Proceedings of the ASME 2015 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference

IDETC/CIE 2015 August 2-5, 2015, Boston, Massachusetts, USA

DETC2015-46755

COMBINATORIAL METHOD FOR CHARACTERIZING SINGULAR CONFIGURATIONS IN PARALLEL MECHANISMS

Avshalom Sheffer School of Mechanical Engineering

Tel Aviv University, Ramat Aviv 69978, Israel

Offer Shai School of Mechanical Engineering,

Tel Aviv University, Ramat Aviv 69978, Israel

ABSTRACT The paper presents a method for finding the different

singular configurations of several types of parallel

mechanisms/robots using the combinatorial method. The main

topics of the combinatorial method being used are:

equimomental line/screw, self-stresses, Dual Kennedy theorem

and circle, and various types of 2D and 3D Assur Graphs such

as: triad, tetrad and double triad. The paper introduces

combinatorial characterization of 3/6 SP and compares it to

singularity analysis of 3/6 SP using Grassmann Line Geometry

and Grassmann-Cayley Algebra. Finally, the proposed method

is applied for characterizing the singular configurations of

more complex parallel mechanisms such as 3D tetrad and 3D

double-triad.

KEYWORDS: Combinatorial method, Equimomental

Line/Screw, Dual Kennedy theorem and circle, Grassmann

Line Geometry and Grassmann-Cayley Algebra, Self-Stress,

Singularity, 3/6 SP

NOMENCLATURE DOF/ 𝛿 Degrees Of Freedom

eqml Equimomental Line

eqms Equimomental Screw

𝑃𝑖 Planar pencil formed by the two leg

lines meeting at 𝐵𝑖

SP Stewart Platform

SS Self-stress

SSt Topological Self-stress

SSg Geometric Self-Stress

2D Two-dimensional

3D Three-dimensional

3/6 SP 3/6 Stewart Platform

1. INTRODUCTION In this paper we present a new combinatorial method for

characterizing singular configurations in parallel mechanisms,

with an emphasis on the six-DOF 3/6 Stewart Platform, called

for short 3/6 SP. The Stewart Platform consists of two bodies

connected by six legs, which can vary their lengths. One of the

bodies is called the base and the other is called the platform.

One of the important problems in parallel robot kinematics is

the characterization of singular or special configurationsI. It is

one of the main concerns in the analysis and design of

manipulators [1]. One of the known singular configurations of

the Stewart Platform is when all the lines meet one line [2]. An

additional configuration is when the moving platform rotates

by around the vertical axis [3]. Merlet [4] classified all

the different singular configurations of 3/6-SP by using

Grassmann geometry. Analysis of wrench singularities was

introduced by using tetrahedrons [5], a figure of four planes,

to identify wrench singularities, i.e. configurations where the

platform can move infinitesimally when all its actuators are

locked. A tetrahedron is non-singular if and only if its four

faces do not have a common point. It has been proven that the

manipulator is at a wrench singularity if and only if the

characteristic tetrahedron is singular. Another method relying

this time on Grassmann–Cayley algebra and the associated

superbracket decomposition introduces the conditions when

the Jacobian (or rigidity matrix) containing these screws is

rank-deficient [6]. The singularity condition of a broad class of

six-DOF three-legged parallel robots that have one spherical

joint somewhere along each leg becomes a check whether the

four planes intersect at a point. Another characterization based

on the kinematical relationship of rigid bodies is that these

I The term “singularity” originates from mathematics and the

term “special configuration” originates from mechanical engineering.

In this paper we chose to adopt and use the term “singularity”.

2 Copyright © 2015 by ASME

four planes include three normal planes of three velocities of

three non-collinear points in the end effector, and the plane is

determined by these three points themselves [7,8,9,10]. When

all the links of 3/6-SP are locked, the three normal planes of

three possible velocities of the platform are the planes created

by the links of the mechanism. Thus, it is possible to

determine the singularity of 3/6-SP and other mechanisms

with a triangle-moving platform. This approach is consistent

with the approach proposed in this paper except that this paper

focuses on statics while in these papers they focus on

kinematics.

In section two we provide a brief explanation of the

combinatorial method used in this paper by introducing the

equimomental lines (eqml) and self-stresses (SS) and how

they are used for characterizing the singular configurations.

The dual Kennedy Theorem is provided and the technique of

drawing the dual Kennedy circle is used for finding all the

eqml. In section three and four we apply the combinatorial

method for finding the singular conditions of 3/6 SP and

compare it to other singular characterization methods. Section

five introduces the applicability of the proposed method, the

combinatorial method, for characterizing the singularity of

more complex parallel mechanisms such as tetrad and double

triad, in part by using equimomental screws.

2. THE COMBINATORIAL METHOD The combinatorial method is founded on discrete

mathematics, in particular on graph theory and rigidity theory.

In this paper this method is applied to characterize the singular

configurations of parallel mechanisms. The main advantage of

the proposed method is that it is also applicable to other types

of parallel mechanisms and not limited to SPs. This section

introduces two concepts underlying the combinatorial method

that are used in this paper: self-stress and the equimomental

line.

2.1 SELF-STRESS Self-Stress is a special feature indicating the existence of

redundant constraints. A self-stress is constructed when there

is a need to insert a rigid body between two joints/vertices

whose distance is constant.

Definition of a self-stress: A set of internal forces in the links

that are dependent on one parameter and satisfy the

equilibrium of forces around each joint.

From this definition it follows that if a mechanism possesses a

self-stress and has joints with degree equal to the dimension d

it should satisfy the following property.

Property 1: If there is a self-stress in all the links incident to a

joint and the degree of the joint (the number of links incident

to a joint) is d in dimension d then all the d links should be

collinear or coplanar in dimension d=2 or d=3, respectively.

The reason for the latter is that these are the only geometries

that can have a self-stress.

From Property 1 it also follows that for any 3D system, when

all the links incident to a joint are not coplanar but have a self-

stress then the degree of the joint is at least four. Similarly, a

singularity condition exists if and only if four faces have a

common point [11].

There are two types of self-stresses: topological and

geometric, designated by SSt and SSg respectively. The

topological self-stresses exist for almost any realization of the

mechanism. On the other hand, the geometric self-stress exists

only at a specific realization, usually the singular

configuration. Thus in this paper only the latter self-stress, the

geometric, is used. These realizations are also called over-

constrained mechanisms [12]. There are several known

geometric constraints that cause the appearance of geometric

self-stresses in 2D mechanisms, such as when three links or

their continuations are constrained to meet at a point, or when

three points should be collinear. For example, the 2D triad

shown in Figure 1 (a) has a geometric SS, SSg, in all the links

if the continuations of the three ground links intersect at the

same point, but for any small changes in the realization there

is no self-stress. The 2D triad in Figure 1(b) is in a special

condition where the links {1,4,3} are collinear, thus only in

these three links there is an internal force, while in the other

links {2,5,6} there is a zero force.

(a) A geometric SS exists in

all the links

(b) A geometric SS exists only

in links, 1, 3 and 4

Figure 1. Examples of 2D triad in special realizations, singular

configurations, both having a geometric self-stress.

2.2 THE EQUIMOMENTAL LINE In kinematics, for every rigid body there exists a point in

the body or on its continuation with a zero velocity, referred to

as the absolute instant center [13]. Analogically, in statics, for

any force there exists a line upon which the moment exerted

by the force is equal to zero. This is the line is along which the

force acts and it is called the absolute equimomental line

(absolute eqml). Note, using the projective geometry duality a

point corresponds to a line, thus the absolute instant center is

dual to the absolute equimomental line. More details about

this duality relation appear in [14].

In kinematics we have in addition the relative instant center

𝐼𝑖,𝑗 : For any two bodies, i and j, there exists a point in the

plane where the velocities of the two bodies at this point is

equal. Relying on the projective geometry duality, for any two

forces there exists a line along which they exert the same

moment at each point on the line. This line is called the

relative equimomental line, or for short relative eqml.

For example, the line 𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼) is a line where the forces 𝐹𝐼

and 𝐹𝐼𝐼 exert the same moment, as shown in Figure 2. Note,

3 Copyright © 2015 by ASME

the eqml of the two forces should pass through the intersection

point of the two forces (the two absolute eqmls).

Figure 2. Forces ⃗⃗ and ⃗⃗ acting along lines of action

( , ) and ( , ).

Now we can transform the Arnohold-Kennedy into statics.

The Arnohold-Kennedy theorem [13] states that the relative

instant centers of any three links i, j and k of the mechanism,

𝐼𝑖𝑗 , 𝐼𝑖𝑘 and 𝐼𝑗𝑘, must lie on a straight line. Applying the duality

relation between the instant centers and the eqml, we derive

the following theorem in statics:

Theorem 1 – The dual Kennedy Theorem in statics [14]: For

any three forces: , and the three relative eqml:

( , ), ( , ) and ( , ) intersect at the same point.

Applying the duality from graph theory [15] it follows that the

absolute eqml is associated to each face, a circuit without

inner edges (called also contours). Each binary link (also

called bar) defines the relative eqml of the two absolute eqml

of the two faces adjacent to this bar [16]. If one of the two

faces next to a link is the ground face (also called reference

face) the absolute eqml of the second face is along this link.

Thus, all the links adjacent to the ground faces define the

absolute eqml, and can thus be called primary absolute eqml.

Following the terminology in mechanisms, the absolute eqml

that are determined by applying the dual Kennedy theorem are

called secondary eqml.

Now we are ready to define the main property that enables us

to determine the singular configuration.

Property 2: For any two adjacent faces if the relative eqml

passes through the meeting point of the two absolute eqml of

these two faces the mechanism is in a singular configuration.

For finding all the eqml in the mechanism we can use the

technique of drawing the dual Kennedy circle [14]. The

procedure to find the secondary eqml is done by drawing a

circle referred to as the dual Kennedy circle and denote the

faces of the mechanism in a clockwise manner on the

circumference of this circle, as shown in Figure 3. The

primary eqml are represented as solid lines and the secondary

eqml as dashed lines in this circle. Note that the dashed line

(𝐼, 𝐼𝐼𝐼) creates the two triangles (𝐼, 𝐼𝐼, 𝐼𝐼𝐼) and (𝐼, 𝐼𝑉, 𝐼𝐼𝐼) in

the quadrangle (𝐼, 𝐼𝐼, 𝐼𝐼𝐼, 𝐼𝑉), see Figure 3. Therefore, the

𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼) must pass through the point of intersection of

𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼) and 𝑒𝑞𝑚𝑙(𝐼𝐼,𝐼𝐼𝐼), namely, between the links 1 and 2,

respectively, and through the point of intersection of 𝑒𝑞𝑚𝑙(𝐼,𝐼𝑉)

and 𝑒𝑞𝑚𝑙(𝐼𝑉,𝐼𝐼𝐼), namely, between the links 4 and 3,

respectively.

(a)

(b)

Figure 3. (a) Mechanism 4 links (b) The dual Kennedy circle

For sake of clarity we show how we use the eqml method for

characterizing the singularity condition of a 2D triad, given in

Figure 4 and 5.

After defining the four faces { , 𝐼, 𝐼𝐼, 𝐼𝐼𝐼} where the reference

face is designated by the number 0, the primary eqml are:

𝑒𝑞𝑚𝑙(𝐼,0) = 6

𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) = 1

𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼) = 5

Due to Property 2, The three eqml: ( , ), ( , ) and

( , ) intersect at a point thus the configuration is in a

singular configuration. The triad in Figure 4 is not in a

singular configuration since the lines 6,1 and 5 don’t intersect

at a point while in Figure 5 the three eqml satisfy property 2,

thus it is in a singular configuration.

Figure 4. Non-singular configuration.

4 Copyright © 2015 by ASME

Figure 5. Singular configuration.

In 3/6-SP in a singular configuration, when there is a self-

stress the platform rotates around a certain axis. In the next

section we introduce several singularity conditions of 3/6 SP.

For each condition we show the eqml defined by three

arbitrary coplanar forces that intersect at a unique point, the

number of zero edges, and the rotation axis of the platform

when there is an infinitesimal movement.

3. APPLYING THE COMBINATORIAL METHOD FOR FINDING THE SINGULARITY CONDITIONS OF 3/6 SP

Merlet [4] studied the singularity of six-DOF 3/6-SP

based on the Grassmann line geometry. He discovered many

new singularities, including 3C, 4B, 4D, 5A, and 5B. Due to

Property 2, we can find the singular configurations of many

types of mechanisms. Relying on the rank of the Jacobian

matrix of SP [17], we conclude that for a mechanism with rank

3, 4 and 5, we will have 2, 1 and 0 zero links, respectively. In

the following section, we will derive all the reported

singularities, this time by applying the combinatorial method

introduced in this paper.

We call 𝑃𝑖 the planar pencil formed by the two leg lines

meeting at 𝐵𝑖 and there are seven faces, but one of them will

be chosen to be the reference frame/face.

Figure 6. The 3/6 Stewart Platform in generic configuration

Bundle Singularity (3C): A singular bundle occurs when

four lines of the six legs intersect at a common point.

Figure 7. Definition of 3C singular configuration.

Let us choose four of the faces to be zero faces, as follows:

(

𝐹𝐼𝐹𝐼𝐼𝐹𝐼𝐼𝐼𝐹𝐼𝑉𝐹𝑉𝐹𝑉𝐼𝐹𝑉𝐼𝐼)

=

(

𝐹𝐼𝐹𝐼𝐼𝐹𝐼𝐼𝐼 )

Since links 3, 5, 7 and 8 are adjacent to zero faces, it follows

that they are absolute eqml. On the other hand, links 4, 6 and 9

are adjacent to two zero faces, i.e. on both of their two sides

there are zero faces, thus they are zero links.

Now that we have all the absolute eqml we can find the

relative eqml by applying Property 2 and the dual Kennedy

circle. The 𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼) must pass through the point of

intersection of 𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼) and 𝑒𝑞𝑚𝑙(𝐼𝐼,𝐼𝐼𝐼), namely, between

links 1 and 2, respectively, and through the point of

intersection of 𝑒𝑞𝑚𝑙(𝐼,0) and 𝑒𝑞𝑚𝑙(0,𝐼𝐼𝐼), namely, between

links 8 and 7, respectively, as shown in Figure 8.

Figure 8. The dual Kennedy circle of a 3C configuration

𝑒𝑞𝑚𝑙(𝐼,0) = 5 = 8

𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) = 3 = 7

𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼) = (1 ∨ 2) ∧ (7 ∨ 8)

For this configuration, links 5 and 8 are on 𝑒𝑞𝑚𝑙(𝐼,0), links 3

and 7 are on 𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) and 𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼) passes through the

meeting point of the links 1-3 and 5. It can be seen that the

four links 1-3 and 5 of the six links intersect at a common

point. Due to Property 2, the three eqml: 𝑒𝑞𝑚𝑙(𝐼,0), 𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0)

and 𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼) intersect at one point, so self-stress occurs in

the links except on 4, 6 and 9 which are zero links, and the

configuration is singular, as shown in Figure 9 and Table 1.

5 Copyright © 2015 by ASME

Figure 9. 3C singular configuration with the eqml where faces

IV, V, VI, VII are zero faces.

Hyperbolic Congruence Singularity (4B): The condition

for this singularity occurs when five lines are concurrent with

two skew lines.

Figure 10. Definition of 4B singular configuration.

Let us choose two of the faces to be zero faces, as follows:

(

𝑽 𝑽 𝑽 𝑽 )

=

(

𝑽 𝑽 )

Since links 1 and 5 are adjacent to zero faces it follows that

they are the absolute eqml of (𝐼, ) and (𝐼𝑉, ), respectively .

On the other hand, link 6 is adjacent to two zero faces, i.e. on

both of its two sides there are zero faces, thus it is a zero link.

Now, that we have all the absolute eqml we can find the

relative eqml by applying Property 2 and dual Kennedy

circle, as follows:

𝑒𝑞𝑚𝑙(𝐼𝑉,0) = 5

𝑒𝑞𝑚𝑙(𝐼𝐼,0) = (1 ∨ 2) ∧ (7 ∨ 8)

𝑒𝑞𝑚𝑙(𝐼𝑉,𝐼𝐼) = (3 ∨ 4) ∧ (8 ∨ )

For this configuration, link 5 is on 𝑒𝑞𝑚𝑙(𝐼𝑉,0), 𝑒𝑞𝑚𝑙(𝐼𝐼,0) passes through the meeting point of links 1, 2, 7 and 8, and in

the same way, 𝑒𝑞𝑚𝑙(𝐼𝑉,𝐼𝐼) passes through the meeting point of

the links 3, 4, 8 and 9. The three links 5, 7 and 9 are coplanar,

due to Property 1 and since link 6 is a zero link. Due to

Property 2, the three eqml: 𝑒𝑞𝑚𝑙(𝐼𝑉,0), 𝑒𝑞𝑚𝑙(𝐼𝐼,0) and

𝑒𝑞𝑚𝑙(𝐼𝑉,𝐼𝐼) intersect at a point, so self-stress occurs in all the

links except for link 6. Thus, 3/6 SP satisfying the above

constraints is in a singular configuration, as shown in Figure

11 and Table 1.

Figure 11. 4B singular configuration with the eqml where

faces V, VI are zero faces.

Degenerate Congruence Singularity (4D): When five

lines are in one plane or pass through one common point in

that plane, they form a singular degenerate congruence.

Figure 12. Definition of 4D singular configuration.

Let us choose three of the faces to be zero faces, as follows:

(

𝐹𝐼𝐹𝐼𝐼𝐹𝐼𝐼𝐼𝐹𝐼𝑉𝐹𝑉𝐹𝑉𝐼𝐹𝑉𝐼𝐼)

=

(

𝐹𝐼𝐹𝐼𝐼𝐹𝐼𝐼𝐼𝐹𝐼𝑉 )

Since links 2, 5, 7 and 8 are adjacent to zero faces it follows

that they are absolute eqml. On the other hand, links 6 and 9

are adjacent to two zero faces, i.e. on both their two sides there

are zero faces, thus they are zero links.

Now, that we have all the absolute eqml we can find the

relative eqml by applying Property 2 and dual Kennedy circle,

as follows:

𝑒𝑞𝑚𝑙(𝐼,0) = 5 = 8

𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) = (1 ∨ 2) ∧ (7 ∨ )

𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼) = (3 ∨ 4) ∧ (7 ∨ 8)

For this configuration, links 5 and 8 are on 𝑒𝑞𝑚𝑙(𝐼,0),

𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) passes through the meeting point of links 1, 2, 7

and 9, and in the same way, 𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼) passes through the

meeting point of links 3, 4, 7 and 8. The three links 1, 2 and 7

are coplanar, due to Property 1, because the other link (9)

incident to that joint is a zero link and the degree of the joint is

three. Links 3-5 pass through 𝑃1 to form a singular degenerate

congruence. In the same way, due to Property 2, the three

eqml: 𝑒𝑞𝑚𝑙(𝐼,0), 𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) and 𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼) intersect at one

point, so self-stress occurs in the links except for links 6 and 9

6 Copyright © 2015 by ASME

that are zero links, and the configuration is singular, as shown

in Figure 13 and Table 1.

Figure 13. 4D singular configuration with the eqml where

faces V, VI, VII are zero faces.

General Complex Singularity (5A): This singularity is

characterized by having more than five skew lines.

Figure 14. Definition of 5A singular configuration.

Let us choose one of the faces to be zero faces, as follows:

(

𝐹𝐼𝐹𝐼𝐼𝐹𝐼𝐼𝐼𝐹𝐼𝑉𝐹𝑉𝐹𝑉𝐼𝐹𝑉𝐼𝐼)

=

(

𝐹𝐼𝐼𝐹𝐼𝐼𝐼𝐹𝐼𝑉𝐹𝑉𝐹𝑉𝐼𝐹𝑉𝐼𝐼)

Since links 1, 6 and 8 are adjacent to zero faces it follows that

they are absolute eqml.

Now, that we have all the absolute eqml we can find the

relative eqml by applying Property 2 and the dual Kennedy

circle, as follows:

𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) = (1 ∨ 2) ∧ (7 ∨ 8)

𝑒𝑞𝑚𝑙(0,𝑉) = (5 ∨ 6) ∧ (8 ∨ )

𝑒𝑞𝑚𝑙(𝑉,𝐼𝐼𝐼) = (3 ∨ 4) ∧ (7 ∨ )

For this configuration, 𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) passes through the meeting

point of links 1, 2, 7 and 8, 𝑒𝑞𝑚𝑙(0,𝑉) passes through the

meeting point of links 5, 6, 8 and 9, and 𝑒𝑞𝑚𝑙(𝑉,𝐼𝐼𝐼) passes

through the meeting point of links 3, 4, 7 and 9. It can be seen

in Figure 15 and Table 1, that the three eqml: 𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0),

𝑒𝑞𝑚𝑙(0,𝑉) and 𝑒𝑞𝑚𝑙(𝑉,𝐼𝐼𝐼) intersect at one point, so based on

Property 2 this is a singularity configuration, as there is an

infinitesimal movement in all the joints along the vertical axis

of the platform.

Figure 15. 5A singular configuration with the eqml where face

I is a zero face.

Special Complex Singularity (5B): This type of

singularity occurs when six segments cross the same line.

Figure 16. Definition of 5B singular configuration.

Let us choose one of the faces to be zero faces, as follows:

(

𝐹𝐼𝐹𝐼𝐼𝐹𝐼𝐼𝐼𝐹𝐼𝑉𝐹𝑉𝐹𝑉𝐼𝐹𝑉𝐼𝐼)

=

(

𝐹𝐼𝐼𝐹𝐼𝐼𝐼𝐹𝐼𝑉𝐹𝑉𝐹𝑉𝐼𝐹𝑉𝐼𝐼)

Since links 1, 6 and 8 are adjacent to zero faces it follows that

they are absolute eqml.

Now, that we have all the absolute eqml we can find the

relative eqml by applying Property 2 and the dual Kennedy

circle, as follows:

𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) = (1 ∨ 2) ∧ (7 ∨ 8)

𝑒𝑞𝑚𝑙(0,𝑉) = (5 ∨ 6) ∧ (8 ∨ )

𝑒𝑞𝑚𝑙(𝑉,𝐼𝐼𝐼) = (3 ∨ 4) ∧ (7 ∨ )

For this configuration, 𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) passes through the meeting

point of links 1, 2, 7 and 8, 𝑒𝑞𝑚𝑙(0,𝑉) passes through the

meeting point of links 5, 6, 8 and 9, and 𝑒𝑞𝑚𝑙(𝑉,𝐼𝐼𝐼) passes

through the meeting point of links 3, 4, 7 and 9. The

intersection line between the planes containing 𝑃3 and 𝑃5

crosses through point 𝐵1. In this case, six segments cross this

line and there is an infinitesimal movement in all the joints

along this axis. In the same way, the three eqml: 𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0),

𝑒𝑞𝑚𝑙(0,𝑉) and 𝑒𝑞𝑚𝑙(𝑉,𝐼𝐼𝐼) intersect at one point, so based on

Property 2 this is a singularity configuration, as shown in

Figure 17 and Table 1.

7 Copyright © 2015 by ASME

Figure 17. 5B singular configuration with the eqml where face

I is a zero face.

In the next section, we compare this method to other

singularity analysis methods listed in the literature.

4. COMPARISON TO OTHER SINGULARITY ANALYSIS METHODS In this section we review the characterization of all the

singular conditions of 3/6 SP according to the various methods

reported in the literature. We compare the method presented in

this paper to the method developed by Merlet [4] based on

Grassmann Line Geometry and to the method developed by

Shoham and Ben-Horin [6] based on Grassmann–Cayley

Algebra. Table 1 shows this comparison. As was mentioned in

section 3, the singular configurations of this structure

according to line geometry analysis include five types: 3C, 4B,

4D, 5A, and 5B.

Shoham and Ben-Horin proved that there is a class of (at least)

144 combinations that include 3/6 SP in which their

singularity condition is delineated to be the intersection of four

planes at one point. They showed that all the singular

configurations of Merlet are particular cases of this condition.

The geometric condition consists of four planes, defined by

the actuator lines and the position of the spherical joints,

which intersect at least at one point. Another point of view of

this singularity appears in [10] this time from kinematics. In

the latter paper they introduced a constraint between the three

velocities of the three joins connecting the legs to the platform

and the plane of the platform.

Bundle Singularity (3C): A singular bundle occurs when

four lines of the six legs intersect at a common point. Table 1

shows this singularity configuration. This occurs when 𝐵1𝐵3

and 𝐴3𝐵3 and also 𝐵1𝐵5 and 𝐴5𝐵5 links lie on the same line. It

is clear that this condition fulfills the four-intersecting-planes

singularity condition, whereby the planes, defined by the

actuator lines and the position of the spherical joints, intersect

at point 𝐵1. In the same way, the three eqml: 𝑒𝑞𝑚𝑙(𝐼,0),

𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) and 𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼) intersect at one point and the

configuration is singular, due to Property 2.

Hyperbolic Congruence Singularity (4B): The condition

for this singularity occurs when five lines are concurrent with

two skew lines. Table 1 shows this singularity configuration. A

singular configuration of this type is when 𝐷1 transversal

𝐵1𝐵3 and 𝐷2 is the intersection line between the planes

containing 𝑃1 and 𝑃3. Arrange 𝑃5 so that one of the leg lines

(5) intersects both 𝐷1 and 𝐷2, 𝐵5 is a generic point on it, and

the other leg line (6) does not have a constraint. The

corresponding leg lines are linearly independent, thus forming

a hyperbolic congruence. In this condition, four planes meet,

𝑃5 and the platform intersects 𝐷2, which is the intersection of

𝑃1 and 𝑃3, at the point that link 5 meets 𝐷2. In the same way,

the three eqml: 𝑒𝑞𝑚𝑙(𝐼𝑉,0), 𝑒𝑞𝑚𝑙(𝐼𝐼,0) and 𝑒𝑞𝑚𝑙(𝐼𝑉,𝐼𝐼) intersect

at this point so the configuration is singular, according to

Property 2.

Note, the drawings of [4,6,18] where they draw 4B seems to

be incorrect, since in all the figures related to 4B they contain

3B, i.e. two planar pencils sharing a line.

Degenerate Congruence Singularity (4D): When five

lines that are in one plane or pass through a common point in

that plane form a singular degenerate congruence. Table 1

shows this singularity configuration. A singular configuration

of this type is when point 𝐵3 lies on 𝑃1, so 𝑃1 and 𝑃3 meet at

point 𝐵3. In addition, 𝐵3𝐵5 and 𝐴5𝐵5 links are on the same

line. In this way, four planes intersect at one point 𝐵3. In the

same way, the three eqml: 𝑒𝑞𝑚𝑙(𝐼,0), 𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) and 𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼) intersect at one point and based on Property 2 the

configuration is singular,.

General Complex Singularity (5A): This singularity is

characterized by having more than five skew lines. Table 1

shows this singularity configuration. The platform is parallel

to the base and a rotation about the vertical axis is performed,

for example, when both platforms are symmetric and the

rotation angle is . Three lines ('𝑎𝐵1 ', '𝑎𝐵3' and '𝑎𝐵5') are

the meeting of planar pencil formed by the two leg lines, with

the platform 𝐵1𝐵3𝐵5. The singularity occurs when the three

lines intersect at one point. Due to Property 2, the three eqml:

𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0), 𝑒𝑞𝑚𝑙(0,𝑉) and 𝑒𝑞𝑚𝑙(𝑉,𝐼𝐼𝐼) intersect at one point, so

this is a singularity configuration. There is an infinitesimal

movement in all the joints along the vertical axis of the

platform.

Special Complex Singularity (5B): This type of

singularity occurs when six segments cross the same line.

Table 1 shows this singularity configuration. There are four

possibilities for a line '𝐵1𝑐' to intersect with this quadruple, all

of them leading to a singular configuration. Without loss of

generality we deal only with one case, the case in which line

'𝐵1𝑐' is the intersection line between the planes containing 𝑃3

and 𝑃5 that cross through point 𝐵1. Once again it is apparent

that the four planes meet, as the intersection of 𝑃3 and 𝑃5 is

line '𝐵1𝑐' itself, which intersects also with planes 𝑃1 and the

platform plane 𝐵1𝐵3𝐵5 at point 𝐵1. In this case, six segments

cross this line and there is an infinitesimal movement in all the

joints along this axis. In the same way, the three eqml:

𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0), 𝑒𝑞𝑚𝑙(0,𝑉) and 𝑒𝑞𝑚𝑙(𝑉,𝐼𝐼𝐼) intersect at one point, so

based on Property 2 this is a singularity configuration.

8 Copyright © 2015 by ASME

Table 1. Comparison to other singularity analysis methods.

Combinatorial Method Grassmann–Cayley Algebra Grassmann Line Geometry

The three eqml: 𝑒𝑞𝑚𝑙(𝐼,0), 𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) and 𝑒𝑞𝑚𝑙(𝐼,𝐼𝐼𝐼)

intersect at a point (𝐵1)

The four planes, defined by the actuator lines

and the position of the spherical joints,

intersect at a point (𝐵1)

Four lines of the six legs intersect at a

common point (𝐵1).

Bundle

Singularity (3C)

Five lines are concurrent with two skew lines

(𝐷1, 𝐷2)

Hyperbolic

Congruence

Singularity (4B)

9 Copyright © 2015 by ASME

All five lines are in one plane or pass

through one common point in that plane

Degenerate

Congruence

Singularity (4D)

General complex:

Generated by five independent skew lines.

Fichter's Singular Configuration:

The platform is parallel to the base and a

rotation about the vertical axis is performed,

for example, when both platforms are

symmetric and the rotation angle is

General

Complex

Singularity (5A)

Hunt's Singular Configuration:

All the lines meeting one given line

Special Complex

Singularity (5B)

10 Copyright © 2015 by ASME

5. FINDING SINGULAR CONFIGURATIONS OF MORE COMPLEX MECHANISMS THROUGH EQUIMOMENTAL SCREW The main advantage of the method introduced in this paper

is that it is applicable to finding the singularity of many types

of mechanisms and is not limited to a particular mechanism. In

this section, we give two examples, out of many that we have,

of more complex mechanisms: characterization of singular

configurations of 3D Tetrad and 3D Double-Triad.

In the same way as in 2D, in 3D we have relative

equimomental screw, for any two absolute equimomental

screws defined as follows:

Let $1 = (𝑆1𝑆01) and $2 = (

𝑆2𝑆02) be two screws acting in two

faces. The relative eqms is a line where the difference between

the two forces, 𝑆1𝑎𝑛𝑑 𝑆2, and the difference between the two

moments of the two forces along this line, 𝑆01 and 𝑆02, are

both in the same direction.

Figure 18. Two absolute equimomental Screw $ and $ and

their relative equimomental screw $ , .

If we have two screws, corresponding to two absolute eqms,

then we find relative eqms as follows:

For the two screws $1 and $2 we find a normal to their axes.

The relative eqms is perpendicular to this normal and the

intersection point can be calculated easily, and will appear in

the forthcoming paper.

The three-dimensional Tetrad consists of a circuit of size four

connected by eight links (legs), which can vary their lengths,

to a bottom body called the base.

As was shown in the previous sections, the space of the

singular configurations of a mechanism can be spanned by the

absolute eqml/s of the faces. From the truth table listed below

we can get all the singular configurations of the mechanism, as

was done in 2D [19].

Table 2. Truth table of the singular configurations of three-

dimensional Tetrad.

Position

No. 𝐹𝐼 𝐹𝐼𝐼 𝐹𝐼𝐼𝐼 𝐹𝐼𝑉 𝐹𝑉 𝐹𝑉𝐼 𝐹𝑉𝐼𝐼 𝐹𝑉𝐼𝐼𝐼 𝐹𝐼𝑋

𝐿1 0 0 0 0 0 0 0 0 1

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 𝐿𝑘 1 1 0 1 1 1 1 1 1

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 𝐿𝑛 1 1 1 1 1 1 1 1 1

Figure 19. The 3D Tetrad in generic configuration.

For the position number 𝐿𝑘 shown in Table 2, we find the

eqml/s by the dual Kennedy circle shown in Figure 20:

Figure 20. The dual Kennedy circle of a 3D Tetrad.

𝑒𝑞𝑚𝑙(𝐼,0) = (1 ∨ 2) ∧ ( ∨ 1 )

𝑒𝑞𝑚𝑙(𝐼,𝑉𝐼𝐼) = (7 ∨ 8) ∧ ( ∨ 12)

𝑒𝑞𝑚𝑙(0,𝑉) = (3 ∨ 4) ∧ (1 ∨ 11)

𝑒𝑞𝑚𝑙(𝑉𝐼𝐼,𝑉) = (5 ∨ 6) ∧ (11 ∨ 12)

𝑒𝑞𝑚𝑙(𝐼𝑋,0) = 1

𝑒𝑞𝑚𝑙(𝐼𝑋,𝑉𝐼𝐼) = 12

𝑒𝑞𝑚𝑠(0,𝑉𝐼𝐼) = ((𝐼, ) ∧ (𝐼, 𝑉𝐼𝐼)) ∨ (( , 𝑉) ∧ (𝑉𝐼𝐼, 𝑉))

∨ ((𝐼𝑋, ) ∧ (𝐼𝑋, 𝑉𝐼𝐼)) From the eqml defined above it follows the singular

characterization of the Tetrad, which is:

Let 𝑛1 be the normal to 𝑒𝑞𝑚𝑙(0,𝑉) and 𝑒𝑞𝑚𝑙(𝑉𝐼𝐼,𝑉), as shown in

Figure 22.

Let 𝑛2 be the normal to 𝑒𝑞𝑚𝑙(𝐼,0) and 𝑒𝑞𝑚𝑙(𝐼,𝑉𝐼𝐼), as shown in

Figure 23.

Let 𝑛3 be the normal to 𝑒𝑞𝑚𝑙(𝐼𝑋,0) and 𝑒𝑞𝑚𝑙(𝐼𝑋,𝑉𝐼𝐼), as shown

in Figure 24.

11 Copyright © 2015 by ASME

The 3D Tetrad is in a singular position if and only if there

exists a line which is perpendicular to both: 𝑛1, 𝑛2 and 𝑛3.

Figure 21. Singular configuration of the middle case shown in

Table 2.

Figure 22. ( ,𝑽) is the planar pencil formed by the two

leg lines 3, 4 and the two leg lines 10, 11. (𝑽 ,𝑽) is the

planar pencil formed by the two leg lines 5, 6 and the two leg

lines 11, 12. is the normal to ( ,𝑽) and (𝑽 ,𝑽).

Figure 23. ( , ) is the planar pencil formed by the two

leg lines 1, 2 and the two leg lines 9, 10. ( ,𝑽 ) is the

planar pencil formed by the two leg lines 7, 8 and the two leg

lines 9, 12. is the normal to ( , ) and ( ,𝑽 ).

Figure 24. is the normal to ( , ) and ( ,𝑽 ).

Figure 25. The 3D Tetrad in a Singular configuration.

The last example upon which we apply the combinatorial

method for characterizing the singular configurations is the 3D

Double-Triad. The 3D Double Triad consists of two platforms

that are connected to the base by nine legs, which can vary in

length.

Table 3. Truth table of the singular positions of three-

dimensional Double Triad.

𝐹𝐼 𝐹𝐼𝐼 𝐹𝐼𝐼𝐼 𝐹𝐼𝑉 𝐹𝑉 𝐹𝑉𝐼 𝐹𝑉𝐼𝐼 𝐹𝑉𝐼𝐼𝐼 𝐹𝐼𝑋 𝐹𝑋 𝐹𝑋𝐼 𝐿1 0 0 0 0 0 0 0 0 0 0 1

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 𝐿𝑘 0 1 1 1 1 1 1 1 1 1 1

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 𝐿𝑛 1 1 1 1 1 1 1 1 1 1 1

12 Copyright © 2015 by ASME

Figure 26. The 3D Double Triad in generic configuration.

For the position number 𝐿𝑘 shown in Table 3, we find the

eqml/s as shown in Figure 27:

𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) = (1 ∨ 2) ∧ (5 ∨ 6)

𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,𝑉) = (3 ∨ 4) ∧ (5 ∨ 7)

𝑒𝑞𝑚𝑙(𝑉,0) = ((𝐼𝐼𝐼, ) ∧ (𝐼𝐼𝐼, 𝑉)) ∨ (6 ∧ 7)

𝑒𝑞𝑚𝑙(𝑋𝐼,0) = (12 ∨ 13) ∧ (14 ∨ 15)

𝑒𝑞𝑚𝑙(𝑋𝐼,𝑉𝐼𝐼) = (1 ∨ 11) ∧ ( ∨ 12)

𝑒𝑞𝑚𝑙(𝑉𝐼𝐼,0) = ((𝑋𝐼, ) ∧ (𝑋𝐼, 𝑉𝐼𝐼)) ∨ (13 ∧ )

From the eqml defined above it follows the singular

characterization of the Double Triad, which is:

Let 𝑛1 be the normal to 𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,0) and 𝑒𝑞𝑚𝑙(𝐼𝐼𝐼,𝑉).

Let 𝑛2 be the normal to 𝑒𝑞𝑚𝑙(𝑋𝐼,0) and 𝑒𝑞𝑚𝑙(𝑋𝐼,𝑉𝐼𝐼).

𝐸𝑞𝑚𝑙(𝑉,𝑉𝐼𝐼) is the relative eqml of the two faces 𝐹𝑉 and 𝐹𝑉𝐼𝐼.

The 3D Double Triad is in a singular position if and only if

there exists a line which is perpendicular to both: 𝑒𝑞𝑚𝑙(𝑉,0),

𝑒𝑞𝑚𝑙(𝑉𝐼𝐼,0) and eqml(V,VII), as shown in Figure 27.

Figure 27. Singular configuration of the middle case shown in

Table 3.

6. CONCLUSIONS AND FUTURE RESEARCH

This paper presents a combinatorial method for

singularity analysis for parallel robots. The method presented

in the paper is consistent with other approaches that appear in

the literature. This paper focuses on statics while in [7,8,9,10],

they focus on kinematics.

It is important to note that 3/6 SP is an Assur Graph of type 3D

triad. In 2D it was proved [20] that the singularity of AG has

special properties and only AGs have the following property in

the singular configuration: the system is both mobile (has an

infinitesimal motion) and has a self-stress. A conjecture says

that the special singularity also exist in 3D Assur Graphs. In

3/6 SP (3D Triad) and in 6/6 SP (3D Body-bar atom) the

conjecture is proved to be correct. For that reason, we used

the self-stresses for characterizing the singularity

configurations of 3/6 SP. It seems that the method introduced

in this paper is applicable in finding the singularity of many

other types of mechanisms and is not limited to a particular

mechanism, such as: 3D Triad (3/6 SP), 3D Tetrad and 3D

Double-Triad, all introduced in this paper. In addition, the

method is based on discrete mathematics thus can be

computerized easily.

The authors believe that eqml/s is a fundamental concept in

statics and have a significant potential in characterizing

singularity of spatial parallel mechanisms. The authors

continue to explore this topic and in the forthcoming paper

they will present new singular conditions of complex

mechanisms in three-dimensional. Future work will also

include analytical techniques to complement the graphical

techniques that are provided in this paper.

ACKNOWLEDGMENT The authors wish to thank Prof. Marco Carricato for his

help to construct correctly the configuration of the hyperbolic

congruence (4B) on 3/6 Stewart platform. His assistance is

greatly appreciated.

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