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Science in China Ser. E Engineering & Materials Science 2004 Vol.47 No.6 641—650 641

Displacement manifold method for type synthe-sis of lower-mobility parallel mechanisms

LI Qinchuan1, HUANG Zhen2 & Jacques Marie Hervé3

1. Center of Integration Engineering, Zhejiang University of Sciences, Hangzhou 310033, China; 2. Robotics Research Center, Yanshan University, Qinhuangdao 066004, China; 3. Ecole Centrale Paris, France

Correspondence should be addressed to Li Qinchuan (email: lqchuanrick@yahoo.com.cn) Received June 29, 2004

Abstract Type synthesis of lower-mobility parallel mechanisms is a hot and frontier

topic in international academic and industrial field. Based on the Lie group theory, a displacement manifold synthesis method is proposed. For all the nine kinds of lower-mobility parallel mechanisms, the mechanism displacement manifold, limb displacement manifold and the geometrical conditions which guarantee that the intersection of the limb displacement manifold is the desired mechanism displacement manifold are enumerated. Various limb kinematic chains can be obtained using the product closure of displacement subgroup. Parallel mechanisms can be constructed with these limbs while obeying the geometrical conditions. Hence, all the nine kinds of lower-mobility parallel mechanisms can be synthesized using this method. Since displacement manifold deals with finite motion, the result mechanism of synthesis have full-cycle mobility. Novel architectures of lower-mobility parallel mechanisms can be obtained using this method.

Keywords: lower-mobility parallel mechanisms, Lie group, displacement manifold, type synthesis. DOI: 10.1360/ 03ye0352

Compared with the general 6-DOF (degrees of freedom) PM (parallel mechanism), lower-mobility PMs have advantages of simple structure, low cost in design, manufac-turing and control. Inventions of new architectures not only mean breakthrough in theory but also help protect independent intellectual properties. Consequently, the potential business value and application can be obtianed. The success of the DELTA robot is such a typical example1). Particularly the symmetrical lower-mobility PM, which character-izes identical limbs, symmetrical arrangement and isotropy, has been a hot and frontier subject in international academic and industrial field.

The history of type synthesis of lower-mobility PMs dates back to 1983, when

1) Bonev, I., Delta parallel robot—the story of success, http://www.parallemic.org/Reviews/Review002.html.

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Hunt[1] proposed some architectures, including some planar PMs, 3-DOF PMs, 6-DOF PMs and some asymmetrical 4- and 5-DOF PMs. Other pioneering work includes 3-DOF translational DELTA robot proposed by Clavel[2] in 1988, the 3-DOF transla-tional PMs with 4-DOF limbs proposed by Hervé[3] in 1991, the 3-UPU translational PM proposed by Tsai[4] in 1996, the 4-DOF H4 robot proposed by Pierrot and Company[5] in 1999, the 4-DOF 4-URU PM proposed by Zhao and Huang[6] in 2000 and the 4-DOF PM proposed by Zlatov and Gosselin[7] in 2001. However, for over 20 years, a universal and effective type synthesis theory has not been well established. The consequence is that there has been a lack of symmetrical 4- and 5-DOF PMs, particularly those without closed-loops in limb.

There are two traditional methods for type synthesis of PMs. One is the enumera-tion method based on the general Grübler-Kutzbach mobility criterion. Tsai[8] applied this method to the type synthesis of a kind of 3-DOF PM. The other is the displacement subgroup synthesis method proposed by Hervé[3,9]. Hervé[3,10] and his colleagues studied the type synthesis of 3-DOF translational and rotational PMs. However, these two methods cannot be applied to all lower-mobility PMs. Merlet1) pointed out that the first method fails to describe the geometrical arrangement of kinematic pairs, so invalid re-sults were often obtained; the effective range of the second method was limited because of the difficulty in retaining the algebraic structures of group. In addition, Yang[11] et al. proposed a type synthesis method based on single-unit open chains and synthesized some mechanisms.

This indicates that the problem of type synthesis of lower-mobility parallel mecha-nism has not been well solved over the last 20 years. The lower-mobility PMs can be sorted into nine categories according to their kinds of mobility. However, the above-mentioned method cannot be applied to all the nine categories. The direct result of such a lack is that few symmetrical 4- and 5-DOF PMs without closed-loop have been proposed. Particularly, most of the kinematic experts were thinking that no symmetrical 5-DOF parallel mechanisms can be discovered. Hunt[1] believed that the symmetrical 5-DOF parallel mechanism does not exist but is instantaneous. Merlet1)[12,13] believed that there were no 4- and 5-DOF parallel robot with identical chains.

Recently, Huang’s group filled this gap and proposed the constraint-synthesis

—

method2)[1416] based on screw theory. The constraint-synthesis method is effective uni-versally and has been applied to the type synthesis of all nine kinds of lower-mobility PMs[16]. Numerous novel PMs have been invented using this method, including the symmetrical 5-DOF PMs[17]. The constraint-synthesis method belongs to the instantane-ous motion field, because screw is instantaneous. Hence, the last indispensable step of

1) Merlet, J. P., Still a long way to go on the road for parallel mechanisms, ASME Design Engineering Tech-nical Conferences, Keynote speech, Montreal, 2002, http://www-sop.inria.fr/coprin/equipe/merlet/ASME/ asme2002.html.

2) Zhao, T. S., Research on analysis and synthesis of lower-mobility parallel mechanisms, PhD Dissertation, Yanshan University, 2000.

Displacement manifold method for type synthesis of lower-mobility parallel mechanisms 643

the constraint-synthesis method is to identify if the synthesized mechanism is only in-stantaneous or not, which sometimes is rather difficult for some complicated architec-tures.

This paper aims to propose a universally effective type synthesis method in the fi-nite motion field. Based on the Lie group theory, a displacement manifold synthesis method is proposed. Limb displacement manifold (LDM) is used to describe the motion of the limb end while mechanism displacement manifold (MDM) is used to describe the motion of the moving platform. The geometrical conditions which guarantee that the intersection of the limb displacement manifold is the desired MDM are enumerated. Various limb kinematic chains can be obtained using the product closure of displacement subgroup. Parallel mechanisms can be constructed with these limbs while obeying the geometrical conditions. Hence, all kinds of lower-mobility parallel mechanisms can be synthesized using this method. 1 Manifold synthesis method 1.1 Displacement subgroup

The set of 6-dimensional rigid motion is endowed with group algebraic structures and forms a Lie group, {D}. Further the motion of a rigid body can be described by a subset of {D}, which can be a group, called displacement subgroup, or a displacement manifold. In 1978, Hervé[9] enumerated all 12 kinds of displacement subgroups, as shown in table 1. It also can be readily proven that the set of relative motion allowed by a lower pair constitutes a displacement Lie subgroup. Hervé[9] also enumerated the in-tersection and composition of different displacement subgroups. The intersection of subgroups follows the rules of intersection of sets.

From table 1, it can be seen that the 12 displacement subgroups cannot explain all the rigid motion in space. Under most conditions, the rigid motion is a displacement manifold included in {D}, as shown in table 2. 1.2 Limb displacement manifold

When analyzing a serial kinematic chain composed of rigid bodies 1, 2, …, i−1, i, the allowed displacements of body i relative to body 1 is a subset of the group of rigid-body motions or displacements. This subset is the composition by implementation of the group product, of all the subgroups associated with the lower kinematic pairs in the kinematic chain. This subset may be a subgroup or only a manifold included in {D} in most cases. For example, a 3P kinematic chain generates the 3-dimensional transla-tional subgroup, namely {T(u)}.{T(v)}.{T(w)}={T}. A RPS kinematic chain only generates a 5-dimensional displacement manifold, namely {R(N1,u)}.{T(v)}.{S(N2)}.

The 1-dimensional displacement subgroup can be generated by 1-DOF pair. Be-cause the multi-DOF pair or joint is equal to the composition of revolute pairs and pris-matic pairs kinematically, the 3-dimensional rotation displacement subgroup {S(N)} is

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Science in China Ser. E Engineering & Materials Science 2004 Vol.47 No.6 641—650

Table 1 Enumeration of displacement subgroups

Displacement subgroup

Dim

Mechanical generator

Note

Displacementsubgroup

Dim Note

Rotation about the axis {R(N, u)} 1 Revolute pairdetermined by the unit

vector u and point N {T(v)} or {T1(v)}

Prismatic pair

Translation along the unit vector v

Rigid connection, no

{E} 0

relative motion {T(Pvw)} or {T2(P)}

Planar motion in plane Pvw determined by two unit vectors v and w

Helical motion deter-

3-dimensional translation

{H(N, u, p)} 1 Helical pair mined by the axis (N, v) {T} 3

in space

and pitch p

Motion formed by planar translation determined by Cylindrical motion

Cylindrical

{Y(w, p)} 3 {C(N, v)} 2 the normal w and the determined by the axis

pair

helical motion with pitch (N, v)

p parallel to w

3-dimensional translation

Planar motion deter- {G(u)} or

3 Planar pair {X(w)} 4 and one rotation about

mined by the normal u {G(Pvw)}

the unit vector w

Spherical Rotation about the 6-dimensional rigid

{S(N)} 3 {D} 6

joint point N motion

Table 2 Rigid motion and displacement subgroups

Rigid motion Subgroups

3-dimensional rotation and 2-dimensional No

translation (3R2T)

2-dimensional rotation and 3-dimensional

translation (2R3T)

1-dimensional rotation and 3-dimensional

{X(w)}

translation (1R3T)

3-dimensional rotation and 1-dimensional

translation (3R1T)

2-dimensional rotation and 2-dimensional

translation (2R2T)

Rigid motion

3-dimensional rotation (3R) 3-dimensional translation (3T)

Subgroups {S(N)} {T}

2-dimensional rotation and

1-dimensional translation (2R1T)

1-dimensional rotation and

{G(u)}

2-dimensional translation (1R2T)

equal to the composition product of three 1-dimensional rotation subgroups, whose axes intersect at a common point, namely {R(N,i)}.{R(N,j)}.{R(N,k)}. In other words, the subgroup is generated by a 3R spherical subchain formed by three revolute pairs whose axes intersect at a common point. Such a 3R spherical subchain is denoted by (iRjRkR)N, where the superscript i, j, k denote the three revolute axes and N the intersec-tion point.

The 3-dimensional subgroup {G(u)} represents 2-dimensional translation in a plane and 1-dimensional rotation about the normal to the plane. The subgroup {G(u)} can be generated by the kinematic chains listed in table 3, where the superscript denotes the axis of kinamatic pair, v and w are two linearly independent unit vectors in the plane. {G2(u)} is a 2-dimensional displacement manifold and can be generated by the kine-matic chain in table 3. The subgroup {S(N)} also includes a 2-dimensional displacement manifold {R(N, i)}.{R(N, j)}, denoted by {S2(N)}. The manifold {S2(N)} can be gener-ated by a 2R spherical subchain (iRjR)N. 1.3 Synthesis procedure

When analyzing a parallel mechanism, the set of the allowed displacements of the

Displacement manifold method for type synthesis of lower-mobility parallel mechanisms

Table 3 Bond and generators of {G(u)} and {G2(u)}

Kinematic bond of {G(u)} {T(v)}.{T(w)}.{R(N,u)}

Generators of {G(u)}

vuv

645

PPR RvPwP

{R(N,u)}.{{T(v)}.{T(w)} {T(v)}.{R(N,u)}.{T(w)} {R(A,u)}.{R(B,u)}.{T(w)} {R(A,u)}.{T(w)}.{R(B,u)} {T(w)}.{R(A,u)}.{R(B,u)}

Kinematic bond of {G2(u)} Generators of {G2(u)}

vw{T(v)}.{T(w)} PP u{R(A,u)}.{T(w)} RwP

PuRwP

RuRwP

uwu

{T(w)}.{R(A,u)} {R(A,u)}.{R(B,u)}

PuR RuR

RwPuR PuRuR RuRuR

{R(A,u)}.{R(B,u)}.{R(C,u)}

moving platform is the intersection of the displacement subgroups or manifolds that are generated by the limbs. The intersection of manifolds is actually the process of finding the intersection of two adjacent subgroups repeatedly, which follows the rules given by Hervé[9]. The motion of the moving platform can be described by MDM and the motion of the ith limb can be described by limb displacement manifold. Hence, the type synthe- sis of PMs can be described as follows:

Given desired {M}, find {Li} and geometrical conditions which make {M}=∩{Li}.

i=1n

The general procedure of displacement manifold synthesis method is as follows: Step 1. Use MDM {M} to describe the motion of the moving platform with de- sired mobility.

The desired manifold {M} can be expressed by the composition product of sub- group {R(N, u)} and {T(v)}.

Step 2. Use manifold {M} to obtain the limb displacement manifold {Li}.

For symmetrical 5-DOF PMs, the MDM is also the LDM namely {M} = {Li}. For symmetrical 4-DOF PMs, the dimension of the MDM is four while the dimen- sion of the limb displacement manifold can be four or five. When the dimension of LDM is four, we have {M} = {Li}. Further, using the product closure and associativity of dis- placement subgroup, the LDM {Li} can be expanded to a 5-dimensional displacement manifold.

For symmetrical 3-DOF PMs, the dimension of the MDM is three while the dimen- sion of the limb displacement manifold can be three, four or five. When the dimension of LDM is three, we have {M} = {Li}. Further, using the product closure and associativ- ity of displacement subgroup, the LDM {Li} can be expanded to a 4- or 5-dimensional displacement manifold.

After {Li} is determined, using the product closure and associativity of dis- placement subgroup, we can obtain numerous kinematic equivalences of {Li}, which can be generated by different limb kinematic chains.

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Science in China Ser. E Engineering & Materials Science 2004 Vol.47 No.6 641—650

Step 3. Find the geometrical conditions which guarantee {M}=∩{Li}. These

i=1

geometrical conditions are also the inherent structural characteristics of the PMs, which will not change. Based on the characteristics[17] of lower-mobility PMs, the geometrical conditions can be obtained.

The MDM, LDM and corresponding geometrical conditions of symmetrical 5-DOF PMs are enumerated in table 4.

Table 4 MDM, LDM and geometrical conditions of symmetrical 5-DOF PMs

5-DOF PM

{M} {Li}

{T(Pvw)}.{S(N)}

Conditions for {M}=∩{Li}

i=1

3R2T {T(Pvw)}.{S(N)} {G(Pivw)}.{S2(Ni)} {G2(Pivw)}.{S(Ni)}

Pivw//Pvw, Ni=N ui//u, vi//v ui//u, vi//v

Pi1//Pj1, u⊥Pi1, Pi2//Pj2, v⊥Pj2

{T}.{R(Ni1,ui)}.{R(Ni2,vi)}

2R3T

{T}.{R(N,u)}{R(N,v)} {X(ui)}.{R(Ni1,vi)}

{G(Pi1)}.{G2(Pi2)}

The MDM, LDM and corresponding geometrical conditions of symmetrical 4-DOF PMs are enumerated in table 5.

Table 5 MDM, LDM and geometrical conditions of symmetrical 4-DOF PMs

4-DOF PM

{M} {Li}

{T(ui)}.{S(Ni)}

Conditions for {M}=∩{Li}

i=1

ui//u, Ni=N Pli//u, Ni=N

3R1T {T(u)}.{S(N)}

{T2(Pi)}.{S(Ni)} {G(Pi)}.{S2(Ni)} {G2(Pi)}.{S(Ni)} {T}.{R(N,ui)} {X(ui)}

ui//u

Pi1//u, Pi2⊥u ui//u, Pi2⊥u

Pi⊥u, N1=N2,N3=N4

1R3T {T}.{R(N,u)}

{G2(Pi1)}.{G(Pi2)} {G(Pi1)}.{G2(Pi2)} {T(ui))}.{G(Pli2)}

2R2T

{T(P)}.{R(N1,u)}.{R(N2,v)}

{G2(Pi)}.{S(Ni)}

P⊥u, v//P

{G(Pi)}.{S2(Ni)}

The MDM, LDM and corresponding geometrical conditions of symmetrical 3-DOF PMs are enumerated in table 6, where Pi1≠Pj1 means the two planes are not parallel and the situation when the limb generates subgroups is not included.

Step 4. Use the limb kinematic chain obtained in Step 2 to construct PMs while obeying the geomerical conditions obtained in Step 3. Since displacement subgroup and

Displacement manifold method for type synthesis of lower-mobility parallel mechanisms 647

manifold deals with finite motion, it is not necessary to identify whether the synthesized PM is instantaneous.

Table 6 MDM, LDM and geometrical conditions of symmetrical 3-DOF PMs

3-DOF PM

3T {T}

{M} {Li}

{G(Pi1)}.{G2(Pi2)}

Conditions for {M}=∩{Li}

i=1

Pi1≠Pj1, Pi2≠Pj2

{S(Ni1)}.{S2(Ni2)}

Ni1≠N, Ni2=N或 Ni2≠N, Ni1=N

Pi≠Pj, Ni=N Pi//w, Pi≠Pj, Ni≠N Pi//w, Pi≠Pj, Ni≠N Pi//Pj, Ni≠N

3R {S(N)}

{G(Pi)}.{S2(Ni)} {G2(Pi)}.{S(Ni)} {G(Pi)}.{S2(Ni)}

2R1T 2T1R

{T(w)}.{R(N,u)}{R(N,v)} {T(w)}.{T(v)}{R(N,u)}

{G2(Pi)}.{S(Ni)} {G2(Pi)}.{S(Ni)}

2 Examples of type synthesis

2.1 Type synthesis of 3R2T 5-DOF parallel mechanims

The target of type synthesis is 5-DOF parallel mechanisms with three rotational DOF and two translational DOF in XY plane. From table 4, it is easy to know that the MDM is {M}={T(Pxy)}.{S(N)}. There are three corresponding LDMs. For simplicity, we only discuss the situation when {Li}={G(Pixy)}.{S2(Ni)}.

The kinematic chain generating {G(Pixy)}{G(Pixy)} can be obtained from table 3 by setting v = x, w = y and u = z. The generator of {S2(Ni)} is a 2R spherical subchain (jRkR)Ni. Linking the kinematic chain generating {G(Pixy)} in table 3 to the 2R

spherical subchain (jRkR)Ni, seven lime kinematic chains can be obtained, for exam- ple, zRzRzR(iRjR)Ni.

The limb kinematic chain obtained above only consists of 1-DOF kinematic pairs. With appropriate arrangement of kinematic pairs in a limb, cylindrical pair and universal joint can be obtained. Obviously, {G(Pixy)} and {S2(Ni)}{S2(Ni)} contain no displace-ment subgroup {C(Ni, vi)} generated by a cylindrical pair. When the last factor in the products that generate {G(Pixy)} is a 1-dimensional translational subgroup {T(yi)}, the {C(N, w)} can be obtained by setting the axis of the first 1-dimensional rotational sub-group {R(Ni, ii)} in {S2(Ni)} parallel to yi, while yi, ji and ki are linearly independent. For example,

{G(Pixy)}.{S2(Ni)}={R(Ai,z)}.{R(Bi,zi)}.{T(yi)}.{R(Ni,yi)}.{R(Ni,ki)}

={R(Ai,z)}.{R(Bi,z)}.{C(Ni,yi)}.{R(Ni,ki)}.

(1)

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Science in China Ser. E Engineering & Materials Science 2004 Vol.47 No.6 641—650

The limb kinematic chain obtained in eq. (1) is zRzRyCNikRNi.

Because of product closure of displacement subgroup, the sequence of kinematic pairs in the chain that generates {C(Ni,yi)} can be changed, namely,

{C(Ni,yi)}={T(yi)}.{R(Ni,yi)}={R(Ni,yi)}.{T(yi)}, (2)

where {C(Ni,yi)} can be generated by

PyRNi or yRNiyP. Replacing yCNi in

limb kinematic chain zRzRyCNikRNi leads to a new limb kinematic chain

RzRyRNiyPkRNi.

3R2T 5-DOF parallel mechanisms can

be constructed using the limb kinematic chain obtained above while obeying the structural conditions in table 4. Fig. 1 shows

Fig. 1. 3-zRzRyCNkRN.

3-zRzRyCNkRN parallel mechanism,

where B denotes the base and M the moving

platform.

2.2 Type synthesis of 3R1T 4-DOF parallel mechanisms

The target of type synthesis is 4-DOF parallel mechanisms with three rotational DOF and one translational DOF in Z direction. From table 5, it is easy to know that the MDM is {M}={T(z)}.{S(N)}. There are three corresponding LDMs. For simplicity, we only discuss the situation when {Li}={G(Pi)}.{S2(Ni)}.

From the structural condition Pi//z in table 5, it is evident that Pi is perpendicular to the XY plane or the base. Hence, the normal of Pi, namely, u, must be parallel to the XY plane or the base. The mechanical generator of {G(Pi)} can be obtained from table 3 by setting v = x, w = z, u = x. Note that at least one 1-dimensional translational subgroup in the product that generates {G(Pi)} must not be parallel to the XY plane. For example, one product that generates {G(Pi)} is {R(Ai,xi)}.{T(pi)}.{R(Bi,xi)}, where pi is not parallel to the XY plane. The corresponding limb kinematic chain is xRpPxR,wherethe superscript p denotes the direction of the prismatic pair. Then, we can set the revo-lute axis of the first rotational subgroup {R(Ni,ii)} perpendicular to xi, namely, ii⊥xi, and let the axis of {R(Ni,ii)} intersect the axis of {R(Bi,xi)}. Consequently, {R(Bi,xi)}.{R(Ni,ii)} is a 2-dimensional displacement manifold in {D} and can be generated by a universal joint. Thus we obtain a limb kinematic chain xRpPxUiNkRN. With four xRpPxUiNkRN limbs, we can construct a 4-xRpPxUiNkRN parallel mecha-Copyright by Science in China Press 2004

Displacement manifold method for type synthesis of lower-mobility parallel mechanisms 649

nism by setting all the limb centers coincident with each other and the revolute pair fixed on the base not parallel to each other, as shown in fig. 2. Various limb kinematic chains can be obtained by using the product closure of displacement subgroup.

Fig. 2. 4-xRpPxUiNkRN.

Fig. 3. 3-xRpPxRiRNkRN.

2.3 Type synthesis of 2R1T 3-DOF parallel mechanism

The target of type synthesis is 3-DOF parallel mechanisms with two rotational DOF about X, Y axis respectively and one translational DOF in Z direction. From table 6, it is easy to know that the MDM is {M}={T(z)}.{R(N,x)}{R(N,y)}. There are two corre-sponding LDMs. For simplicity, we only discuss the situation when {Li}= {G(Pi)}.{S2(Ni)}.

Comparing the structural conditions of 3R1T 4-DOF parallel mechanisms in table 5 with the structural conditions of 2R1T 3-DOF parallel mechanisms, we can find that the difference is only whether the limb centers are coincident with each other or not. Thus, the limb kinematic chains of 3R1T 4-DOF parallel mechanisms can be used to construct the 2R1T 3-DOF parallel mechanism by setting all the limb centers not coincident. For example, the xRpPxRiRNikRNi limbs generate {R(Ai,xi)}.{T(pi)}. {R(Bi,xi)}.{R(Ni,ii)}.{R(Ni,ki)}. We can construct a 3-xRpPxRiRN3kRN3 parallel

mechanism, where the subscript N3 denotes that there exist three limb centers. From i

xpxi

ref. [16], we know that the 3-RPRRNkRN parallel mechanism has two rotational DOF about X, Y axis respectively and one translational DOF in Z direction, as shown in fig. 3.

3 Conclusions

Displacement manifold method for type synthesis of lower-mobility PMs is based

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Science in China Ser. E Engineering & Materials Science 2004 Vol.47 No.6 641—650

on the algebraic structure properties of Lie group. Furthermore, in this method, the dis-placement manifolds are characterized by geometrical entities that are intrinsically de-fined (like points, vectors etc.) instead of matrix subsets depending on the reference frame. Consequently, the architectures of PMs, including limb kinematic chains and structural conditions, can be obtained in a straightforward manner. It is shown that this method is applicable to all nine kinds of lower-mobility parallel mechanisms.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 50075074).

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