1. COMBINATORIAL METHOD FOR CHARACTERIZING
SINGULAR CONFIGURATIONS IN PARALLEL MECHANISMS

2.  INTRODUCTION
 THE EQUIMOMENTAL LINE AND EQUIMOMENTAL SCREW
 THE KENNEDY AND DUAL KENNEDY METHOD
 APPLYING THE COMBINATORIAL METHOD FOR FINDING
SINGULARITY CONDITIONS
 CONCLUSION
TABLE OF CONTENTS

3. INTRODUCTION
Parallel manipulators have a specific mechanical architecture where all the links are
connected both at the base.
3/6 Stewart Platform-
3D Triad
6/6 Stewart Platform 3D Double Triad 3D Tetrad

4.  Hunt (1978) found a singular configuration of Stewart Platform
when all the lines meeting one given line
 Merlet (1989) 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
 Shoham and Ben-Horin (2006) proved that there is a class of
144 combinations in which their singularity condition is
delineated to be the intersection of four planes at one point
 Huang et al. (2014) introduced, based on the kinematical
relationship of rigid bodies, that these 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
3C
4D 5A
5B
4B

5. THE EQUIMOMENTAL LINE AND EQUIMOMENTAL SCREW
Two absolute equimomental Screw $ 𝟏 and
$ 𝟐 and their relative equimoment screw $ 𝟏,𝟐
Forces 𝑭 𝑰 and 𝑭 𝑰𝑰 acting along lines of
action 𝒆𝒒𝒎𝒍 𝑰,𝟎 and 𝒆𝒒𝒎𝒍 𝑰𝑰,𝟎
Absolute equimomental line- a line upon which the moment exerted
Relative equimomental line- for any two forces there exists a line along
which they exert the same moment at each
point on the line by the force is equal to zero
Relative equimomental screw- The relative eqms is a line where the
difference between the two forces, 𝑆1 𝑎𝑛𝑑 𝑆2, and the difference between
the two moments of the two forces along this line, 𝑆0
1 and 𝑆0
2, are both
in the same direction

6. Property 1: 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
Non-singular configuration Singular configuration

7. THE KENNEDY AND DUAL KENNEDY METHOD
The Arnohold-Kennedy theorem 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
The dual Kennedy Theorem in statics: For any
three forces: 𝑭𝒊, 𝑭𝒋 and 𝑭 𝒌 the three relative
eqml: 𝒆𝒒𝒎𝒍 𝒊,𝒋 , 𝒆𝒒𝒎𝒍 𝒋,𝒌 and 𝒆𝒒𝒎𝒍 𝒊,𝒌
intersect at the same point

8. APPLYING THE COMBINATORIAL METHOD FOR FINDING SINGULARITY
CONDITIONS
𝑒𝑞𝑚𝑙 𝐼,0 = 5 = 8
𝑒𝑞𝑚𝑙 𝐼𝐼𝐼,0 = 3 = 7
𝑒𝑞𝑚𝑙 𝐼,𝐼𝐼𝐼 = 1 ∨ 2 ∧ 7 ∨ 8
The dual Kennedy circle
of a 3C configuration
Bundle Singularity (3C): A singular bundle
occurs when four lines of the six legs
intersect at a common point
3C singular configuration with the eqml where faces IV, V, VI,
VII are zero faces.

9. 𝑒𝑞𝑚𝑙 𝐼,0 = 1 ∨ 2 ∧ 9 ∨ 10
𝑒𝑞𝑚𝑙 𝐼,𝑉𝐼𝐼 = 7 ∨ 8 ∧ 9 ∨ 12
𝑒𝑞𝑚𝑙 0,𝑉 = 3 ∨ 4 ∧ 10 ∨ 11
𝑒𝑞𝑚𝑙 𝑉𝐼𝐼,𝑉 = 5 ∨ 6 ∧ 11 ∨ 12
𝑒𝑞𝑚𝑙 𝐼𝑋,0 = 10
𝑒𝑞𝑚𝑙 𝐼𝑋,𝑉𝐼𝐼 = 12
𝑒𝑞𝑚𝑠 0,𝑉𝐼𝐼 = 𝐼, 0 ∨ 𝐼, 𝑉𝐼𝐼 ∧ 0, 𝑉 ∨ 𝑉𝐼𝐼, 𝑉 ∧
𝐼𝑋, 0 ∨ 𝐼𝑋, 𝑉𝐼𝐼

10. 𝑛1 be the normal to 𝑒𝑞𝑚𝑙 0,𝑉 and 𝑒𝑞𝑚𝑙 𝑉𝐼𝐼,𝑉
𝑛2 be the normal to 𝑒𝑞𝑚𝑙 𝐼,0 and 𝑒𝑞𝑚𝑙 𝐼,𝑉𝐼𝐼
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
𝑛3 be the normal to 𝑒𝑞𝑚𝑙 𝐼𝑋,0 and 𝑒𝑞𝑚𝑙 𝐼𝑋,𝑉𝐼𝐼

11. CONCLUSION
 The method presented is consistent with other approaches that
appear in the literature
 It seems that the method introduced is applicable in finding the
singularity of many other types of mechanisms and is not limited to a
particular mechanism
 The method is based on discrete mathematics thus can be
computerized easily
 I believe that equimomental line/ screw is a fundamental concept in
statics and have a significant potential in characterizing singularity of
spatial parallel mechanisms