; n–the numbers of pairs; ns, nu, nc, np, nr – the numbers of S, U, C, P, R; ν – the numbers of redundant constraints; ζ–passive DoF; (b, t q, p, h) – binary link, ternary link, quaternary link, pentagonal link, hexagonal link; ni (i=2, …, 6) – the number of (b, t, q, p, h); s L1 – sub-serial limb with 1 actuator; 1L1 – 1-DoF planar closed mechanism with 1 actuator; 3 L2 – 3-DoF planar closed mechanism with 2 actuators; L3I – 3-DoF sub-PM; L4I – 4-DoF sub-PM.

6. Conclusions The conditions for the type synthesis of parallel mechanisms using the digital topology graphs and sub- mechanisms are determined. The 32 digital topology graphs and their revised digital topology graphs can be derived from associated linkages for synthesizing the parallel mechanisms with sub-mechanisms. The sub-mechanisms in the parallel mechanisms can be transformed into the simple equivalent limbs, and their equivalent relations and merits are determined. The 17 novel parallel mechanisms with different sub-mechanisms or their combinations can be synthesized using the digital topology graphs and revised digital topology graphs. They can be simplified by replacing complicated sub-mechanisms with their simple equivalent limbs. The degrees of freedom of all synthesized parallel mechanisms are verified to be the correction. More novel parallel mechanisms can be synthesized by utilizing different digital topology graphs, different the sub- mechanisms and their combinations. Acknowledgements The authors would like to acknowledge (1) Project (E2016203379) supported by Natural Science Foundation of Hebei (2) Project (51175447) supported by National Natural Science Foundation of China (NSFC) and (3) Project (JX2014-02) supported by Yanshan University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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