# Singularity Analysis of Redundant Space Robot with the Structure of Canadarm2

Authors: Gang Chen; Long Zhang; Qingxuan Jia; Hanxu Sun
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

A novel method of singularity analysis for redundant space robot with the structure of Canadarm2 is proposed in this paper. This kind of structure has the characteristics of three consecutive parallel axes. First, the “virtual manipulator” method is employed to transfer the singularity problem of a space robot to that of a ground one. By choosing an appropriate reference system and a reference point of the end-effector, Jacobian matrix is greatly simplified and then it is reconstructed according to a new standard. On this basis, the Jacobian matrix can be partitioned into four submatrixes whose degradation conditions are put forward; thereafter, the singularity conditions and singular directions of the redundant space robot are obtained. The effectiveness of the proposed singularity analysis method is verified through simulation.


---

## Body

## 1. Introduction

Singularity is the inherent characteristic of robots. At a singular configuration, the end-effector of the robot loses the ability to move along a certain direction, which is called the singular direction. In this case, if there is still a velocity component of the end-effector in singular direction, the joint angular velocities will become unacceptably large, and the end-effector will deviate from its expected trajectory. Moreover, it is possible that a robot will be out of control [1, 2]. Therefore, in order to ensure the stability and reliability of robot, it is necessary to do the research on singularity analysis.Robot singularity analysis, as the basis of singularity avoidance, is used to determine the singularity conditions. For nonredundant robots, the determinant value of Jacobian matrix can directly determine robot singularity conditions. However, for redundant robots, because its Jacobian matrix is not square, the singularity analysis becomes much more complex. Much effort in research community has been paid on dealing with singularity analysis of redundant robots. Whitney [3] determined singularity conditions by calculating the determinant value of matrixJ
J
T. The pseudoinverse of Jacobian matrixJis given byJ
†
=
J
T
(
J
J
T
)
-
1. Singular configurations occur when the determinant value of the J
J
T portion ofJ
†is equal to zero. Although the matrix formed by J
J
T is a square matrix, the determinant expression could be quite complicated, making it very difficult to obtain the analytical solution. Chou et al. [4] defined the concept of null motion differential dynamic system, and then the local topology and singularity avoidability of redundant robots are studied according to nonlinear singularity theory. Podhorodeski et al. [5] proposed using six-joint subgroups of Jacobian matrix to determine the singularity conditions of redundant robots. Conditions that cause the determinant values of all possible six-joint subgroups to simultaneously equal zero are singularity conditions. The method works well, but the entire solving process is much complicated. Nokleby and Podhorodeski [6, 7] used the properties of reciprocal screws to determine the single-DOF-loss and the multi-DOF-loss conditions of redundant robots and also work out the singular directions. This method is also used in the singularity analysis of Canadian Space Agency (CSA) Canadarm2 [8]. The reciprocal method is effective for the singularity analysis of all redundant serial manipulators. However, with the increase of DOF, its calculation amount is little large. For a 7-DOF robot with the structure of the last, three axes perpendicular to each other, Waldron et al. [9, 10], based on matrix partitioning, converted the segmented3
×
4submatrix to a4
×
4submatrix with an additional relationship between joint angular velocities. And then the singularity conditions can be achieved through the determinant value of square submatrixes. Due to the introduction of an additional relationship, the method may encounter algorithm singular problems. Cheng et al. [11] suggested decomposing singularities into position singularities and orientation singularities by partitioning the6
×
7Jacobian matrix. This method only needs to determine each singularity condition of the submatrixes including two3
×
4matrixes, one3
×
3matrix, and one3
×
3zero matrix to obtain the singularity conditions. It can greatly reduce the computation complexity, but it can only be applied to the singularity analysis of the 7-DOF robots with the structure of the last three axes perpendicular to each other.All the singularity analysis methods mentioned above are aiming at ground robots or space robots in fixed-base mode. For free-floating or free-flying space robots, some scholars utilized the method in [2], only replacing the Jacobian matrixJbyJ
ginJ
†
=
J
T
(
J
J
T
)
-
1; here,J
gis called the generalized Jacobian matrix. Similarly, it is very difficult to obtain the analytical solution. Xu et al. [12] presented a singularity separation method which separates the singularity parameters from the inverse of the Jacobian to simplify the analysis. However, sometimes the parameters separation is hard. Nenchev et al. [13] utilized the singular-value decomposition (SVD) of the Jacobian matrix to judge whether the singularity happens, and it is really effective for numerical computation not for analytical solution.A novel method of singularity analysis for redundant space robots with the structure of Canadarm2 is proposed in this paper. First, the “virtual manipulator” method is employed to transfer the singularity problem of a space robot to that of a ground one. Second, Jacobian matrix is reconstructed according to a new standard on the basis of simplification. Thereafter, the Jacobian matrix can be partitioned into four submatrixes whose degradation conditions are put forward. Finally, the singularity conditions and singular directions of redundant space robots with the structure of Canadarm2 are obtained.This paper is organized as follows. Section1 gives a brief overview of the research on singularity analysis. Section 2 employs the “virtual manipulator” method to transfer the singularity problem of a space robot to that of a ground one. The Jacobian matrix is simplified and the equivalence of singularity is analyzed in detail in Section 3. Section 4 obtains the singularity conditions and singular directions. Section 5 is a simulation example and Section 6 concludes the paper.


## 2. Virtual Manipulator

For a free-floating space robot as shown in Figure1, the linear momentum of the system is conserved, which is the holonomic constraint. According to “virtual manipulator” concept proposed in [14, 15], the kinematic problem in this mode can be simplified to that belonging to a ground robot. By this concept, the base of free-floating manipulator is regarded as a rod linked to the virtual ground by a passive sphere joint. The relationship between system mass center and the each link’s mass center is as follows (without special instruction, all variables are expressed in inertial frame):
(1)
M
r
g
=
∑
i
=
0
n
m
i
r
i
.Figure 1
A general model of free-floating space robot.Σ
I
,
Σ
E: the inertia frame and the end-effector frame, respectively. Σ
i
(
i
=
1
,
…
,
n
): the frame of link i. C
i
(
i
=
1
,
…
,
n
): the mass center position of link i. J
i
(
i
=
1
,
…
,
n
): the position of joint i. a
i
,
b
i
(
i
=
1
,
…
,
n
): the position vectors fromJ
itoC
iandC
itoJ
i
+
1, respectively. r
i
(
i
=
1
,
…
,
n
): the position vector ofC
i. r
g: the position vector of the system’s mass center. p
i
(
i
=
1
,
…
,
n
): the position vector ofJ
i. p
e: the position vector of the end-effector. Θ: the joint angle vector. m
i
(
i
=
1
,
…
,
n
): the mass of link i. M: the total mass of the system. E
3: 3
×
3identity matrix.And the base’s mass center is(2)
r
0
=
r
g
-
(
m
1
+
⋯
+
m
n
)
(
b
0
+
a
1
)
M
-
⋯
-
m
n
(
b
n
-
1
+
a
n
)
M
.The position of the end-effector is(3)
p
e
=
r
g
+
b
^
0
+
∑
i
=
1
n
(
a
^
i
+
b
^
i
)
,
wherea
^
i
=
(
(
∑
k
=
0
i
m
k
)
/
M
)
a
i,b
^
i
=
(
(
∑
k
=
0
i
m
k
)
/
M
)
b
i,i
=
1
,
…
n. Vectorsa
^
iandb
^
iare called “virtual link vectors,” which are parallel to vectorsa
iandb
i, respectively.Differentiating two sides of (3), (4) can be obtained as follows:
(4)
v
e
=
v
g
+
b
^
˙
0
+
∑
i
=
1
n
(
a
^
˙
i
+
b
^
˙
i
)
,
wherev
grepresents the linear velocity of the system’s mass center. Assuming that the initial momentum is zero,v
g
=
0can be obtained due to the holonomic constraint. Moreover,
(5)
a
^
˙
i
=
d
(
a
^
i
)
d
t
=
d
(
A
i
a
^
i
i
)
d
t
=
d
(
A
i
)
d
t
(
a
^
i
i
)
=
(
ω
i
×
A
i
)
(
a
^
i
i
)
=
ω
i
×
a
^
i
.In the same way,(6)
b
^
˙
i
=
ω
i
×
b
^
i
,
whereA
iis the rotation matrix from frame i to the inertial frame and ω
iis the angular velocity of ith body, and combining (4), (5), and (6),
(7)
v
e
=
ω
0
×
b
^
0
+
∑
i
=
1
n
ω
i
×
(
a
^
i
+
b
^
i
)
.Meanwhile, the angular velocity of linki is
(8)
ω
i
=
ω
0
+
∑
k
=
1
i
k
k
θ
˙
k
.Substituting (8) into (7),
(9)
v
e
=
-
p
~
g
e
ω
0
+
∑
k
=
1
n
[
k
k
×
(
p
e
-
p
^
k
)
]
θ
˙
k
,
wherep
~
g
eis the antisymmetric matrix of vector(
p
e
-
r
g
).Combining (8) and (9), the following relationship can be obtained:
(10)
[
v
e
ω
e
]
=
J
^
b
ω
0
+
J
^
m
Θ
˙
,
where
(11)
J
^
b
=
[
-
p
~
g
e
E
3
]
∈
R
6
×
3
,
J
^
m
=
[
z
1
×
(
p
e
-
p
^
1
)
⋯
z
n
×
(
p
e
-
p
^
n
)
z
1
⋯
z
n
]
∈
R
6
×
n
,
and z
iis the unit vector ofz-axis of coordinate systemi.When a space robot is in free-flying mode, its attitude is kept unchanged; namely,ω
0
=
[
0
0
0
]
T; thus
(12)
[
v
e
ω
e
]
=
J
^
m
Θ
˙
.Thereby, the singularity analysis of a free-flying space robot is only related toJ
^
m, which only contains the virtual link parameters.When a space robot is in free-floating manipulator, the base angular velocity can be measured by its carried sensor; namely,ω
0is a known term, so
(13)
[
v
e
ω
e
]
-
J
^
b
ω
0
=
J
^
m
Θ
˙
,
(14)
[
v
^
e
ω
^
e
]
=
J
^
m
Θ
˙
,
where[
v
^
e
ω
^
e
]
=
[
v
e
ω
e
]
-
J
^
b
ω
0.In the same way, the singularity analysis of a free-floating space robot is also only related toJ
^
m, which only contains the virtual link parameters.According to the derivation above, it can be found that the singularity problem of space robots can be transferred to that of a ground one through the “virtual manipulator” method.


## 3. Simplifying the Jacobian Matrix

For a 7-dof space robot with the structure of Canadarm2, its axis 3, axis 4, and axis 5 are parallel to each other. Link-pole coordinate system based on the method of Denavit-Hartenberg is established in Figure2. The relevant parameters of the space robotic system are given in Table 1, and the corresponding parameters of virtual manipulator are shown in Table 2.Table 1
The relevant parameters of the space robotic system.
Base
Link 1
Link 2
Link 3
Link 4
Link 5
Link 6
Link 7
Mass/kg
m
0
m
1
m
2
m
3
m
4
m
5
m
6
m
7
a
i/m
a
/
2
c
/
2
d
/
2
e
/
2
f
/
2
h
/
2
k
/
2
b
i/m
l
a
/
2
c
/
2
d
/
2
e
/
2
f
/
2
h
/
2
k
/
2Table 2
The corresponding parameters of virtual manipulator of the space robotic system.
Base
Link 1
Link 2
Link 3
Link 4
Link 5
Link 6
Link 7
Mass/kg
m
0
m
1
m
2
m
3
m
4
m
5
m
6
m
7
a
^
i/m
(
m
0
+
m
1
)
a
2
M
∑
k
=
0
2
m
k
c
2
M
∑
k
=
0
3
m
k
d
2
M
∑
k
=
0
4
m
k
e
2
M
∑
k
=
0
5
m
k
f
2
M
∑
k
=
0
6
m
k
h
2
M
∑
k
=
0
7
m
k
k
2
M
b
^
i/m
m
0
l
M
(
m
0
+
m
1
)
a
2
M
∑
k
=
0
2
m
k
c
2
M
∑
k
=
0
3
m
k
d
2
M
∑
k
=
0
4
m
k
e
2
M
∑
k
=
0
5
m
k
f
2
M
∑
k
=
0
6
m
k
h
2
M
∑
k
=
0
7
m
k
k
2
MFigure 2
DH coordinate system of the studied robot.The Jacobian matrix of the studied robot is very complicated in inertial coordinate system, making it very tough to analyze the singularity characteristics using matrixJ
^
mdirectly; therefore, it is necessary to simplify the analytical form of J
^
m. Usually, the Jacobian matrix can be simplified greatly by choosing an appropriate reference system and a proper reference point of the end-effector [8, 10]. For the robot configuration studied in this paper, the fifth joint coordinate systemΣ
4in Figure 2 is chosen as the reference systemΣ
refand select the intersection formed by the enlarged end-effector and the origin ofΣ
refas the end-effector reference point. And the simplified Jacobian matrix can be obtained as follows:(15)
J
^
ref
ref
=
[
C
2
(
a
^
C
345
+
e
^
S
5
+
d
^
S
45
)
-
(
c
^
+
f
^
)
S
2
S
345
-
(
c
^
+
f
^
)
C
345
e
^
S
5
+
d
^
S
45
e
^
S
5
0
0
-
h
^
C
6
-
(
c
^
+
f
^
)
S
2
C
345
+
C
2
(
-
a
^
S
345
+
e
^
C
5
+
d
^
C
45
)
(
c
^
+
f
^
)
S
345
e
^
C
5
+
d
^
C
45
e
^
C
5
0
0
0
-
S
2
(
a
^
-
d
^
S
3
-
e
^
S
34
)
d
^
C
3
+
e
^
C
34
0
0
0
0
-
h
^
S
6
C
345
S
2
-
S
345
0
0
0
0
-
S
6
-
S
2
S
345
-
C
345
0
0
0
-
1
0
C
2
0
1
1
1
0
C
6
]
,whereC
i
,
S
i
,
C
i
j
,
S
i
j
,
C
i
j
k
,
and
S
i
j
k represent Cos
⁡
(
θ
i
)
,
Sin
⁡
(
θ
i
)
,
Cos
⁡
(
θ
i
+
θ
j
)
,
Sin
⁡
(
θ
i
+
θ
j
)
,
Cos
⁡
(
θ
i
+
θ
j
+
θ
k
), and Sin
⁡
(
θ
i
+
θ
j
+
θ
k
), respectively. a
^
=
(
(
m
0
+
m
1
)
a
)
/
M,  c
^
=
(
∑
k
=
0
2
m
k
c
)
/
M,  d
^
=
(
∑
k
=
0
3
m
k
d
)
/
M,  e
^
=
(
∑
k
=
0
4
m
k
e
)
/
M,   f
^
=
(
∑
k
=
0
5
m
k
f
)
/
M,h
^
=
(
∑
k
=
0
6
m
k
h
)
/
M,  and  k
^
=
(
∑
k
=
0
7
m
k
k
)
/
M.In the following, the singular consistency betweenJ
^
ref
refandJ
^
mis proved in detail. With respect to the reference frame, the actual end point vector p
e
ref and the end reference point vector p
ref
ref have the following relationship (superscript “ref” means vectors described in reference system):
(16)
p
e
ref
=
p
ref
ref
+
p
e
ref
ref
,
where p
e
ref
ref is the vector from the end reference point to the actual end point.On this basis, the relationship between the end reference point velocityv
ref
ref and actual end point velocity v
e
ref can be expressed as
(17)
v
e
ref
=
v
ref
ref
+
ω
ref
ref
×
p
e
ref
ref
=
v
ref
ref
-
p
~
e
ref
ref
·
ω
ref
ref
,
where ω
ref
ref is the angular velocity of the end reference point.Since the end reference point and the actual end point attach to the same rigid body, their angular velocities are the same:(18)
ω
ref
ref
=
ω
e
ref
.Combining (17) with (18), the actual end point velocity is
(19)
p
˙
e
ref
=
[
v
e
ref
ω
e
ref
]
=
[
E
3
-
p
~
e
ref
ref
O
3
E
3
]
[
v
ref
ref
ω
ref
ref
]
=
J
e
ref
ref
·
p
˙
ref
ref
.Combining (12) with (19), the relationship between the original Jacobian matrix J
^
m and the simplified Jacobian matrix J
^
ref
ref can be represented as
(20)
J
^
m
=
J
ref
·
J
^
m
ref
=
J
ref
·
J
e
ref
ref
·
J
^
ref
ref
,
whereJ
ref
=
[
R
ref
O
3
O
3
R
ref
],R
refis the rotation matrix fromΣ
reftoΣ
I, and J
^
m
ref is the Jacobian matrix expressed in reference system. Because both the determinant values of J
ref and J
e
ref
ref are 1, the singularity of J
^
mis equivalent to that of J
^
ref
ref. The simplification will provide the foundation for robot singularity analysis in the following section.


## 4. Singularity Analysis

On the basis of the Jacobian matrix simplification, it can be found that a3
×
3zero matrix exists in J
^
ref
ref. The zero matrix resulted from the structure with three consecutive parallel axes.


### 4.1. The Characteristics of the Structure of Canadarm2

For the structure of Canadarm2, which has three consecutive parallel axes, the orientations of the joint coordinate axes corresponding to three consecutive parallel axes are consistent. Without any loss of generality, take the structure of axis 3, axis 4, and axis 5 parallel to each other as an example; the orientations of coordinate axesz
2, z
3, and z
4are the same as shown in Figure 2. Since reference systemΣ
refand joint coordinate systemΣ
4coincide with each other, the vector of axesz
2, z
3, andz
4can be represented as{
0,0
,
1
}
TinΣ
ref. Generally speaking, the first three rows of J
^
ref
refare called linear velocity Jacobian matrix, and the last three rows are called angular velocity Jacobian matrix. Therefore, the angular velocity Jacobian submatrix corresponding to the three consecutive parallel axes can be obtained as follows:
(21)
J
345
ω
=
[
0
0
0
0
0
0
1
1
1
]
.As for the linear velocity Jacobian submatrix corresponding to the three consecutive parallel axes, vector(
p
e
-
p
^
i
) in (10) needs to be replaced by (
p
ref
ref
-
p
^
i
ref
). Both the vectors (
p
ref
ref
-
p
^
2
ref
)
,
(
p
ref
ref
-
p
^
3
ref
) are in the X
O
Y plane of Σ
ref, so there are no components inz-axis direction. Then assume that (
p
ref
ref
-
p
^
2
ref
)
=
{
a
x
,
a
y
,
0
}
T, (
p
ref
ref
-
p
^
3
ref
)
=
{
b
x
,
b
y
,
0
}
T, and (
p
ref
ref
-
p
^
4
ref
)
=
{
0,0
,
0
}
T are obvious. Meanwhile, considering the expression z
2
ref
=
z
3
ref
=
z
4
ref
=
{
0,0
,
1
}
T, the linear velocity Jacobian submatrix corresponding to the three consecutive parallel axes is as follows:
(22)
J
345
v
=
[
-
a
y
-
b
y
0
a
x
b
x
0
0
0
0
]
.Thus, by merging (21), (22) and (23) can be constructed:
(23)
J
345
=
[
-
a
y
a
x
-
b
y
b
x
0
0
0
0
0
0
0
0
0
0
0
1
1
1
]
T
.Through the derivation above, it can safely draw the conclusion that a3
×
3zero matrix exists in the simplified Jacobian matrix of any robot with the structure of Canadarm2.


### 4.2. The Reconstruction of Jacobian Matrix

The characteristics of the structure of Canadarm2 can be used to simplify the singularity analysis. In order to decompose the Jacobian matrix to simplify the singularity analysis, matrixJ
^
ref
ref is reconstructed according to a novel standard, whose basic idea is to put the3
×
3zero matrix in the corner. And the new Jacobian matrix is shown as follows:(24)
J
^
ref
ref
′
=
[
C
2
0
C
6
0
1
1
1
C
2
(
a
^
C
345
+
e
^
S
5
+
d
^
S
45
)
-
(
c
^
+
f
^
)
S
2
S
345
-
(
c
^
+
f
^
)
C
345
-
h
^
C
6
0
e
^
S
5
+
d
^
S
45
e
^
S
5
0
-
(
c
^
+
f
^
)
S
2
C
345
+
C
2
(
-
a
^
S
345
+
e
^
C
5
+
d
^
C
45
)
(
c
^
+
f
^
)
S
345
0
0
e
^
C
5
+
d
^
C
45
e
^
C
5
0
-
S
2
(
a
^
-
d
^
S
3
-
e
^
S
34
)
d
^
C
3
+
e
^
C
34
-
h
^
S
6
0
0
0
0
C
345
S
2
-
S
345
-
S
6
0
0
0
0
-
S
2
S
345
-
C
345
0
-
1
0
0
0
]
.Since the elementary transformation of matrix does not change its singularity, the singularity characteristic of the matrixJ
^
ref
ref
′ will be analyzed instead in the following discussion. According to (12) and (14), the relationship between the velocity of the end reference point and the joint angular velocity can be described as
(25)
p
˙
ref
ref
′
=
J
^
ref
ref
′
·
θ
˙
′
,
where, p
˙
ref
ref
′
=
{
ω
^
z
ref
,
υ
^
x
ref
,
υ
^
y
ref
,
υ
^
z
ref
,
ω
^
x
ref
,
ω
^
y
ref
}
Tand θ
˙
′
=
{
θ
˙
1
,
θ
˙
2
,
θ
˙
7
,
θ
˙
6
,
θ
˙
3
,
θ
˙
4
,
θ
˙
5
}
T.


### 4.3. Singularity Analysis

On the basis of matrix reconstruction,J
^
ref
ref
′ can be decomposed to facilitate the singularity analysis. Considering (24) having a 3
×
3zero matrix in the right bottom corner, the decomposed matrix can be written as
(26)
J
^
ref
ref
′
=
[
J
11
J
12
J
21
O
3
]
.And the joint angular velocities can be obtained according to (26):
(27)
θ
˙
′
(
5,6
,
7
)
=
(
J
12
)
-
1
[
p
˙
ref
(
1,2
,
3
)
ref
′
-
J
11
θ
˙
′
(
1,2
,
3,4
)
]
,
θ
˙
′
(
1,2
,
3,4
)
=
(
J
21
)
†
p
˙
ref
(
4,5
,
6
)
ref
′
.Now we know that the robot singularity is only relevant to matrixesJ
12andJ
21. In this way, singularity analysis of redundant space robots with the structure of Canadarm2 will be simplified greatly. Consequently, in the following, the singularity conditions of matrixesJ
12andJ
21will be discussed.


#### 4.3.1. The Singularity Condition of Matrix J12

Because matrixJ
12is a3
×
3square matrix, the singularity condition can be directly obtained through its determinant value d
^
e
^
S
4. Hence, Cond1: S
4
=
0is the singularity condition ofJ
12.SubstitutingS
4
=
0 into matrixJ
^
ref
ref
′, we can get the base solution vector. Then, the singular direction corresponding to the singularity condition is
(28)
U
s
1
ref
=
{
0
,
-
C
5
S
6
,
S
5
S
6
,
N
a
,
-
h
^
N
a
+
h
^
C
5
C
6
,
0
}
T
,
where N
a
=
(
-
(
c
^
+
f
^
)
C
34
S
6
+
h
^
S
345
C
5
C
6
)
/
(
h
^
S
345
+
e
^
C
34
+
d
^
C
3
).


#### 4.3.2. The Singularity Condition of Matrix J21

SinceJ
21
∈
R
3
×
4is not square and the determinant value expression of matrixJ
21
·
J
21
Tis very complicated, the singularity condition of matrix J
21will be analyzed by using its four 3 × 3 submatrixes [16]. All the four submatrix determinant values are listed in the following:
(29)
Det
(
J
21
(
1,2
,
3
)
)
=
S
2
S
6
(
h
^
+
a
^
C
345
+
e
^
S
5
+
d
^
S
45
)
,
Det
(
J
21
(
1,2
,
4
)
)
=
S
2
(
e
^
C
5
+
d
^
C
45
-
a
^
S
345
)
,
Det
(
J
21
(
1,3
,
4
)
)
=
-
S
2
S
6
(
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
)
,
Det
(
J
21
(
2,3
,
4
)
)
=
S
6
(
d
^
C
3
+
e
^
C
34
+
h
^
S
345
)
,
where J
21
(
i
,
j
,
k
) represents the matrix composed of column i
,
j
,
k ofJ
21.Set each determinant value of (29) to zero simultaneously, and the singularity conditions are
(30)
Cond
2


:
S
2
=
0
,
S
6
=
0
,
Cond
3


:
S
2
=
0
,
d
^
C
3
+
e
^
C
34
+
h
^
S
345
=
0
,
Cond
4


:
S
6
=
0
,
e
^
C
5
+
d
^
C
45
-
a
^
S
345
=
0
,
Cond
5


:
d
^
C
3
+
e
^
C
34
+
h
^
S
345
=
0
,
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
=
0
.All the conditions except Cond5 can be easily obtained. Considering the multiplication factors exceptS
2andS
6in (29), the following relationships can be found:
(31)
h
^
+
a
^
C
345
+
e
^
S
5
+
d
^
S
45
=
S
345
(
d
^
C
3
+
e
^
C
34
+
h
^
S
345
)
+
C
345
(
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
)
,
e
^
C
5
+
d
^
C
45
-
a
^
S
345
=
C
345
(
d
^
C
3
+
e
^
C
34
+
h
^
S
345
)
-
S
345
(
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
)
.Clearly, the linear combination condition ofh
^
+
a
^
C
345
+
e
^
S
5
+
d
^
S
45
=
0 and e
^
C
5
+
d
^
C
45
-
a
^
S
345
=
0 is equivalent to the condition of d
^
C
3
+
e
^
C
34
+
h
^
S
345
=
0 and a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
=
0. Thus, Cond5 can be obtained.Substituting the four singularity conditions above into matrixJ
^
ref
ref
′, respectively, we can get the base solution vectors. Then, the singular direction corresponding to each singularity condition is listed as
(32)
U
s
1
ref
=
{
0,0
,
0
,
S
345
,
d
^
C
3
+
e
^
C
34
,
0
}
T
,
U
s
2
ref
=
{
0,0
,
0,1
,
-
h
^
,
0
}
T
,
U
s
3
ref
=
{
0,0
,
0
,
S
345
,
d
^
C
3
+
e
^
C
34
,
0
}
T
,
U
s
4
ref
=
{
0,0
,
0,1
,
-
h
^
,
0
}
T
.


## 4.1. The Characteristics of the Structure of Canadarm2

For the structure of Canadarm2, which has three consecutive parallel axes, the orientations of the joint coordinate axes corresponding to three consecutive parallel axes are consistent. Without any loss of generality, take the structure of axis 3, axis 4, and axis 5 parallel to each other as an example; the orientations of coordinate axesz
2, z
3, and z
4are the same as shown in Figure 2. Since reference systemΣ
refand joint coordinate systemΣ
4coincide with each other, the vector of axesz
2, z
3, andz
4can be represented as{
0,0
,
1
}
TinΣ
ref. Generally speaking, the first three rows of J
^
ref
refare called linear velocity Jacobian matrix, and the last three rows are called angular velocity Jacobian matrix. Therefore, the angular velocity Jacobian submatrix corresponding to the three consecutive parallel axes can be obtained as follows:
(21)
J
345
ω
=
[
0
0
0
0
0
0
1
1
1
]
.As for the linear velocity Jacobian submatrix corresponding to the three consecutive parallel axes, vector(
p
e
-
p
^
i
) in (10) needs to be replaced by (
p
ref
ref
-
p
^
i
ref
). Both the vectors (
p
ref
ref
-
p
^
2
ref
)
,
(
p
ref
ref
-
p
^
3
ref
) are in the X
O
Y plane of Σ
ref, so there are no components inz-axis direction. Then assume that (
p
ref
ref
-
p
^
2
ref
)
=
{
a
x
,
a
y
,
0
}
T, (
p
ref
ref
-
p
^
3
ref
)
=
{
b
x
,
b
y
,
0
}
T, and (
p
ref
ref
-
p
^
4
ref
)
=
{
0,0
,
0
}
T are obvious. Meanwhile, considering the expression z
2
ref
=
z
3
ref
=
z
4
ref
=
{
0,0
,
1
}
T, the linear velocity Jacobian submatrix corresponding to the three consecutive parallel axes is as follows:
(22)
J
345
v
=
[
-
a
y
-
b
y
0
a
x
b
x
0
0
0
0
]
.Thus, by merging (21), (22) and (23) can be constructed:
(23)
J
345
=
[
-
a
y
a
x
-
b
y
b
x
0
0
0
0
0
0
0
0
0
0
0
1
1
1
]
T
.Through the derivation above, it can safely draw the conclusion that a3
×
3zero matrix exists in the simplified Jacobian matrix of any robot with the structure of Canadarm2.


## 4.2. The Reconstruction of Jacobian Matrix

The characteristics of the structure of Canadarm2 can be used to simplify the singularity analysis. In order to decompose the Jacobian matrix to simplify the singularity analysis, matrixJ
^
ref
ref is reconstructed according to a novel standard, whose basic idea is to put the3
×
3zero matrix in the corner. And the new Jacobian matrix is shown as follows:(24)
J
^
ref
ref
′
=
[
C
2
0
C
6
0
1
1
1
C
2
(
a
^
C
345
+
e
^
S
5
+
d
^
S
45
)
-
(
c
^
+
f
^
)
S
2
S
345
-
(
c
^
+
f
^
)
C
345
-
h
^
C
6
0
e
^
S
5
+
d
^
S
45
e
^
S
5
0
-
(
c
^
+
f
^
)
S
2
C
345
+
C
2
(
-
a
^
S
345
+
e
^
C
5
+
d
^
C
45
)
(
c
^
+
f
^
)
S
345
0
0
e
^
C
5
+
d
^
C
45
e
^
C
5
0
-
S
2
(
a
^
-
d
^
S
3
-
e
^
S
34
)
d
^
C
3
+
e
^
C
34
-
h
^
S
6
0
0
0
0
C
345
S
2
-
S
345
-
S
6
0
0
0
0
-
S
2
S
345
-
C
345
0
-
1
0
0
0
]
.Since the elementary transformation of matrix does not change its singularity, the singularity characteristic of the matrixJ
^
ref
ref
′ will be analyzed instead in the following discussion. According to (12) and (14), the relationship between the velocity of the end reference point and the joint angular velocity can be described as
(25)
p
˙
ref
ref
′
=
J
^
ref
ref
′
·
θ
˙
′
,
where, p
˙
ref
ref
′
=
{
ω
^
z
ref
,
υ
^
x
ref
,
υ
^
y
ref
,
υ
^
z
ref
,
ω
^
x
ref
,
ω
^
y
ref
}
Tand θ
˙
′
=
{
θ
˙
1
,
θ
˙
2
,
θ
˙
7
,
θ
˙
6
,
θ
˙
3
,
θ
˙
4
,
θ
˙
5
}
T.


## 4.3. Singularity Analysis

On the basis of matrix reconstruction,J
^
ref
ref
′ can be decomposed to facilitate the singularity analysis. Considering (24) having a 3
×
3zero matrix in the right bottom corner, the decomposed matrix can be written as
(26)
J
^
ref
ref
′
=
[
J
11
J
12
J
21
O
3
]
.And the joint angular velocities can be obtained according to (26):
(27)
θ
˙
′
(
5,6
,
7
)
=
(
J
12
)
-
1
[
p
˙
ref
(
1,2
,
3
)
ref
′
-
J
11
θ
˙
′
(
1,2
,
3,4
)
]
,
θ
˙
′
(
1,2
,
3,4
)
=
(
J
21
)
†
p
˙
ref
(
4,5
,
6
)
ref
′
.Now we know that the robot singularity is only relevant to matrixesJ
12andJ
21. In this way, singularity analysis of redundant space robots with the structure of Canadarm2 will be simplified greatly. Consequently, in the following, the singularity conditions of matrixesJ
12andJ
21will be discussed.


### 4.3.1. The Singularity Condition of Matrix J12

Because matrixJ
12is a3
×
3square matrix, the singularity condition can be directly obtained through its determinant value d
^
e
^
S
4. Hence, Cond1: S
4
=
0is the singularity condition ofJ
12.SubstitutingS
4
=
0 into matrixJ
^
ref
ref
′, we can get the base solution vector. Then, the singular direction corresponding to the singularity condition is
(28)
U
s
1
ref
=
{
0
,
-
C
5
S
6
,
S
5
S
6
,
N
a
,
-
h
^
N
a
+
h
^
C
5
C
6
,
0
}
T
,
where N
a
=
(
-
(
c
^
+
f
^
)
C
34
S
6
+
h
^
S
345
C
5
C
6
)
/
(
h
^
S
345
+
e
^
C
34
+
d
^
C
3
).


### 4.3.2. The Singularity Condition of Matrix J21

SinceJ
21
∈
R
3
×
4is not square and the determinant value expression of matrixJ
21
·
J
21
Tis very complicated, the singularity condition of matrix J
21will be analyzed by using its four 3 × 3 submatrixes [16]. All the four submatrix determinant values are listed in the following:
(29)
Det
(
J
21
(
1,2
,
3
)
)
=
S
2
S
6
(
h
^
+
a
^
C
345
+
e
^
S
5
+
d
^
S
45
)
,
Det
(
J
21
(
1,2
,
4
)
)
=
S
2
(
e
^
C
5
+
d
^
C
45
-
a
^
S
345
)
,
Det
(
J
21
(
1,3
,
4
)
)
=
-
S
2
S
6
(
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
)
,
Det
(
J
21
(
2,3
,
4
)
)
=
S
6
(
d
^
C
3
+
e
^
C
34
+
h
^
S
345
)
,
where J
21
(
i
,
j
,
k
) represents the matrix composed of column i
,
j
,
k ofJ
21.Set each determinant value of (29) to zero simultaneously, and the singularity conditions are
(30)
Cond
2


:
S
2
=
0
,
S
6
=
0
,
Cond
3


:
S
2
=
0
,
d
^
C
3
+
e
^
C
34
+
h
^
S
345
=
0
,
Cond
4


:
S
6
=
0
,
e
^
C
5
+
d
^
C
45
-
a
^
S
345
=
0
,
Cond
5


:
d
^
C
3
+
e
^
C
34
+
h
^
S
345
=
0
,
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
=
0
.All the conditions except Cond5 can be easily obtained. Considering the multiplication factors exceptS
2andS
6in (29), the following relationships can be found:
(31)
h
^
+
a
^
C
345
+
e
^
S
5
+
d
^
S
45
=
S
345
(
d
^
C
3
+
e
^
C
34
+
h
^
S
345
)
+
C
345
(
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
)
,
e
^
C
5
+
d
^
C
45
-
a
^
S
345
=
C
345
(
d
^
C
3
+
e
^
C
34
+
h
^
S
345
)
-
S
345
(
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
)
.Clearly, the linear combination condition ofh
^
+
a
^
C
345
+
e
^
S
5
+
d
^
S
45
=
0 and e
^
C
5
+
d
^
C
45
-
a
^
S
345
=
0 is equivalent to the condition of d
^
C
3
+
e
^
C
34
+
h
^
S
345
=
0 and a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
=
0. Thus, Cond5 can be obtained.Substituting the four singularity conditions above into matrixJ
^
ref
ref
′, respectively, we can get the base solution vectors. Then, the singular direction corresponding to each singularity condition is listed as
(32)
U
s
1
ref
=
{
0,0
,
0
,
S
345
,
d
^
C
3
+
e
^
C
34
,
0
}
T
,
U
s
2
ref
=
{
0,0
,
0,1
,
-
h
^
,
0
}
T
,
U
s
3
ref
=
{
0,0
,
0
,
S
345
,
d
^
C
3
+
e
^
C
34
,
0
}
T
,
U
s
4
ref
=
{
0,0
,
0,1
,
-
h
^
,
0
}
T
.


## 4.3.1. The Singularity Condition of Matrix J12

Because matrixJ
12is a3
×
3square matrix, the singularity condition can be directly obtained through its determinant value d
^
e
^
S
4. Hence, Cond1: S
4
=
0is the singularity condition ofJ
12.SubstitutingS
4
=
0 into matrixJ
^
ref
ref
′, we can get the base solution vector. Then, the singular direction corresponding to the singularity condition is
(28)
U
s
1
ref
=
{
0
,
-
C
5
S
6
,
S
5
S
6
,
N
a
,
-
h
^
N
a
+
h
^
C
5
C
6
,
0
}
T
,
where N
a
=
(
-
(
c
^
+
f
^
)
C
34
S
6
+
h
^
S
345
C
5
C
6
)
/
(
h
^
S
345
+
e
^
C
34
+
d
^
C
3
).


## 4.3.2. The Singularity Condition of Matrix J21

SinceJ
21
∈
R
3
×
4is not square and the determinant value expression of matrixJ
21
·
J
21
Tis very complicated, the singularity condition of matrix J
21will be analyzed by using its four 3 × 3 submatrixes [16]. All the four submatrix determinant values are listed in the following:
(29)
Det
(
J
21
(
1,2
,
3
)
)
=
S
2
S
6
(
h
^
+
a
^
C
345
+
e
^
S
5
+
d
^
S
45
)
,
Det
(
J
21
(
1,2
,
4
)
)
=
S
2
(
e
^
C
5
+
d
^
C
45
-
a
^
S
345
)
,
Det
(
J
21
(
1,3
,
4
)
)
=
-
S
2
S
6
(
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
)
,
Det
(
J
21
(
2,3
,
4
)
)
=
S
6
(
d
^
C
3
+
e
^
C
34
+
h
^
S
345
)
,
where J
21
(
i
,
j
,
k
) represents the matrix composed of column i
,
j
,
k ofJ
21.Set each determinant value of (29) to zero simultaneously, and the singularity conditions are
(30)
Cond
2


:
S
2
=
0
,
S
6
=
0
,
Cond
3


:
S
2
=
0
,
d
^
C
3
+
e
^
C
34
+
h
^
S
345
=
0
,
Cond
4


:
S
6
=
0
,
e
^
C
5
+
d
^
C
45
-
a
^
S
345
=
0
,
Cond
5


:
d
^
C
3
+
e
^
C
34
+
h
^
S
345
=
0
,
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
=
0
.All the conditions except Cond5 can be easily obtained. Considering the multiplication factors exceptS
2andS
6in (29), the following relationships can be found:
(31)
h
^
+
a
^
C
345
+
e
^
S
5
+
d
^
S
45
=
S
345
(
d
^
C
3
+
e
^
C
34
+
h
^
S
345
)
+
C
345
(
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
)
,
e
^
C
5
+
d
^
C
45
-
a
^
S
345
=
C
345
(
d
^
C
3
+
e
^
C
34
+
h
^
S
345
)
-
S
345
(
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
)
.Clearly, the linear combination condition ofh
^
+
a
^
C
345
+
e
^
S
5
+
d
^
S
45
=
0 and e
^
C
5
+
d
^
C
45
-
a
^
S
345
=
0 is equivalent to the condition of d
^
C
3
+
e
^
C
34
+
h
^
S
345
=
0 and a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
=
0. Thus, Cond5 can be obtained.Substituting the four singularity conditions above into matrixJ
^
ref
ref
′, respectively, we can get the base solution vectors. Then, the singular direction corresponding to each singularity condition is listed as
(32)
U
s
1
ref
=
{
0,0
,
0
,
S
345
,
d
^
C
3
+
e
^
C
34
,
0
}
T
,
U
s
2
ref
=
{
0,0
,
0,1
,
-
h
^
,
0
}
T
,
U
s
3
ref
=
{
0,0
,
0
,
S
345
,
d
^
C
3
+
e
^
C
34
,
0
}
T
,
U
s
4
ref
=
{
0,0
,
0,1
,
-
h
^
,
0
}
T
.


## 5. Simulation Results

In order to verify the effectiveness of the singularity analysis method proposed in this paper, the following verifications are carried out. Assign the robot parameters in Figure2: a
=
c
=
f
=
h
=
k
=
l
=
0.5
m, d
=
e
=
2.5
m, m
0
=
1000
kg, m
1
=
m
2
=
m
5
=
m
6
=
m
7
=
30
kg, and m
3
=
m
4
=
100
kg. The corresponding parameters of virtual manipulator are a
^
=
0.38
m, c
^
=
0.39
m, d
^
=
2.15
m,  e
^
=
2.33
m,  f
^
=
0.48
m,  h
^
=
0.49
m,  and  k
^
=
0.5
m. List five groups of robot joint angles satisfying the five singularity conditions above, respectively, and solve the corresponding determinant values of J
^
m
·
(
J
^
m
)
T. Then the relative results are shown in Table 3.Table 3
The summary of examples.
Number
Singularity condition
Joint angle
Det
(
J
^
m
·
(
J
^
m
)
T
)
Cond1
S
4
=
0
{
2
5
°
,
0
°
,
9
0
°
,
0
°
,
4
5
°
,
9
0
°
,
0
°
}
−1.06608 × 10−14
Cond2
S
2
=
,
S
6
=
0
{
2
5
°
,
0
°
,
3
4
°
,
3
0
°
,
4
5
°
,
0
°
,
0
°
}
−6.80933 × 10−14
Cond3
S
2
=
0
,
d
^
C
3
+
e
^
C
34
+
h
^
S
345
=
0
{
2
5
°
,
0
°
,
2
0
°
,
14
0
°
,
-
139
.
1
°
,
4
5
°
,
0
°
}
2.43766 × 10−30
Cond4
S
6
=
0
,
e
^
C
5
+
d
^
C
45
-
a
^
S
345
=
0
{
2
5
°
,
1
0
°
,
-
132
.
9
°
,
14
0
°
,
2
0
°
,
0
°
,
0
°
}
2.97193 × 10−12
Cond5
d
^
C
3
+
e
^
C
34
+
h
^
S
345
=
0,
a
^
+
h
^
C
345
-
d
^
S
3
-
e
^
S
34
=
0
{
2
5
°
,
1
0
°
,
2
4
°
,
18
0
°
,
-
44
.
9
°
,
4
5
°
,
0
°
}
−1.60952 × 10−16From Table3, it can be found that all the determinant values of J
^
m
·
(
J
^
m
)
T approach to zero. Thus, it can be judged that robot singularity happens.According to the analytical expressions of singular directions given above, the singular configuration and singular direction corresponding to each singularity condition are shown in Figures3, 4, 5, 6, and 7.Figure 3
The singular configuration corresponding to Cond1.Figure 4
The singular configuration corresponding to Cond2.Figure 5
The singular configuration corresponding to Cond3.Figure 6
The singular configuration corresponding to Cond4.Figure 7
The singular configuration corresponding to Cond5.In the process of singularity analysis, because the simplified Jacobian matrix shows the relationship between the velocity of the end reference point and joint angular velocities in reference systemΣ
ref, the singular direction given in Figures 3 to 7 describes the degradation direction of the end reference point expressed in reference systemΣ
ref.The simulation results verify the effectiveness of the proposed method of singularity analysis for redundant space robots with the structure of Canadarm2.


## 6. Summary and Conclusions

A novel singularity analysis method for redundant space robots with the structure of Canadarm2 is proposed in the paper. Different from ground robots, the singularity analysis of space robots needs to consider the dynamic parameters because of the moving base. Therefore, the “virtual manipulator” method is employed to transfer the singularity problem to that of a ground one. As the generalized Jacobian matrix is very complex, it is greatly simplified by choosing a proper reference system and an end reference point and the equivalence of singularity is analyzed in detail. Then on the basis of novel reconstruction, the simplified Jacobian matrix can be decomposed to reduce the complexity of singularity analysis. Finally, through the analysis of the submatrixes, singularity conditions and singular directions of the redundant space robot with the structure of Canadarm2 are obtained. The method proposed in the paper is effective and needs less calculation.



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*Source: 735030-2014-06-26.xml*