Enumeration of Spatial Manipulators by Using the Concept of Adjacency Matrix

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AKKARAPAKAM SUNEESH JACOB1,*, BHASKAR DASGUPTA1and RITUPARNA

DATTA2

1Indian Institute of Technology Kanpur, Kanpur, India

2Capgemini Technology Services India Limited, Bengaluru, India

e-mail: suneeshjacob@gmail.com; dasgupta@iitk.ac.in; rituparna.datta@capgemini.com

Abstract. This study is on the enumeration of spatial robotic manipulators, which is an essential basis for a

companion study on dimensional synthesis, both of which together present a wider utility in manipulator synthesis.

The enumeration of manipulators is done by using adjacency matrix concept. In this paper, a novel way of applying

adjacency matrix to spatial manipulators with four types of joints, namely revolute, prismatic, cylindrical and

spherical joints, is presented. The limitations of the applicability of the concept to 3D manipulators are discussed.

1-DOF (Degree Of Freedom) manipulators of four links and 2-DOF, 3-DOF and 4-DOF manipulators of three

links, four links and ﬁve links, are enumerated based on a set of conventions and some assumptions. Finally, 96

1-DOF manipulators of four links, 696 2-DOF manipulators of 5 links, 4 2-DOF manipulators of three links, 8

3-DOF manipulators of four links and 15 4-DOF manipulators of ﬁve links are presented.

Keywords. enumeration, structural synthesis, mechanisms, spatial manipulators, adjacency matrix

1 Introduction and Literature Review

Synthesis of a mechanism is conducted in many stages that

include type synthesis, number synthesis, structural synthe-

sis and dimensional synthesis. Type synthesis deals with de-

signing the type of mechanism (such as types of joints, etc.)

for a required task. Number synthesis deals with designing

things such as the number of links, the number of joints and

the number of Degrees Of Freedom (DOF) for the required

task (also used for the study of determining the number of

manipulators that can be enumerated for a given number of

links and a given DOF). Dimensional synthesis deals with

designing the dimensions of links, etc. Structural synthesis

is the enumeration of all possible distinct mechanisms with

the given number of links for a given DOF requirement.

One of the earliest studies on number synthesis of mech-

anisms in literature is done by Crossley [1] who introduced a

simple algorithm to solve Gruebler’s equation for constrained

kinematic chains. In his paper, the mobility is chosen, and the

Gruebler’s equation is used to ﬁnd how many binary links,

ternary links, etc., would be needed to produce the chosen

mobility. Correspondingly, the possible linkages would be

enumerated and grouped. Franke [2] used a notation to trans-

form a mechanism into a graph of nodes and edges. The

nodes represent the non-binary links (ternary links, quater-

nary links, etc.), and each node bears the number of joints

the corresponding link is connected to. The edges with mul-

tiple parallel lines (single line, double line, triple line, etc.)

bear sequences of numbers that represent the number of con-

nections between the corresponding two links. Graph theory

is extensively used in the literature, in which the links of a

mechanism are represented by vertices, joints by edges and

joint connections by edge connections. Damir et al. [3] pre-

sented an application of graph theory to the kinematic syn-

thesis of mechanisms. Manolescu [4] presented a method

based on Baranov Trusses to enumerate planar kinematic chains

and mechanisms. Another famously used method in the lit-

erature for enumeration is the method of Assur groups [5].

Assur groups are formed by removing a link from Baranov

Trusses. Assur group is a group of mechanisms that do not

alter the DOF of that mechanism if added to a mechanism.

Jinkui et al. [6] presented Assur groups with multiple joints.

Many more methods were used in literature to enumerate

planar mechanisms. Mruthyunjaya [7] presented a review

of methods for structural synthesis of planar mechanisms.

Raicu [8] used the concept of adjacency matrix to capture

the topological information. This matrix has all diagonal el-

ements as zeroes and all oﬀ-diagonal elements as either ze-

roes or ones. The diagonal elements signify the links of the

mechanism. The number 0 in each oﬀ-diagonal element sig-

niﬁes no connection between the two corresponding links.

The number 1 signiﬁes a joint that connects the two corre-

sponding links. This type of representation of mechanisms

can be useful in implementing the enumeration process on a

computer, as permuting the oﬀ-diagonal elements of a ma-

trix of a given size n×nwould include the representation

of all the possible mechanisms of nlinks that are connected

with joints. Mruthyunjaya et al. [9] proposed a generalised

matrix notation to facilitate the representation and analysis

of multiple-jointed chains. In their paper, multiple-joints are

considered for planar mechanisms. Each of the oﬀ-diagonal

elements in the matrix consists of a value that represents the

number of links the joint is connected to. The concept of

adjacency matrix for planar mechanisms is extensively used

in the literature, especially for computerised enumeration of

planar mechanisms. Li et al. [10] presented application of

arXiv:2210.03327v2 [cs.RO] 5 May 2024

2Jacob, Dasgupta and Datta

adjacency matrix to compliant mechanisms. However, these

were designed for planar mechanisms. Wenjian et al. [11]

presented a review paper on the structural synthesis of planar

mechanisms that covered the history of structural synthesis

and its recent research progress. There are many studies in

the literature [12–18] that used various methods to present

enumerated mechanisms. But even though these were very

successful methods, unfortunately, these methods cannot be

applied to spatial mechanisms. Moreover, these methods did

not consider the distinction amongst mechanisms based on

base and end-eﬀector links.

Amongst studies on spatial manipulators, enumeration

of serial manipulators is straightforward but enumeration of

parallel manipulators is non-trivial. Pierrot et al. [19] pre-

sented a new family of 4-DOF parallel robots by choosing

diﬀerent types of limbs of the end-eﬀector plate. Dimiter et

al. [20] also developed a family of new parallel manipulators

of 4-DOF by analysing the inverted mechanism, i.e., con-

sidering the end-eﬀector plate to be ﬁxed and analysing the

relative motion of the base link. An extension of Pierrot’s

work is proposed by Fang and Tsai [21] that includes de-

velopment of a systematic method to enumerate 4-DOF and

5-DOF overconstrained parallel manipulators with identical

serial limbs using screw theory. But this enumeration is lim-

ited to identical limb structures. Extending this, Hess-Coelho

[22] presented an alternative procedure for type synthesis of

2D and 3D parallel manipulators by employing asymmetric

(non-identical) limbs. Li et al. [23] used a kind of adja-

cency matrix to describe metamorphic mechanisms with di-

agonal elements representing the joints and the oﬀ-diagonal

elements representing the links. Bai et al. [24] used adja-

cency matrix to describe scaling mechanisms in which oﬀ-

diagonal elements represent links. Siying et al. [25] pro-

posed a method to perform type synthesis of 6-DOF manip-

ulators based on screw theory. Even though these studies

on spatial manipulators were very useful, they did not use

the advantages of the adjacency matrix concept. Qiang et

al. [26] used 12-bit string matrix representation of manipu-

lators to perform structural synthesis of serial-parallel hybrid

mechanisms. Extending this work, Zhang et al. [27] used a

string-matrix based geometrical and topological representa-

tion of mechanisms. Their paper presents an extension of the

2D adjacency matrix concept to 3D by using 16 bits in each

matrix element. Although the study opens up the applica-

tion of the adjacency matrix concept to spatial manipulators,

a proper study on the applicability of the concept to three di-

mensions is still needed for the justiﬁcation of its usage and

to understand its limitations for enumerating spatial manip-

ulators. Moreover, there is no distinction based on base and

end-eﬀector links that is vital for enumerating robotic ma-

nipulators. There exist studies [28] on enumeration of spa-

tial parallel manipulators, however their scope is apparently

limited to platform manipulators with limbs connected from

the base link, which does not take into account many com-

plex structures that are beyond platform manipulators (such

as serial-parallel hybrid manipulators [29, 30], etc.)

To summarise, many methods such as contracted graph

methods, adjacency matrix methods and Assur groups ex-

ist in the literature and are used to enumerate 2D kinematic

chains. Spatial manipulators have been enumerated using

the adjacency matrix concept, but the direct application of

adjacency matrix for 3D manipulators has not been studied,

and moreover, the distinction based on base and end-eﬀector

links has not been made in the studies. The method presented

in the present study extends the adjacency matrix application

directly from 2D to 3D manipulators and takes into consider-

ation the distinction based on base and end-eﬀector links for

four types of joints, namely revolute, prismatic, cylindrical

and spherical.

An important issue in mechanism synthesis is isomor-

phism detection and elimination. Isomorphism detection is

generally considered to be a diﬃcult problem due to the num-

ber of permutations it involves for a large number of nodes

and there are several studies on handling isomorphism [15,

31, 32]. However, for reduction in computational load, ma-

nipulators only up to 5 links are considered for enumeration.

For 4-node and 5-node graph adjacency matrices, the per-

mutations involved are 4! And 5! respectively. Moreover,

in the current study, the permutations are required for links

other than the ﬁrst and the last links, and hence the total per-

mutations involved for each adjacent matrix are just 2! and

3! for 4-link and 5-link robots, respectively. Hence, brute-

force search is considered to detect isomorphism, which is

basically to produce all possible isomorphic adjacency ma-

trices of a given adjacency matrix and compare it with other

adjacency matrices in the enumerated list, to check for any

match.

2 Analysis and Methodology

2.1 Adjacency matrix representation

The adjacency matrix notation used to describe manipula-

tors in this study is an n×nsymmetric matrix where nis

the number of links of the manipulator. The diagonal ele-

ments represent the links. Moreover, the ﬁrst diagonal ele-

ment represents the base link and the last diagonal element

represents the end-eﬀector link. Each oﬀ-diagonal element

represents the joint connecting the two links corresponding

to its indices. A typical adjacency matrix structure is shown

in Figure 1.

2.2 On the applicability of adjacency matrix representation

of spatial manipulators

The concept of adjacency matrix representation for enumer-

ating 2D manipulators is extended to 3D in this study. Re-

garding the compatibility of adjacency matrix representation

with 3D manipulators, the main constraint of adjacency ma-

trix representation is that it allows a maximum of one joint

to be the connection between any two links. The joints con-

sidered in this study are Revolute, Prismatic, Cylindrical and

Spherical joints. Additionally, in this study, it is assumed

that only prismatic and revolute joints are capable of serving

as actuators. Since this study considers four types of joints,

Enumeration of spatial manipulators by using the concept of Adjacency Matrix 3

Figure 1: Typical adjacency matrix structure

there would be 42−4

2! +4=10 possible cases of links sharing

two joints, namely the revolute-revolute case, the revolute-

prismatic case, the revolute-cylindrical case, the revolute-

spherical case, the prismatic-prismatic case, the prismatic-

cylindrical case, the prismatic-spherical case, the cylindrical-

cylindrical case, the cylindrical-spherical case and the spherical-

spherical case. In all the cases except the spherical-spherical

case, the relative motion is either impossible or not guaran-

teed for arbitrary positions and orientations of axes of joints,

and therefore these cases are omitted in the enumeration. In

the spherical-spherical case, two links connected with two

spherical joints at arbitrary positions can have relative mo-

tion. The kind of relative motion in such a case would be

equivalent to the relative motion of two links connected with

a single revolute joint, where the axis of the revolute joint

would be the line passing through the two centres of the two

spherical joints. The above cases show that two links con-

nected by two joints (at arbitrary locations and orientations)

can have motion only in the spherical-spherical case. Since

the motion in this case is equivalent to the motion of two

links connected by a single revolute joint, this case is omit-

ted in the enumeration such that the entire possible enumer-

ations that are not omitted would consist of at most one joint

connecting any two links. Thus, such enumeration can be

made by permutating the oﬀ-diagonal elements of an adja-

cency matrix.

2.3 Methodology to enumerate manipulators

To enumerate n-link manipulators, the concept of adja-

cency matrix is used in this study. All the possible adjacency

matrices of nlinks with the four kinds of joints are generated

using Python, and the criteria, listed below, are used to elim-

inate invalid and isomorphic adjacency matrices. The set of

all the possible n×nadjacency matrices would capture all the

possible n-link manipulators. Since an n×nadjacency ma-

trix is symmetric with the diagonal elements representing the

links, there would be n2−n

2independent places of the matrix

that can be ﬁlled with joints. Since each of these places can

be either ﬁlled with one of the four joints or left empty, there

would be 4+1 possible types of connection between any two

links. By ﬁlling the places with all the possible types of con-

nection, 5n2−n

2distinct adjacency matrices can be formed,

among which some would not qualify and some would be

isomorphic.

The criteria listed below are used to eliminate invalid and

isomorphic mechanisms.

•DOF should be greater than or equal to 1.

Since the enumerated adjacency matrices would have struc-

tures (including indeterminate ones) included, these are re-

moved by identifying the DOF using the Kutzbach criterion.

•The mechanism should have at least one revolute

joint or prismatic joint.

Only revolute and prismatic joints are considered for actua-

tion, and therefore at least one prismatic or revolute joint is

needed in a mechanism.

•The sum of the numbers of prismatic and revolute

joints should be greater than or equal to the DOF.

Since the actuation is given through only prismatic and revo-

lute joints, the sum of the numbers of prismatic and revolute

joints must be greater than or equal to the DOF of the manip-

ulator.

•The mechanism should not have any link that is en-

tirely unconnected from all the other links.

One possible adjacency matrix for a ﬁve-link manipulator is

shown in Equation (1) in which, the third link, i.e., L3, has no

connection with any other link and therefore is not part of the

mechanism, and hence such mechanisms should be removed.

A=

L1O O R P

O L2O C S

O O L3O O

R C O L4O

P S O O L5



(1)

•The mechanism should not have open-chains that

do not have a connection from the base-link to the end-

eﬀector link.

In ﬁgure 2, the mechanism has the open-chain of links 3 and

4 that do not have a connection from the base-link to the end-

eﬀector link. Such mechanisms are removed.

•The mechanism should not have non-contributing

loops.

In ﬁgure 3, the mechanism has the L3−L4−L5−L6loop

non-contributing and therefore cannot contribute, either ac-

tively or passively, to the transmission of velocity to the end-

eﬀector. Hence such mechanisms are to be removed.

•The mechanisms should not be isomorphic.

Since two isomorphic adjacency matrices represent the same

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EnumerationofSpatialManipulatorsbyUsingtheConceptofAdjacencyMatrixAKKARAPAKAMSUNEESHJACOB1,*,BHASKARDASGUPTA1andRITUPARNADATTA21IndianInstituteofTechnologyKanpur,Kanpur,India2CapgeminiTechnologyServicesIndiaLimited,Bengaluru,Indiae-mail:suneeshjacob@gmail.com;dasgupta@iitk.ac.in;rituparna.datta@capg...

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