# Type Synthesis of 5-DoF Parallel Mechanisms with Different Submechanisms

Authors: Yi Lu; Yang Lu; Lijie Zhang; Nijia Ye; Bo Hu; Yang Liu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The type synthesis of the 5-DoF (degree of freedom) parallel mechanisms with different submechanisms is studied by utilizing digital topology graphs (DTGs). The conditions for synthesizing the 5-DoF parallel mechanisms with different submechanisms using DTGs are determined. Many valid DTGs are derived from 17 different 5-DoF associated linkages, and the valid DTGs are transformed into revised DTGs. The subplanar and/or spatial parallel mechanisms in the 5-DoF parallel mechanisms are transformed into some simple equivalent limbs, and their equivalent relations and merits are analyzed. Using the derived valid DTGs and revised DTGs, many 5-DoF parallel mechanisms with different subserial or parallel mechanisms are synthesized, and they are simplified by replacing the complicated subparallel mechanisms with their simple equivalent limbs. Finally, their DoFs are calculated to verify the correctness and effectiveness of the proposed approach.


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## Body

## 1. Introduction

Type synthesis of mechanisms is a well-known technology for creating and designing novel mechanisms [1–5]. In this aspect, Gogu [1–3] studied the type synthesis of parallel mechanisms (PMs) and presented morphological and evolutionary approaches. Huang conducted the type synthesis of PM by utilizing Lie group and screw-theory [4]. Yang studied the topology structure design of mechanisms [6]. Johnson derived a planar associated linkage (ALs) by utilizing a determining tree and synthesized many planar mechanisms by AL [5]. Merlet proposed a design methodology for conception of PMs [7]. Ceccarelli studied design methodology for compliant binary actuated PM with flexure hinges and design and evaluation of a discretely actuated multimodule PM [8]. A TG (topology graph) has been applied widely for the type synthesis of closed mechanisms [5, 6]. A contracted graph (CG) without any binary link is applied for deriving TG. In this aspect, Vucina and Freudenstein [9] and Tsai and Norton [10] proposed TG and CG and used them to synthesize mechanisms. Yan and Kang [11] studied the configuration synthesis of mechanismsby changing types and/or motion orientations of joints. Hervé [12] studied type synthesis of mechanisms using Lie group. Pucheta and Cardona [13, 14] synthesized planar linkages based on constrained subgraph isomorphism detection and existing mechanisms. Saxena and Ananthasuresh [15] selected the best configuration based on design specifications. Lu et al. explained conditions for deriving the valid DTGs by utilizing submechanisms, discovered the relations between AL, redundant constraint, and DoFs of PMs [16], and conducted type synthesis of 3-DoF PMs by utilizing revised DTGs [17].Generally, the tool axis of a machine tool is required to be perpendicular to the 3D free-form surfaces during the machining of workpieces (the injection moulds, dies, models of automobile windshields, impeller blades of ships or airplanes, launches, and turbines) in order to improve the machining quality and reduce the machining forces. Therefore, the 5-DoF PMs have more industrial applications and become the best option of developing the parallel machine tools for the following reasons [18–21]:(
1
) The tool axis of 5-DoF parallel machine tool formed by the 5-DoF PMs can be kept perpendicular to the 3D free-form surfaces during machining 3D free-form surfaces.(
2
) Comparing with the existing serial-parallel 5-DoF machine tool which is formed by a 3-DoF PM connected with a 2-DoF spindle actuator in series, the 5-DoF parallel machine tool formed by the 5-DoF PMs has higher rigid and precision because its actuators are close to the base and has more active limbs for supporting m.(
3
) Comparing with the 6-DoF PMs, the 5-DoF PMs are simpler in structure and easier in control.Let “R, P, C, U, and S” represent “revolute, prismatic, cylinder, universal, and spherical” pairs, respectively. In this aspect, Gao et al. developed a 5-DoF parallel machine tool with 4 PSS limbs and a composite 3UU limb [18]. Liu et al. proposed a coupling 3-PSR/PSU 5-axis compensation mechanism [19]. Fang and Tsai synthesized a class of 5-DoF overconstrained PMs with identical serial limbs [20]. Kong and Gosselin synthesized several 5-DoF PMs based on screw theory and the concept of virtual chains [21]. These studies have their merits and different focuses and have laid a theoretical foundation for this study.Up to now, several 3- and 4-DoF PMs have been synthesized [1–6, 13]. When they are used as one submechanism of 5-DoF PMs, many novel 5-DoF PMs with high stiffness can be synthesized. Therefore, it is a significant and challenging issue to synthesize novel 5-DoF PMs with special functions and/or better characteristics (large position and orientation workspace, large capability of load bearing, high stiffness, simple structure, and easy control). For this reason, this paper focuses on the type synthesis of the 5-DoF PMs with subserial mechanisms, subplanar mechanisms, and sub-PMs or their combinations by utilizing valid DTGs (digital topology graph). The following problems are solved: (1) constructing various DTGs of 5-DoF PMs by utilizing 17 ALs; (2) determining the relationship between the valid DTGs and the equivalent limbs of the subplanar mechanisms and sub-PMs; (3) synthesizing the 5-DoF PMs with the subplanar or sub-PMs by utilizing valid DTGs.


## 2. Concepts and Conditions

A parallel mechanism (PM) may be composed of various links connected by several kinds of kinematic pairs such asR, P, C, U, and S (revolute, prismatic, cylinder, universal, and spherical) pairs. A Kutzbach-Grübler formula had been widely used to calculate DoF of the closed mechanisms [1–6]. By considering both the redundant constraints and the passive DoF in the PMs, Huang [4] revised the Kutzbach-Grübler formula as follows:(1)M=6N-n-1+∑fj+ν-ζ,∑fj=3ns+2nu+2nc+np+nr.Here, M 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; fj is the local DoF of the jth kinematic pair; ν is the number of all the redundant constraints; ζ is the number of passive DoFs which does not influence the moving characteristics of m; “ns,nu,nc,np,andnr” are the numbers of “S,U,C,P,andR” pairs, respectively.Let “b,t,q,p,h” be a “binary, ternary, quaternary, pentagonal, hexagonal” link in a mechanism, respectively. Let J be a point of connection with one-DoF. Let ni (i=2,…,6) be the number of “b, t, q, p, and h,” respectively. A topology graph (TG) is formed by some links connected by several J [5]; see Figure 1(a). TG is a basic tool for type synthesis of mechanisms [1–3, 5]. An associated linkage (AL) may include the acceptable group of links “b,t,q,p,h,….” AL with given DoF is a basic tool for deriving various TGs with the same DoF [5].Figure 1
A TG withq+2t+18b and 23J (a) and a DTG with q+2t+18b and an array t1144q4445t2541 (b).
(a)
(b)The type synthesis of PMs using TG generally is quite complicated because TG includes manyb. When all links of “t,q,p,h,…” in TG are replaced by the vertices connected by “3,4,5,6,…” edges, respectively, and each of the edges in TG is marked by a digit which represents the number of b connected in series by several J on this edge, the representation of TG can be simplified greatly. Therefore, a digital topology graph (DTG) is used to simplify the representation of TG; see Figure 1(b).An array is an aided tool for deriving and representing DTGs and is also used to identify the isometric DTGs and the invalid DTGs. Thus, the type synthesis of PMs and their isometric identification become quite easy. Generally, a DTG includes several vertices connected by several edges. “t,q,p,h,…” in a DTG are represented by the vertices connected by “3,4,5,6,…” edges, respectively. Each of the edges in DTG is marked by a digit. Each digit in the DTG represents the number of b connected in series by several J on this edge. For instance, a TG representing 5-DoF PMs includes 1q, 2t, and 18b which are connected by 23J; see Figure 1(a). The complicated representation of TG can be simplified by a DTG; see Figure 1(b). In this DTG, 1q is represented by 1 vertex which are connected by 4 edges ej (j=1,2,4,5) marked by digits (4,4,4,5), respectively. 2t are represented by 2 vertices which are connected by 5 edges ej (j=1,…,5) marked by digits (4,4,1,4,5), respectively. Here, “4,4,1,4,5” represent “4,4,1,4,5” b connected in series by “5,5,2,5,6” J in ei (i=1,…,5), respectively. An array t1144q4445t2541 is used for representing this DTG. In this array, t1144 represents the case that t1 is connected with 3 edges ei (i=1,2,3). Here, “e1, e2, and e3” are formed by “4,4,1” b connected in series, respectively.LetPMi (i≤2; M=1,2,3) be the subplanar closed mechanisms with M DoFs and i actuators. Four typical PMi are represented in [17]; see Figure 2. In fact, 5-DoF PMs may include some other different PMi (i≤2; M=1,2,3).Figure 2
Four typicalPMi; P10 (a1), P11 for increasing force (b1) and a P11 for enlarging displacement (b2), P22 (c1), and P32 (d1) and their simple representations (a2), (b3), (c2), and (d2).A 5-DoF PM generally includesB, m, and several limbs. These limbs may include several links which are connected by several kinematic pairs (R,P,C,U,andS). The following 5 conditions must be satisfied in order to derive the valid DTGs and synthesize novel 5-DoF PMs:(
1
) If “ν=0; ζ=0” are satisfied in a 5-DoF PM, the number of the links in any closed loop chain must be 7 or more in order to avoid any local structure.(
2
) If “ν=0 and ζ=0” are satisfied in a 5-DoF PM, the number of b on any edge must be 6 or less in order to avoid any local structure.(
3
) The number of J connected in series in each limb from B to m must be 5 or more.(
4
) The number of the links in a closed loop chain of DTG for constructing P1i (i=0,1) must be 7 in order to avoid any local structure.(
5
) The number of the links in a closed loop chain of DTG for constructing P22 must be 8 in order to avoid any local structure.(
6
) The number of the links in a closed loop chain of DTG for constructing P32 must be 9 in order to avoid any local structure.The theoretical bases of the above 6 conditions have been explained in the Appendix. Another three auxiliary conditions [17] should be satisfied as follows:(
7
) It is permissible to replace any one J in the TG with R or P pair in the mechanism.(
8
) It is permissible to replace any 2J connected in series in the TG with U or C pair in the mechanism.(
9
) It is permissible to replace any 3J connected in series in the TG with S pair in the mechanism.In the light of the 5-DoF PMs, 17 ALk (k=1,…,17) with 5-DoF and ni (i=2,…,6) links have been derived based on (1). ni in 17 ALs with 5-DoF can be solved in [17] and listed in Table 1.Table 1
The number of the binary links and basic links in 17 spatial 5-DoF associated linkages (ALs).
Number
AL
n
2, M=5
n
3
n
4
n
5
n
6
1.1
AL1
5
+
ζ
-
ν
+
9
2
0
0
0
2.3
AL2
5
+
ζ
-
ν
+
14
0
2
0
0
3.7
AL3
5
+
ζ
-
ν
+
19
0
0
2
0
3.8
AL4
5
+
ζ
-
ν
+
18
2
0
0
1
2.1
AL5
5
+
ζ
-
ν
+
12
4
0
0
0
2.2
AL6
5
+
ζ
-
ν
+
13
2
1
0
0
3.3
AL7
5
+
ζ
-
ν
+
17
2
2
0
0
3.2
AL8
5
+
ζ
-
ν
+
16
4
1
0
0
3.5
AL9
5
+
ζ
-
ν
+
17
3
0
1
0
3.6
AL10
5
+
ζ
-
ν
+
18
1
1
1
0
3.1
AL11
5
+
ζ
-
ν
+
15
6
0
0
0
4.3
AL12
5
+
ζ
-
ν
+
20
4
2
0
0
4.2
AL13
5
+
ζ
-
ν
+
19
6
1
0
0
4.11
AL14
5
+
ζ
-
ν
+
20
5
0
1
0
3.4
AL15
5
+
ζ
-
ν
+
18
0
3
0
0
4.6
AL16
5
+
ζ
-
ν
+
21
4
0
0
1
5.7
AL17
5
+
ζ
-
ν
+
23
7
0
1
0


## 3. Valid DTGs and Revised DTGs

The valid DTG can be derived from the AL by distributing allb into the contracted graph [16]. Based on conditions (1)–(6), some DTGs with different arrays for synthesizing the 5-DoF PMs are derived and displayed in Figure 3.Figure 3
The 22 DTGs with their arrays.The 4 DTGs with2t+14b and different arrays t446t644, t356t653, t455t554, and t347t743 are derived from AL1 in Table 1; see Figures 3(a1)–3(a4). The 2 DTGs with 2q+19b and different arrays q4555q5554 and q4546q6454 are derived from AL2 in Table 1; see Figures 3(b1) and 3(b2). The 2 DTGs with 2p+24b and different arrays p45456p65454 and p45555p55554 are derived from AL3 in Table 1; see Figures 3(c1) and 3(c2). The 2 DTGs with q+2t+18b and different arrays t1144q4445t2541 and t1244q4444t1442 are derived from AL6 in Table 1; see Figures 3(d1) and 3(d2). The 18 DTGs are derived from AL5 and AL7→AL17 in Table 1; see Figures 3(e)–3(p).If the 5-DoF PMs includePMi (i≤2; M=1,2,3), then the DTGs must be revised into r-DTGs by utilizing approach in [17]. For instance, a DTG with 4t+17b in Figure 4(a1) is revised into an r-DTG with 4T+(17-3)B in Figure 4(a2). Similarly, the other DTGs in Figures 4(b1)–4(i1) can be revised into the r-DTGs in Figures 4(b2)–4(i2). Thus, all the r-DTGs can be used to synthesize the 5-DoF PMs with PMi (i≤2; M=1,2,3).Figure 4
The 9 DTGs with their arrays and r-DTGs.


## 4. Submechanisms and Their Equivalent Limbs

### 4.1. Characteristics and Functions of Submechanisms

LetLsi (1≤i≤3) be a subserial mechanism with i actuators. Let Li (2≤i≤4) be a subspatial i-DoF PM with i actuators. The submechanisms of the 5-DoF PMs may be Lsi, PMi (i≤2; M=1,2,3), and Li. Each of them has different merits. These merits can bring benefits for the 5-DoF PMs as follows.Wh300enLsi is taken as the limb for connecting m with B in the 5-DoF PMs, and the total number of the required limbs must be reduced. Thus, the interference between the limbs and m may be avoided easily, and the workspace of the 5-DoF PMs can be enlarged. In addition, since the other active limbs of the 5-DoF PMs can be transformed into the SPU-type linear active limbs which are not sensitive to manufacturing error, not only can the tiny self-motion of the PMs be removed effectively, but also the capability of the load bearing can be increased. When a spatial PM includes one subplanar mechanism or more, it may have merits [17] such as simplicity in structure, ease in manufacturing compliant mechanism and increasing the mechanical advantage, and being larger in capability of the pulling force and rotation angle. When different PMi (i≤2; M=1,2,3) and Li (2≤i≤4) are applied to constructing one limb of the 5-DoF PM, its stiffness can be increased.Generally, the above-mentioned submechanisms can be built into different standard units with high precision by a special company. Therefore, many novel 5-DoF PMs with high precision can be built and assembled easily by including these standard units.


### 4.2. Equivalent Limbs of Submechanisms

Similarly, each ofLsi (1≤i≤3) and Li (2≤i≤4) in the 5-DoF PMs can be replaced by its equivalent limb and represented by a line marked with Lsi or Li.The equivalent symbols of various kinematic pairs are represented in Figure5. Generally, each of Lsi in the 5-DoF PMs includes a lower link ll, an upper link lu, and i actuators which are connected in series from B to m; see Figure 5(a1). The equivalent limb of Lsi is represented by a line with Lsi; see Figure 5(a2). Lsi can be selected from an edge with 4 or 5 in the DTG. The number of actuators can be 2 or more. The actuators may be translational ones or rotational ones. The number of Lsi can be one or more. For instance, a DTG with 2t+14b and an array t1455t2554 is proposed to synthesize the 5-DoF PMs; see Figure 4(a3). The 6 different 5-DoF PMs with Lsi are synthesized using the proposed DTG; see Figure 5(b). Each of the 6 PMs has 3 limbs, the first 4 PMs include 2 Ls2, and the rest 2 PMs include 1 Ls3. Generally, if a DTG includes one closed loop which is formed by 4 links connected in series by 4J, then this DTG can be revised into an r-DTG. After that, the 5-DoF PMs with P10 can be synthesized by utilizing the r-DTG.Figure 5
DTG with2t+14b and array t1455t2554 (a) and the six 5-DoF PMs with serial mechanisms (b).For instance, a DTG withp+7t+28b and an array t1304t2430t3002t4242t5200t6034t7403p33433 is proposed for the type synthesis of the 5-DoF PMs; see Figure 6(a). Since the proposed DTG includes a closed loop which is formed by 4 links connected in series by 4J for constructing P10, the proposed DTG can be revised into an r-DTG with p+7t+(28-6)b and an array t1301t2130t3002t4242t5200t6031t7103p33433; see Figure 6(b). Thus, a novel 5-DoF PM with 2 P10 can be synthesized using the r-DTG; see Figure 6(c).Figure 6
A DTG withp+7t+28b (a), an r-DTG with p+7t+(28-6)b (b), and a 5-DoF PMs with 2 P10 (c).Based on condition(4) in Section 2, P11 can be selected from a DTG with a closed loop which is formed by 2t and 5b, and the equivalent limb of P11 is represented by a line with P11; see Figure 7(b). In addition, the DTG for synthesizing the 5-DoF PMs with P11 must be transformed into a revised DTG (r-DTG) by reducing ν=3 from the number of b in the closed chain for P11. For instance, a DTG with 4t+17b and an array t1132t2232t3245t4541 is proposed for the type synthesis of 5-DoF PMs; see Figure 7(a). Since the proposed DTG includes a closed loop which is formed by 2t+5b for constructing P11, the proposed DTG must be revised into an r-DTG with 4t+(17-3)b and an array t1120t2022t3245t4541; see Figure 7(b). Thus, two novel 5-DoF PMs with P11 and Ls2 are synthesized using the r-DTG; see Figure 7(c).Figure 7
DTG with4t+17b (a), r-DTG with 4t+(17-3)b (b), and two novel 5-DoF PMs with P11 and Ls2 (c1, c2).Based on condition(5) in Section 2, the equivalent limb of P22 can be constructed by utilizing a DTG with a closed loop which is formed by 2t+6b. The DTG for synthesizing 5-DoF PMs with P22 must be revised into an r-DTG by reducing ν=3 from the number of b in the closed loop for constructing P22. For example, a DTG with 4t+17b and an array t1233t2331t3144t4442 is proposed for the type synthesis of 5-DoF PMs; see Figure 8(a). Since the proposed DTG includes a closed loop formed by 2t and 7b, it must be revised into an r-DTG with 4t+(17-3)b and an array t1221t2121t3144t4442; see Figure 8(b). Thus, 4 novel 5-DoF hybrid PMs with P22 and Ls2 are synthesized by utilizing the r-DTG; see Figures 8(c1)–8(c4). Here, P22 and Ls2 can be represented by the lines marked by P22 and Ls2, respectively.Figure 8
DTG with4t+17b (a), an r-DTG with 4t+(17-3)b (b), four 5-DoF hybrid PMs with 1 P22 and 1 Ls2 (c1–c4), a DTG with 4t+17b (d), an r-DTG with 4t+(17-3)b (e), and four 5-DoF hybrid PMs with 1 P32 and 1 Ls2 (f1–f4).It is known that the number of the links in a spatial closed 3-DoF PM must be 9 or more. Based on condition(6) in Section 2, the equivalent limb of P32 can be constructed by utilizing a DTG with a closed loop which is formed by 2t and 7b and is represented by a line marked by P32; see Figure 8(g). In addition, the DTG for synthesizing 5-DoF PMs with P32 must be revised into an r-DTG by reducing ν=3 from the number of b in the closed loop for constructing P32. For example, a DTG with 4t+17b and an array t1134t2431t3144t4441 is proposed for the type synthesis of 5-DoF PMs; see Figure 8(d). Since the proposed DTG includes a closed loop formed by 2t and 7b, it must be revised into an r-DTG with 4t+(17-3)b and an array t1122t2221t3144t4441; see Figure 8(e). Thus, the 4 novel 5-DoF hybrid PMs with P32 and L2s are synthesized by utilizing the r-DTG with 4t+(17-3)b; see Figures 8(f1)–8(f4).A subspatial 3-DoF PML3 generally includes an upper platform lu, a lower platform ll, and 3 active limbs. Since ll and lu are required to have 4 connection points: three connection points for connecting with three active limbs and 1 for connection point (al and au) at the two ends of L3 for connecting with 5-DoF PMs, both ll and lu must be quaternary links. Therefore, the equivalent limb of L3 can be constructed using a DTG with 2q which are connected by 3 edges in parallel with 12b. The 8 different DTGs for the type synthesis of various 3-DoF PMs can be obtained; see Figure 9(a). Their equivalent limb is represented by a line with L3; see Figure 9(b). If L3 has symmetry in structure, the first DTG in Figure 9(a) can be used for synthesizing symmetry L3, and the 26 different symmetry kinematic chains [16] can be applied to the construction of the 26 different symmetry L3. When L3 is asymmetry in structure, the other 7 DTGs in Figure 9(a) can be used for synthesizing asymmetry L3, and more asymmetry kinematic chains can be obtained. Thus, many novel 5-DoF PMs can be synthesized.Figure 9
The 8 different DTGs ofL3 (a1–a8) and their equivalent limb (b).A sub-4-DoF PML4 generally includes an upper platform lu, a lower platform ll, and 4 active limbs; see Figure 10. Since ll and lu are required to provide 5 connection points: 4 connection points are used for connecting with 4 active limbs and 1 is used as the connection point (al and au) at the 2 ends of L4 for connecting with 5-DoF PMs, both ll and lu must be pentagonal links. Therefore, the equivalent limb of L4 can be constructed using a DTG with 2p which are connected by 4 edges in parallel with 18b. The 14 different DTGs with 2q+18b for synthesizing various 4-DoF spatial PMs can be obtained. Three different promised DTGs are proposed for synthesizing 7 different subspatial 4-DoF PMs; see Figures 10(a)–10(c). Their equivalent limb is represented by a line with L4; see Figure 10(d). Thus, many novel 5-DoF PMs can be synthesized.Figure 10
The 3 DTGs ofL4 ((a), (b), and (c)) and the 9 subspatial 4-DoF PM L4 and their equivalent limb (d).For instance, a DTG with2p+24b and an array p45456p65454 is proposed for the type synthesis of the 5-DoF PMs with L4; see Figure 11(a). Two equivalent mechanisms I and II with L4 are derived from the proposed DTG; see Figures 11(b) and 11(d). Three novel 5-DoF PMs with the p-type base are synthesized from the equivalent mechanism I; see Figures 11(c1)–11(c3). Three novel 5-DoF PMs with the p-type moving platform are synthesized from the equivalent mechanism II; see Figures 11(e1)–11(e3). In fact, some novel 5-DoF PMs with PMi and Li or their combinations can be synthesized. Some examples are illustrated as follows.Figure 11
A DTG with2p+24b (a), equivalent mechanism I with L4, (b), 3 novel 5-DoF PMs with p-type base (c1–c3), equivalent mechanism II with L4 (d), and 3 novel 5-DoF PMs with p-type moving platform (e1–e3).Example 1.
A DTG with2q+2t+22b and an array t1134t2432q12444q24441 is proposed for the type synthesis of the 5-DoF PMs with L3 and P32; see Figure 12(a). Since the proposed DTG includes both one parallel closed loop formed by 2q+12b and one closed loop formed by 2t+9b, it must be revised into an r-DTG with 2q+2t+(22-3)b and an array t1122t2222q12444q24441; see Figure 12(b). An equivalent mechanism with 1 L3 and 1 P32 is constructed from the r-DTG; see Figure 12(c). Finally, two 5-DoF PMs are synthesized from equivalent mechanism; see Figures 12(d1) and 12(d2).Figure 12
A DTG with2q+2t+22b (a), an r-DTG with 2q+2t+(22-3)b (b), equivalent mechanism with both L3 and P32 (c), two novel 5-DoF PMs (d1, d2).Example 2.
A DTG with2q+2t+25b and an array t1034t2431q11451t3123t4321q21540 is proposed for synthesizing the 5-DoF PMs with P11 and P32; see Figure 13(a). Since the proposed DTG includes both one closed loop chain formed by 2t and 9b and one closed loop chain formed by 2t and 7b, it must be revised into an r-DTG with 2q+2t+(25-6)b and an array t1022t2211q11451t3102t4201q21540; see Figure 13(b). Thus, the equivalent mechanism of a novel 5-DoF PM with 1 P11 and 1 P32 (see Figure 13(c)) can be constructed from the r-DTG in Figure 13(b). Similarly, a DTG with q+6t+24b and an array t1011t2144t3123t4321t5134t6430q1441 is proposed for synthesizing the 5-DoF PMs with P11 and P32; see Figure 13(d). It can be revised into an r-DTG with q+6t+(24-6)b and an array t1011t2144t3102t4201t5122t6220q1441; see Figure 13(e). Thus, the equivalent mechanism of a novel 5-DoF PM with 1 P11 and 1 P32 is constructed from the r-DTG in Figure 13(e); see Figure 13(f) (q+6t+(24-6)b (e), equivalent mechanism with P11 and P32 (f)).Figure 13
A DTG with2q+2t+25b (a), an r-DTG with 2q+2t+(25-6)b (b), equivalent mechanism with P11 and P32 (c), DTG with q+6t+24b (d), and r-DTG with P11 and P32 (f).Example 3.
A DTG with6t+20b and an array t1040t2034t3431t4141t5134t6430 is proposed for synthesizing the 5-DoF PMs with 2 P32; see Figure 14(a). Since the proposed DTG includes 2 closed loop chains formed by 2t and 9b, it must be revised into an r-DTG with 6t+(20-6)b and an array t1040t2022t3221t4141t5122t6220; see Figure 14(b). Finally, the equivalent mechanism of a novel 5-DoF PM with 2 P32 is constructed from the r-DTG in Figure 14(b); see Figure 14(c). Similarly, a DTG with 2q+4t+25b and an array t1134t2430q10450t3034t4431q21541 is proposed for synthesizing the 5-DoF PMs with 2 P32; see Figure 14(d). Since the proposed DTG includes 2 closed loop chains formed by 2t and 9b, it must be revised into an r-DTG with 2q+4t+(25-6)b and an array t1122t2220q10450t3022t4221q21541; see Figure 14(e). Finally, the equivalent mechanism of a novel 5-DoF PM with 2 P32 is constructed from the r-DTG in Figure 14(e); see Figure 14(f).Figure 14
A DTG with6t+20b (a), an r-DTG with 6t+(20-6)b (b), equivalent mechanism with 2 P32 (c), a DTG with 2q+4t+25b (d), an r-DTG with 2q+4t+(25-6)b (b), and equivalent mechanism with 2 P32 (f).Example 4.
A DTG with2q+2t+22b and an array t1043t2430q10555q25550 is proposed for the type synthesis of the 5-DoF PMs with P32; see Figure 15(a). Since the proposed DTG includes one closed loop formed by 2t+7b and one parallel closed loop formed by 2q+15b, it must be revised into an r-DTG with 2q+2t+(22-3)b and an array t1022t2220q10555q25550; see Figure 15(b). An equivalent mechanism with 1 P32 is constructed from the r-DTG; see Figure 15(c). Finally, one 5-DoF PM is synthesized from equivalent mechanism; see Figure 15(d).Figure 15
A DTG with2q+2t+22b (a), an r-DTG with 2q+2t+(22-3)b (b), a novel 5-DoF PM with P32 (c), equivalent mechanism of novel 5-DoF PM with P32 (d), and prototype of novel 5-DoF PM with P32 (e).A prototype of the novel 5-DoF PM withP32 is built up from an existing prototype of the PM with 5 SPS type limbs in Yanshan University; see Figures 15(e) and 15(f). Here, each S pair can be transformed into U pair or R pair by adding constraint. Therefore, after an upper constraint and a lower constraint between two closed SPS limbs in the prototype of PM are added by utilizing four beam-blocks, the upper 2S pairs of the two closed SPS limbs are transformed into 3R pairs, in which one R is connected with m and is crossed with another two parallel R. Similarly, the lower 2S pairs of the two closed SPS limbs are transformed into 3R pairs, in which one R is connected with B and is crossed with other two parallel R. It is verified by experiment that this novel 5-DoF PM with P32 can be moved well.The DoFs of the 30 different 5-DoF PMs with different submechanisms synthesized from the 10 valid DGTs in Figures3(b)–15(c) are verified by utilizing (1) and listed in Table 2.Table 2
DoF verifications of the 5-DoF PMs with different submechanisms synthesized from the 10 valid DGTs in Figures3(b)–15(c).
Number
Figure
n
2
n
3
n
4
n
5
n
6
N
n
ζ
ν
L
i
s,
P
i
M,
L
i
1
5(b1)
14
2
0
0
0
8
9
0
0
2
L
s
2
2
5(b2)
14
2
0
0
0
8
9
0
0
2
L
s
2
3
5(b3)
14
2
0
0
0
8
9
0
0
2
L
s
2
4
5(b4)
14
2
0
0
0
8
9
0
0
2
L
s
2
5
5(b5)
14
2
0
0
0
8
9
0
0
L
s
3
6
5(b6)
14
2
0
0
0
9
10
0
0
L
s
3
7
6(c)
28
-
6
5
0
1
0
16
21
0
6
2
P
1
0
8
7(e1)
17
-
3
4
0
0
0
4
+
6
6
+
6
0
3
L
s
2
+
P
3
2
9
7(e2)
17
-
3
4
0
0
0
5
+
6
7
+
6
0
3
L
s
2
+
P
3
2
10
8(c1)
17
-
3
4
0
0
0
7
+
5
9
+
5
0
3
L
s
2
+
P
2
2
11
8(c2)
17
-
3
4
0
0
0
7
+
5
9
+
5
0
3
L
s
2
+
P
2
2
12
8(c3)
17
-
3
4
0
0
0
7
+
5
9
+
5
0
3
L
s
2
+
P
2
2
13
8(c4)
17
-
3
4
0
0
0
7
+
5
9
+
5
0
3
L
s
2
+
P
2
2
14
8(f1)
17
-
3
4
0
0
0
6
+
6
8
+
6
0
3
L
s
2
+
P
3
2
15
8(f2)
17
-
3
4
0
0
0
7
+
6
9
+
6
0
3
L
s
2
+
P
3
2
16
8(f3)
17
-
3
4
0
0
0
7
+
6
9
+
6
0
3
L
s
2
+
P
3
2
17
8(f4)
17
-
3
4
0
0
0
7
+
6
9
+
6
0
3
L
s
2
+
P
3
2
18
11(c1)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
19
11(c2)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
20
11(c3)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
21
11(e1)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
22
11(e2)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
23
11(e3)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
24
12(d1)
22
-
3
2
2
0
0
16
19
0
3
P
3
2
+
L
3
25
12(d2)
22
-
3
2
2
0
0
16
19
0
3
P
3
2
+
L
3
26
13(c)
25
-
6
4
2
0
0
16
20
0
6
P
1
1
+
P
3
2
27
13(f)
24
-
6
6
1
0
0
7
+
10
11
+
10
0
6
P
1
1
+
P
3
2
28
14(c)
20
-
6
6
0
0
0
4
+
12
7
+
12
0
6
2
P
3
2
29
14(f)
25
-
6
4
2
0
0
6
+
12
10
+
12
0
6
2
P
3
2
30
15(d)
22
-
3
2
2
0
0
14
17
3
3
P
3
2
M
=
6
N
-
n
-
1
+
∑
f
i
+
ν
-
ζ
=
5, N: the numbers of links; n: the numbers of pairs; ν: the numbers of redundant constraints; ζ: passive DoF; fi: local DoF of kinematic pair; ∑fi=3ns+2nu+2nc+np+nr; ns, nu, nc, np, and nr: the numbers of S, U, C, P, and R.


## 4.1. Characteristics and Functions of Submechanisms

LetLsi (1≤i≤3) be a subserial mechanism with i actuators. Let Li (2≤i≤4) be a subspatial i-DoF PM with i actuators. The submechanisms of the 5-DoF PMs may be Lsi, PMi (i≤2; M=1,2,3), and Li. Each of them has different merits. These merits can bring benefits for the 5-DoF PMs as follows.Wh300enLsi is taken as the limb for connecting m with B in the 5-DoF PMs, and the total number of the required limbs must be reduced. Thus, the interference between the limbs and m may be avoided easily, and the workspace of the 5-DoF PMs can be enlarged. In addition, since the other active limbs of the 5-DoF PMs can be transformed into the SPU-type linear active limbs which are not sensitive to manufacturing error, not only can the tiny self-motion of the PMs be removed effectively, but also the capability of the load bearing can be increased. When a spatial PM includes one subplanar mechanism or more, it may have merits [17] such as simplicity in structure, ease in manufacturing compliant mechanism and increasing the mechanical advantage, and being larger in capability of the pulling force and rotation angle. When different PMi (i≤2; M=1,2,3) and Li (2≤i≤4) are applied to constructing one limb of the 5-DoF PM, its stiffness can be increased.Generally, the above-mentioned submechanisms can be built into different standard units with high precision by a special company. Therefore, many novel 5-DoF PMs with high precision can be built and assembled easily by including these standard units.


## 4.2. Equivalent Limbs of Submechanisms

Similarly, each ofLsi (1≤i≤3) and Li (2≤i≤4) in the 5-DoF PMs can be replaced by its equivalent limb and represented by a line marked with Lsi or Li.The equivalent symbols of various kinematic pairs are represented in Figure5. Generally, each of Lsi in the 5-DoF PMs includes a lower link ll, an upper link lu, and i actuators which are connected in series from B to m; see Figure 5(a1). The equivalent limb of Lsi is represented by a line with Lsi; see Figure 5(a2). Lsi can be selected from an edge with 4 or 5 in the DTG. The number of actuators can be 2 or more. The actuators may be translational ones or rotational ones. The number of Lsi can be one or more. For instance, a DTG with 2t+14b and an array t1455t2554 is proposed to synthesize the 5-DoF PMs; see Figure 4(a3). The 6 different 5-DoF PMs with Lsi are synthesized using the proposed DTG; see Figure 5(b). Each of the 6 PMs has 3 limbs, the first 4 PMs include 2 Ls2, and the rest 2 PMs include 1 Ls3. Generally, if a DTG includes one closed loop which is formed by 4 links connected in series by 4J, then this DTG can be revised into an r-DTG. After that, the 5-DoF PMs with P10 can be synthesized by utilizing the r-DTG.Figure 5
DTG with2t+14b and array t1455t2554 (a) and the six 5-DoF PMs with serial mechanisms (b).For instance, a DTG withp+7t+28b and an array t1304t2430t3002t4242t5200t6034t7403p33433 is proposed for the type synthesis of the 5-DoF PMs; see Figure 6(a). Since the proposed DTG includes a closed loop which is formed by 4 links connected in series by 4J for constructing P10, the proposed DTG can be revised into an r-DTG with p+7t+(28-6)b and an array t1301t2130t3002t4242t5200t6031t7103p33433; see Figure 6(b). Thus, a novel 5-DoF PM with 2 P10 can be synthesized using the r-DTG; see Figure 6(c).Figure 6
A DTG withp+7t+28b (a), an r-DTG with p+7t+(28-6)b (b), and a 5-DoF PMs with 2 P10 (c).Based on condition(4) in Section 2, P11 can be selected from a DTG with a closed loop which is formed by 2t and 5b, and the equivalent limb of P11 is represented by a line with P11; see Figure 7(b). In addition, the DTG for synthesizing the 5-DoF PMs with P11 must be transformed into a revised DTG (r-DTG) by reducing ν=3 from the number of b in the closed chain for P11. For instance, a DTG with 4t+17b and an array t1132t2232t3245t4541 is proposed for the type synthesis of 5-DoF PMs; see Figure 7(a). Since the proposed DTG includes a closed loop which is formed by 2t+5b for constructing P11, the proposed DTG must be revised into an r-DTG with 4t+(17-3)b and an array t1120t2022t3245t4541; see Figure 7(b). Thus, two novel 5-DoF PMs with P11 and Ls2 are synthesized using the r-DTG; see Figure 7(c).Figure 7
DTG with4t+17b (a), r-DTG with 4t+(17-3)b (b), and two novel 5-DoF PMs with P11 and Ls2 (c1, c2).Based on condition(5) in Section 2, the equivalent limb of P22 can be constructed by utilizing a DTG with a closed loop which is formed by 2t+6b. The DTG for synthesizing 5-DoF PMs with P22 must be revised into an r-DTG by reducing ν=3 from the number of b in the closed loop for constructing P22. For example, a DTG with 4t+17b and an array t1233t2331t3144t4442 is proposed for the type synthesis of 5-DoF PMs; see Figure 8(a). Since the proposed DTG includes a closed loop formed by 2t and 7b, it must be revised into an r-DTG with 4t+(17-3)b and an array t1221t2121t3144t4442; see Figure 8(b). Thus, 4 novel 5-DoF hybrid PMs with P22 and Ls2 are synthesized by utilizing the r-DTG; see Figures 8(c1)–8(c4). Here, P22 and Ls2 can be represented by the lines marked by P22 and Ls2, respectively.Figure 8
DTG with4t+17b (a), an r-DTG with 4t+(17-3)b (b), four 5-DoF hybrid PMs with 1 P22 and 1 Ls2 (c1–c4), a DTG with 4t+17b (d), an r-DTG with 4t+(17-3)b (e), and four 5-DoF hybrid PMs with 1 P32 and 1 Ls2 (f1–f4).It is known that the number of the links in a spatial closed 3-DoF PM must be 9 or more. Based on condition(6) in Section 2, the equivalent limb of P32 can be constructed by utilizing a DTG with a closed loop which is formed by 2t and 7b and is represented by a line marked by P32; see Figure 8(g). In addition, the DTG for synthesizing 5-DoF PMs with P32 must be revised into an r-DTG by reducing ν=3 from the number of b in the closed loop for constructing P32. For example, a DTG with 4t+17b and an array t1134t2431t3144t4441 is proposed for the type synthesis of 5-DoF PMs; see Figure 8(d). Since the proposed DTG includes a closed loop formed by 2t and 7b, it must be revised into an r-DTG with 4t+(17-3)b and an array t1122t2221t3144t4441; see Figure 8(e). Thus, the 4 novel 5-DoF hybrid PMs with P32 and L2s are synthesized by utilizing the r-DTG with 4t+(17-3)b; see Figures 8(f1)–8(f4).A subspatial 3-DoF PML3 generally includes an upper platform lu, a lower platform ll, and 3 active limbs. Since ll and lu are required to have 4 connection points: three connection points for connecting with three active limbs and 1 for connection point (al and au) at the two ends of L3 for connecting with 5-DoF PMs, both ll and lu must be quaternary links. Therefore, the equivalent limb of L3 can be constructed using a DTG with 2q which are connected by 3 edges in parallel with 12b. The 8 different DTGs for the type synthesis of various 3-DoF PMs can be obtained; see Figure 9(a). Their equivalent limb is represented by a line with L3; see Figure 9(b). If L3 has symmetry in structure, the first DTG in Figure 9(a) can be used for synthesizing symmetry L3, and the 26 different symmetry kinematic chains [16] can be applied to the construction of the 26 different symmetry L3. When L3 is asymmetry in structure, the other 7 DTGs in Figure 9(a) can be used for synthesizing asymmetry L3, and more asymmetry kinematic chains can be obtained. Thus, many novel 5-DoF PMs can be synthesized.Figure 9
The 8 different DTGs ofL3 (a1–a8) and their equivalent limb (b).A sub-4-DoF PML4 generally includes an upper platform lu, a lower platform ll, and 4 active limbs; see Figure 10. Since ll and lu are required to provide 5 connection points: 4 connection points are used for connecting with 4 active limbs and 1 is used as the connection point (al and au) at the 2 ends of L4 for connecting with 5-DoF PMs, both ll and lu must be pentagonal links. Therefore, the equivalent limb of L4 can be constructed using a DTG with 2p which are connected by 4 edges in parallel with 18b. The 14 different DTGs with 2q+18b for synthesizing various 4-DoF spatial PMs can be obtained. Three different promised DTGs are proposed for synthesizing 7 different subspatial 4-DoF PMs; see Figures 10(a)–10(c). Their equivalent limb is represented by a line with L4; see Figure 10(d). Thus, many novel 5-DoF PMs can be synthesized.Figure 10
The 3 DTGs ofL4 ((a), (b), and (c)) and the 9 subspatial 4-DoF PM L4 and their equivalent limb (d).For instance, a DTG with2p+24b and an array p45456p65454 is proposed for the type synthesis of the 5-DoF PMs with L4; see Figure 11(a). Two equivalent mechanisms I and II with L4 are derived from the proposed DTG; see Figures 11(b) and 11(d). Three novel 5-DoF PMs with the p-type base are synthesized from the equivalent mechanism I; see Figures 11(c1)–11(c3). Three novel 5-DoF PMs with the p-type moving platform are synthesized from the equivalent mechanism II; see Figures 11(e1)–11(e3). In fact, some novel 5-DoF PMs with PMi and Li or their combinations can be synthesized. Some examples are illustrated as follows.Figure 11
A DTG with2p+24b (a), equivalent mechanism I with L4, (b), 3 novel 5-DoF PMs with p-type base (c1–c3), equivalent mechanism II with L4 (d), and 3 novel 5-DoF PMs with p-type moving platform (e1–e3).Example 1.
A DTG with2q+2t+22b and an array t1134t2432q12444q24441 is proposed for the type synthesis of the 5-DoF PMs with L3 and P32; see Figure 12(a). Since the proposed DTG includes both one parallel closed loop formed by 2q+12b and one closed loop formed by 2t+9b, it must be revised into an r-DTG with 2q+2t+(22-3)b and an array t1122t2222q12444q24441; see Figure 12(b). An equivalent mechanism with 1 L3 and 1 P32 is constructed from the r-DTG; see Figure 12(c). Finally, two 5-DoF PMs are synthesized from equivalent mechanism; see Figures 12(d1) and 12(d2).Figure 12
A DTG with2q+2t+22b (a), an r-DTG with 2q+2t+(22-3)b (b), equivalent mechanism with both L3 and P32 (c), two novel 5-DoF PMs (d1, d2).Example 2.
A DTG with2q+2t+25b and an array t1034t2431q11451t3123t4321q21540 is proposed for synthesizing the 5-DoF PMs with P11 and P32; see Figure 13(a). Since the proposed DTG includes both one closed loop chain formed by 2t and 9b and one closed loop chain formed by 2t and 7b, it must be revised into an r-DTG with 2q+2t+(25-6)b and an array t1022t2211q11451t3102t4201q21540; see Figure 13(b). Thus, the equivalent mechanism of a novel 5-DoF PM with 1 P11 and 1 P32 (see Figure 13(c)) can be constructed from the r-DTG in Figure 13(b). Similarly, a DTG with q+6t+24b and an array t1011t2144t3123t4321t5134t6430q1441 is proposed for synthesizing the 5-DoF PMs with P11 and P32; see Figure 13(d). It can be revised into an r-DTG with q+6t+(24-6)b and an array t1011t2144t3102t4201t5122t6220q1441; see Figure 13(e). Thus, the equivalent mechanism of a novel 5-DoF PM with 1 P11 and 1 P32 is constructed from the r-DTG in Figure 13(e); see Figure 13(f) (q+6t+(24-6)b (e), equivalent mechanism with P11 and P32 (f)).Figure 13
A DTG with2q+2t+25b (a), an r-DTG with 2q+2t+(25-6)b (b), equivalent mechanism with P11 and P32 (c), DTG with q+6t+24b (d), and r-DTG with P11 and P32 (f).Example 3.
A DTG with6t+20b and an array t1040t2034t3431t4141t5134t6430 is proposed for synthesizing the 5-DoF PMs with 2 P32; see Figure 14(a). Since the proposed DTG includes 2 closed loop chains formed by 2t and 9b, it must be revised into an r-DTG with 6t+(20-6)b and an array t1040t2022t3221t4141t5122t6220; see Figure 14(b). Finally, the equivalent mechanism of a novel 5-DoF PM with 2 P32 is constructed from the r-DTG in Figure 14(b); see Figure 14(c). Similarly, a DTG with 2q+4t+25b and an array t1134t2430q10450t3034t4431q21541 is proposed for synthesizing the 5-DoF PMs with 2 P32; see Figure 14(d). Since the proposed DTG includes 2 closed loop chains formed by 2t and 9b, it must be revised into an r-DTG with 2q+4t+(25-6)b and an array t1122t2220q10450t3022t4221q21541; see Figure 14(e). Finally, the equivalent mechanism of a novel 5-DoF PM with 2 P32 is constructed from the r-DTG in Figure 14(e); see Figure 14(f).Figure 14
A DTG with6t+20b (a), an r-DTG with 6t+(20-6)b (b), equivalent mechanism with 2 P32 (c), a DTG with 2q+4t+25b (d), an r-DTG with 2q+4t+(25-6)b (b), and equivalent mechanism with 2 P32 (f).Example 4.
A DTG with2q+2t+22b and an array t1043t2430q10555q25550 is proposed for the type synthesis of the 5-DoF PMs with P32; see Figure 15(a). Since the proposed DTG includes one closed loop formed by 2t+7b and one parallel closed loop formed by 2q+15b, it must be revised into an r-DTG with 2q+2t+(22-3)b and an array t1022t2220q10555q25550; see Figure 15(b). An equivalent mechanism with 1 P32 is constructed from the r-DTG; see Figure 15(c). Finally, one 5-DoF PM is synthesized from equivalent mechanism; see Figure 15(d).Figure 15
A DTG with2q+2t+22b (a), an r-DTG with 2q+2t+(22-3)b (b), a novel 5-DoF PM with P32 (c), equivalent mechanism of novel 5-DoF PM with P32 (d), and prototype of novel 5-DoF PM with P32 (e).A prototype of the novel 5-DoF PM withP32 is built up from an existing prototype of the PM with 5 SPS type limbs in Yanshan University; see Figures 15(e) and 15(f). Here, each S pair can be transformed into U pair or R pair by adding constraint. Therefore, after an upper constraint and a lower constraint between two closed SPS limbs in the prototype of PM are added by utilizing four beam-blocks, the upper 2S pairs of the two closed SPS limbs are transformed into 3R pairs, in which one R is connected with m and is crossed with another two parallel R. Similarly, the lower 2S pairs of the two closed SPS limbs are transformed into 3R pairs, in which one R is connected with B and is crossed with other two parallel R. It is verified by experiment that this novel 5-DoF PM with P32 can be moved well.The DoFs of the 30 different 5-DoF PMs with different submechanisms synthesized from the 10 valid DGTs in Figures3(b)–15(c) are verified by utilizing (1) and listed in Table 2.Table 2
DoF verifications of the 5-DoF PMs with different submechanisms synthesized from the 10 valid DGTs in Figures3(b)–15(c).
Number
Figure
n
2
n
3
n
4
n
5
n
6
N
n
ζ
ν
L
i
s,
P
i
M,
L
i
1
5(b1)
14
2
0
0
0
8
9
0
0
2
L
s
2
2
5(b2)
14
2
0
0
0
8
9
0
0
2
L
s
2
3
5(b3)
14
2
0
0
0
8
9
0
0
2
L
s
2
4
5(b4)
14
2
0
0
0
8
9
0
0
2
L
s
2
5
5(b5)
14
2
0
0
0
8
9
0
0
L
s
3
6
5(b6)
14
2
0
0
0
9
10
0
0
L
s
3
7
6(c)
28
-
6
5
0
1
0
16
21
0
6
2
P
1
0
8
7(e1)
17
-
3
4
0
0
0
4
+
6
6
+
6
0
3
L
s
2
+
P
3
2
9
7(e2)
17
-
3
4
0
0
0
5
+
6
7
+
6
0
3
L
s
2
+
P
3
2
10
8(c1)
17
-
3
4
0
0
0
7
+
5
9
+
5
0
3
L
s
2
+
P
2
2
11
8(c2)
17
-
3
4
0
0
0
7
+
5
9
+
5
0
3
L
s
2
+
P
2
2
12
8(c3)
17
-
3
4
0
0
0
7
+
5
9
+
5
0
3
L
s
2
+
P
2
2
13
8(c4)
17
-
3
4
0
0
0
7
+
5
9
+
5
0
3
L
s
2
+
P
2
2
14
8(f1)
17
-
3
4
0
0
0
6
+
6
8
+
6
0
3
L
s
2
+
P
3
2
15
8(f2)
17
-
3
4
0
0
0
7
+
6
9
+
6
0
3
L
s
2
+
P
3
2
16
8(f3)
17
-
3
4
0
0
0
7
+
6
9
+
6
0
3
L
s
2
+
P
3
2
17
8(f4)
17
-
3
4
0
0
0
7
+
6
9
+
6
0
3
L
s
2
+
P
3
2
18
11(c1)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
19
11(c2)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
20
11(c3)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
21
11(e1)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
22
11(e2)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
23
11(e3)
24
0
0
2
0
3
+
10
4
+
12
0
0
L
4
24
12(d1)
22
-
3
2
2
0
0
16
19
0
3
P
3
2
+
L
3
25
12(d2)
22
-
3
2
2
0
0
16
19
0
3
P
3
2
+
L
3
26
13(c)
25
-
6
4
2
0
0
16
20
0
6
P
1
1
+
P
3
2
27
13(f)
24
-
6
6
1
0
0
7
+
10
11
+
10
0
6
P
1
1
+
P
3
2
28
14(c)
20
-
6
6
0
0
0
4
+
12
7
+
12
0
6
2
P
3
2
29
14(f)
25
-
6
4
2
0
0
6
+
12
10
+
12
0
6
2
P
3
2
30
15(d)
22
-
3
2
2
0
0
14
17
3
3
P
3
2
M
=
6
N
-
n
-
1
+
∑
f
i
+
ν
-
ζ
=
5, N: the numbers of links; n: the numbers of pairs; ν: the numbers of redundant constraints; ζ: passive DoF; fi: local DoF of kinematic pair; ∑fi=3ns+2nu+2nc+np+nr; ns, nu, nc, np, and nr: the numbers of S, U, C, P, and R.


## 5. Conclusions

The conditions for the type synthesis of 5-DoF parallel mechanisms with subserial or subparallel mechanisms are determined. The 31 valid DTGs (digital topology graphs) and revised DTGs can be derived from 17 different spatial 5-DoF associated linkages.The subplanar mechanisms and/or parallel mechanisms in the 5-DoF parallel mechanisms can be transformed into simple equivalent limbs, and their equivalent relations and merits are determined.The 30 different 5-DoF parallel mechanisms with different subserial or parallel mechanisms can be synthesized by utilizing the valid DTGs and revised DTGs. They can be simplified by replacing complicated subparallel mechanisms with their simple equivalent limbs.The DoFs of all the synthesized parallel mechanisms are verified to be correct.Many novel 5-DoF parallel mechanisms with subserial or parallel mechanisms can be synthesized by utilizing different valid DTGs, or by varying the order of the kinematic pairs and the orientations of the kinematic pairs.



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*Source: 6021702-2018-02-15.xml*