Type Synthesis of 3-DOF RPR-Equivalent Parallel Mechanisms

IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 6, DECEMBER 2014

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Type Synthesis of 3-DOF RPR-Equivalent Parallel Mechanisms Qinchuan Li, Member, IEEE, and Jacques Marie Herv´e

Abstract—The moving platform of an RPR-equivalent parallel mechanism (PM) can undergo a 3-degree-of-freedom (DOF) motion that is the product of a rotation followed by a translation and another rotation. A 5-DOF hybrid parallel manipulator can be developed by adding an x-y gantry or an RR serial mechanism to an RPR-equivalent PM, which is suitable for manipulations requiring high rigidity and accuracy with good dexterity along surfaces of the 3-D space. This paper investigates the type synthesis of the RPR-equivalent PM. First, the RPR motion is briefly discussed. Then, the kinematic bonds of limb chains and their mechanical generators are presented. Structural conditions for constructing an RPR-equivalent PM are presented. Furthermore, the RPRequivalent PMs are classified into several categories, depending on the DOF of its limb chains. Numerous new architectures of the RPR-equivalent PMs are synthesized. Index Terms—Lie group theory, parallel robots, type synthesis.

I. INTRODUCTION HE 3-degree-of-freedom (DOF) parallel mechanisms (PM) with two rotational DOF and one translational DOF are particularly useful. They are often denoted 1T2R or 2R1T, where T indicates a translational DOF and R a rotational DOF. The most famous 1T2R PM may be the 3-RPS PM proposed by Hunt in 1983 [1], where P denotes a prismatic pair, R a revolute pair, and S a spherical joint. The 3-RPS PM and its variants have attracted a lot of attention and can be implemented in many applications [2]–[7]. One typical example is the Z3 head [2], which is based on a 3-PRS PM. Another example is the parallel part of Exechon [3], which is a 2-RPS-SPR PM. However, the notion 1T2R or 2R1T is not rigorous. First, it ignores the information on axes of rotation. Second, the motion of the end-effector cannot simply be regarded as a commutative addition of rotations and translations. Considering this fact, we defined the concepts of general aTbR motion and special aTbR motion [8]. A general aTbR motion cannot always be decomposed into a product of (a + b) factors. The orientation of the moving body is represented by the motion in the spherical indicatrix; it has b

T

Manuscript received December 12, 2013; revised April 18, 2014; accepted July 22, 2014. Date of publication September 10, 2014; date of current version December 3, 2014. This paper was recommended for publication by Associate Editor N. Simaan and Editor C. Torras upon evaluation of the reviewers’ comments. This work was supported by the National Natural Science Foundation of China under Grant 51135008 and Grant 51475431 and by the Natural Science Foundation of Zhejiang Province under Grant LZ14E050005. Q. Li is with the Mechatronic Institute, Zhejiang Sci-Tech University, Hangzhou 310018, China (e-mail: [email protected]). J. M. Herv´e, retired, was with the Ecole Centrale Paris, Paris, France. He resides in F-85470, Brem sur Mer, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TRO.2014.2344450

Fig. 1. Serial chains generating special 3-DOF motion. (a) PU chain (b) UP chain (c) RPR chain.

DOF, and a is the difference: total DOF minus b. Thus, parasitic motion (translation coupled with the orientation changes) occurs in the 3-DOF PM with general 1T2R motion [8], such as the Z3 head based on the 3-PRS PM. A special aTbR motion can be decomposed into a product of (a + b) factors. The product consists of a translations parallel to a linearly independent vectors followed by b rotations around b axes passing through one point undergoing the a translations. The motion generated by an RPR open chain belongs to the special 1T2R category and can be decomposed into a product of three factors. There are three special categories of 3-DOF motion that include two rotational DOFs and one translational DOF. The first category is a special 1T2R motion, which can be produced by a serial PU chain, as shown in Fig. 1(a). A new family of PM that has the special 1T2R motion is proposed in [8]. Using Kong’s virtual chain approach [9], we call this family of 3-DOF PMs as PU-equivalent PM, where U denotes a universal joint. The second category is a special 2R1T motion, which can be generated by a serial UP chain, as shown in Fig. 1(b). The UP motion is the kinematic inverse of the PU motion. The UPequivalent PMs were first proposed by Kong and Gosselin [9]. The third category is a special motion, which can be realized by a serial RPR chain, as shown in Fig. 1. A PM that generates such a motion is called an RPR-equivalent PM. The displacement of the moving platform of an RPR-equivalent PM can be represented by a product of a rotation followed by a translation and then a rotation. The two axes of rotation generally do not intersect and remain perpendicular to each other. The RPR-equivalent PMs are suitable for many manipulations along a curved surface when high rigidity and accuracy are

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IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 6, DECEMBER 2014

required with good dexterity. For example, a five-axis hybrid parallel kinematic machine (PKM) or medical robot can be constructed by integrating an X–Y gantry or a 2-DOF RR chain to an RPR-equivalent PM. The five- or six-axis hybrid PKM based on an RPR-equivalent PM can also be used in friction stir welding of large panels in 3-D surfaces like aircraft fuselage segments. In 2011, Li et al. proposed an RPR-equivalent PM, 2RPU-UPR [11], which was used to construct a six-axis hybrid PKM. In 2012, Jin et al. proposed another RPR-equivalent PM, 2RPU-SPR, for aircraft wing assembly [10]. One obvious advantage of the RPR-equivalent PM or UP/PUequivalent PM is their two specified axes of rotation compared with its general 1T2R or 2R1T counterpart. With two encoders attached to the axes of rotation, one can easily obtain the orientation of the moving platform, which is particularly helpful to simplify the control or calibration of a PM. Such a feature is not achieved by the general 1T2R PMs, and thus, the control of Z3 head or Exechon is much more complicated. In addition, it is also believed that higher stiffness and accuracy can be obtained in the RPR-equivalent PM, compared with the existing five-axis hybrid PKM [10]. Much progress has been made in general methods for type synthesis of lower mobility PMs [12]–[24] and inventing 3T or 3T1R PMs with specified kinematic properties [24]–[29]. However, few architectures of RPR-equivalent PMs have been invented [10], [11]. In this paper, we attempt to explain and fill the gap by systematically addressing the type synthesis of RPR-equivalent PMs. New architectures of RPR-equivalent PMs are disclosed. Importantly, some of them have encouraging potentials in practice. The organization of this paper is as follows. Section II presents a brief introduction of notations and properties of RPR motion. Section III discusses the limb bond of the RPR-equivalent PM. Section IV discusses parallel arrangements of three limbs. Section V presents the type synthesis of overconstrained RPRequivalent PMs. Section VI presents the type synthesis of nonoverconstrained RPR-equivalent PMs. II. RPR MOTION

TABLE I DISPLACEMENT SUBGROUPS Subgroup

Dim

MG (mechanical generator)

Motion

{R(N,u)}

1

Revolute pair

{T(v)} or {T1(v)} {T(⊥u)} or {T2(v, w)}

1

Prismatic pair

{C (N,u)}

2

Cylindrical pair

{T} {G (u)}

3 3

Planar pair

{S(N )} {X(u)}

3 4

{E}

0

{D }

6

Rotation about the axis determined by a unit vector u and a point N Translation parallel to the unit vector v Planar translation perpendicular to u v⊥u and w⊥u Cylindrical motion of axis (N,u) 3-D translation in space Planar gliding motion determined by the normal u Rotation about the point N Schoenflies motion (special 3T1R motion), axes of rotation parallel to u Rigid connection, no relative motion General spatial motion

2

Spherical joint

a vector given without using a couple of points. Unit vectors are labeled using bold lower case letters: u, v, w. An axis characterized by (N, u) is the set of points P = N + pu, where p is a real parameter called abscissa of P in the frame (N, u). Because lower mobility PMs usually require specified geometrical condition, we use superscripts to denote the direction of the kinematic pair in a limb. Let (O, u, v, w) denote a Cartesian frame of reference with an origin O and an orthonormal vector base (u, v, w). For example, a prismatic (P) pair parallel to u is denoted u P, and a revolute (R) pair with an axis parallel to u is denoted u R. uv U denotes a universal (U) joint in which the first revolute axis is parallel to u, and the second revolute axis is parallel to v. The spherical (S) joint is denoted by N S, where the superscript N denotes its center. For clarity and practical concerns, we neglect limbs containing two or three P pairs, helical pairs, and parallelograms. However, one can synthesize RPR-equivalent PMs with these pairs easily following the rules presented in this paper.

A. Notation The Lie group algebraic properties of the set of rigid-body displacements are used in this paper. When the points of the Euclidean space are referenced in an orthonormal frame, the displacement group is represented by the matrix group often denoted SE(3) in textbook on matrix Lie groups. For readers’ sake, we give a short explanation of the notations of displacement Lie subgroups used in the following sections. For detailed information on using Lie groups in type synthesis of PMs, see [8], [12]–[14], and [28]–[32]. The displacement subgroups used in this paper are listed in Table I. The notation employed in the paper has to be specified. The points that are elements of the Euclidean affine space are denoted with capital letters: O, A, B, N, M, etc. The (free) vectors derived from an ordered couple of points as A and B are denoted (AB) or AB. One can write: B = A + (AB) and AB = B − A. A single bold capital letter as V designates

B. RPR Motion An RPR motion can be generated by a u Rw Pv R serial chain and can be represented by a product {R(O, u)}{T(w)}{R (B,v)}. A displacement of {R(O, u)}{T(w)}{R(B, v)} is obtained by the composition of three displacements. The first displacement is a rotation of angle ψ around an axis (B, v); any point M is transformed into a point MR1 : M → MR1 = B + exp(ψv × )(BM ).

(1)

where B belongs to the axis (B, v), ψ is the angle of rotation around the axis (B, v), (BM ) is the vector M−B. This rotation belongs to the group {R(O, v)}, and the subscript R1 denotes the first rotation. The term exp(ψv × ) is the exponential series of the skew-symmetric linear operator (ψv × ) of the vector product by the vector ψv. The term exp(ψv × )(BM ) is a vector.

´ TYPE SYNTHESIS OF 3-DOF RPR-EQUIVALENT PARALLEL MECHANISMS LI AND HERVE:

The second displacement is a translation parallel to w MR1 → MT = tw + MR1

∀ point MR1 .

(2)

where the real number t is the amplitude of translation, and the subscript T stands for translation. This translation is an element of the group {T(w)}. Hence, the transformation of M into MT is expressed by M → MT = B + tw + exp(ψv ×)(BM ).

(3)

The third displacement is a rotation of angle ϕ around the axis (O, u), which belongs to the group {R(O, u)}. The point MR1 is transformed into the final position M of M MT → M = O + exp(ϕu ×)(OM T ) = O + exp(ϕu ×)(OB + BM T ) = O + exp(ϕu ×)(bw + tw + exp(ψv ×)(BM )) = O + exp(ϕu ×)[(b + t)w + exp(ψv ×)(BM )] = O + (b + t)exp(ϕu ×)w + exp(ϕu ×)exp(ψv ×)(BM ).

(4)

where OB = bw, and exp(ϕu ×)w = −sin ϕv + cos ϕw. Equation (4) expresses the transformation M → M , which belongs to the product {R(O, u)}{T(w)}{R(B, v)}. The kinematic inversion of the u Rw Pv R chain is the v Rw Pu R chain and generates [{R(O, u)}{T(w)}{R(B, v)}]−1 = {R(B, v)}−1 {T(w)}−1 { R(O, u)}−1 = {R(B, v)}{T(w)} {R(O, u)}. Note that {R(B, v)}{T(w)}{R(O, u)} and {R(O, u)}{T(w)}{R(B, v)} are not equal sets but represent equivalent motion types.

{Li } = {R(O, u)}{T(w)}{R(B, v)}{T(t)} = {R(O, u)}{G(v)} (if t⊥v).

(6)

Note that adding an 1-D rotational subgroup {R(C, v)} to the right end of {R(O, u)}{T(w)}{R(B, v)} leads to the same result because {T(w)}{R(B, v)}{R(C, v)} = {G(v)}. Using the product closure in the displacement subgroup {G(v)}, {G(v)} can be equated to various equivalent products. One is = {R(O, u)}{R(O, v)}{T(s)}{R(C, v)} × ( with s⊥v)

A. Displacement Set of the RPR-Equivalent Parallel Mechanism {R(O, u)}{T(w)}{R(B, v)} must be the intersection of all the limb kinematic bonds produced by all limb chains in a RPRequivalent PM. Here, we consider parallel manipulators with three limbs. Each limb generates a set of feasible displacements (or kinematic bond) denoted {Li }, i = 1, 2 and 3. Hence, we can write {Li } = {R(O, u)}{T(w)}{R(B, v)}.

Note that limb bond can be {D} itself because {R(O, u)}{T(w)}{R(B, v)} ⊂ {D}. Mechanical generators (MGs) of {D} are numerous. If one or two limbs are sufficient to produce an RPR motion, one or two generators of {D} can be the second and third limb. For example, one can construct an RPR-equivalent PM using two 6-DOF SPS or UPS limbs and one u Rw Pv R chain. This is straightforward and will not be discussed in this paper. We focus on the type synthesis of the RPR-equivalent PM without 6-DOF limbs. Three families of limb bond can be obtained based on the dimension (Dim) of the limb bond. 1) Family 1: Dim ({Li }) = 3 When Dim ({Li }) = 3, apparently, we have only {Li } = {R(O, u)}{T(w)}{R(O, v)}. The corresponding MG is the 3-DOF u Rw Pv R chain. 2) Family 2: Dim ({Li }) = 4 In order to obtain limbs with 4-DOFs, we add 1-D subgroups as factors in the product {R(O, u)}{T(w)} {R(B, v)}. We consider the results that yield subgroups of {D}. The added factor can be either a 1-D group of rectilinear translations or a 1-D group of rotations. Adding an 1-D translational subgroup {T(t)} to the right end of {R(O, u)}{T(w)}{R(B, v)} yields

{Li } = {R(O, u)}{G(v)}

III. LIMB BOND OF RPR-EQUIVALENT PARALLEL MECHANISMS

3 

1335

(5)

i=1

B. Limb Bond of an RPR-Equivalent Parallel Mechanism Necessarily, the displacement set {Li } produced by the ith limb chain must contain {R(O, u)}{T(w)}{R(B, v)}, that is, {R(O, u)}{T(w)}{R(B, v)} ⊆ {Li }. We have to seek after subsets of the group {D} of all displacements, which contain the product {R(O, u)}{T(w)}{R(B, v)}.

(7)

which can be generated by a uv Us Pv R chain. Adding an 1-D translational subgroup {T(t)} to the left end of {R(O, u)}{T(w)}{R(B, v)} yields {Li } = {T(t)}{R(O, u)}{T(w)}{R(B, v)} = {G(u)}{R(B, v)}

(if t⊥u).

(8)

Adding an 1-D rotational subgroup {R(E, u)} to the left end of {R(O, u)}{T(w)}{R(B, v)} also leads to the same result because {R(E, u)}{R(O, u)}{T(w)} = {G(u)}. The combinations {C(O, u)} {T(w)}{R(B, v)}, {R(O, u)}{C(C, w)}{R(B, v)}, and {R(O, u)}{T(w)}{C(B, v)} are discarded because they involve idle pairs in a parallel generator of RPR motion. In summary, we obtain the two types of 4-D products: {R(O, u)}{G(v)} and {G(u)}{R(B, v)}. Table II lists the 4-D limb bonds and their MG for RPR-equivalents PMs. 3) Family 3: Dim {Li } = 5

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TABLE II FOUR-DIMENSIONAL LIMB BOND FOR RPR-EQUIVALENT PM {L i }

TABLE III FIVE-DIMENSIONAL LIMB BONDS OF UP-EQUIVALENT PM

MG of {L i }

{R(O, u)}{G (v)}

u

{G (u)}{R(B, v)}

u

{L i } {X(u)}{X(v)}

Rv RPR uv Uv PR u Rv PRR uv Uv RP u v R RRP uv Uv RR u Rv RRR v

u

uv

u

v

u

MG of {L i } See [15] and [35] ij

uv

RPR R PR U PRR R RP U u RRPv R u RRuv U u RRRv R

Note: The underline in v RPR, v PRR,v RRP, and v RRR denotes that they generate {G (v)}.

{S(D )}{G (v)} = {R(D , i)}{R(D , j)}{G (v)} = {S(D )}{G 2(v)} ={S(D)}{R(B , v)}{T(r )},r⊥v = {S(D)}{T(r)}{R(B , v)} = {S(D)}{R(B , v)}{R(C , v)} with D ∈ axis(O , u)

Uv RRP Uv RPR ij v U RRR ij v U PRR D v S RP D v S PR D v S RR

{G (u)}{S (F )} = {G (u)}{R(F , i)}{R(F , j)} = {G 2(u)}{S (F) } = {R(B,u)}{T(r)}{S(F)},r⊥v = {T(r)}{R(B,u)}{S(F)} = {R(B,u)}{R(C,u)}{S(F)} with F ࢠ axis(B,v)

RRPij U RPRij U u RRRij U u PRRij U u RPF S u PRF S u RRF S

ij

u

Adding a group of translations {T(v)} to the right end of the 4-D product {R(O, u)}{G(v)}, we obtain {R(O, u)}{G(v)}{T(v)} = {R(O, u)}{X(v)} = {R(O, u)}{T}{R(B, v)} = {X(u)}{R(B, v)}

u

Note: In {R(C,u)}{R(D,v)}, the points C and D can coincide and then {R(C,u)} {R(C, v)} is embodied by uv U.

= {X(u)}{T}{R(B, v)} = {X(u)}{X(v)}.

(9)

{X(u)}{X(v)} is a 5-D displacement set called double Schoenflies motion or X-X motion for brevity. An X-X motion includes three independent translations and two independent sequential rotations. Angeles [15] also proved that {X(u)}{X(v)} = {G(u)}{G(v)}. {G(u)}{G(v)} is also called a G-G bond. Note that the products {X(u)}{X(v)} and {G(u)}{G(v)} are both reducible. The detailed enumeration of MGs of {X(u)}{X(v)} or {G(u)}{G(v)} can be found in [15] and [31] and is not included in this paper for succinctness. Other 5-D product can also be obtained. Noticing that {R(O, u)} ⊂ {S (D)} provided that the point D belongs to the axis (O, u), we can write {S (D) }{T(w)}{R(B, v)} ⊃ {R(O, u)}{T(w)}{R(B, v)} where {S (D) }{T(w)}{R(B, v)} is one of the irreducible representations of the reducible product {S(D)}{G(v)}. {S(D)}{G(v)} is reducible because {S(D)} ∩ {G(v)} = {R(D, v)}. Note that {S(D)}{G(v)} has two main categories of irreducible representations. One is {R(D, i)}{R(D, j)}{G(v)}, where (i, j, v) is a vector base (generally, i = w and j = u). The other is {S (D) }{G2(v)}, where {G2(v)} denotes one of the 2-D submanifolds of the 3-D group {G(v)}. The {G2(v)} manifolds are not equal and can be obtained from products equal to {G(v)} by removal of one factor. A {G2(v)} manifold can be either {R(A, v)}{R(B, v)}, {T(r)}{R(A, v)}, or {R(A, v)}{T(r)}; r is any unit vector that is orthogonal to v. In a similar way, we have {R(B, v)} ⊂ {S (F) } provided that F ∈ axis(B, v), and, consequently {R(O, u)}{T(w)}{S(F)} ⊃ {R(O, u)}{T(w)}{R(B, v)} where {R(O, u)}{T(w)}{S(F)} is an irreducible representation of {G(u)}{S(F)}. {G(u)}{S(F)} are reducible because {G(u)} ∩ {S(F)} = {R(F, u)}. It is straightforward to

see that the two irreducible representations of {G(u)}{S(F)} are {G(u)}{R(F, i)}{R(F, j)} and {G2(u)}{S(F)}. In summary, we obtain three types of 5-D products: {X(u)}{X(v)}, {S(D)}{G(v)}, and {G(u)}{S(F)}. Table III lists the 5-D limb bonds and their MGs used in this paper. IV. PARALLEL ARRANGEMENTS OF THREE LIMBS An RPR-equivalent PM can be constructed by connecting the moving platform and the fixed base by three limb chains listed in Tables II and III. We denote the type of RPR-equivalent PMs by the dimensions of the three limb bonds, that is, Dim({L1 })Dim({L2 })-Dim({L3 }). To obtain the maximum symmetry, we focus on RPR-equivalent PMs in which the Dims of two limbs are equal at least. Furthermore, 3-DOF limb bonds are neglected intentionally due to their less practical interest. If one limb generates {R(O, u)}{T(w)}{R(B, v)}, then the platform motion is constrained to be RPR equivalent only by this limb, and the other two limbs can generate any kinematic bond that contains {R(O, u)}{T(w)}{R(B, v)}. This corresponds to the categories X-Y-3, where X = 3, 4 or 5 and Y = 3, 4 or 5. Two or three limbs generating {R(O, u)}{T(w)}{R(B, v)} do not correspond to truly PMs. Moreover, if we assume that two limbs generate bonds having the same dimension, then the possible categories are 4–4–3 and 5–5–3. If the parallel arrangement of two limbs 1 and 2 generates {L1 } ∩ {L2 } = {R(O, u)}{T(w)}{R(B, v)}, then the platform motion is constrained only by these two limbs. The third limb can generate any kinematic bond that contains {R(O, u)}{T(w)}{R(B, v)}.{L1 } ∩ {L2 } = {R(O, u)}{T(w)}{R(B, v)} can be obtained with Dim {L1 } = 4 and Dim {L2 } = 4, as well as with Dim {L1 } = 4 and Dim {L2 } = 5. The case Dim {L1 } = 5 and Dim {L2 } = 5 produces a bond {L1 } ∩ {L2 } of Dim ≥ (5 + 5 −

´ TYPE SYNTHESIS OF 3-DOF RPR-EQUIVALENT PARALLEL MECHANISMS LI AND HERVE:

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The kinematic bond of limb 1 is {L1 } = {R(A1 , u)}{T(r)}{R(B1 , u)}{R(B1 , v)} with r⊥u = {G(u)}{R(B1 , v)}.

(11)

The kinematic bond of limb 2 is {L2 } = {R(A2 , u)}{R(A2 , v)}{T(s)}{R(B2 , v)} with s⊥v = {R(A2 , u)}{G(v)} = {R(O, u)}{G(v)} Fig. 2.

because A2 ∈ axis(O, u). (12)

2-vu Uu PR/u RPuv U PM.

The kinematic bond of limb 3 is 6) = 4 > 3 = Dim {R(O, u)}{T(w)}{R(B, v)} and does not correspond to the assumption. That way, we classify the RPR-equivalent PMs into two categories: 4–4-X, and 5–5-X, where X can be 4 or 5. The second category 4–4-X includes subcategories of 4–4–4 and 4–4–5. The third category 5–5-X includes subcategories of 5–5–4 and 5–5–5. Only the 5–5–5 combinations are not overconstrained. The following sections focus on the cases in which two limbs are sufficient to constrain the RPR motion, including the 4–4–4, 4–4–5, 5–5–4, and 5–5–5 subcategory.

{L3 } = {R(A3 , u)}{R(A3 , v)}{T(s)}{R(B3 , v)} with s⊥v = {R(A3 , u)}{G(v)} = {R(O, u)}{G(v)}

because A3 ∈ axis(O, u). (13)

Obviously, we have {L2 } = {L3 }. Hence, the displacement set of the moving platform is given by {L1 } ∩ {L2 } = {G(u)}{R(B1 , v)} ∩ {R(O, u)}{G(v)}

V. OVERCONSTRAINED RPR-EQUIVALENT PARALLEL MECHANISMS

= {R(O, u)}({G(u)} ∩ {G(v)}){R(B1 , v)}

A. Subcategory 4–4–4 Using the 4-DOF limbs in Table II, one can construct RPRequivalent PMs in subcategory 4–4–4. Note that one limb must be a MG of {R(O, u)}{G(v)}, and another limb must be a MG of {G(u)}{R(B, v)}. We can verify that the intersection set of {R(O, u)}{G(v)} and {G(u)}{R(B, v)} is an RPR motion {R(O, u)}{G(v)} ∩ {G(u)}{R(B, v)} = {R(O, u)}({G(v)} ∩ {G(u)}){R(B, v)} because {R(O, u)} ⊂ {G(u)} and {R(B, v)} ⊂ {G(v)} = {R(O, u)}{T(w)}{R(B, v)}.

(10)

Therefore, the RPR-equivalent PMs in 4–4–4 subcategory can be obtained by two combinations, respectively. The first combination is {L1 } = {G(u)}{R(B, v)}, and {L2 } = {L3 } = {R(O, u)}{G(v)}. The second combination is {L1 } = {R(O, u)}{G(v)} and {L2 } = {L3 } = {G(u)}{R(B, v)}, which corresponds to the kinematic inversion of the previous family. For example, Fig. 2 shows a 2-UPR/RPU PM. Limb 2 and limb 3 are UPR chains. Limb 1 is a RPU chain. Ai denotes a point of the first revolute pair axis in the ith limb. Bi denotes a point of the last revolute pair axis in the ith limb. Moreover, A2 , A3 , and B1 are located at the centers of the three U joints.

= {R(O, u)}{T(w)}{R(B1 , v)} = {R(O, u)}{T(w)}{R(B, v)}

B ∈ axis(B1 , v). (14)

It is shown in (14) that the two axes of rotation are axis (O, u) and axis (B, v). The 2-UPR/RPU PM is an RPR-equivalent PM. Its kinematic inversion is a 2-RPU/UPR PM that is also an RPR-equivalent PM. For conciseness, Table IV only enumerates 49 RPRequivalent PMs belonging to the first family in 4–4–4 category. Figs. 3 and 4 illustrate 12 architectures that have potentials in practice. Although kinematic inversions are neglected in Table IV, Fig. 5 shows six kinematic inversions for reader’s better understanding. In our view, the 4–4–4 RPR-equivalent PMs with identical limbs, such as the 2-UPR/RPU, 2-URR/RRU, and their kinematic inversions, are of particularly important interest for practical application. For example, compared with the 2-UPR-SPR PM in Exechon robot, the 2-UPR/RPU PM has a simpler structure because the one spherical joint is replaced by a universal joint. Another advantage is the two specified axes of rotation that will simplify the control and calibration of the PKM. However, the workspace of the 2-UPR/RPU PM is smaller than that of the 2-UPR-SPR PM in Exechon robot because one rotation in the 2-UPR/RPU PM is near the moving platform and is not amplified by link length. This disadvantage can be

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TABLE IV FORTY-NINE RPR-EQUIVALENT PMS IN 4–4–4 CATEGORY 4–4–4

{L 1 } = {G (u)}{R(B,v)} and {L 2 } = {L 3 } = {R(O, u)}{G (v)}

2-u Rv RPR/u RPRv R 2-u Rv RPR/ u PRuv U 2-u Rv RPR/ u PRRv R 2-u Rv RPR/ u RPuv U 2-u Rv RPR/ u RRPv R 2-u Rv RPR/ u RRuv U 2-u Rv RPR/ u RRRv R

2- uv Uv PR/u RPRv R 2- uv Uv PR/ u PRuv U 2- uv Uv PR/ u PRRv R 2- uv Uv PR/ u RPuv U 2- uv Uv PR/ u RRPv R 2- uv Uv PR/ u RRuv U 2- uv Uv PR/ u RRRv R

2- u Rv PRR/u RPRv R 2- u Rv PRR/ u PRuv U 2-u Rv PRR/ u PRRv R 2- u Rv PRR/ u RPuv U 2- u Rv PRR/ u RRPv R 2- u Rv PRR/ u RRuv U 2-u Rv PRR/ u RRRv R

2- uv Uv RP/u RPRv R 2- uv Uv RP/ u PRuv U 2- uv Uv RP/u PRRv R 2- uv Uv RP/u RPuv U 2- uv Uv RP/ u RRPv R 2- uv Uv RP/ u RRuv U 2- uv Uv RP/ u RRRv R

2- u Rv RRP/u RPRv R 2- u Rv RRP/ u PRuv U 2-u Rv RRP/u PRRv R 2- u Rv RRP/u RPuv U 2- u Rv RRP/ u RRPv R 2- u Rv RRP/ u RRuv U 2-u Rv RRP/ u RRRv R

2- uv Uv RR/u RPRv R 2- uv Uv RR/ u PRuv U 2- uv Uv RR/u PRRv R 2- uv Uv RR/u RPuv U 2- uv Uv RR/ u RRPv R 2- uv Uv RR/ u RRuv U 2- uv Uv RR/ u RRRv R

2- u Rv RRR/u RPRv R 2- u Rv RRR/ u PRuv U 2- u Rv RRR/ u PRRv R

2- u Rv RRR/ u RPuv U 2- u Rv RRR/ u RRPv R

2- u Rv RRR/ u RRuv U 2- u Rv RRR/u RRRv R

Fig. 4. 4–4–4 RPR-equivalent PMs with two uv Uv RR limbs. (a) 2-uv Uv RR/u RPRv R. (b) 2-uv Uv RR/u PRuv U. (c) 2-uv Uv RR/u PRRv R. (d) 2-uv Uv RR/u RPuv U. (e) 2- U uv Uv RR/u RRRv R. (f) 2-uv Uv RR/u RRuv U.

is used to move the working table along the short side of the workspace of the 2-UPR/RPU PM. B. Subcategory 4–4–5

Fig. 3. 4–4–4 RPR-equivalent PMs with two uv Uv PR limbs. (a) 2-uv Uv PR/u RPRv R. (b) 2-uv Uv PR/u PRuv U. (c) 2-uv Uv PR/u PRRv R. (d) 2-uv Uv PR/u RPuv U. (e) 2- U uv Uv PR/u RRRv R. (f) 2-uv Uv PR/u RRuv U.

compensated by integrating a linear guide to move the PM or the working table. Fig. 6 shows a six-axis hybrid PKM that is being built in Zhejiang Sci-Tech University. The PKM consists of a 2-UPR/RPU PM and an articulated RR serial mechanism. A linear guide

RPR-equivalent PMs in subcategory 4–4–5 can be constructed with two 4-DOF limbs in Table II and one 5-DOF limbs in Table III. We neglect the architectures with limbs generating {X(u)}{X(v)} because it would be very lengthy to enumerate RPR-equivalent PMs with limbs generating {X(u)}{X(v)}. Then, the RPR-equivalent PMs in 4–4–5 subcategory can be obtained by two combinations, respectively. The first combination is {L1 } = {G(u)}{S(F)} and {L2 } = {L3 } = {R(O, u)}{G(v)}. The second combination is {L1 } = {S(D)}{G(v)} and {L2 } = {L3 } = {G(u)}{R(B, v)}, which results from the kinematic inversion of the previous family. For conciseness, Table V only enumerates 49 RPR-equivalent PMs belonging to the first family in 4–4–5 category. Fig. 7 shows four architectures belonging to the first family, and Fig. 8 shows four architectures belonging to the second family. C. Subcategory 5–5–4 RPR-equivalent PMs in subcategory 5–5–4 can be constructed with two 5-DOF limbs in Table II and one 4DOF limb in Table III. The first combination is {L1 }

´ TYPE SYNTHESIS OF 3-DOF RPR-EQUIVALENT PARALLEL MECHANISMS LI AND HERVE:

Fig. 6.

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Six-axis hybrid PKM based on a 2-UPR/RPU PM.

TABLE V FORTY-NINE RPR-EQUIVALENT PMS IN THE FIRST FAMILY IN 4–4–5 CATEGORY 4–4–5 u

Fig. 5 4–4–4 RPR-equivalent PMs that are kinematic inversions of the first family. (a) 2-u PRuv U/uv Uv PR. (b) 2-u PRuv U/u Rv PRR. (c) 2-u RPuv U/uv Uv PR. (d) 2-u RPuv U/u Rv PRR. (e) 2-u RRuv U/uv Uv PR. (f) 2-u RRuv U/u Rv PRR.

= {R(O, u)}{G(v)} and {L2 } = {G(u)}{S(Fa )} {L3 } = {G(u)}{S(Fb )}, with Fa ∈ axis(B, u) and Fb ∈ axis(B, v). The second combination is {L1 } = {G(u)}{R(B, v)}, and {L2 } = {S(Da )}{G(v)}, {L3 } = {S(Db )}{G(v)} with Da ∈ axis(O, u) and Db ∈ axis(O, u), which is a kinematic inversion of the first family. Architectures with limbs generating {X(u)}{X(v)} are also neglected. Fig. 9 shows six RPR-equivalent PMs in this category. Note that Fig. 9(e) and (f) belongs to the second family. Table VI enumerates 49 RPR-equivalent PMs in the first family in 5–5–4 category. VI. NONOVERCONSTRAINED RPR-EQUIVALENT PARALLEL MECHANISMS The dimension of the three limb bonds of a nonoverconstrained RPR-equivalent PM must be five. Using the 5-D limb bond in Table III, one can construct a nonoverconstrained RPRequivalent PM A. {L1 } = {G(u)}{S(F)}, {L2 } = {S(Da )}{G(v)}, and {L3 } = {S(Db )}{G(v)} Like the 4–4–4 subcategory, the PM in 5–5–5 subcategory can be constructed by two combinations, respectively. The first combination is {L1 } = {G(u)}{S(F)}

{L 1 } = {G (v)}{S(F)} and {L 2 } = {L 3 } ={R(O, u)}{G (v)}

v

2- R RPR/ u RRPij U 2-u Rv RPR/ u RPRij U 2-u Rv RPR/ u RRRij U 2-u Rv RPR/ u PRRij U 2-u Rv RPR/ u RPF S 2-u Rv RPR/ u PRF S 2-u Rv RPR/ u RRF S

2- uv Uv PR/u RRPij U 2- uv Uv PR/u RPRij U 2- uv Uv PR/u RRRij U 2- uv Uv PR/u PRRij U 2- uv Uv PR/u RPF S 2- uv Uv PR/u PRF S 2- uv Uv PR/u RRF S

2- u Rv PRR/u RRPij U 2- u Rv PRR/u RPRij U 2-u Rv PRR/u RRRij U 2- u Rv PRR/u PRRij U 2- u Rv PRR/u RPF S 2- u Rv PRR/u PRF S 2-u Rv PRR/u RRF S

2- uv Uv RP/u RRPij U 2- uv Uv RP/u RPRij U 2- uv Uv RP/u RRRij U 2- uv Uv RP/u PRRij U 2- uv Uv RP/u RPF S 2- uv Uv RP/u PRF S 2- uv Uv RP/u RRF S

2- u Rv RRP/u RRPij U 2- u Rv RRP/u RPRij U 2-u Rv RRP/u RRRij U 2- u Rv RRP/u PRRij U 2- u Rv RRP/u RPF S 2- u Rv RRP/u PRF S 2-u Rv RRP/u RRF S

2- uv Uv RR/u RRPij U 2- uv Uv RR/u RPRij U 2- uv Uv RR/u RRRij U 2- uv Uv RR/u PRRij U 2- uv Uv RR/u RPF S 2- uv Uv RR/u PRF S 2- uv Uv RR/u RRF S

2- u Rv RRR/u RRPij U 2- u Rv RRR/u RPRij U 2- u Rv RRR /u RRRij U

2- u Rv RRR/u PRRij U 2- u Rv RRR/u RPF S

2- u Rv RRR/u PRF S 2- u Rv RRR/u RRF S

and {L2 } = {S(Da )}{G(v)}, {L3 } = {S(Db )}{G(v)} with Da ∈ axis(O, u) and Db ∈ axis(O, u). The second combination is {L1 } = {S(D)}{G(v)} and {L2 } = {G(u)}{S(Fa )}, {L3 } = {G(u)}{S(Fb )} with Fa ∈ axis(B, v), and Fb ∈ axis(B, v). The second family results from the kinematic inversion of the first family. Fig. 10 shows a 2-SPR/RPS PM. The realization is obtained by replacing in Fig. 1, the three U joints by S pairs with the same centers. Limbs 2 and 3 are SPR chains, in which the S joint is embodied by three revolute pairs with intersecting axes. Limb 1 is a RPS chain. D1 denotes the center of the spherical joint in limb 1. F2 and F3 denote the centers of the spherical joint in limbs 2 and 3, respectively. Bi denotes the center of the last revolute pair in the ith limb.

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Fig. 7. 4–4–5 RPR-equivalent PMs in the first family. (a) 2-uv Uv PR/u RPF S. (b) uv Uv PR/u PRF S. (c) 2-uv Uv PR/u RRF S. (d) 2-uv Uv RR/u RRF S.

Fig. 9. 5–5–4 RPR-equivalent PMs with one U joint. (a) 2-u PRS/uv Sv PR. (b) 2-u RPS/uv Sv PR. (c) 2-u RRS/uv Sv PR. (d) 2-u RRS/uv Sv RR. (e) 2-Su PR/v RPuv U. (f) 2-Su RR/v RRuv U.

= {S(F3 )}{G(v)},F3 ∈ (O,u).

(17)

The displacement set of the moving platform is given by {L1 } ∩ {L2 } ∩ {L3 } = {G(u)}{S(D1 )} ∩ {S(F2 )}{G(v)} ∩ {S(F3 )}{G(v)} = {G(u)}{S(D1 )} ∩ ({S(F2 )}{G(v)} ∩ {S(F3 )}{G(v)}). (18) One can verify that Fig. 8. 4–4–5 RPR-equivalent PMs in the second family. (a) 2u RPuv U/D Sv PR. (b) 2-u RRuv U/D Sv PR. (c) 2-u RPuv U/D Sv RR. (d) 2u RRuv U/D Sv RR.

= {R(O,u}{G(v)}.

{L1 } = {R(A1 , u)}{T(r)}{S(D1 )} with r⊥u (15)

{L3 } = {S(F3 )}{T(s3 )}{R(B3 , v)} with s3 ⊥v

Substituting (19) into (18) yields

= {G(u)}{S(D1 )} ∩ {R(O, u)}{G(v)}

{L2 } = {S(F2 )}{T(s2 )}{R(B2 , v)}with s2 ⊥v

The kinematic bond of limb 3 is

(19)

{L1 } ∩ {L2 } ∩ {L3 }

The kinematic bond of limb 2 is

= {S(F2 )}{G(v)},F2 ∈ (O,u).

= ({S(F2 )} ∩ {S(F3 )}){G(v)} = {R(F2 , (F2 F3 )/||F2 F3 ||)}

The kinematic bond of limb 1 is

= {G(u)}{S(D1 )}, D1 ∈ (B, v).

{S(F2 )}{G(v)} ∩ {S(F3 )}{G(v)}

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= {R(O, u)}{T2(⊥u)}{S(D1 )} ∩ {R(O, u)}{G(v)} = {R(O, u)}({T2(⊥u)}{S(D1 )} ∩ {G(v)}).

(20)

Using the product closure in subgroups, we have {G(v)} = {T2(⊥v)}{R(D1 , v)} = {T2(⊥v)}{R(B, v)}

´ TYPE SYNTHESIS OF 3-DOF RPR-EQUIVALENT PARALLEL MECHANISMS LI AND HERVE:

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TABLE VI FORTY-NINE RPR-EQUIVALENT PMS IN THE FIRST FAMILY IN 5–5–4 CATEGORY 5–5–4

{L 1 } ={R(O, u)}{G (v)} and {L 2 } = {G (u)}{S(F a )}, {L 3 } = {G (u)}{S(F b )}, Fa ࢠ axis(B,v) and Fb ࢠ axis(B,v)

2- u RRPij U/u Rv RPR 2- u RPRij U/u Rv RPR 2- u RRRij U/u Rv RPR 2- u PRRij U/u Rv RPR 2- u RPF S/u Rv RPR 2- u PRF S/u Rv RPR 2- u RRF S/u Rv RPR

2- u RRPij U/uv Uv PR 2- u RPRij U/uv Uv PR 2- u RRRij U/uv Uv PR 2- u PRRij U/uv Uv PR 2- u RPF S/uv Uv PR 2- u PRF S/uv Uv PR 2- u RRF S/uv Uv PR

2- u RRPij U/u Rv PRR 2- u RPRij U/u Rv PRR 2- u RRRij U/u Rv PRR 2- u PRRij U/u Rv PRR 2- u RPF S/u Rv PRR 2- u PRF S/u Rv PRR 2- u RRF S/u Rv PRR

2- u RRPij U/uv Uv RP 2- u RPRij U/uv Uv RP 2- u RRRij U/uv Uv RP 2- u PRRij U/uv Uv RP 2- u RPF S/uv Uv RP 2- u PRF S/uv Uv RP 2- u RRF S/uv Uv RP

2- u RRPij U/u Rv RRP 2- u RPRij U/u Rv RRP 2- u RRRij U/u Rv RRP 2- u PRRij U/u Rv RRP 2- u RPF S/u Rv RRP 2- u PRF S/u Rv RRP 2- u RRF S/u Rv RRP

2- u RRPij U/uv Uv RR 2- u RPRij U/uv Uv RR 2- u RRRij U/uv Uv RR 2- u PRRij U/uv Uv RR 2- u RPF S/uv Uv RR 2- u PRF S/uv Uv RR 2- u RRF S/uv Uv RR

2- u RRPij U/u Rv RRR 2- u RPRij U/u Rv RRR 2- u RRRij U/u Rv RRR

2- u PRRij U/u Rv RRR 2- u RPF S/u Rv RRR

2- u PRF S/u Rv RRR 2- u RRF S/u Rv RRR

Fig. 10.

Fig. 11. 5–5–5 RPR-equivalent PMs with idle pairs. (a) 2-u RPRij U / ij Uv RRR. (b) 2-u RPRij U / ij Uv PRR

TABLE VII FORTY-NINE RPR-EQUIVALENT PMS IN THE FIRST FAMILY IN 5–5–5 CATEGORY 5-5-5

{L 1 } = {G (u)}{S(F)} and {L 2 } = {S(D a )}{G (v)}, {L 3 } = {S(D b )}{G (v)}, D a ∈ axis(O, u) and D b ∈ axis(O, u).

2- ij Uv RRP/u RRPij U 2-ij Uv RPR/u RRPij U 2-ij Uv RRR/u RRPij U 2- ij Uv PRR/u RRPij U 2- D Sv RP/u RRPij U 2- D Sv PR/u RRPij U 2- D Sv RR/u RRPij U

2- ij Uv RRP/u RPRij U 2-ij Uv RPR/u RPRij U 2-ij Uv RRR/u RPRij U 2- ij Uv PRR/u RPRij U 2- D Sv RP/u RPRij U 2- D Sv PR/u RPRij U 2- D Sv RR/u RPRij U

2- ij Uv RRP/u PRRij U 2-ij Uv RPR/u PRRij U 2-ij Uv RRR/u PRRij U 2- ij Uv PRR/u PRRij U 2- D Sv RP/u PRRij U 2- D Sv PR/u PRRij U 2- D Sv RR/u PRRij U

2- ij Uv RRP/u RRRij U 2-ij Uv RPR/u RRRij U 2-ij Uv RRR/u RRRij U 2-ij Uv PRR/u RRRij U 2- D Sv RP/u RRRij U

2- ij Uv RRP/u RPF S 2-ij Uv RPR/u RPF S 2-ij Uv RRR/u RPF S 2- ij Uv PRR/u RPF S 2- D Sv RP/u RPF S

2- ij Uv RRP/u PRF S 2-ij Uv RPR/u PRF S 2-ij Uv RRR/u PRF S 2- ij Uv PRR/u PRF S 2- D Sv RP/u PRF S

2- D Sv PR/u RRRij U 2- D Sv RR/u RRRij U

2- D Sv PR/u RPF S 2- D Sv RR/u RPF S

2- D Sv PR/u PRF S 2- D Sv RR/u PRF S

2- ij Uv RRP/u RRF S 2-ij Uv RPR/u RRF S 2-ij Uv RRR/u RRF S

2- ij Uv PRR/u RRF S 2- D Sv RP/u RRF S

2- D Sv PR/u RRF S 2- D Sv RR/u RRF S

2-SPR/RPS PM.

which shows that the 2-SPR/RPS PM is an RPR-equivalent PM. The arrangements of the two revolute axes of ij U are because B ∈ axis(D1 , v), and worth addressing. In the u RPRij U limb, the unit vectors i, j, u are linearly independent, and generally, i and j can belong {S(D1 )} = {R(D1 , u)}{R(D1 , w)}{R(D1 , v)} to the vector plane orthogonal to u without being v and w. For = {R(D1 , u)}{R(D1 , w)}{R(B, v)}. the same reason, in the ij Uv RRR limb, the unit vectors i, j, v Hence, the displacement set of the moving platform is given are linearly independent, and generally i and j can belong to the vector plane orthogonal to v without being u and w. However, by for example, one can set i = v and j = w for a u RPRij U limb and i = w and j = u or a ij Uv RRR or a ij Uv RPR limb. A 2{L1 } ∩ {L2 } ∩ {L3 } u RPRvw U/wu Uv RRR and a 2-u RPRvw U/wu Uv RPR PM is, thus, = {R(O, u)}{T2(⊥u)}{R(D1 , u)}{R(D1 , w)}{R(B, v)} obtained as shown in Fig. 11. We can readily verify that the w R ∩ {T2(⊥v)}{R(B, v)} pairs in vw U and wu U are idle, and the 2-u RPRvw U/wu Uv RRR works like an overconstrained 2-u RPRv R/u Rv RRR PM. = {R(O, u)}{T2(⊥u)}{R(D1 , u)}{R(D1 , w)} Table VII enumerates 49 RPR-equivalent PMs in this category ∩ {T2(⊥v)}{R(B, v)} Fig. 12 shows four RPR-equivalent PMs belonging to the first family in 5–5–5 category. For the readers’ sake, Fig. 13 shows = {R(O, u)}({T2(⊥u)} ∩ {T2(⊥v)}{R(B, v)} four RPR-equivalent PMs belonging to the second family in the = {R(O, u)}{T(w)}{R(B, v)} (21) 5–5–5 category.

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Fig. 14.

Sketch of a D Sv PR-u RPF S-u RRv RRR PM.

Let {L3 } = {S(D)}{G(v)} with D ∈ axis(O, u). The intersection of three limb bonds is given by Fig. 12. 5–5–5 RPR-equivalent PMs in the first family. (a) 2-D Sv PR/u RPF S. (b) 2-D Sv RR/u RPF S. (c) 2-D Sv PR/u RRF S. (d) 2-D Sv RR/u RRF S

{L1 } ∩ {L2 } ∩ {L3 } = {G(u)}{R(B, v)} ∩ {S(D)}{G(v)} = {R(O, u)}{T2(⊥u)}{R(B, v)} ∩{R(O, u)}{R(D, i)}{R(D, j)}{T2(⊥v)}{R(B, v)} = {R(O, u)}({T2(⊥u)} ∩ {R(D, i)}{R(D, j)} {T2(⊥v)}{R(B, v)}.

(23)

Note that {T2 ( ⊥ u)} ∩ {R (D, i)} {R (D, j)} {T2 (⊥v)} = {T(w)}. Therefore, we have

Fig. 13. 5–5–5 RPR-equivalent PMs in the second family. (a) 2-u RPF S/D Sv PR. (b) 2-u PRF S/D Sv PR. (c) 2-u RRF S/D Sv RR. (d) 2-u RPRij U/D Sv RR

B. {L1 } = {X(u)}{X(v)} and {L2 } = {G(u)}{S(F )} With Fࢠ(B,v) and {L3 } = {S(D)}{G(v)} With D ∈ axis(O, u) When {L1 } = {X(u)}{X(v)} and {L2 } = {G(u)}{S (F)}, the intersection of {L1 } and {L2 } is given by {L1 } ∩ {L2 } = {X(u)}{X(v)} ∩ {G(u)}{S (F) } = {G(u)}{R(B, v)}{T(u)} ∩ {G(u)}{R(B, v)} × {R(F, i)}{R(F, j)}

VII. CONCLUSION A new family of RPR-equivalent PMs is disclosed using the Lie group algebraic properties of the set of rigid-body displacements. The motions generated by the limb chains of the PMs are products of Lie subgroups that contain the RPR motion. Numerous novel RPR-equivalent PMs are enumerated. The results fill one gap in the type synthesis and mobility analysis theory of lower mobility PM. Some architectures in this family have the potential for practical applications with the addition of an x–y gantry or an articulated RR serial mechanism. REFERENCES

= {G(u)}{R(B, v)}({T(u)} ∩ {R(F, i)}{R(F, j)}) = {G(u)}{R(B, v)}.

{L1 } ∩ {L2 } ∩ {L3 } = {R(O, u)}{T(w)}{R(B, v)}. (24) The above analysis shows that the end-effector motion is RPR equivalent with limb 1 generating {X(u)}{X(v)}, limb 2 generating {G(u)}{S(F)} with F ࢠ(B, v), and limb 3 generating {S(D)}{G(v)} with Dࢠ(O, u). Fig. 14 shows the sketch of a D Sv PR-u RPF S-u RRv RRR PM in this subcategory. There are 863 RPR-equivalent PMs in this subcategory totally, which are not enumerated here for simplicity. Moreover, their potential in practice is limited due to three different limb chains.

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[1] K. H. Hunt, “Structural kinematics of in-parallel-actuated robot arms,” ASME J. Mech. Trans. Autom. Des., vol. 105, pp. 705–712, 1983. [2] J. Wahl, “Articulated tool head,” WIPO Patent WO/2000/025976, 2000.

´ TYPE SYNTHESIS OF 3-DOF RPR-EQUIVALENT PARALLEL MECHANISMS LI AND HERVE:

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Qinchuan Li (M’13) received the B.Eng., M.Eng., and Ph.D. degrees in mechanical engineering from Yanshan University, Hebei, China, in 1997, 2000, and 2003, respectively. He is currently a Professor with Zhejiang Sci-Tech University, Hangzhou, China. His research interests include type synthesis, kinematics, and application of parallel mechanisms. He has authored more than 30 publications in journals and conference proceedings.

Jacques Marie Herv´e was born in France in 1944. He received the Dipl.Ing. degree from Ecole Centrale Paris, Paris, France, in 1968 and the Ph.D. degree from the University of Paris 6 in 1976. He began his academic career in 1968, and in 1983, he was appointed as a Professor and became responsible for a research team in mechanical design with Ecole Centrale Paris. He has been an Invited Researcher in the U.S., Canada, and Japan and is also a consultant for several companies. His professional interest is teaching and research in mechanism and machine science.