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IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING 1
Kinematic Control With Singularity Avoidance for
Teaching-Playback Robot Manipulator System
Yanjiang Huang, Student Member, IEEE, Yoon Seong Yong, Ryosuke Chiba, Tamio Arai, Member, IEEE,
Tsuyoshi Ueyama, Member, IEEE, and Jun Ota, Member, IEEE
Abstract—A teaching-playback robot manipulator system
whereby the user controls the manipulator through a teaching
pendant has been used widely in industrial applications. Kine-
matic singularity issue becomes an important problem in the
control of robot with a teaching-playback system. In this paper, we
propose and investigate three singularity avoidance methods for
a teaching-playback robot manipulator system. Nonredundancy
singularity avoidance (NRSA) attempts to reduce both the position
and orientation errors of the end-effector with the same priority.
Redundancy singularity avoidance (RSA) attempts to reduce
the position error of the end-effector with the first priority and
reduce the orientation error of the end-effector with the second
priority; Both NRSA and RSA are based on a modification of a
Jacobian matrix. Point-to-point singularity avoidance (PTPSA)
makes the end-effector pass through a singular region based
on joint-interpolated control without maintaining the position
and orientation of the end-effector. Experimental case studies
are developed to investigate the manipulator performance when
the end-effector approaches the wrist and shoulder singularity.
The maximal end-effector trajectory error and users' feelings
are statistically evaluated and analyzed in the experiment. The
results of the experiment show the effectiveness and practice of
the proposed methods.
Manuscript received September 21, 2014; revised November 24, 2014; ac-
cepted January 04, 2015. This paper was recommended for publication by As-
sociate Editor K. Fregene and Editor J. Wen upon evaluation of the reviewers'
comments. This work was supported in part by the National Natural Science
Foundation of China under Grant 91223201, and in part by the Natural Science
Foundation of Guangdong Province under Grant S2013030013355. This paper
was presented at the IEEE/ASME International Conference on Advanced Intel-
ligent Mechatronics, Wollongong, Australia, July 2013.
Y. Huang is with the Guangdong Provincial Key Laboratory of Precision
Equipment and Manufacturing Technology, School of Mechanical and Automo-
tive Engineering, South China University of Technology, Guangdong 510640,
China (e-mail: mehuangyj@scut.edu.cn).
Y. S. Yong, and J. Ota are with Research into Artifacts, Center for Engi-
neering, University of Tokyo, Chiba 277-8568, Japan (e-mails: yong, ota@race.
u-tokyo.ac.jp).
R. Chiba is with the Research Center for Brain Function and Medical En-
gineering, Asahikawa Medical University, Hokkaido 078-8510, Japan (e-mail:
rchiba@asahikawa-med.ac.jp)
T. Arai is with the Department of Mechanical Engineering, College of En-
gineering, Shibaura Institute of Technology, Tokyo 135-8548, Japan (e-mail:
arai-t@shibaura-it.ac.jp).
T. Ueyama is with Denso Wave Incorporated, Aichi 470-2297, Japan (e-mail:
tsuyoshi.ueyama@denso-wave.co.jp).
This paper has supplementary downloadable multimedia material available
at http://ieeexplore.ieee.org provided by the authors. The Supplementary Mate-
rial includes an MP4 audio/video file (.mp4) for illustrating the performance of
robot manipulator under different singularity avoidance methods in the experi-
ment and an excel file (.xlsx) for illustrating the biographical information of 24
participants in the experiment. This material is 3.82 MB in size.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TASE.2015.2392095
Note to Practitioners—This paper is motivated by the problem
of proposing appropriate methods to avoid singularities for a
teaching-playback robot manipulator system. Three singularity
avoidance methods are proposed and investigated from the
perspective of the movement of the end-effector when it passes
through a singular region. With NRSA method, both the posi-
tion and orientation errors of the end-effector are reduced with
the same priority. With RSA method, the position error of the
end-effector is reduced with the first priority and the orientation
error of the end-effector is reduced with the second priority.
With PTPSA method, the end-effector passes through a singular
region under the joint-interpolated control without maintaining
the position and orientation of the end-effector. NRSA, RSA, and
PTPSA have been successfully tested with wrist and shoulder
singularity through the experimental case studies. Experiment
results show that the proposed methods can automatically avoid
a singularity for a teaching-playback robot manipulator, which
would enhance the capability of robots for industrial automation.
In future research, the proposed methods can be extended to
simultaneously realize the singularity and obstacle avoidance.
Index Terms—end-effector trajectory, Jacobian matrix, point-
to-point motion, singularity avoidance, teaching-playback robot
manipulator system.
I. INTRODUCTION
TEACHING-PLAYBACK robot manipulator systems
which use a teaching pendant to control the robot motion
are widely used in industrial applications such as assembly,
painting, and inspection tasks [1]. The kinematic singularities
of a robot manipulator are a set of singular configurations in
which the end-effector locality loses the ability to move in
an arbitrary direction [2]. When the manipulator is at or is in
the neighborhood of a kinematic singularity, the usual inverse
differential kinematics solutions based on the Jacobian matrix
give rise to some undesirable conditions, and this results in
very high joint velocities and large control deviations [3], [4].
In a teaching-playback robot manipulator system, the user
normally uses the teaching pendant to control the movement
of the end-effector in the X, Y, and Z directions to make the
end-effector move from an initial position and orientation
to a goal position and orientation. During the movement of
the end-effector, a singularity may occur, as shown in Fig. 1,
which may surprise the user and result in a failure to reach
the goal position and orientation. Therefore, it is necessary to
automatically avoid such kinematic singularities when using
a teaching-playback robot manipulator system to complete a
task. In this paper, singularity avoidance is defined as dealing
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2 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING
Fig. 1. A teaching playback robot manipulator system which does not consider
singularity avoidance. is the position and orientation of the end-effector.
with a kinematic singularity by causing the end-effector to
bypass it. A teaching-playback robot manipulator system with
a serial robot is considered in this paper, as shown in Fig. 1.
In previous studies, the singularities were avoided either in
trajectory planning or motion planning with predefined initial
and final configurations of the robot manipulator. The Jaco-
bian-based methods, such as pseudoinverse method, Jacobian
transpose, and damped least-squares inverse of the Jacobian
matrix, have been proposed to avoid the ill-conditioned of the
Jacobian matrix [5]–[7]. However, the pseudoinverse method
is computationally time consuming; the Jacobian transpose
and damped least-squares inverse are sensitive to parameter
selection and may cause large tracking errors. In recent, some
researchers have proposed other Jacobian-based methods to
avoid kinematic singularities [8]–[10]. In [8], a new filter was
proposed for the singular values of the Jacobian matrix and a
continuous task priority strategy with selective damping was
proposed to generate smooth trajectory. Nevertheless, this
method is computationally time consuming and may result in
algorithmic singularities. In [9], an inverse kinematics method
that transforms the inverse kinematics matrix equations into
eight pure algebraic equations was proposed to solve the wrist
singularity problem for a PUMA560-structured robot manip-
ulator. An elementary Jacobian transformation was used to
identify the singularity conditions of the Canadarm2 in [10].
These methods were proposed for specific robot manipulators.
Some researchers focused on the control algorithm or scheme
to avoid singularities [11]–[19]. Several tracking controllers
have been proposed. In [11], an augmented hybrid impedance
control scheme consisted of a Cartesian-level potential differ-
ence controller and a joint-space inverse dynamic controller
was proposed. An energy-based controller incorporated with
fuzzy neural network compensation was proposed to avoid
singularities in [12]. A voltage-based control scheme and a
force-based control scheme were proposed in [13] and [14],
respectively. However, additional device was required to obtain
the force, energy, and voltage in these controllers. In other
controllers, the singularity was avoided based on task space
control. In [15], a method with Lyapunov stability theory was
proposed. In [16], a continuous task transition algorithm was
applied to a task space control framework. Tasks were defined
for dealing with singularities and joint limits, and they were
activated or deactivated by using the continuous task transition
algorithm. In [17], the source of the discontinuity was discussed
and null motion was proposed to control the end-effector over
the singularity. Motion planning was integrated into control
scheme in some researches [18], [19]. A fuzzy motion planning
mapping method and a path planning technique were proposed
to avoid singularities in [18] and [19], respectively. Many
control algorithms have been proposed to control robot manip-
ulator to avoid obstacles [20]–[23]. Many of these algorithms
can be extended to avoid singularities. However, it is difficult
to tune the parameters for these control algorithms and schemes
to make them perform optimally.
Other researchers formulated the singularity avoidance
problem as an optimization problem [24]–[27]. In [24], a mo-
tion tracking of robot manipulator based on a genetic algorithm
was proposed. In [25], the singularity avoidance problem was
formulated as minimization of the measure of manipulability
and a nonlinear programming method was proposed to solve
the optimization problem. A hierarchical constraint-based
singularity avoidance was proposed in [26], which can be
considered as a constraint-optimization problem. By solving
the optimization problem, much computation time is required
to obtain an optimal or near-optimal solution. Although an
appropriate small search region [24] and an analytical method
[27] can reduce computation time, it is difficult to determine the
appropriate small search region and the analytical method nor-
mally depend on robot structure. A large number of researchers
have focused on the singularity avoidance of robot manipulator,
however, singularity avoidance for a teaching-playback robot
manipulator system was not considered because the robot does
not know where to move until a move command is sent from
the teaching pendant.
In factories, singularity avoidance for a teaching-playback
robot manipulator is realized based on the repetitive adjustment
of joint angles. However, this is not intuitive and relies on the
repetitive adjustment of joint angles, which may surprise the
user because the end-effector does not move in the direction de-
sired by that user.
In this study, we set out to propose appropriate methods to
avoid kinematic singularities for a teaching-playback robot ma-
nipulator system. We propose and investigate three singularity
avoidance methods for a teaching-playback robot manipulator
system and compare the performance of the robot manipulator
with these three methods through experimental case studies.
Two of these three singularity avoidance methods, which are
based on a modified Jacobian matrix, were proposed in our re-
cent study [28]. The other, based on the point-to-point jumping
motion of the end-effector, is originally proposed in this paper.
This study differs from our previous studies in following three
aspects. (1) We propose a novel singularity avoidance method
for a teaching-playback robot manipulator system. (2) We in-
vestigate three singularity avoidance methods for a teaching-
playback robot manipulator system. (3) We investigate and sta-
tistically analyze users' feelings under these three singularity
avoidance methods. The contribution of this paper is to compare
the user's experience of the different methods for singularity
avoidance for a teaching-playback robot manipulator system,
and implementation of the point-to-point singularity avoidance
method.
The remainder of this paper is organized as follows. In
Section II, we formulate and analyze the singularity avoidance
problem. The proposed methods are described in Section III.

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HUANG et al.: KINEMATIC CONTROL WITH SINGULARITY AVOIDANCE FOR TEACHING-PLAYBACK ROBOT MANIPULATOR SYSTEM 3
The experiment and results are presented in Section IV. Finally,
this paper is concluded in Section V.
II. PROBLEM FORMULATION
In this section, we formulate the singularity avoidance
problem for a teaching-playback robot manipulator system. A
description of the teaching-playback robot manipulator system,
kinematic singularity analysis, end-effector trajectory tracking,
and the feelings of user are presented.
A. Teaching-Playback Robot Manipulator System
In this study, we consider a teaching-playback robot ma-
nipulator system which has 6 degrees of freedom (6-DOFs)
and a teaching pendant, as shown in Fig. 1. Let denote the
-dimensional position and/or orientation of the end-effector,
and denote the -dimensional configuration. Because roll,
pitch and yaw angles are general parameters to describe the ori-
entation of the end-effector and they are straight forward [29],
we describe the orientation of the end-effector via roll, pitch
and yaw angles. When , the robot manipulator is non-re-
dundant. In the case of
and . Where, is
the position vector of the end-effector;
is the orientation vector of the end-effector, which are defined
as yaw, pitch, and roll relative to the robot base coordinate
frame; are the joint angles. When
, the robot manipulator is redundant. In this paper, a
task was defined as moving the end-effector from an initial
to a goal by using a teaching pendant. The input pa-
rameter to the inverse kinematics of the manipulator is the
change in the position and orientation of the end-effector
(i.e., ) set by the user
through a teaching pendant, which is determined by the end-ef-
fector velocity and sampling time. The output parameter of
the inverse kinematics is the change in the configuration (i.e.,
caused by .
After several steps, the end-effector can be moved from the
initial to the goal .
B. Kinematic Singularity Analysis
According to the forward kinematics of a robot manipulator,
the relationship between and can be described as
(1)
(2)
where is a vector function expressing the forward kine-
matics relation; is the end-effector velocity vector;
is the joint velocity vector; is the Jacobian ma-
trix.
To minimize the joint velocities when the robot manipulator
is at or is in the neighborhood of a kinematic singularity, we
choose the Moore–Penrose pseudoinverse to calculate joint ve-
locities. Although there are many other criteria in solving the
singularity avoidance problem, such as dynamically consistent
pseudoinverse [30], which may complicate the problem since
it requires additional force, torque information and energy ma-
trix. According to the inverse kinematics of a robot manipulator,
when rank is the full rank, the joint velocities can be rep-
resented as
(3)
where, is the inverse of the Jacobian or the
pseudo-inverse of the Jacobian .
When rank is not the full rank and the determinant
of is zero, singularity occurs in this configuration. This
configuration is called “singular configuration.”
The manipulator degenerates not only at the exact singular
configuration, but also in the neighborhood of the singular con-
figuration. In this paper, we define a singular region as the area
in which the end-effector locality loses the ability to move in an
arbitrary direction. The positions and orientations of the end-ef-
fector in the singular region were mapped to the singular con-
figuration and its neighborhood. Normally, the size of singular
region is calculated based on the estimation of the smallest sin-
gular value of Jacobian matrix [31], [32]. The robot manipulator
is checked inside or outside the singular region by comparing
the estimation of smallest singular value to a threshold value.
If the estimation of smallest singular value is smaller than the
threshold value, then the robot is singular. Otherwise, the robot
is not singular. The estimation of smallest singular value can be
realized based on the singular value decomposition of the Jaco-
bian matrix [31, 33], the threshold value is defined by a user.
There are some criteria for singularity detection [29], such
as singular value, measurement of manipulability, and condi-
tion number. However, it is difficult to determine the threshold
value for these criteria and much computation time is required
to evaluate the criteria. In this paper, we use a different method
to detect singularity by checking whether the calculated joint
velocity exceed the maximal joint velocity. Low computational
time is required to calculate the joint velocity and the maximal
joint velocity is easy to know. The details of the proposed sin-
gularity detection method is described in Section III-A.
C. End-Effector Trajectory Tracking and Feelings of User
Because the joint angles change largely and a deviation in
the end-effector trajectory occurs when the end-effector enters a
singular region, we investigated the end-effector trajectory error
including the position error and orientation error.
The position error and orientation error can be defined as fol-
lows [34], [35]:
(4)
(5)
where is the position error; is the actual position;
is the desired position. is the orientation error, which
is restricted in the range of , and
are the orthonormal unit vectors which specify the orientation
of the end-effector. are the orthonormal vectors
which specify the desired orientation of the end-effector. Note
that when calculating the maximal position error and orien-
tation error, the absolute value of (4) and (5) should be used,
respectively.

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4 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING
Fig. 2. Overview of singularity avoidance for a teaching playback robot ma-
nipulator.
Because a robotic task is completed by a user through a
teaching pendant, we investigate the users' feelings about
their robot control intentions, smoothness in the movement
of the end-effector, efficiency with which a task can be com-
pleted, sense of danger, and tension, from the viewpoint of
human-robot interaction [36], [37]. The analysis of joint trajec-
tory and end-effector trajectory is an objectiveness analysis in
the evaluation of the singularity avoidance methods. The us-
ability of a teaching playback robot manipulator system can be
evaluated through the joint trajectory, end-effector trajectory,
control intention, and moving smoothness of the end-effector.
III. PROPOSED METHODS FOR SINGULARITY AVOIDANCE
In this section, we describe the proposed singularity avoid-
ance methods for a teaching-playback robot manipulator
system. Fig. 2 shows an overview of singularity avoidance
for a teaching-playback robot manipulator. Because a singular
region is not established in advance by using a teaching pen-
dant to control the robot, we check whether the end-effector
is about to a singular region in real time after the user enters
the command , but before moving the end-effector. If the
end-effector is found to be entering a singular region after
the user enters the command , the singularity avoidance
algorithm is applied. Otherwise, the robot will move based on
an inverse kinematics solution. The inverse kinematics solution
is that configuration which requires the least time to move
from the previous configuration [38]. When the end-effector
bypasses a singular region, its movement will differ from the
user. This reduces the manipulability. We therefore propose
three singularity avoidance methods from the perspective
of a user's intention: (1) nonredundancy singularity avoid-
ance (NRSA); (2) redundancy singularity avoidance (RSA);
(3) point-to-point singularity avoidance (PTPSA). The move-
ment of the end-effector when avoiding or bypassing a singular
region through the application of these three methods is shown
Fig. 3. Movement of end-effector when avoiding singular region with three
singularity avoidance methods. (a) Movement of end-effector when avoiding
singular region in NRSA. (b) Movement of end-effector when avoiding singular
region in RSA. (c) Movement of end-effector when avoiding singular region in
PTPSA.
in Fig. 3. With NRSA, both the position and orientation of
the end-effector are important and the position and orienta-
tion errors need to be reduced with the same priority. The
end-effector passes through the singular region by following a
trajectory crosses over that region. With RSA, the position of
the end-effector is performed as a subtask with the first priority
and a singularity is avoided by using redundancy. The position
error is reduced with first priority and the orientation error is
reduced with second priority. With PTPSA, neither the position
nor the orientation of the end-effector are maintained. Rather,
the joints are controlled to make the end-effector pass through
the singular region.
When the end-effector is in a singular region, a discontinuity
occurs because the singular components of the robot task space
are removed and the control algorithms are calculated with a
different task space specification [17]. Due to the ill-conditioned
inverse kinematics and dynamics of a robot manipulator system,
motion in the singular direction within the singular region is not

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HUANG et al.: KINEMATIC CONTROL WITH SINGULARITY AVOIDANCE FOR TEACHING-PLAYBACK ROBOT MANIPULATOR SYSTEM 5
Fig. 4. Singularity avoidance. is the number of steps through which the robot joints move from the current configuration to the saved configuration when they
cannot move from the current configuration to the saved configuration in one step. is the current end-effector position and orientation; is the next end-effector
position and orientation; is the current configuration; is the next configuration; is the velocity limit for the th joint; is the configuration in which
the robot joints move from the current configuration to the saved configuration. is the joint change limit of the th joint in sampling time; , is the
angle change for the th joint; is the th joint angle; is the temporary angle of the th joint. IK is the inverse kinematic function; FK is the forward
kinematic function. Ceil() is the ceiling function which maps a real number to the smallest following integer. The difference between non-redundancy singularity
avoidance and redundancy singularity avoidance is on “calculate joint” velocity as shown in green box.
controllable, which results in tracking errors. Therefore, the po-
sition and orientation errors of the end-effector owing to the dis-
continuity of singularity avoidance algorithms should be elim-
inated when the end-effector escapes from the singular region.
The details of singularity detection, NRSA, RSA, and PTPSA
are described in Sections III-A, III-B1, III-B2, and III-B3, re-
spectively. The overall implementation of NRSA and RSA is
shown in Fig. 4. The difference between NRSA and RSA is
on “calculate joint velocity,” which is described in detail in
Sections III-B1 and III-B2, respectively.
A. Singularity Detection
If the end-effector were to enter a singular region, the joint
velocity would change considerably, exceeding the allowable
maximum. To prevent the end-effector from entering a singular
region, therefore, we calculate each joint's velocity before the
robot moves from its current configuration to the next config-
uration. This velocity is called the “calculated joint velocity,”
described as follows:
(6)
where, is the calculated joint velocity for the th joint;
is the angle change of the joint in the sampling time; is the
sampling time.
In this study, we consider that the singular region is an area
that makes the exceed the maximal joint velocity. It is diffi-
cult to calculate the exact value of the size of the singular region.
If is larger than the joint velocity limit for the th joint, it
is determined that a singularity has been detected, and the sin-
gularity avoidance algorithm is applied to prevent the end-ef-
fector from entering the singular region. We can detect the po-
sition and orientation at which the end-effector is about to enter
a singular region. We define this position and orientation of the

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6 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING
end-effector as “critical singular position” and “orientation,” re-
spectively.
1) Non-Redundancy Singularity Avoidance: NRSA is pro-
posed for use with nonredundant robot manipulators. With this
method, we consider . The position and orien-
tation of the end-effector are both important when the end-ef-
fector passes through a singular region. Damped least-squares
(DLS) method is regarded as being an effective strategy for
making a nonredundant manipulator avoid kinematic singular-
ities [32]. By using DLS, both the position and orientation er-
rors of the end-effector are attempted to be reduced with the
same priority when determining the motion of the end-effector
to bypass a singular region. In this study, we implement DLS
for a teaching-playback robot manipulator system, and propose
a method for eliminating the end-effector trajectory error when
the end-effector leaves the singular region.
DLS can be described as
(7)
where is a damping factor and is a (6 6) identity matrix.
The solution to (7) can be written as
(8)
When , the solution to (7) is reduced to a regular matrix
inversion which produces unfavorable conditions close to the
singular region. Any solution obtained by using DLS is sensi-
tive to the damping factor. To obtain a solution with tradeoff ac-
curacy and robustness, the smallest singular value [31], [32] or
manipulability measure [39] is usually used to adjust the values
of . In this study, we calculate the value of according to the
smallest singular value, as shown in (9) [32]
(9)
where is the smallest singular value; is a user defined
number; is the maximal value of the damping factor set
by the user.
Because the robot manipulator is controlled by a teaching
pendant, it is difficult for a user to eliminate the position and
orientation error through the teaching pendant. We propose a
method to automatically eliminate the errors by moving the
end-effector to a specified position and orientation. When a sin-
gularity is detected, the current position and orientation of the
end-effector (i.e., critical singular position and orientation) are
saved, as the “Save Data” shown in Fig. 4. After the end-effector
escapes the singular region, the robot control is returned to its
full degree-of-freedom and the end-effector is returned to the
specified position and orientation by loading the saved data as
the “Load Data” shown in Fig. 4. A specified position and ori-
entation is defined as a modified critical singular position and
orientation that replaces the saved critical singular position in
the desired end-effector moving direction by the next position
in the desired end-effector moving direction when the end-ef-
fector escapes the singular region, as shown in Fig. 5. When the
robot joints cannot move from the current configuration to the
specified configuration (i.e., inverse kinematics solution corre-
sponding to the specified position and orientation) in one step,
Fig. 5. Error accumulated inside the singular region. is the position and ori-
entation of the end-effector.
several steps are required to make the robot joint velocity
be less than their limits. It is worthy to note that when the speci-
fied position and orientation of the end-effector is in the singular
region, the singularity avoidance algorithm will be called again
to make the end-effector pass through the singular region. With
NRSA, when a singularity is detected, the (8) is used to calcu-
late the joint velocity, as the “calculate joint velocity” shown in
Fig. 4. The elimination of the position and orientation errors is
newly implemented for NRSA in this study, relative to NRSA
in [28].
2) Redundancy Singularity Avoidance: RSA is proposed for
redundant manipulators. This method is based on the concept of
subtasks with order of priority for a robot manipulator with re-
dundancy [40]. Normally, a complicated task given to the robots
with many degrees of freedom can be decomposed into several
subtasks with order of priority. The subtask with the first priority
is first performed, and then for the second subtask as much as
possible, and so on. By using the RSA for a redundant robot ma-
nipulator, the subtask with first priority is first performed (e.g.,
position of end-effector or orientation of end-effector), and the
redundancy is used to avoid singularities. In this paper, we con-
sider a 6R robot manipulator, and assume the first subtask as to
move the end-effector along a desired trajectory, while trying
to avoid the singularity by utilizing the redundancy. Therefore,
as an example to describe the RSA, only the end-effector po-
sition is performed, which means and . Singu-
larity avoidance can be realized by using the redundancy. We
implement a potential approach which uses a potential function
to control the position of the end-effector to avoid singularities
for the redundant manipulator, because the potential approach
is normally used to avoid singularity in the case of redundant
manipulators [40], [41]. The potential approach corresponds to
the solution to the following equation:
(10)
where is a positive constant and is a potential function,
which can be formulated as [32]:
(11)
The implementation of RSA in a teaching-playback robot ma-
nipulator system is also shown in Fig. 4. With RSA, when a
singularity is detected, equation (10) is used to calculate the

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HUANG et al.: KINEMATIC CONTROL WITH SINGULARITY AVOIDANCE FOR TEACHING-PLAYBACK ROBOT MANIPULATOR SYSTEM 7
joint velocity, as the “Calculate joint velocity” shown in Fig. 4.
The elimination of the end-effector trajectory error is integrated
into the RSA as the “Save Data” and “Load Data” shown in
Fig. 4. The concept of error elimination used in the RSA is the
same as that used for the NRSA. A specified position and ori-
entation is calculated based on the saved critical singular po-
sition and orientation. After the end-effector exits the singular
region, the position and orientation errors can be eliminated by
moving the end-effector to the specified position and orientation
based on joint-interpolated motion. It is notable that the poten-
tial approach can be modified for application to other redundant
manipulator cases by, for example, only performing the end-ef-
fector orientation with the first priority. With RSA in this paper,
because the position of the end-effector is performed first and
the redundancy is used to avoid a singularity, the orientation of
the end-effector is considered as subtask with the second pri-
ority. We therefore say that the orientation of the end-effector
is sacrificed by using RSA. While with NRSA, the position and
orientation of the end-effector are performed with the same pri-
ority. From the perspective of task constraints, we can consider
the subtask with the first priority as the first task constraint and
the subtask with the second priority as the second task con-
straint. If the subtasks are with the same priority, they can be
considered as the task constraints with the same priority and be
solved simultaneously. Considering the example described for
RSA, the position of the end-effector can be considered as the
task constraint with the first priority and the orientation of the
end-effector can be considered as the task constraint with the
second priority. While with NRSA, we can consider the posi-
tion and orientation of the end-effector as the task constraints
with the same priority.
3) Point-to-Point Singularity Avoidance: PTPSA is pro-
posed to avoid singularity based on the concept of making the
end-effector pass through the singular region under joint-inter-
polated control. As described in Section III-A, we can obtain
the critical singular position and orientation. If we can find a
subposition and orientation in the desired end-effector moving
direction which is outside the singular region, then we can
make the end-effector pass through the singular region based on
point-to-point jumping motion. When the end-effector passes
through the singular region based on PTPSA, only the critical
singular position and orientation, and the subposition and orien-
tation are exactly identified. The trajectory of the end-effector
from the critical singular position and orientation to the sub-
position and orientation is undefined due to the point-to-point
jumping motion. In this case, the position and orientation of the
end-effector can be considered as unconstrained axes because
the motion of robot manipulator is under the joint-interpolated
control. Therefore, the position and orientation errors of the
end-effector are not guaranteed when the end-effector passes
through the singular region based on PTPSA. However, be-
cause the subposition and orientation is exactly defined, the
position and orientation errors occur in passing through the
singular region can be eliminated. Before entering and after
escaping the singular region, the position and orientation of the
end-effector can be considered as the task constraints. Which
axis is unconstrained and can be sacrificed? This depends on
the task requirement and the used method.
Fig. 6. Singularity avoidance with PTPSA.
Fig. 6 shows the concept of PTPSA with an example which
assumes that the end-effector is moving in the Z-direction. If
the current position and orientation is the critical singular
position and orientation, singularity would be detected when
the end-effector were to move to the next position and orien-
tation given by the teaching pendant. To make
the end-effector pass through the singular region, we add a rea-
sonable distance to the critical singular position in the desired
end-effector moving direction to specify the subposition and
orientation. Therefore, the subposition and orientation can
be represented by , where is a 6-dimensional
vector. We calculate the configurations and corresponding
to and , respectively, based on inverse kinematics. Then,
we made the manipulator joint move from to in several
steps based on joint-interpolated motion. Because and are
exactly determined, the end-effector can be moved from the crit-
ical singular position and orientation to the subposition and ori-
entation, even though the end-effector passes through the sin-
gular region. Because the changes in each joint may be different,
the number of steps taken by each joint between and may
differ. We compare the number of steps needed for each joint
and choose the maximal one for all the joints to ensure that all
the joints arrive at at the same time. The PTPSA procedure
is shown in Fig. 7. The kinematic control of the teaching-play-
back robot manipulator system with PTPSA can be considered
as a hybrid of Cartesian trajectory control and joint-interpolated
control. The motion from the critical singular position and ori-
entation to the subposition and orientation is under the joint-in-
terpolated control, while in others, the robot is under the Carte-
sian trajectory control with the teaching pendant.
Because the motion relationship between the end-effector and
joint is nonlinear, and the boundary of the singular region de-
pends on the motion of the end-effector and joint, it is very dif-
ficult to determine to ensure that the subposition and orienta-
tion of the end-effector is on the boundary of the singular region.
It is reasonable to assume that the subposition and orientation
of the end-effector is outside the singular region. Therefore, we
can calculate several under the conditions that the end-effector
moves from a different initial to a different goal , and choose
the maximal to guarantee that the subposition and orientation
of the end-effector are outside the singular region.

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8 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING
Fig. 7. PTPSA procedure. is the maximal number of steps through
which the robot joints move from to is the angle through
which the th joint moves; is the th joint angle.
IV. EXPERIMENTS
A. Experiment Conditions
We evaluated the performance of NRSA, RSA, and PTPSA
through experimental case studies. These three methods were
compared with a method which does not consider singularity
avoidance, which is called “no singularity avoidance” (NSA).
A teaching-playback robot manipulator system with a six-rev-
olute-joint (6R) manipulator was used in the experiment, as
shown in Fig. 8. The Denavit–Hartenberg parameters, joint
limit, and joint velocity are listed in Table I [42]. The wrist,
shoulder, and elbow singularities are three typical singularities
for a 6R manipulator [43]. They are often used as case studies
in solving the problem of singularity avoidance [9], [17], [31],
[43]. Because the elbow singularity will not occur for the
manipulator considered in this study, given the limitation on
this joint (the limitation of joint 3), we only investigated the
performance of the proposed methods regarding wrist and
shoulder singularity. Because the singular region is taken into
account in this study, we think that using two typical singulari-
ties is enough to investigate the singularity avoidance methods.
We set up two robotic tasks based on the actual applications,
as shown in Fig. 9. In task I, a wrist singularity occurs when
the end-effector moves from the initial to the goal in the
Fig. 8. Teaching-playback robot manipulator system.
TABLE I
DENAVIT-HARTENBERG PARAMETERS, JOINT RANGE AND VELOCITY LIMITS
Fig. 9. Robotic tasks. (a) Wrist singularity test. (b) Shoulder singularity test.
direction. We referred to this task as the “wrist singularity
test.” In task II, the end-effector moves from the initial to the
first goal and then to the second goal . Shoulder singularity
occurs when the end-effector moves from the first goal to the
second goal in the direction. We referred to this task as the
“shoulder singularity test.” Both tests were completed by using
a teaching pendant to control the motion of the end-effector.
We used a heuristic approach to determine the parameter values
for PTPSA, NRSA, and RSA in the wrist singularity test and
shoulder singularity test. The details of parameter setting was
discussed in [28]. For parameter setting in NRSA, was set to

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HUANG et al.: KINEMATIC CONTROL WITH SINGULARITY AVOIDANCE FOR TEACHING-PLAYBACK ROBOT MANIPULATOR SYSTEM 9
Fig. 10. Joint trajectory in wrist singularity test. (a) Trajectory of joint 1. (b) Trajectory of joint 2. (c) Trajectory of joint 3. (d) Trajectory of joint 4. (e) Trajectory
of joint 5. (f) Trajectory of joint 6.
0.01; the damping factor was set to 0.01. For parameter
setting in RSA, was set to 0.01; the positive constant was set
to 1. For parameter setting in PTPSA, was set to 50 mm. The
velocity of the end-effector was set 250 mm/s. The sampling
time for the robot manipulator was 0.008 s.
To investigate the performance of the robot manipulator when
the proposed methods are applied, twenty-four participants who
had no experience with controlling a robot manipulator were
asked to complete two tests by using a teaching pendant. First,
the users were given three minutes to read a manual and then
two minutes to practice the robot control. Each test consisted of
four trials which were based on NSA, PTPSA, NRSA, and RSA,
respectively. To evaluate the four methods (i.e., NSA, PTPSA,
NRSA, and RSA) fairly, we therefore chose 24 persons to par-
ticipate the experiment based on the concept of counter balance
[44]. The sequence of trials for each user differed by consid-
ering the counter balance. Under these two tests carried out by
each participant, the posture of robot manipulator that results
in singularity may be different. We statistically analyzed the
end-effector trajectory error and users' feelings derived by the
24 participants.
B. Results and Discussion
In both wrist and shoulder singularity test, the end-effector
avoided the singular region and reached the goal by using
PTPSA, NRSA, and RSA. While by using NSA, the end-ef-
fector stopped in the singular region due to the singularity,
which made joints reach their limits. From these results, we can
find that NRSA, RSA, and PTPSA provided an effective means
of avoiding singularity. Because the difference between NSA
and our proposed methods was obvious, we focused on the
differences between NRSA, RSA, and PTPSA in this section.
The joint trajectories in the wrist and shoulder singularity
tests derived by a participant are shown in Figs. 10 and 11, re-
spectively. Due to the robot structure, the wrist singularity can
make the joint angle of joint 4 and 6 change largely. Considering
the trajectory of joints 4 and 6 shown in Fig. 10, the trajectory
derived by PTPSA is smooth. There are fewer inflection in the
trajectory derived by PTPSA than that in the trajectory derived
by NRSA or RSA. The changed amplitude of joint angle derived
by PTPSA is smaller than that derived by NRSA or RSA. Due to
the robot structure, the shoulder singularity can make the joint
angle of joints 1 and 6 change largely. From Fig. 11, we can find
that PTPSA can make the joints 1 and 6 rotate smoothly when
the end-effector passes through the singular region. The joint
trajectory derived by PTPSA is smooth, this is because PTPSA
makes the end-effector pass through the singular region based
on the joint-interpolated motion.
In the wrist singularity test, singularity occurred when the
end-effector was moving in the direction; we therefore an-
alyzed the end-effector position error for the X- and Y-axes,
and the orientation. In the shoulder singularity test, singularity
occurred when the end-effector was moving in the direc-
tion; we therefore analyzed the end-effector position error for
the Y- and Z-axes, and orientation error. Fig. 12 shows the max-
imal end-effector position and orientation error derived when
using NRSA, RSA, and PTPSA in the wrist singularity test.
Fig. 13 shows the maximal end-effector position and orienta-
tion error derived through the use of NRSA, RSA, and PTPSA
in the shoulder singularity test. The significant difference in the
maximal position and orientation errors between NRSA, RSA,
and PTPSA was tested by applying an analysis of variance and
the significant difference between each of the two methods was
tested by using Tukey's honest significant difference (Tukey's
HSD) [45]. In the wrist singularity test, the maximal position

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10 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING
Fig. 11. Joint trajectory in shoulder singularity test. (a) Trajectory of joint 1. (b) Trajectory of joint 2. (c) Trajectory of joint 3. (d) Trajectory of joint 4. (e)
Trajectory of joint 5. (f) Trajectory of joint 6.
error for the X- and Y-axes, as derived with PTPSA was larger
than that derived with either NRSA or RSA. The maximal po-
sition error for both the X- and Y-axes, as derived by RSA was
larger than that derived with NRSA. Considering the orientation
error, the maximal error derived with PTPSA was smaller than
that derived with either NRSA or RSA. The maximal orientation
error derived with NRSA was larger than that derived with RSA.
In the shoulder singularity test, the position error occurs mainly
along the Y-axis. The maximal position error in the Y-axis as
derived by PTPSA was smaller than that derived by both NRSA
and RSA. Considering the position error for the Z-axis, the max-
imal error as derived with PTPSA was larger than that derived
with either NRSA or RSA. Considering the orientation error, the
maximal error derived by RSA was large, but there were only
small orientation errors when using either NRSA or PTPSA.
From these results, we can conclude that PTPSA produced
smooth joint motion and a small orientation error when the end-
effector was controlled so as to bypass the singular region. This
is because, with PTPSA, the end-effector is controlled so as to
move from a critical singular position and orientation to a sub-
position and orientation with joint-interpolated motion control.
The end-effector orientation in the critical singular position and
orientation is the same as that in the subposition and orientation.
The use of RSA resulted in an orientation error in both tests; this
is because the emphasis is on reducing the position error of the
end-effector. With NRSA, both a position error and orientation
error occurred in the wrist singularity test, but only a small ori-
entation error occurred in the shoulder singularity test. This is
because both the end-effector position and orientation errors are
attempted to be reduced with NRSA. The position error and ori-
entation error are affected by both the singularity type and the
singularity avoidance methods.
C. Statistical Analysis of Users' Feelings
To analyze users' feelings regarding the performance of the
robot manipulator with PTPSA, NRSA, and RSA, a ques-
tionnaire was used to evaluate the users' impressions of the
performance of the robot manipulator using the three different
methods. The questionnaire posed the following five questions:
• Question 1: Did the robot manipulator move with your in-
tentions? (Control intention)
• Question 2: Did the robot manipulator end-effector move
smoothly? (Smoothness)
• Question 3: Was the test completed efficiently? (Effi-
ciency)
• Question 4: Did you have any sense of danger when you
were controlling the robot manipulator? (Danger)
• Question 5: Did you feel nervous while you were control-
ling the robot manipulator? (Tension)
The questionnaire set out to determine users' feelings. To
evaluate them quantitatively, we defined a point score for each
feeling, as shown in Table II. The results of the experiments
were quantified by using the average and standard deviation for
the points.
Fig. 14 shows the results obtained for users' feelings re-
garding control intention, smoothness, efficiency, sense of
danger, and tension. The significant difference of results de-
rived between NRSA, RSA, and PTPSA was tested by applying
an analysis of variance and the significant difference between
every method pair was tested by using the Tukey's HSD [45].
In robot control intention, the users' feelings about the perfor-
mance of NRSA, RSA, and PTPSA are similar. This is because
the difference between the maximal end-effector position errors
as derived by these three methods is less than 2 mm, as shown in

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HUANG et al.: KINEMATIC CONTROL WITH SINGULARITY AVOIDANCE FOR TEACHING-PLAYBACK ROBOT MANIPULATOR SYSTEM 11
Fig. 12. Maximal end-effector position and orientation in wrist singularity test.
Significant difference was indicated if . (a) Maximal end-ef-
fector position error along X-axis. (b) Maximal end-effector position error along
Y-axis. (c) Maximal end-effector orientation error.
Figs. 12 and 13. Given that the difference is so small, the users
felt that the robot control intention under these three methods
was similar. Considering the smoothness of the end-effector,
the users felt that the end-effector moved more smoothly with
PTPSA than that with NRSA in the shoulder singularity test.
This is because the maximal Y-axis position error for NRSA
is larger than that for PTPSA, as shown in Fig. 13(a). Consid-
ering the efficiency with which a test was completed, PTPSA
performed better than either NRSA or RSA. The efficiency
derived with PTPSA was, on average, 24.93% and 20.70%
better than that derived with NRSA and RSA, respectively.
With PTPSA, the end-effector passed through a singular region
under joint-interpolated control, which made the users feel that
the test was completed more quickly. Considering the users'
feelings about danger and tension, the results derived by using
NRSA, RSA, and PTPSA were similar. This is because the
Fig. 13. Maximal end-effector position and orientation in shoulder singularity
test. Significant difference was indicated if . (a) Maximal end-
effector position error along Y-axis. (b) Maximal end-effector position error
along Z-axis. (c) Maximal end-effector orientation error.
TABLE II
POINTS OF USER'S FEELING
end-effector can avoid the singular region with any of NRSA,
RSA, and PTPSA, which gave the users similar impressions
about danger and tension. The motion of the robot with the
application of NSA gave the users a greater sense of danger
and tension than that with any of NRSA, RSA, and PTPSA.
This is because, with NSA, the positions of the joints change
considerably when the end-effector enters the singular region.
From these results, we can conclude that the users felt that
the performances of NRSA, RSA, and PTPSA were similar in a

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12 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING
Fig. 14. Results of users' feelings about control intention, smoothness, effi-
ciency, sense of danger, and tension. Significant difference was indicated if
. (a) Results of users' feelings about robot control intention. (b) Re-
sults of users' feelings about moving smoothness of end-effector. (c) Results of
users' feelings about efficiency of completing a test. (d) Results of users' feel-
ings about sense of danger. (e) Results of users' feelings about tension.
simple wrist singularity test. In a more complex test involving
shoulder singularity, however, PTPSA performed better than
NRSA and RSA in terms of smoothness and efficiency. The per-
formance of NRSA and RSA was similar since both caused the
end-effector to avoid a singular region based on a modification
of the Jacobian matrix. When the end-effector was requested to
quickly pass through a singular region, PTPSA proved to be a
better choice than either NRSA or RSA.
Based on these results in the end-effector trajectory tracking
and feelings of users, we can conclude that the users can choose
the PTPSA for their task in terms of the end-effector trajectory
error, moving smoothness of the end-effector, and efficiency in
completing a task. When a task can be decomposed into several
subtasks with order of priority, then the RSA is a better choice.
V. CONCLUSION AND FUTURE WORK
In this study, we proposed a novel singularity avoidance
method named PTPSA, whereby the end-effector passes
through a singular region with point-to-point jumping motion,
and compared the use of PTPSA, NRSA, and RSA for a
teaching-playback robot manipulator system. These methods
have been successfully tested with wrist and shoulder singu-
larity through experimental case studies. The results of these
experiments showed that PTPSA, NRSA, and RSA can suc-
cessfully avoid a singular region. PTPSA and NRSA resulted
in large position but small orientation errors in the end-effector
trajectory. RSA mainly incurred orientation errors. A statistical
analysis of the users' feelings showed that PTPSA performed
better than NRSA, and RSA in terms of smoothness of the
movement of the end-effector and the efficiency with which
the test could be completed. The efficiency impression derived
with PTPSA was improved on average by 24.93% and 20.70%
relative to that derived with NRSA and RSA, respectively. The
performance of PTPSA, NRSA, and RSA were all similar in
terms robot control intention, sense of danger and tension.
In future work, the manipulator dynamics and obstacle avoid-
ance will be investigated when the end-effector approaches a
singular region. The obstacle avoidance can be considered as
a task constraint. The relationship between singularity and jerk
for each trajectory will be studied.
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tical Application. Cambridge, U.K.: Cambridge Univ. Press, 1988.
[45] S. Hardeo, The Analysis of Variance: Fixed, Random and Mixed
Models. New York, NY, USA: Springer, 2012.
Yanjiang Huang received the B.S. degree in me-
chanical engineering from Xi'an Jiaotong University,
Xi'an, China, in 2005, the M.S. degree in mechanical
engineering from Xiamen University, Xiamen,
China, in 2008, and the Ph.D. degree in precision
mechanical engineering from the University of
Tokyo, Tokyo, Japan, in 2013.
From April 2013, he became a Project Researcher
at Research Into Artifacts, Center for Engineering,
University of Tokyo. From September 2014, he be-
came a Lecturer with the South China University of
Technology, China. His main research interests are robot manipulator system
optimization, robot manipulator kinematics, and multirobot coordination.
Yoon Seong Yong received the B.S. degree in
mechanical engineering from Technical University
of Malaysia Malacca in 2009 and the M.S. degree in
precision mechanical engineering from the Univer-
sity of Tokyo, Tokyo, Japan, in 2013. He is currently
an engineer in KAWATA INDUSTRIES, Inc.
His main research interest is robot manipulator
kinematics.
Ryosuke Chiba received the B.S., M.S., and Ph.D.
degrees in precision mechanical engineering from the
University of Tokyo, Tokyo, Japan, in 1999, 2001,
and 2004, respectively.
In 2002, he visited the University of Edinburgh
and researched a coevolutional design process. From
2004 to 2008, he was a Project Assistant Researcher
with the Department of Precision Engineering,
University of Tokyo. From 2008 to 2014, he was
an Assistant Professor at Tokyo Metropolitan Uni-
versity. Currently, he is an Associate Professor with
Asahikawa Medical University. His research interests are multiple mobile
robot systems, transportation systems, environmental design for robot systems
and coevolutional design processes.

14. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
14 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING
Tamio Arai received the B.S., M.S., and Ph.D. de-
grees in precision mechanical engineering from the
University of Tokyo, Tokyo, Japan, in 1970, 1972,
and 1977, respectively.
From 1979 to 1981, he was with the Department
of Artificial Intelligence, University of Edinburgh, as
a Visiting Researcher. From 2000 to 2005, he was
Director of Research Into Artifacts, Center for Engi-
neering, University of Tokyo, in which he proposed
service engineering. From 1987 to 2012, he was a
Professor with the Department of Precision Mechan-
ical Engineering, University of Tokyo. He is currently a Professor with the
Shibaura Institute of Technology, Tokyo. His current research subjects are ser-
vice engineering, skills in production and robotics for human support.
Prof. Arai is a Fellow of CIRP, Robotics Society of Japan and the Japan So-
ciety for Precision Engineering (JSPE). He was a former President of JSPE and
the Asian Society for Precision Engineering and Nanotechnology, and Honorary
President of the Japan Association for Automation Advancement.
Tsuyoshi Ueyama received the B.S., M.S., and
Ph.D. degrees in engineering from the Nagoya
University, Nagoya, Japan, in 1989, 1991, and 1994,
respectively.
He is currently an Engineer in Denso Wave Inc.,
Aichi, Japan. His main research interests are robot
manipulator system control.
Jun Ota received the B.S., M.S., and Ph.D. degrees
in precision mechanical engineering from the Univer-
sity of Tokyo, Tokyo, Japan, in 1987, 1989, and 1991,
respectively.
From 1989 to 1991, he was with Nippon Steel
Cooperation. In 1991, he was a Research Associate
with the University of Tokyo. In 1996, he was an
Associate Professor. From April 2009, he was a
Professor at the Graduate School of Engineering,
the University of Tokyo. From June 2009, he was
a Professor at Research Into Artifacts, Center for
Engineering (RACE), University of Tokyo. From 1996 to 1997, he was a
Visiting Scholar at Stanford University. His research interests are multi-agent
robot systems, design support for large-scale production/material handling
systems, mobiligence, human behavior analysis and support.
Prof. Ota is the recipient of awards and honors including the Inoue Research
Award for Young Scientists, Intelligent Manufacturing Systems Japan Prize for
Excellent Papers, FANAC FA and Robot Foundation Excellent Papers, Best
Paper Award of the IEEE International Workshop on Robot and Human Interac-
tive Communication, and Advanced Engineering Informatics Top Cited Article.