A class of novel 2T2R and 3T2R parallel mechanisms with large decoupled output rotational angles

Mechanism and Machine Theory 114 (2017) 156–169

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Research paper

A class of novel 2T2R and 3T2R parallel mechanisms with large decoupled output rotational angles Xiaodong Jin, Yuefa Fang∗, Haibo Qu, Sheng Guo Department of Mechanical Engineering, Beijing Jiaotong University, Beijing 100044, PR China

a r t i c l e

i n f o

Article history: Received 11 October 2016 Revised 17 March 2017 Accepted 5 April 2017

Keywords: Parallel mechanism Type synthesis Articulated moving platform Rotational capability Parallelogram Lie group theory

a b s t r a c t This article focuses on the synthesis and analysis of a class of novel 2T2R (T denotes Translation and R denotes Rotation) and 3T2R parallel mechanisms (PMs) with large decoupled output rotational angles. Two articulated moving platforms (AMPs) composed by revolute joints are proposed. A common parallelogram and an evolution parallelogram which are used to orient the rotation axes of the AMPs are designed and analyzed. By means of Lie group theory, the limbs connected in the AMPs are enumerated and two types of 2T2R PMs are synthesized. Then, based on the proposed 2T2R PMs, a family of 3T2R PMs are obtained through adding a translational degree of freedom (DOF). The fact that the orthogonal arrangement of the revolute joints in the AMPs eliminates the interference between the two rotations guarantees the high rotational performance of the manipulators. Finally, the inverse kinematics of the example PM is conducted and the workspace is obtained to illustrate the high rotational capability. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction PMs with lower-mobility have the advantages of simple architecture, lower cost of manufacturing compared with its 6DOF counterpart. Meanwhile, it is suitable for many tasks requiring less than 6-DOF. Therefore, synthesis and analysis about lower-mobility PMs have drawn extensive interests. Many lower-mobility PMs are proposed and applied to productions, such as the famous Delta robot [1], the PMs for ankle rehabilitation [2,3], the asymmetrical 2T3R PM for surgical operation [4], and parallel manipulators applied in bionics of spinal column [5]. However, the traditional parallel mechanisms suffer from problems of small workspace, limited rotational angles. To overcome the drawbacks of the traditional PMs, researchers focus on the design and analysis of the generalized PMs, which are not limited to the definition that the rigid moving platform is connected with the base with multiple independent kinematic chains. Scholars make a breakthrough from two aspects. One is to design spatial mechanisms with coupling chains [6–8], which have higher rigidity and greater load-carrying capacity. The other is to construct PMs with non-rigid moving platform to obtain larger workspace and perform specific objective [9–11]. There are two types of PMs with non-rigid moving platforms. One is the PM with configurable platform, which was studied by Yi [12] first and generalized by Gosselin [13] in 2005. A distinct characteristic for the PM with configurable platform is that a closed-loop chain is used as the end-effector, which can be used to grasp irregular and large objects. The other kind is the PMs with articulated moving platform. Typically the articulated moving platform (AMP) can be seen as an

∗

Corresponding author. E-mail addresses: [email protected] (X. Jin), [email protected] (Y. Fang), [email protected] (H. Qu), [email protected] (S. Guo).

http://dx.doi.org/10.1016/j.mechmachtheory.2017.04.003 0094-114X/© 2017 Elsevier Ltd. All rights reserved.

X. Jin et al. / Mechanism and Machine Theory 114 (2017) 156–169

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open-loop parallel combination of rigid bodies and joints. Since an APM has some relatively independent DOF, it can greatly reduce the interference between the platform and limbs, as a result, achieve larger workspace, especially the orientationworkspace. The H4 mechanisms series [14–17] is the typical example of using the AMP. Since different components in the articulated platform have relative rotations which are amplified by the gear system or other amplifying mechanisms, the H4 series possess large rotational angles. With the similar principle, Sun [18,19] designed an AMP using the helix pairs and synthesized a class of 3R1T PMs. Wang and Fang [20,21] synthesized four types of AMPs using revolute joints only, based on which obtained a family of 2T2R, 3T2R, 3T3R PMs with high rotational capability. The reason why the PMs with AMPs possess relatively large rotational angles is that these PMs obtained the property of partially-decoupled through adding coaxial kinematic pairs in the platform. Based on the idea of partially-decoupled, Liu [22] designed a class of two to six DOFs PMs in which the three DOFs possess the advantage of high rotational capability. Using the parallelograms to realize the kinematic decoupling, Salgado [23,24] presented two types of 4-DOF PM which can generate the Schönflies motion. Based on the theory of linear transformations and the evolutionary morphology, Gugo [25] proposed a family of kinematic-decoupled and fully-isotropic PMs, whose rotations are generated by the relative motion of two groups of limbs. Synthesis approaches of PMs include the enumeration approach based on the general Chebyshev–Grübler–kutzbach mobility formula [26], the constraint approach based on the reciprocal screw theory [27,28], the synthesis method based on the Lie group theory [29,30]. The reciprocal screw theory is effective for the synthesis of the symmetrical and isotropic PMs, such as the symmetrical lower-mobility PMs presented by Huang [31] and the 4&5-DOF PMs with identical limb structures proposed by Fang [32]. The Lie group theory is more intuitive to characterize the limbs structure, as a result, is a convenient method to synthesize the asymmetrical and non-isotropic PMs. For example, Refaat and Hervé [33] Proposed four families of 3-DOF translational-rotational PMs which are asymmetric by Lie group theory. Fan [34] obtained a large number of 2T2R, 1T2R and 2R PMs based on the integration of configuration evolution and Lie group theory. This paper devotes to synthesize and analyze a class of parallel mechanisms with large orientation-workspace. Two kinds of AMPs constructed by revolute joints only are designed, which guarantee the proposed PMs to be partially decoupled. The common parallelogram is used and an evolution is created to orient the rotation axes of the AMPs. By means of Lie group theory, the basic limb structure to prompt a rotation is deduced and the non-identical limb structures connected with the AMPs are obtained. Using the proposed limbs, two types of 2T2R and a type of 3T2R PMs are presented. The PMs presented in this paper are constructed only by lower pairs with single DOF to reduce the interference and guarantee the high rotational performance. The proposed PMs possess large workspace and high rotational capability, which are proved by the inverse kinematics and the workspace analysis. The structure of this article is as follows. Section 2 introduces the concept of the Lie group theory and proposes two kinds of AMPs as well as a common and an evolution parallelogram. The motion property of the parallelograms is analyzed in this part. Two types of 2T2R PMs are synthesized and their limb structures are listed by means of Lie group theory in the Section 3. Then based on the proposed 2T2R PMs, a class of 3T2R PMs is obtained in Section 4 through adding a translational degree of freedom. Section 5 conducts the inverse kinematics of the example PMs and depicts the constantorientation workspace and the orientation-workspace to analyze the performance of the proposed PMs. Finally, conclusion is presented in Section 6. 2. Design of two novel AMPs and two parallelograms 2.1. Basic conceptions about lie group theory The set of 6-dimensional rigid motion can be endowed with the algebraic structure of a group, represented by {D} as Lie group. Further any motion of a rigid body can be described by a subset of {D}, which may be either a group, called a displacement subgroup (DSG) or a displacement submanifold (DSM). According to Hervé [35], {T} denotes the set of 3-DOF translation, {T2 (u)} denotes 2-DOF planar translation whose plane is perpendicular to u, {T(u)} represents linear translations parallel to u, and {R(N,u)} is the representation of one-dimensional rotational subgroup, in which (N, u) represents the axis determined by the unit vector u and point N. In other words, {T(u)} and {R(N,u)} are associated with the lower pairs of prismatic pairs P and revolute pairs R. The displacement set of a limb is the product of DSGs of all pairs in this limb. The product of groups is closed [36,37]. Assume that the rigid bodies constructed a limb in a parallel mechanism are 1, 2, 3, … , j-1, j in turn and the DSGs or DSMs (DSGs/DSMs) of corresponding pairs are {D1 },{D2 },{D3 },…, {Dj +1 }, the DSG/DSM of the end of the limb is the product of all the DSGs/DSMs, i.e.

{Li } = {D1 }{D2 }{D3 } · · · {D j+1 }

(1)

where {Li } is the DSG/DSM of the end of the limb. The intersection of two subgroups is always a subgroup. The DSG/DSM of the moving platform of a parallel mechanism is the intersection of the DSGs/DSMs of all limbs, i.e.

{M } =

n 

{L i }

i=1

where {M} denotes the DSG/DSM of the moving platform.

(2)

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X. Jin et al. / Mechanism and Machine Theory 114 (2017) 156–169

R2j

AL2

j

i

R2j

R2i i2

R3j

R1j

R3j

AL3

AL1 j2(3)

R1j

j2(3)

R1i

j1

j1

i3 R3i

i1 (a) one axis configuration about j

(b) two axes configuration about i, j k R1k

R2i

i

AL2

i2

R1k k1

i1 R2k

R2k

AL1

k2 (c) one axis configuration about k

R3i

k1

R1i

k2

i3

(d) two axes configuration about i, k

Fig. 1. Two kinds of AMPs.

Assume that {A} and {B} are two DSGs or DSMs included in a subgroup {Q}, i.e.{A}⊆{Q},{B}⊆{Q}, and Dim({A}{B}) = Dim{Q}, it can be deduced that the product {A}{B} is equivalent to subgroup {Q}, i.e. {A}{B} = {Q}. 2.2. Two kinds of AMPs In order to achieve large rotational angles about two independent axes, only revolute joints are used to construct the AMPs instead of using universal, spherical joints depending on the higher displacement range of the revolute joints. The construction of AMPs is divided into two steps. A group of revolute joints are combined first to provide rotation axes parallel to j, as shown in Fig 1(a). In other words, after connecting appropriate limbs with R1 j , R2 j , R3 j , the platform is capable of rotating about any axis which is parallel to j within the workspace with a possible large angles. Generalizations about this step is to construct 1-dimension unfixed rotation axis through two groups of parallel rotating axes, which are provided by two groups of parallel revolute joints. Next is to create a rotation axis about i by using another group of revolute joints. As shown in Fig 1(b), three revolute pairs R1 i , R2 i , R3 i are connected to R1 j , R2 j , R3 j orthogonally and the link connecting Rxi and Rxj is defined as Articulated Link x (ALx ), where x = 1, 2, 3, …, n. A notable feature for the orthogonal arrangement of the rotational axes is that it can maximum reduce interference between these two rotations, which simplifies the synthesis process and realizes the partially decoupled control. With the same method, another kind of AMP is synthesized as illustrated in Fig 1(c) and (d), where the platform possesses the capability of rotating about i and k axes. We name the limb which is connected by ALx and the subchain connected in Rx i as Lx . Since all jx provided by Rx j are located in the platform and parallel to each other inherently, it is relatively simple to construct L1 , L2 , L3 to ensure the platform rotate about any axis parallel to j. However, since ix axes provided by joints R1 i , R2 i , R3 i are not parallel constantly due to the movement of R1 j , R2 j , R3 j and R1 k , R2 k , the rotation about i1 would be deprived if the parallel relationship is broken. Therefore, some constraints should be applied to maintain such parallel relations. In addition, since only i1 provided by R1 i passes through the platform and i2 , i3 provided by R2 i , R3 i are beyond the platform, i1 is selected as the only rotational axis for the platform’s i-direction rotation. In other words, the rotational axis along i-direction is unique about i1 . 2.3. Analysis of the planar parallelogram To restrict i1 provided by R1 i to parallel with i2 , i3 produced by R2 i and R3 i , we orient the link AL1 to make it unable to rotate about R1 j and R1 k . In general, only translations can be implemented in AL1 , but no rotations. According to Ref [38], the planar parallelogram is an effective choice due to its special motion property — one translation output — which allows the output link to remain at a fixed orientation with respect to the input link. A common parallelogram is shown in

X. Jin et al. / Mechanism and Machine Theory 114 (2017) 156–169

Output link

159

Output link

Pa2

Subordinate link Intput link Intput link

Base

Pa1

Base

(a)

(b)

Fig. 2. Planar parallelograms: (a) a common parallelogram; (b) an evolution parallelogram.

Output link

R2j

R1i

R3j

Output link

R1i

R1i

R2i Subchain

R3i

R2i (a)

R2i (b)

(c)

Fig. 3. Simplified models.

Fig 2(a). The motion of the parallelogram is a 1-D translational submanifold included in the 2-D planar translational subgroup {T2 (w)} and the 3-D planar subgroup {G(w)}. As a result, the displacement of the output link can be written as {T (u)}, in which u is a vector that represents the translational direction. An evolution parallelogram can be obtained through connecting two common parallelograms in series as illustrated in Fig 2(b). The first parallelogram, named as Pa1 , associates with the submanifold of {T (u)} and the subordinate one, named as Pa2 , associates with the submanifold of {T (v)}. It should be mentioned that u and v are two vectors which are intersect but not coincide in the general position and w is a vector that perpendicular to the plane determined by uv. Obviously, {T (u)}⊆{T2 (w)}, {T (v)}⊆{T2 (w)} and Dim({T (u)}{T (v)}) = Dim{T2 (w)}. As a result, {T (u)}{T (v)} is equivalent to {T2 (w)}, which can be written as {T (u)}{T (v)} = {T2 (w)}. This deduction illustrates that the evolution parallelogram output a 2-D planar translation subgroup, although both of the two substructures are translational submanifolds. If connecting these parallelograms with AMP1 and AMP2, AL1 will loss the rotations about R1 j and R1 k and be maintained the orientation with respect to the base.

3. Type synthesis of 2T2R PMs with AMPs 3.1. The basic limb structure for one rotational mobility For the articulated moving platforms shown in Fig 1, a striking characteristic is that rotations about i and j are orthogonal and relatively independent, which provides the limb structure analysis a shortcut that the rotation about an axis can be simplified as a planar rotation. For example, it can be seen as planar displacement when the AMP rotating about i1 provided by R1 i , this rotation can be simplified as a 2-dimensional model and R1 i can be seen as fixed as shown in Fig 3(a). Due to the symmetry, this model can be further simplified as Fig 3(b), where R2 i is used to connect a subchain to provide the output link a rotation about R1 i .

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Ti

Rj

Ri

Rk

G(j)

G(i)

G(k)

Tk

Tk

Tj

Tj

Ti

Fig. 4. Relations between rotational directions and planar motion subgroups.

Since only basic lower pairs (revolute pairs and prismatic pairs) are used in the PMs presented in this paper, the formula of DOF of the output link shown in Fig 3(c) is:

F = 3n − 2 pl

(3)

where F denotes DOF of the output link, n the number of movable links, pl the number of the lower pairs and here n = pl − 1. It is known that the output link rotates about R1 i and its DOF is 1, as a result, F = 1.

1 = 3 ( pl − 1 ) − 2 pl

(4)

It turns out that pl = 4, which means four lower pairs are needed to guarantee the output link to rotate about R1 i . The Lie group expression associated with the output link can be written as

{R(N1 , i )} = {R(N1 , i )} ∩ {MSCi }{R(N2 , i )}

(5)

where {R(N1 ,i)} and {R(N2 ,i)} is the Lie group expressions associated with joints R1 i and R2 i , {MSC i } the Lie group expression of the subchain shown in Fig 3(c). Because there are four lower pairs in the closed-loop chain and R1 i and R2 i are counted as two, as a result, the subchain should be connected by two lower pairs. The possible subchains’ groups and their corresponding joints structures are listed as

⎧ {R(N3 , i )}{R(N4 , i )} → R3i R4i ⎪ ⎪ ⎪ ⎨{T (k )}{R(N , i )} → P R 3 k 3i {MSCi } = ⎪ { R ( N , i ) }{ T ( k ) } → R 3 3i Pk ⎪ ⎪ ⎩ {T ( j )}{T (k )} → Pj Rk

(6)

where {R(N3 ,i)}, {R(N4 ,i)} associate with the parallel revolute joints in the subchain, N3 , N4 denote the points on the joints axes and {T(j)}, {T(k)} associate with the prismatic joints in the subchain. For the limb constituted by the subchain and R2 i ,

{MSCi }{R(N2 , i )} = {R(N4 , i )}{R(N3 , i )}{R(N2 , i )} = {T (k )}{R(N3 , i )}{R(N2 , i )} = {R(N3 , i )}{T (k )}{R(N2 , i )} = {T (k )}{T ( j )}{R(N, i )} = {G(i )}

(7)

It can be thrown up a result form Eq. (7) that a planar motion {G(i)} should be included in the limbs displacement groups. Similarly, for the rotation about j and k, the planar motions {G(j)} and {G(k)} are needed respectively. As a result, the rotational direction of a platform is closely relevant to the planar motion subgroup. As illustrated in Fig 4, Ri , Rj , Rk denote the rotational mobility needed for a platform, {G(i)}, {G(j)}, {G(k)} are the corresponding motion subgroup needed in the relevant limbs and Ti , Tj , Tk are the translational DOFs included in the limbs. 3.2. Two types of 2T2R PMs According to Fig 4, if the platform would rotate about i and j axes, corresponding to AMP1, L2 and L3 have to include translational DOFs of Tj , Tk and L1 have to include translational DOFs of Ti , Tk . If the platform would rotate about i and k axes, corresponding to AMP2, L2 and L3 still have to include Tj , Tk and L1 includes DOFs Ti , Tj . It can be observed that Tk is existed in the situation of rotating about i&j and Tj is shared in the situation of rotating about i&k. Thus, when synthesizing the 2T2R PMs, we first let the 2-D translational DOFs are Ti &Tk for AMP1 and Ti &Tj for AMP2, which can obtain the first kind of 2T2R PMs. Another case is assuming the 2-D translational DOFs are Tj &Tk for the both AMPs, which can synthesize the second type of 2T2R PMs. 3.2.1. The first kind of 2T2R PMs Providing the translations are along i&k axes for the PMs with AMP1 and i&j axes for the PMs with AMP2, the Lie group expressions associated with the end-effectors can be written as

{M} = {T (i )}{T (k )}{R(N, i )}{R(N, j )} for AMP1 {M} = {T (i )}{T ( j )}{R(N, i )}{R(N, k )} for AMP2

(8)

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Table 1 Feasible limbs and the 2T2R PMs. AMP type

Number

L1s L11

L12

AMP1

2Tik 2Rij -1 2Tik 2Rij -2 2Tik 2Rij -3 2Tik 2Rij -4 2Tik 2Rij -5

Pi Pa

Pi Pa Pi Rj Rj P# a Rj Rj Rj Pi Pt

2Tij 2Rik -1 2Tij 2Rik -2 2Tij 2Rik -3 2Tij 2Rik -4 2Tij 2Rik -5

Pi Pa

AMP2

P# a Pi Pt

P# a Pi Pt

Pi Pa Pi Rk Rk P# a Rk Rk Rk Pi Pt

L2s

L3s

PM

Pj Pi a Ri

Pj Pi a Ri

Ri Pi a Ri

Ri Pi a Ri

Pj Pi a Pt

Pj Pi a Pt

[2Pi Pa -2Pj Pi a Ri ]-AMP1 [[Pi Pa -Pi Rj Rj ]−2Pj Pi a Ri ]-AMP1 [2P# a -2Ri Pi a Ri ]-AMP1 [[P# a -Rj Rj Rj ]−2Ri Pi a Ri ]-AMP1 [2Pi Pt -2Pj Pi a Pt ]-AMP1

Pj Pi a Ri

Pj Pi a Ri

Ri Pi a Ri

Ri Pi a Ri

Pj Pi a Pt

Pj Pi a Pt

[2Pi Pa -2Pj Pi a Ri ]-AMP2 [[Pi Pa -Pi Rk Rk ]−2Pj Pi a Ri ]-AMP2 [2P# a -2Ri Pi a Ri ]-AMP2 [[P# a -Rk Rk Rk ]−2Ri Pi a Ri ]-AMP2 [2Pi Pt -2Pj Pi a Pt ]-AMP2

Note that Pa represents the common parallelogram which is used to orient the rotation axis, Pi a is the common parallelogram which provides a translational mobility for the limb and P# a denotes the evolution parallelogram. The displacement directions are denoted as subscripts for the R and P pairs. The pair symbols with underline denote the drive joints.

Besides,

{M } = {ML1 } ∩ {ML2 } ∩ {ML3 }

(9)

where {ML 1 }, {ML 2 } and {ML 3 } are the group expressions of the limbs and Lx denotes the limb connected in Rx j (x = 1, 2, 3). According to above analysis, {G(i)} is needed for L2 and L3 to guarantee the rotation about i1 , as a result, {T(j)} should be multiplied into the first formula of Eq. (8) and {T(k)} should be multiplied into the second one. Therefore, {ML 2 } and {ML 3 } can be deduced from Eq. (8):

{ML2 } = {ML3 } = {T (i )}{G(i )}{R(N, j )} = {X (i )}{R(N, j )} for AMP1 {ML2 } = {ML3 } = {T (i )}{G(i )}{R(N, k )} = {X (i )}{R(N, k )} for AMP2

(10)

It is best to construct four limbs for the 4-DOF PMs, considering the structure of AL1 , we let {ML 1 } be constructed by two subsets, i.e.

{ML1 } = [{ML11 } ∩ {ML12 }] ∪ [{R(N, i )}{R(N, j )}] for AMP1 {ML1 } = [{ML11 } ∩ {ML12 }] ∪ [{R(N, i )}{R(N, k )}] for AMP2

(11)

where,

{ML11 } ∩ {ML12 } = {T (i )}{T (k )} = {T2 ( j )} for AMP1 {ML11 } ∩ {ML12 } = {T (i )}{T ( j )} = {T2 (k )} for AMP2

(12)

Thus, the limb connected in R1 i output planar translation. There are two choices for {ML 11 } and {ML 12 } to construct this motion. If the two DSGs are both planar translations, the intersection of these groups will still be planar translation, i.e. {ML 11 } = {ML 12 } = {T2 (j)} for AMP1 and {ML 11 } = {ML 12 } = {T2 (k)} for AMP2. There are three kinds of subchain structures feasible for L11 and L12 , which can be denoted as Pi Pa , P# a , Pi Pt for AMP1 and Pi Pa , P# a , Pi Pt for AMP2. If L11 is planar translation and L12 is planar motion, i.e., {ML 11 } = {T2 (j)}, {ML 12 } = {G(j)} for AMP1 and {ML 11 } = {T2 (k)}, {ML 12 } = {G(k)} for AMP2, the intersection of {ML 11 } and {ML 12 } is planar translation. There are two kinds of feasible structures for L12 corresponding to such group combination, i.e. Pi Rj Rj , Rj Rj Rj for AMP1 and Pi Rk Rk , Rk Rk Rk for AMP2, and structure of L11 is remain unchanged compared with the previous situation. The feasible subchains of L11 and L12 are listed in Table 1, where the 1-D translation and the 2-D planar translation are constructed by the common parallelogram and the evolution parallelogram respectively for the sake of orientating AL1 . In Table 1, Lx s denotes the subchain connected in the joint Rx i and L1s is synthesized by L11 and L12 . Based on Eq (10), there are three types of combinations for L2s and another three identical structures for L3s . When connecting L1s , L2s , L3s with the AMPs, a class of 2T2R PMs can be obtained. In order to keep isomorphism for the PMs as much as possible, ten kinds of limb collocations are selected, as a result, ten novel PMs are taken shape as illustrated in Table 1, five of which are PMs with AMP1 and the other five are PMs with AMP2. The CAD models of the example PMs are drawn in Fig 5. For the [[2Pi Pt -2Pj Pi a Pt ]-AMP1 PM and the [[2Pi Pt -2Pj Pi a Pt ]-AMP2 PM, it can be observed that there is no need for parallelograms to guarantee the orientation of AL1 depending on their own structures. As a result, these two PMs are relatively simple in manufacturing and assembly. 3.2.2. The second kind of 2T2R PMs For the first kind of 2T2R PMs, their translations are along i&k axes and i&j axes, which depend on the 3-dimension translations of L2 and L3 . Now we assume that L1 outputs 3-D translations but L2 and L3 output 2-D translations along j&k

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Fig. 5. CAD models of the first type of 2T2R PMs. Table 2 Feasible limbs and the 2T2R PMs. AMP type

Number

L1s

L2s

L3s

PM

AMP1

2Tjk 2Rij -1 2Tjk 2Rij -2 2Tjk 2Rij -3

Pi Ri Pa Ri P# a Pi Ri Pt

Pj Ri Ri Ri Pj Pt

Pj Ri Ri Ri Pj Pt

[Pi Ri Pa -2Pj Ri ]-AMP1 [Ri P# a -2Ri Ri ]-AMP1 [2Pi Ri Pt -2Pj Pt ]-AMP1

AMP2

2Tjk 2Rik -1 2Tjk 2Rik -2 2Tjk 2Rik -3

Pi Ri Pa Ri P# a Pi Ri Pt

Pj Ri Ri Ri Pj Pt

Pj Ri Ri Ri Pj Pt

[Pi Ri Pa -2Pj Ri ]-AMP2 [Ri P# a -2Ri Ri ]-AMP2 [2Pi Ri Pt -2Pj Pt ]-AMP2

Note that the pair symbols with double underline denote only one of the two identical limbs install this type of actuator.

axes, the end-effector would output 2-D translations along j&k axes. The group expressions can be written as:

{M} = {T ( j )}{T (k )}{R(N, i )}{R(N, j )} for AMP1 {M} = {T ( j )}{T (k )}{R(N, i )}{R(N, k )} for AMP2

(13)

Similar to the first kind of PMs, the group expressions of L2 and L3 are:

{ML2 } = {ML3 } = {G(i )}{R(N, j )} for AMP1 {ML2 } = {ML3 } = {G(i )}{R(N, k )} for AMP2

(14)

The group expression of L1 can be written as:

{ML1 } = {T (i )}{T ( j )}{T (k )}{R(N, i )}{R(N, j )} = {X (i )}{R(N, j )} for AMP1 {ML1 } = {T (i )}{T ( j )}{T (k )}{R(N, i )}{R(N, k )} = {X (i )}{R(N, k )} for AMP2

(15)

Combining Eq. (14), structure of L2s and L3s can be figured out, and L1s can be constructed by Eq. (15). The limbs are listed in Table 2, where enumerates three identical structures for L2s and L3s , three structures for L1s . Although there is a great deal of combinations for the feasible pairs to construct a limb, only the uncomplicated structures are selected. For example, L1s can be connected by Pi Ri Pa or Pi Pj a Pa , both of which make L1 output the same motion type of {X(j)}{R(N, i)} when connected in joint R1 i shown in Fig 1, but the Pi Pj a Pa isn’t selected as L1s for the reason that it should avoid the use of too many parallelograms in a limb. Subsequently, a type of 2T2R PMs shown in Table 2 can be obtained through connecting the new class of limbs with the AMPs. Fig 6 shows the CAD models which construct a translation along j-axis through adding a revolute joint Ri closed to the base. Note that a universal joint is used to amalgamate the two revolute joints Ri and Rj which are near to the base, but they are computed and represented independently to display the indispensable parallelogram in the limb. For example, we use Pi Ri Pa to represent L1 in the [Pi Ri Pa -2Pj Ri ]-AMP1 PM shown in Fig 6 but not Pi UURj Rj . Considering that the parallelogram

X. Jin et al. / Mechanism and Machine Theory 114 (2017) 156–169

163

Fig. 6. CAD models of the second type of 2T2R PMs.

is a 2-leg mechanism which is connected or near to the base in the PMs proposed in this type, we deprive the limb L12 and install an actuator along i-axis in the universal joint as shown in Fig 6. For the [2Pi Ri Pt -2Pj Pt ]-AMP1 and the [2Pi Ri Pt 2Pj Pt ]-AMP2, there are still two subchains constructing L1 , for the reason that these PMs don’t use parallelograms to orient the rotation axes and each subchain is a 1-leg chain. Both prismatic pairs in the subchain which is install electric machinery are passive, however a hydraulic drive denoted by Pt is used in another subchain. 4. Type synthesis of 3T2R PMs with AMPs After obtained 2T2R PMs, the 3T2R PMs can be synthesized through adding a translational mobility. The group expressions of the 3T2R PMs can be written as:

{M} = {T }{R(N, i )}{R(N, j )} = {T (i )}{T ( j )}{T (k )}{R(N, i )}{R(N, j )} for AMP1 {M} = {T }{R(N, i )}{R(N, k )} = {T (i )}{T ( j )}{T (k )}{R(N, i )}{R(N, k )} for AMP2

(16)

Compared with Eqs. (8) and (13), it can be drawn that to construct 3T2R PMs, one translation along j-axis is needed for the first type of 2T2R PMs and one translation along i-axis is needed for the second one. Specifically, the translational mobility should be added in L1 for the first type of 2T2R PMs and be added in L2 and L3 for the second type. Thus,

{ML1 } = {X (i )}{R(N, j )} for AMP1 {ML1 } = {X (i )}{R(N, k )} for AMP2

(17)

{ML2 } = {ML3 } = {X (i )}{R(N, j )} for AMP1 {ML2 } = {ML3 } = {X ( j )}{R(N, k )} for AMP2

(18)

Similarly as the first type 2T2R PMs, L1 is hybrid constructed by L1s and R1 i R1 j , where L1s is intersected by L11 , L12 . The group expressions can still be written as Eq. (11). However, there are two different selections for L11 and L12 in the 3T2R category, i.e.{ML11 } ∩ {ML12 } = {T } or {ML11 } ∩ {ML12 } = {T2 ( j )}{R(N  , i )} for the two AMPs for the reason that {T }{R(N, i )}{R(N, j )} = {T2 ( j )}{R(N , i )}{R(N , i )}{R(N, j )}. If taking {T} as the group expression of L1s , at least one passive prismatic pair, which should be avoided as far as possible, have to be constructed in L11 under the premise that the number of pairs in L11 is equal to its DOF. It should be mentioned that a common parallelogram is counted as one kinematic pair and an evolution parallelogram is regarded as two. Thus, only {T2 (j)}{R(N, i)} is selected as the group expression of L1s . Since the parallelograms is connected in L11 , there is no need to embed parallelogram in L12 to orient the rotation axis, we construct single chain for L12 , whose group expression is {G(j)}{R(N, i)}. L2s and L3s are same as the corresponding limbs in Table 1. The limbs and the resulting 3T2R PMs are listed in Table 3. Fig 7 is the CAD models of the selected 3T2R PMs. Note that there are only four limbs for the 5-DOF PMs, an electric rotating machinery should be installed along i-axis of the universal joints in L11 or L12 . The other four actuators are installed in the four limbs homogeneously.

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X. Jin et al. / Mechanism and Machine Theory 114 (2017) 156–169 Table 3 Feasible limbs and the 3T2R PMs. AMP type

Number

L1s L11

L12

AMP1

3Tijk 2Rij -1 3Tijk 2Rij -2 3Tijk 2Rij -4

Pi Ri Pa Ri P# a Pi Ri Pt

AMP2

3Tijk 2Rik -1 3Tijk 2Rik -2 3Tijk 2Rik -4

Pi Ri Pa Ri P# a Pi Ri Pt

L2s

L3s

PM

Pi Ri Rj Rj Rj Ri Rj Rj Pi Ri Pt

Pj Pi a Ri Ri Pi a Ri Pj Pi a Pt

Pj Pi a Ri Ri Pi a Ri Pj Pi a Pt

[[Pi Ri Pa -Pi Ri Rj Rj ]−2Pj Pi a Ri ]-AMP1 [[Ri P# a -Rj Ri Rj Rj ]−2Ri Pi a Ri ]-AMP1 [2Pi Ri Pt -2Pj Pi a Pt ]-AMP1

Pi Ri Rk Rk Rk Ri Rk Rk Pi Ri Pt

Pj Pi a Ri Ri Pi a Ri Pj Pi a Pt

Pj Pi a Ri Ri Pi a Ri Pj Pi a Pt

[[Pi Ri Pa -Pi Ri Rk Rk ]−2Pj Pi a Ri ]-AMP2 [[Ri P# a -Rk Ri Rk Rk ]−2Ri Pi a Ri ]-AMP2 [2Pi Ri Pt -2Pj Pi a Pt ]-AMP2

Fig. 7. CAD models of the 3T2R PMs.

5. Verification of the high rotational capability This section takes the [[Pi Ri Pa -Pi Ri Rj Rj ]−2Pj Pi a Ri ]-AMP1 PM as an example to prove the large rotational angles of the proposed PMs.

5.1. Inverse kinematics The geometrical model of the [[Pi Ri Pa -Pi Ri Rj Rj ]−2Pj Pi a Ri ]-AMP1 PM is depicted in Fig 8. The global reference frame Ob Xb Yb Zb is attached at the intersection of the sliding guides. Xb -axis is along the sliding direction of A1 and Yb -axis is along the sliding direction of A3 , A4 . A moving reference frame O-XYZ is established at the center of O3 O4 . In the moving reference frame, X-axis and Y-axis are fixed in the AMP where X-axis points to the center of joint O1 and Y-axis is along the axis of joint O4 . The position of O expressed in the global frame can be written as ob =(x y z)T and the orientation of the moving frame can be expressed as (α , β ), where α denotes the Euler angle rotating about i-axis and β the Euler angle rotating about j-axis. For the inverse kinematics of this PM, it can be described as: given the position and orientation values (x y z α β ) of the moving reference frame, compute the displacements (p1 p2 p3 p4 p5 ) of the actuators. It is worth mentioning that the displacements p1 ∼p4 are actuated by prismatic pairs whereas p5 is driven by electric rotating machinery. The orientation matrix R can be written as follows:

R = Ri ( α )R j ( β )

(19)

X. Jin et al. / Mechanism and Machine Theory 114 (2017) 156–169

165

Z

O3

O

O4

Y(j)

X(i) O1

C3 B3

C4

O2 C2

B4

C1

Zb

A3 B2

p3

A2

Yb

A4

Xb p2

p5

B1

Ob

p4

A1 p1

Fig. 8. The geometrical model of the example PM.

The coordinates of O1 , O3 , O4 represented in the moving frame can be separately written as: p

o1 = ( r1

0

p

o3 = ( 0

−r1

p

o4 = ( 0

r1

0 )T 0 )T 0)

(20)

T

Then the coordinates of O1 , O3 , O4 represented in the global frame can be deduced as: b

om = ob + R · p om

m = 1, 2, 3

(21)

And the coordinates of O2 represented in the global frame is: b

o2 = b o1 + ( r2

0 0 )T

(22)

The coordinates of Cm represented in the global frame can be calculated as: b

c1 = b o2 + ( 3r3

0 0 )T

b

c2 = b o2 + ( r3

0 0 )T

b

c3 = b o3 + ( 0

−r4 sin α

−r4 cos α )T

b

c4 = b o4 + ( 0

−r4 sin α

−r4 cos α )T

(23)

Since Am located in the plane Xb Yb , the coordinates of point Am expressed in the global frame can be obtained directly, depend on which the coordinates of point Bm can be computed as follows: b

a1 = ( p1

0

0 )T b

b1 = ( p1 + r5

b

a2 = ( p2

0

0 )T b

b2 = ( p2

0

b

a3 = ( 0

−p3

b3 = ( x

−p3

b

a4 = ( 0

p4

b4 = ( x

p4

0 )T b 0 )T b

0

a1 )T

a1 )T a1 + a1 +





l2 − x2 )T 2

l2 − x2 )T 2

(24)

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X. Jin et al. / Mechanism and Machine Theory 114 (2017) 156–169 Table 4 The architectural parameters. L

OO1 /OO3 /OO4

O1 O2

O2C2

O2C1

O3C3 /O4C4

B1 B11

B1C1 /B2C2

Pa1 Pa2

B3C3 /B4C4

A2 B2

P V

r1 50

r2 20

r3 30

3r3 90

r4 20

2r5 30

l1 200

l2 150

l3 150

a1 10

Note that L denotes the links between different joints, P the parameters of these links and V the values whose unit is mm.

200

150

z/mm 100

50 150 100

0 - 100

0 - 50

y/mm

- 100

0 x/mm

50

100

- 150

Fig. 9. The Constant-orientation workspace.

Owing to the geometrical constraints, the following relations can be deduced:

  b c m − b b m  = l 1 m = 1 , 2   b c m − b b m  = l 3 m = 3 , 4

(25)

If Bm is represented as b bm = (b bxm leads to:



b x bm b y bm

b by m

b bz )T m

2

2

2

2

x and Cm is expressed as b cm = (b cm

y x z = ± l1 − (b bym − b cm ) − (b bzm − b cm ) + b cm 2



= ± l1 − (

b bx 1

2

b bx m

r5 , b bx2

−

b cx m

) −(

y p3 , b b3

b bz m

−

b cz m

) +

b y cm

m = 1, 2

b cy m

b c z )T , m

the above relations

(26)

m = 3, 4

y −p3 , b b4

where = p1 + = = = p4 . Thus, the values of p1 ∼p4 which positive values should be selected can be calculated out. p5 is the angular displacement of the joint Rj embedded the universal joint. According to the geometrical relationship, the following relation can be satisfied: b y b1 b bz 1

− b c1y − b c1z

= tan p5

(27)

b y b1 b bz 1

(28)

As a result:

p5 = arctan

− b c1y − b c1z

5.2. Constant-orientation workspace Since the rotational angles are variable with changing of the positions, it is necessary to trace out the constantorientation workspace of the proposed PMs. According to the inverse kinematics provided above and the architectural parameters given in Table 4, the constant-orientation workspace of the [[Pi Ri Pa -Pi Ri Rj Rj ]−2Pj Pi a Ri ]-AMP1 PM can be depicted in Fig 9 under the constraint condition that the parameters α and β are set to zero and the actuators stroke are set as p1 → (180, 450), p2 → (−200, 55 ), p3 → (30, 390), p4 → (30, 390), p5 → (−π /2, π /2 ). It can be deserved that this class of

X. Jin et al. / Mechanism and Machine Theory 114 (2017) 156–169

150

200

150

150

100

100

z mm

z mm

z mm

100

50

50 0 - 180 - 120 - 60 0 α

60

60 120 - 60 180 (x, y)=(-100, 50)

0

167

50

- 60

β

α

60

0

0

60

120 - 60 (x, y)=(-100, 100)

- 180 - 120 - 60

120 60 0 60 α - 60 β 120 180 - 120 (x, y)=(-50, 50)

β

0

200 200

150 z mm100

200

150

150

z mm 100

z mm100

50 50

- 60 α

0 60 120

- 120

- 60

120 0 60

- 180 - 120 - 60 0 α

β

0 - 60 60 - 120 120 - 180 180

180 120 60

50 - 180 - 120 - 60 α

β

200

200

150

150

150

z mm 100

z mm 100

z mm 100

50

50

200

50 0

α

0 - 60 120 120- 180 (x, y)=(50, 100)

180 120 60

60

β

β

(x, y)=(50, 50)

(x, y)=(0, 0)

(x, y)=(-50, 100)

- 60

0 0 60 - 60 120 - 120 180 - 180

180 120 60

- 180 - 120 - 60 α

0 - 60 180 60120

0 60 - 60 0 120 β - 120 180- 180 (x, y)=(100, 50)

0 α

180 120 60 0 60 - 60 - 120 β 120 - 180 (x, y)=(100, 100)

Fig. 10. The orientation-workspace.

3T2R PMs possess large translational workspace which is similar to the famous Delta. The constant-orientation workspace is separated symmetrical about the Xb Zb -plane and the Yb Zb -plane. In other words, the reachable translational workspace is symmetrical distribution around Zb -axis, which is crucial for many manufactural processes. Generally, the translational workspace map of a fully decoupled PM is a cuboid. However, workspace of the [[Pi Ri Pa Pi Ri Rj Rj ]−2Pj Pi a Ri ]-AMP1 is approximate hemispherical, which indicate that these 3-D translations are uncoupled on the edge of the workspace and decoupled inside the workspace because the approximate hemispherical workspace can inscribed a variable volume cuboid. Hence, the translation mobility of this type of PMs is partially decoupled. 5.3. Orientation-workspace analysis As mentioned above that the rotational capability of the PM depends on the concrete position, we investigate the reachable rotational workspaces in different positions in this part. Due to the fact that the translational workspace is symmetrical distribution around Zb -axis, the relationship between the rotational angles α , β and the vertical displacement z is studied as shown in Fig 10, where lists nine representative (x, y) positions. Constraints on the orientation workspace are same as the constraints on the constant-orientation workspace. Some conclusions can be obtained through these workspace diagrams. • The reachable rotational angle can achieve ± 180° at most positions and the maximum rotation amplitude is obtained in the position (0, 0).

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X. Jin et al. / Mechanism and Machine Theory 114 (2017) 156–169

• The rotational capability is declining with the displacement increase along Zb -axis. • The reachable rotation angles are on the progressive decrease accompanied by the distance from the Zb -axis to become far. • Under given values z and β , α almost locates in a symmetrical positive and negative area benefit from the symmetrical structure of L2 and L3 , which play a major role of controlling the rotation about i-axis. However, the reachable rotational angle β shows a significant difference between the positive and negative area under the given values z and α , especially when the value z is larger. The reason is that the limb is asymmetrical arrangement for the rotation about j-axis. 6. Conclusion Parallel mechanisms with large orientation-workspace have a widespread application value. This paper focuses on the proposal of a class of novel 2T2R and 3T2R PMs, which can be used to achieve larger rotational angles. 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