topology of serial and parallel manipulators and topological diagrams 2008 mechanism and machine...

8/3/2019 Topology of Serial and Parallel Manipulators and Topological Diagrams 2008 Mechanism and Machine Theory

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Topology of serial and parallel manipulatorsand topological diagrams

Xiaoyu Wang, Luc Baron *, Guy Cloutier

Ecole Polytechnique of Montreal, Department of Mechanical Engineering, C.P. 6079, Succ. Centre-Ville, Montreal, QC, Canada H3C 3A7

Received 3 November 2006; received in revised form 2 May 2007; accepted 15 May 2007Available online 12 July 2007

Abstract

This paper deals with manipulator topology. The term topology was first introduced into robotics to characterize thekinematic structure of a manipulator without reference to its dimensions. Its dimension-independent aspect does not pose aconsiderable problem to planar manipulators, but makes it no longer appropriate to describe spatial manipulators espe-cially spatial parallel manipulators, because such important properties as the degree of freedom of a manipulator and thedegree of mobility of its end-effector as well as the nature of the mobility are highly dependent on some dimensional ele-ments. In order to address this fundamental issue in kinematic synthesis, we introduced essential constraints into the con-cept of topology through the analysis of representative architectures, proposed a topological representation which providesa better correspondence between the representation and the intended manipulators, a necessity for topological synthesis

and documentation. Introducing the essential constraints can also improve the efficiency of some synthesis methods. 2007 Elsevier Ltd. All rights reserved.

Keywords: Manipulator; Topology; Geometry; Representation; Diagram

1. Introduction

As far as the kinematics is concerned, a mechanism is a kinematic chain with one of its links designated asthe base and another as the end-effector. A manipulator is a mechanism with all or some of its joints actuated.Driven by the actuated joints, the end-effector and all links undergo constrained motions with respect to the

base [1]. A parallel manipulator is a mechanism whose end-effector is connected to its base by at least twoindependent kinematic chains [2]. The early works in the manipulator research mostly dealt with a particulardesign; each design was described in a particular way. With the number of designs increasing, the consistency,preciseness and conciseness of topological description of manipulators become more and more problematic.To describe how a manipulator is kinematically constructed, no normalized term and definition have been

0094-114X/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mechmachtheory.2007.05.005

* Corresponding author. Tel.: +1 514 340 4711x4744; fax: +1 514 340 5867.E-mail address: [email protected] (L. Baron).

Available online at www.sciencedirect.com

Mechanism and Machine Theory 43 (2008) 754770

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MechanismandMachine Theory
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proposed. The words architecture [3], structure [4], topology [5], and type [6,7] all found their way into the lit-erature, describing kinematic chains without reference to dimensions. However, the kinematic properties ofspatial manipulators are highly sensitive to some dimensional elements. The problem is that with the originaldefinition of the term topology (the term topology is preferred here to other terms), manipulators of the sametopology are often too different to even be classified in the same category. It is therefore necessary to take into

account some dimensional elements when dealing with topology.Words used to define kinematic details are equally diverse. Parameter [8], dimension [9,10] and geometry [11]are among the terms used to this end. As for the kinematic representation of parallel manipulators, one canhardly find a method which is adequate for a wide range of manipulators and commonly accepted and used inthe literature. However, in the classification [12,13], comparison studies [14,15] (equivalence, isomorphism,similarity, difference, etc.) and manipulator kinematic synthesis, an effective kinematic representation isessential.

This work is aimed at identifying and introducing essential geometry elements into topology and propos-ing a new topological representation accordingly. The representation should work appropriately for most ofthe manipulators and be as precise and concise as possible. It should also be adaptable to facilitate auto-matic kinematic synthesis. The rest of this paper is organized as follows. In Section 2, some basic conceptsand definitions are presented, the importance of geometric constraints in determining the basic kinematic

properties are discussed. The concepts of kinematic composition, essential constraints, topology, and geom-etry are introduced in Section 3. In Section 4, the kinematic representations are reviewed and their limita-tions identified. Based on the review and the analysis of geometric constraints, a topological diagramis proposed. Topological diagrams of representative manipulators are presented in Section 5. Howthe efficiency of kinematic synthesis can be improved by introducing essential constraints is discussed inSection 6.

2. Preliminary

Some basic concepts and definitions about kinematic chains are necessary to review as a starting point ofour discussion on topology and topological representation. A kinematic chain is a set of rigid bodies, also

called links, coupled by kinematic pairs. A kinematic pair is, then, the coupling of two rigid bodies so as toconstrain their relative motion. We distinguish upper pairs and lower pairs. An upper kinematic pair con-strains two rigid bodies such that they keep a line or point contact; a lower kinematic pair constrains two rigidbodies such that a surface contact is maintained [16]. A joint is a particular mechanical implementation of akinematic pair [17]. As shown in Fig. 1, there are six types of joints corresponding to the lower kinematic pairs

spherical (S), cylindrical (C), planar (E), helical (H), revolute (R) and prismatic (P) [18]. Since all these jointscan be obtained by combining the revolute and prismatic ones, it is possible to deal only with revolute andprismatic joints in kinematic modeling. Moreover, all these joints can be represented by elementary geometricelements, i.e., point and line. To characterize links, the notions of simple link, binary link, ternary link, quater-nary linkand n-linkwere introduced to indicate how many other links is connected to a link. Similarly, binaryjoint, ternary joint and n-joint indicate how many links are connected to a joint. A similar notion is the con-nectivity of a link or a joint [19]. These basic concepts constitute a basis for kinematic analysis and kinematicsynthesis.

For kinematic analysis, the kinematic description of a mechanism includes basic information and param-eters. The basic information is which link is connected to which other links by what types of joints. This basicinformation is referred to as structure, architecture, topology, or type, respectively, by different authors. Thisbasic information alone is of little interest, because the kinematic properties of the corresponding mechanismscan vary too much to characterize a mechanism. This can be demonstrated by the single-loop 4-bar mecha-nisms shown in Fig. 2. Without reference to dimensions, all mechanisms shown in Fig. 2 are of the same kine-matic structure but have very distinctive kinematic properties and therefore are used for different applications

mechanism a: generates planar motion; mechanism b: generates spherical motion; mechanism c: is a Bennettmechanism [20], while mechanism d: permits no relative motion at any joints. Fig. 3 shows an example of par-allel mechanisms having the same kinematic structure mechanism a: has 3 degrees of freedom (DOFs) whose

end-effector has no mobility; mechanism b: has 3 DOFs whose end-effector has 3 degrees of mobility (DOMs)

X. Wang et al. / Mechanism and Machine Theory 43 (2008) 754770 755


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in translation; mechanism c: permits no relative motion at any joints. A particular mechanism is thus

described, in addition to the basic information, by a set of parameters which define the relative position

Prismatic (P)

Revolute (R)

Spherical (S)

Cylindrical (C)

Helical (H)

Planar (E)

Fig. 1. Lower kinematic pairs.

Fig. 2. 4-Bar mechanisms of different geometries.

Fig. 3. 3-PRRR parallel mechanisms.

756 X. Wang et al. / Mechanism and Machine Theory 43 (2008) 754770


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and orientation of each joint with respect to its neighbors. For complex closed-loop mechanisms, an oftenignored problem is that certain parameters must take particular values in order for the mechanism to be func-tional and have the intended kinematic properties. More attention should be payed to these parameters whichplay a qualitative role in determining some important kinematic properties of the mechanism.

For kinematic synthesis, not only do the eligible mechanisms have particular kinematic structures, but also

they feature some particular relative positions and orientations between certain joints. If this particularity isnot taken into account when formulating the synthesis model, a great number of mechanisms generated withthe model will not have the required kinematic properties and have to be discarded.

In Section 3, we investigate the special joint dispositions and give a new definition of the term topology byintroducing the essential constraints.

3. Topology

Since the 1960s, a very large number of manipulator designs have been proposed in the literature or dis-closed in patent files. The kinematic properties of these designs were studied mostly on a case by case basis;characteristics of their kinematic structure were often not investigated explicitly; the constraints on the relative

joint locations which are essential for a manipulator to meet the kinematic requirements were rarely put into

the kinematic representation.Constraints are introduced mainly to meet the functional requirements, to simplify the kinematic model, to

optimize the kinematic performances, or from manufacturing considerations. These constraints can berevealed by investigating the underlying design ideas.

For a serial manipulator to generate planar motion, all its revolute joints need to be parallel and all its pris-matic joints should be perpendicular to the revolute joints. For a serial manipulator to generate sphericalmotion, the axes of all its revolute joints must be concurrent [21]. For a parallel manipulator with three iden-tical legs to produce only translational motion, the revolute joints of the same leg must be arranged in one ortwo directions [22].

A typical example of simplifying the kinematic model is the decoupling of the position and orientation ofthe end-effector of a 6-joint serial manipulator. This is realized by having three consecutive revolute joint axes

concurrent. A comprehensive study was presented in [23] on the inverse kinematic solutions of 6-joint serialmanipulators. The study reveals how the inverse kinematic problem is simplified by making joint axes parallel,perpendicular or intersect.

Optimizing the kinematic performance also gives rise to special relative joint locations. Isotropic parallelmanipulators having orthogonal actuated joint axes proposed in [24,25] are examples of this kind. In sum-mary, these special constraints are the special relative joint locations, i.e. parallelism, perpendicularity, inter-section, coplanarity, etc.

To facilitate the kinematic analysis, kinematic synthesis, classification and comparison studies of manipu-lators, the terms kinematic composition, essential constraint, topology, geometry and their definitions are pro-posed as follows:

the kinematic composition of a manipulator is the essential information about the number of its links, whichlink is connected to which other links by what types of joints and which joints are actuated;

the essential constraints are the minimum conditions for a manipulator of given kinematic composition tohave the required DOF and DOM;

the topology of a manipulator is its kinematic composition plus the essential constraints; The geometry of a manipulator is a set of constraints on the relative locations of its joints which are unique

to each of the manipulators of the same topology.

Hence, topology also has a geometrical aspect such as parallelism, perpendicularity, coplanar, and evennumerical values and functions on the relative joint locations which used to be considered as geometry. Byour definition, geometry no longer includes relative joint locations which are common to all manipulatorsof the same topology because the later are the essential constraints and belong to the topology category. A

manipulator can thus be much better characterized by its topology.



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4. Topological diagrams

4.1. Topological representation review

The basic notions about link and joint introduced in Section 2 were used to describe manipulator kinemat-

ics. With these notions, the general spatial arrangement of links and joints of a manipulator can be describedwith ease. For example, the kinematic chain associated with a serial manipulator is constructed with binarylinks, two simple links and binary joints; while that associated with a parallel manipulator is composed of bin-ary links, two multiple links and binary joints. The limitation of such description is obvious since the infor-mation about the joint types and which joints are actuated can not be provided without lengthy notations.Alphanumeric representation, 3-UPU or 3-RRR for example, may be the most effective one for kinematiccomposition, which was widely used in the literature. In general, the number indicates the number of identicalkinematic chains, the sequence of upper case letters indicates the joint types along the kinematic chain and theactuated joint is identified by an underline. One of the more visual representation methods is the kinematicdiagram. A kinematic diagram is a drawing of a mechanism showing its essential elements in simplified formwith graphical symbols (Fig. 4). Although a great deal of effort has been made to normalize the graphical sym-bols for kinematic diagrams [27,26,28], the practice in the literature is just far from normalized. Similar to the

kinematic diagram, the kinematic layout graph (Fig. 5) was proposed in [29]. With the kinematic layout graph,each joint is represented by an uppercase letter indicating the joint type surrounded by a rectangle; joints car-ried by the same link are connected by a line. The kinematic layout graph or its variants (Fig. 6) can be foundin a number of recent publications. The common problem with the above kinematic representations is thepoor correspondence between the representation and the intended manipulators. One kinematic diagram orlayout graph may correspond to many manipulators with quite different kinematic properties.

Revolute

Prismatic

Cylindrical

Spherical

a

b

c

d

Fig. 4. Kinematic joint symbols [26].

RepresentationJoint name

Passive joint Actuated joint

Revolute

Prismatic P

RR

P

Fig. 5. Layout graph symbols [29].



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For planar mechanisms, there exist three similar representations, i.e., structural schema, graph, and func-tional schema (Fig. 7). The graph representation proposed in [30,31] is based on an abstract kinematic repre-sentation similar to the symbolic representation of chemical compound. With graph representation, thevertices denote links while edges denote joints. To distinguish different joints, the edges can be labeled or num-bered. Since a graph can be easily represented by matrices, automatic synthesis algorithms can be implementedand kinematic chains can be systematically enumerated using graph theory and combinatorial analysis. Whenused with spatial mechanisms, the graph representation suffers from the same drawbacks as the kinematic dia-gram and the kinematic layout graph. For spatial manipulators, joint and link locations are often so complexthat perspective views and section views have to be used to represent the kinematic chains. Although the per-spective views and section views are simplified as much as possible, the special features of the joint locations,

parallelism and perpendicularity for example, can hardly be recognized by anyone not familiar with spatialmechanism. The topology features and geometry features can not be distinguished without complementarynotations. A new representation of layout graph with graphical symbols for geometric constraints was pro-posed in [32]. Only one example of this representation was given, whether it is suitable for a wide range ofmanipulators remains to be seen. DenavitHartenberg notation [8] and the initial configuration matrix [22]were used for implicit topological representation when emphasis was put on geometric synthesis and visualrepresentation could be compromised. In summary, the existing kinematic representations are only suitablefor kinematic composition or kinematic geometry. There is a lack of topological representation.

4.2. Kinematic composition

Taking the basic ideas of graph and layout graph, we propose that the kinematic composition be repre-sented by a diagram having the graph structure so as to be eventually adapted for automatic synthesis. The

joint type is designated as an upper case letter, i.e., R for revolute, P for prismatic, H for helical, C for cylin-drical, S for spherical and E for planar. Actuated joints are identified by a line under the corresponding joint.The letters denoting joint types are placed at the vertices of the diagram, while the links are represented byedges. But there is a problem for non serial mechanisms. If a link carries more than two joints, then this linkcorresponds to more than one edge. From the diagram, edges corresponding to one link can not be identifiedand the fact that no motion is permitted between them is not reflected. A simple way to solve this problem,while keeping the graph structure, is the use of an arrow at the edge end. Each joint has two joint elements, towhich element a link is connected is indicated by the presence or absence of the arrow. Any link connected tothe same joint element is actually rigidly attached and no relative motion is possible. ! R ! R ! R ! is a

serial mechanism while ! RM RM R or RRR is a single link having three revolute joints. In

0 1 22

R7

R2

R3 3

6

4 5R5R1R6

R4

Fig. 6. Layout graph symbols [19].

Structural schema Graph Functional schema

a b c

Fig. 7. Structural schema, graph, and functional schema.



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this way, the diagram always has a graph structure. The kinematic composition of a planar parallel manipu-lator is shown in Fig. 8. The most left column represents the base carrying three actuated revolute joints whilethe most right column the end-effector. The end-effector is connected to the base by three identical kinematic

chains composed of three revolute joints respectively. It is noteworthy that a very different manipulator shownin Fig. 9 has exactly the same kinematic composition. The diagram must bear additional information in orderto appropriately represent the topology.

4.3. Essential constraints

Essential constraints are the relative locations between lines. Constraints on relative joint axis locations canbe summarized as six and only six possible situations, which are depicted in Fig. 10.

Superimposing the essential constraint symbols on the kinematic composition diagrams shown in Figs. 8and 9, we get the diagrams shown in Figs. 11 and 12. We can see that the topology of the planar parallelmanipulator is fully represented by the diagram. The diagram shown in Fig. 13 still does not reflect the fact

that two joint axes intersect the third joint axis at the same point.In order to characterize the relative situations of non-adjacent joint axes, we introduce subscripts andsuperscripts into the diagram. Axes of the joints with the same subscript are concurrent while those withthe same superscript are parallel. Figs. 13 and 14 with subscripts and superscripts.

4.4. Simplified diagram

For the purpose of representing mechanisms of a subtopology, complex constraints are introduced andexpressed by constraint vertices. A constraint vertex is indicated by a rectangle. Inside the rectangle, eithera meaningful symbol or a notation index can be placed. Some symbols are proposed as shown in Fig. 15.An example is shown in Fig. 16.

Physical manipulator Diagram

a b

Fig. 8. Kinematic composition of a planar 3-RRR parallel manipulator.


a b

Fig. 9. Kinematic composition of a spherical 3-RRR parallel manipulator.



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Some special constraints on relative locations of revolute joints can be found in a large number of manip-ulator designs, introducing the following complex joints can greatly simplify their representations. These com-plex joints are shown in Fig. 17, where the PI joint is proposed by Angeles [33]. Two examples are shown inFigs. 18 and 19.

4.5. Topological diagram

The diagram containing information on both the kinematic composition and the essential constraints is

referred to as topological diagram. A summary of topological diagram is given in the following paragraphs.

Fig. 11. Diagram of a planar parallel manipulator with essential constraints on the relative joint locations.

Fig. 12. Diagram of a spherical parallel manipulator with essential constraints on relative joint locations.

Joint axes are not parallel and do not intersect;

Joint axes are parallel and do not intersect;

Joint axes are perpendicular and do not intersect;

Joint axes intersect at an arbitrary angle;

Joint axes intersect at zero angle or aligned;

Joint axes intersect at right angle.

Fig. 10. Graphic symbols for essential constraints.



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Fig. 13. Diagram of a spherical parallel manipulator with constraints on relative non-adjacent joint locations.

Physical manipulator Topological diagram

a b

Fig. 14. Diagram of 3-UPU.

Coplanar

Concurrent

Intersect at equal angle

Axes (points) forming the edges (vertices) of an equilateral polygon

Equal link length

Fig. 15. Constraint complex symbols.

Physical manipulator Diagram of a subtopology

a b

Fig. 16. Diagram with constraint vertices.



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A topological diagram has vertices and edges. There are joint vertices and constraint vertices. Joint verticesare marked by an upper case letter or a circled graphic symbol denoting a joint, while constraint vertices areindicated by a rectangle denoting special constraints on joints connected to it. Edges connecting joint verticesdenote links. The leftmost edge corresponds to the base while the rightmost edge to the end-effector. Links

have two kinds of extremities, i.e. with an arrow and without arrow. Links connected to the same joint element

Two consecutive parallel revolute joints

Three parallel revolute joints

PI joint

Tow revolute joints with general relative location

Two intersecting revolute joints

Two revolute joints intersecting at right angle

Tree concurrent revolute joints

Universal joint

Fig. 17. Complex joints.


Fig. 18. Diagram of delta [34].

Fig. 19. Diagram of parallel manipulator proposed by [35].



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of a joint have the same kind of extremities pointing to this joint, they are rigidly attached. The ways to indi-cate the essential constraints are:

(1) superimpose graphical symbols on edges;(2) add subscripts to joint vertices to indicate the concurrent joint axes;(3) add superscripts to joint vertices to indicate which joints are parallel;(4) add constraint vertices.

Joints can be grouped into complex joints to simplify the topological diagram. Actuated joints will not begrouped with non actuated joints. When different grouping is possible, the order from high priority to lowpriority is shown in Fig. 20.

5. Examples of topological diagrams

5.1. Serial manipulators

Two examples are shown in Figs. 21 and 22.

Fig. 20. Joint grouping priority order.

Fig. 21. PUMA manipulator.

Fig. 22. A serial manipulator designed by Victor Scheinman.



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5.2. Parallel manipulators

Three examples are shown in Figs. 2325.

Fig. 23. 4-DOF flight simulator by Koevermans et al.

Fig. 24. 5-DOF by Hesselbach et al.

Fig. 25. 6-DOF decoupled manipulator by Nabla, drawing by Merlet.



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6. Kinematic synthesis methodology

There exist three major synthesis approaches for spatial manipulators. The first one consists of three steps:

(1) Determine the possible kinematic compositions;

(2) For each kinematic composition, a parametrization is performed;(3) With some of the parameters as design variables, the search for suitable designs is performed.

The choice of the design variables is usually done by resorting to experience and intuition. As shown inSection 2, given a kinematic composition, particular geometric constraints must be satisfied in order for themanipulator to have required kinematic properties. This approach inevitably suffers from the poor correspon-dence between the suitable design space and the searched design space (illustrated in Fig. 26). With the secondapproach, synthesis based on screw theory or instantaneous kinematics for instance, the synthesis model isformulated as implicit functions with joint types and link geometries as unknowns. It is obvious that the syn-thesis can not be performed without the implication of all link geometries, making it not suitable for qualita-tive synthesis. The third approach is based on group theory. This approach is limited to kinematic chainswhose displacements form a Lie group.

The systematic analysis of the essential constraints can improve these synthesis approaches. Since the essen-tial constraints are relative locations between lines or between points and lines, they can be easily formulatedinto the kinematic model. Only parameters which are not under the essential constraints are taken as designvariables, a good correspondence between the intended manipulators and the synthesis model can therefore beachieved. With the kinematic compositions and essential constraints, qualitative synthesis can be performed.

6.1. Numerical topological representation

To implement geometric and topological synthesis algorithms the topology of a serial kinematic chain com-posed of only revolute and prismatic joints is represented by 6 integers, i.e.

n: number of joints. x0: kinematic composition. Its bits 0 to n 1 represent respectively the joint type of joints 1 to n with 1 for

revolute and 0 for prismatic. x1: bits 0 to n 2 indicate respectively whether the axes of joints 2 to n 1 intersect the immediate preced-

ing joint axis. x2: each two consecutive bits characterize the orientation of the corresponding joint relative to the imme-

diate preceding joint with 00 for parallel, 01 for perpendicular, and 10 for the general case. x3: supplementary constraint identifying joints whose axes are concurrent. All joint axes whose correspond-

ing bits are set to 1 are concurrent.

:all design space :eligible design space

:searched design space

Fig. 26. Illustration of design spaces.



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x4: supplementary constraint identifying joints whose axes are parallel. All joint axes whose correspondingbits are set to 1 are parallel.

With this numerical representation, topological constraint can be imposed on a general kinematic model tocarry out geometric synthesis to ensure that the search is performed in designs with the intended motion type.

The binary form makes the representation very compact. No serial kinematic chain should have more than 3prismatic joints, so all values for x0 of 6 joint kinematic chains take only 42B (byte) storage. Those for x1 take31B while those for x2 243B. Without supplementary constraints which are applied between non adjacent

joints, the maximum number of topologies is 316386 (some topologies, those with two consecutive parallelprismatic joints for example, will not be considered for topological synthesis purpose). All topologies withoutsupplementary constraint can be stored in a list, making the walk through quite straightforward. Applyingsupplementary constraints while walking through the list provides a systematic way for topological synthesis.In the following sections, the main steps to search for geometries and topologies of parallel manipulators withcertain characteristics are presented.

6.2. Isotropic design of translational parallel manipulators of a given topology

Parallel manipulators (PM) composed of three identical sub-chains of the topology shown in Fig. 27 can bea translational PM [36,15,37].

The numerical topological representation of this sub-chain is

n = 5. x0 = 111112. x1 = 01012. x2 = 000100012. x3 = 110012. x4 = 000002.

Under these explicit constraints, the geometries of the PM can be generated. Applying stochastic searchmethod, similar results as presented in [25] are obtained. An interesting geometry with continuous isotropicconfigurations is shown in Fig. 28.

Fig. 27. A sub-chain topology of translational parallel manipulators.

Fig. 28. PM with isotropic continuum.



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6.3. Topological synthesis examples

The motion of the end-effector (EE) is characterized in its displacement tangent space. The motion type isthen the structure of this space. If this space is represented by range(GE) where GE is a 6 n matrix with n asthe degree of freedom of the EE, then the motion type can be characterized by the ranks of the upper three

rows (GEU

) and the lower three rows (GEL

) ofGE

. Letrange

(Gi

) represent the EE motion under the constraintsof the ith sub-chain, then

rangeGE rangeG1 \ rangeG2 \ \ rangeGn 1

Given the motion type of the EE, range(Gi) can be derived by using either Eq. (1) or the methods presented in[3638]. The synthesis problem is then transformed into the search for appropriate sub-chain topologies whichis done by walking through the topology list and applying supplementary constraints. Figs. 2932 show some

Fig. 29. 2-DOF translational PM.

Fig. 30. PM with 1 DOF in translation and 3 DOF in rotation.

Fig. 31. Spherical PM.



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examples. From the best knowledge of the authors, the topologies shown in Figs. 29 and 30 are newtopologies.

7. Conclusion

By introducing essential geometric constraints, kinematic chains of serial and parallel manipulators can bebetter characterized. This is essential for both topology synthesis and geometry synthesis. On the one hand,topology synthesis of spatial manipulator is no longer dimension-independent; most of the topology synthesesare actually the search for some special geometric constraints which play a key role in determining the funda-mental kinematic properties. On the other hand, it is necessary to identify the essential constraints when per-forming geometry synthesis in order for the design space to correspond appropriately to the intendedtopology. The proposed topological diagram is comprehensive which enables the kinematic composition tobe described precisely, essential constraints to be reflected appropriately and the elements of geometry tobe isolated completely from the topological representation. The graph structure makes it possible to imple-ment computer algorithms in order to perform systematic enumeration, comparison and classification of serialand parallel manipulators. The topological diagram is also a useful tool for manual or automatic synthesis. Itis highly flexible and can be extended with ease by introducing new joint or constraint symbols.

Acknowledgements

The authors acknowledge the financial support of NSERC (National Science and Engineering ResearchCouncil of Canada) under grants OGPIN-203618 and RGPIN-138478.

References

[1] L.-W. Tsai, Mechanism design: enumeration of kinematic structures according to function, Mechanical Engineering Series: CRCMechanical Engineering Series, CRC Press, 2001.

[2] J.-P. Merlet, Les Robots Paralleles, Hermes, Paris, 1997.[3] K.H. Hunt, Geometry of robotics devices, Mechanical Engineering Transactions 7 (4) (1982) 213220, record Number: 2700 1982.[4] K.H. Hunt, Structural kinematics of in parallel actuated robot arms, record Number: 2710, in: Proceedings Title: Design and

Production Engineering Technical Conference Place of Meeting: Washington.[5] I. Powell, The kinematic analysis and simulation of the parallel topology manipulator, Marconi Rev. (UK) 45 (226) (1982) 1238.[6] F. Freudenstein, E. Maki, On a theory for the type synthesis of mechanism, in: Proceedings of the 11th International Congress of

Applied Mechanics, Springer, Berlin, 1965, pp. 420428.[7] D. Yang, T. Lee, Feasibility study of a platform type of robotic manipulator from a kinematic viewpoint, Journal of Mechanisms,

Transmissions and Automation in Design 106 (1984) 191198.[8] J. Denavit, R.S. Hartenberg, Kinematic notation for lower-pair mechanisms based on matrices, in: American Society of Mechanical

Engineers (ASME), 1954.[9] P. Chen, B. Roth, A unified theory for finitely and infinitesimally separated position problems of kinematic synthesis, ASME Journal

of Engineering for Industry, Series B 91 (1969) 203208, record Number: 2800.

x

y

z

Fig. 32. Spherical PM.



17/17

[10] P. Chedmail, Optimization of multi-dof mechanisms, in: J. Angeles, E. Zakhariev (Eds.), Computational Methods in MechanismsSystem, Springer-Verlag, 1998, pp. 97130.

[11] F.C. Park, J.E. Bobrow, Geometric optimization algorithms for robot kinematic design, Journal of Robotic Systems 12 (6) (1995)453463.

[12] T. Balkan, M. Kemal Ozgoren, M. Sahir Arikan, H. Murat Baykurt, A kinematic structure-based classification and compactkinematic equations for six-dof industrial robotic manipulators, Mechanism and Machine Theory 36 (7) (2001) 817832.

[13] H. Su, C. Collins, J. McCarthy, Classification of rrss linkages, Mechanism and Machine Theory 37 (11) (2002) 14131433.[14] C.M. Gosselin, R. Ricard, M.A. Nahon, Comparison of architectures of parallel mechanisms for workspace and kinematic properties,

American Society of Mechanical Engineers, Design Engineering Division (Publication) DE 82 (1) (1995) 951958.[15] L.-W. Tsai, S. Joshi, Comparison study of architectures of four 3 degree-of-freedom translational parallel manipulators, Proceedings

IEEE International Conference on Robotics and Automation 2 (2001) 12831288.[16] J. Angeles, Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms, 2nd ed., Mechanical Engineering

series: Mechanical Engineering Series (Springer), Springer, New York, 2003.[17] IFToMM, Iftomm terminology, Mechanism and Machine Theory 38 (2003) 912913.[18] J. Angeles, Spatial Kinematic Chains, Analysis, Synthesis, Optimization, Springer-Verlag, Berlin, 1982.[19] L. Baron, Contributions to the estimation of rigid-body motion under sensor redundancy, Ph.D. thesis, McGill University, c1997.[20] G. Bennett, A new mechanism, Engineering.[21] J.M. McCarthy, An Introduction to Theoretical Kinematics, The MIT Press, Cambridge, Massachusetts, London, England, 1990.[22] X. Wang, L. Baron, G. Cloutier, Design manifold of translational parallel manipulators, in: lAgence spatiale canadienne (Ed.),

Proceedings of 2003 CCToMM Symposium on Mechanisms, Machines, and Mechatronics, Montreal, Quebec, Canada, 2003, pp.

231239.[23] M. Ozgoren, Topological analysis of 6-joint serial manipulators and their inverse kinematic solutions, Mechanism and MachineTheory 37 (5) (2002) 511547.

[24] P. Wenger, D. Chablat, Kinematic analysis of a new parallel machine tool: The orthoglide, in: J. Lenarcic, V. Parenti-Castelli (Eds.),Recent Advances in Robot Kinematics, Kluwer Academic Publishers, 2000, pp. 305314.

[25] X. Wang, L. Baron, G. Cloutier, Kinematic modelling and isotropic conditions of a family of translational parallel manipulators,IEEE International Conference on Control and Automation (2003) 173177.

[26] ISO, Kinematic diagrams graphical symbols, ISO 3952-1, 1981.[27] R. Smith, Diagrams of kinematics, Engineer, 1886.[28] The Association of German Engineers, Symbols for the kinematic structure description of industrial robots, Guideline 2681, 1993.[29] F. Pierrot, Robots pleinement paralleles legers: Conception, modelisation et commande, Ph.D. thesis, Universite Montpellier II,

Montpellier, France, 1991.[30] F. Crossley, Contribution to grueblers theory in number synthesis of plane mechanisms, American Society of Mechanical Engineers

Papers, 1962, p. 5.

[31] F. Crossley, Permutations of kinematic chains of eight members or less from graph theoretic viewpoint, Developments inTheoretical and Applied Mechanics 2 (1965) 467486.

[32] X. Wang, L. Baron, G. Cloutier, The design of parallel manipulators of delta topology under isotropic constraint, in: T. Huang (Ed.),Proceedings of the 11th World Congress in Mechanism and Machine Science, Vol. 1, China Machinery Press, Tianjin, China, 2004,pp. 202206.

[33] J. Angeles, The qualitative synthesis of parallel manipulators, in: Proceedings of the WORKSHOP on Fundamental Issues andFuture Research Directions for Parallel Mechanisms and Manipulators, Quebec City, Quebec, Canada, 2002.

[34] R. Clavel, Delta, a fast robot with parallel geometry, in: Proceedings of the 18th International Symposium on Industrial Robots,1988, pp. 91100.

[35] H.S. Kim, L.-W. Tsai, Evaluation of a cartesian parallel manipulator, in: J. Lenarcic, F. Thomas (Eds.), Advances in RobotKinematics Theory and Applications, Kluwer Academic Publishers, 2002, pp. 1928.

[36] A. Frisoli, D. Checcacci, F. Salsedo, M. Bergamasco, Synthesis by screw algebra of translating in-parallel actuated mechanisms, in: J.Lenarcic, V. Parenti-Castelli (Eds.), Recent Advances in Robot Kinematics, Kluwer Academic Publishers, 2000.

[37] X. Kong, C. Gosselin, Generation of parallel manipulators with three translational degrees of freedom based on screw theory, in:T.C.S. Agency (Ed.), Symposium 2001 sur les mecanismes,les machines et la mecatronique de CCToMM, Saint-Hubert, Montreal,Quebec, Canada, 2001.

[38] S. Leguay-Durand, C. Reboulet, Design of a 3-dof parallel translating manipulator with u-p-u joints kinematic chains, in: Proceedingsof the 1997 IEEE/RSJ International Conference on Intelligent Robot and Systems. Innovative Robotics for Real-World Applications.IROS 97, 1997, pp. 16371642.


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