kinematic control with singularity avoidance for teaching-playback robot manipulator system

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IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING 1

Kinematic Control With Singularity Avoidance forTeaching-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 systemwhereby the user controls the manipulator through a teachingpendant has been used widely in industrial applications. Kine-matic singularity issue becomes an important problem in thecontrol of robot with a teaching-playback system. In this paper, wepropose and investigate three singularity avoidance methods fora teaching-playback robot manipulator system. Nonredundancysingularity avoidance (NRSA) attempts to reduce both the positionand orientation errors of the end-effector with the same priority.Redundancy singularity avoidance (RSA) attempts to reducethe position error of the end-effector with the first priority andreduce the orientation error of the end-effector with the secondpriority; Both NRSA and RSA are based on a modification of aJacobian matrix. Point-to-point singularity avoidance (PTPSA)makes the end-effector pass through a singular region basedon joint-interpolated control without maintaining the positionand orientation of the end-effector. Experimental case studiesare developed to investigate the manipulator performance whenthe end-effector approaches the wrist and shoulder singularity.The maximal end-effector trajectory error and users' feelingsare statistically evaluated and analyzed in the experiment. Theresults of the experiment show the effectiveness and practice ofthe 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 ScienceFoundation of China under Grant 91223201, and in part by the Natural ScienceFoundation of Guangdong Province under Grant S2013030013355. This paperwas 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 andManufacturing Technology, School ofMechanical and Automo-tive Engineering, South China University of Technology, Guangdong 510640,China (e-mail: [email protected]).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, [email protected]).R. Chiba is with the Research Center for Brain Function and Medical En-

gineering, Asahikawa Medical University, Hokkaido 078-8510, Japan (e-mail:[email protected])T. Arai is with the Department of Mechanical Engineering, College of En-

gineering, Shibaura Institute of Technology, Tokyo 135-8548, Japan (e-mail:[email protected]).T. Ueyama is with DensoWave Incorporated, Aichi 470-2297, Japan (e-mail:

[email protected]).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 ofrobot manipulator under different singularity avoidance methods in the experi-ment and an excel file (.xlsx) for illustrating the biographical information of 24participants 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 problemof proposing appropriate methods to avoid singularities for ateaching-playback robot manipulator system. Three singularityavoidance methods are proposed and investigated from theperspective of the movement of the end-effector when it passesthrough a singular region. With NRSA method, both the posi-tion and orientation errors of the end-effector are reduced withthe same priority. With RSA method, the position error of theend-effector is reduced with the first priority and the orientationerror of the end-effector is reduced with the second priority.With PTPSA method, the end-effector passes through a singularregion under the joint-interpolated control without maintainingthe position and orientation of the end-effector. NRSA, RSA, andPTPSA have been successfully tested with wrist and shouldersingularity through the experimental case studies. Experimentresults show that the proposed methods can automatically avoida singularity for a teaching-playback robot manipulator, whichwould enhance the capability of robots for industrial automation.In future research, the proposed methods can be extended tosimultaneously realize the singularity and obstacle avoidance.

Index Terms—end-effector trajectory, Jacobian matrix, point-to-point motion, singularity avoidance, teaching-playback robotmanipulator system.

I. INTRODUCTION

T EACHING-PLAYBACK robot manipulator systemswhich 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 singularitiesof a robot manipulator are a set of singular configurations inwhich the end-effector locality loses the ability to move inan arbitrary direction [2]. When the manipulator is at or is inthe neighborhood of a kinematic singularity, the usual inversedifferential kinematics solutions based on the Jacobian matrixgive rise to some undesirable conditions, and this results invery high joint velocities and large control deviations [3], [4].In a teaching-playback robot manipulator system, the usernormally uses the teaching pendant to control the movementof the end-effector in the X, Y, and Z directions to make theend-effector move from an initial position and orientationto a goal position and orientation. During the movement ofthe end-effector, a singularity may occur, as shown in Fig. 1,which may surprise the user and result in a failure to reachthe goal position and orientation. Therefore, it is necessary toautomatically avoid such kinematic singularities when usinga teaching-playback robot manipulator system to complete atask. 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 considersingularity avoidance. is the position and orientation of the end-effector.

with a kinematic singularity by causing the end-effector tobypass it. A teaching-playback robot manipulator system witha 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 initialand final configurations of the robot manipulator. The Jaco-bian-based methods, such as pseudoinverse method, Jacobiantranspose, and damped least-squares inverse of the Jacobianmatrix, have been proposed to avoid the ill-conditioned of theJacobian matrix [5]–[7]. However, the pseudoinverse methodis computationally time consuming; the Jacobian transposeand damped least-squares inverse are sensitive to parameterselection and may cause large tracking errors. In recent, someresearchers have proposed other Jacobian-based methods toavoid kinematic singularities [8]–[10]. In [8], a new filter wasproposed for the singular values of the Jacobian matrix and acontinuous task priority strategy with selective damping wasproposed to generate smooth trajectory. Nevertheless, thismethod is computationally time consuming and may result inalgorithmic singularities. In [9], an inverse kinematics methodthat transforms the inverse kinematics matrix equations intoeight pure algebraic equations was proposed to solve the wristsingularity problem for a PUMA560-structured robot manip-ulator. An elementary Jacobian transformation was used toidentify 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 controllershave been proposed. In [11], an augmented hybrid impedancecontrol scheme consisted of a Cartesian-level potential differ-ence controller and a joint-space inverse dynamic controllerwas proposed. An energy-based controller incorporated withfuzzy neural network compensation was proposed to avoidsingularities in [12]. A voltage-based control scheme and aforce-based control scheme were proposed in [13] and [14],respectively. However, additional device was required to obtainthe force, energy, and voltage in these controllers. In othercontrollers, the singularity was avoided based on task spacecontrol. In [15], a method with Lyapunov stability theory wasproposed. In [16], a continuous task transition algorithm wasapplied to a task space control framework. Tasks were definedfor dealing with singularities and joint limits, and they wereactivated or deactivated by using the continuous task transitionalgorithm. In [17], the source of the discontinuity was discussedand null motion was proposed to control the end-effector overthe singularity. Motion planning was integrated into control

scheme in some researches [18], [19]. A fuzzy motion planningmapping method and a path planning technique were proposedto avoid singularities in [18] and [19], respectively. Manycontrol algorithms have been proposed to control robot manip-ulator to avoid obstacles [20]–[23]. Many of these algorithmscan be extended to avoid singularities. However, it is difficultto tune the parameters for these control algorithms and schemesto 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 algorithmwas proposed. In [25], the singularity avoidance problem wasformulated as minimization of the measure of manipulabilityand a nonlinear programming method was proposed to solvethe optimization problem. A hierarchical constraint-basedsingularity avoidance was proposed in [26], which can beconsidered as a constraint-optimization problem. By solvingthe optimization problem, much computation time is requiredto obtain an optimal or near-optimal solution. Although anappropriate small search region [24] and an analytical method[27] can reduce computation time, it is difficult to determine theappropriate small search region and the analytical method nor-mally depend on robot structure. A large number of researchershave focused on the singularity avoidance of robot manipulator,however, singularity avoidance for a teaching-playback robotmanipulator system was not considered because the robot doesnot know where to move until a move command is sent fromthe teaching pendant.In factories, singularity avoidance for a teaching-playback

robot manipulator is realized based on the repetitive adjustmentof joint angles. However, this is not intuitive and relies on therepetitive adjustment of joint angles, which may surprise theuser 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 singularityavoidance methods for a teaching-playback robot manipulatorsystem and compare the performance of the robot manipulatorwith these three methods through experimental case studies.Two of these three singularity avoidance methods, which arebased on a modified Jacobian matrix, were proposed in our re-cent study [28]. The other, based on the point-to-point jumpingmotion of the end-effector, is originally proposed in this paper.This study differs from our previous studies in following threeaspects. (1) We propose a novel singularity avoidance methodfor 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 singularityavoidance methods. The contribution of this paper is to comparethe user's experience of the different methods for singularityavoidance for a teaching-playback robot manipulator system,and implementation of the point-to-point singularity avoidancemethod.The remainder of this paper is organized as follows. In

Section II, we formulate and analyze the singularity avoidanceproblem. The proposed methods are described in Section III.


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 avoidanceproblem for a teaching-playback robot manipulator system. Adescription 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, pitchand yaw angles. When , the robot manipulator is non-re-dundant. In the case ofand . Where, isthe position vector of the end-effector;is the orientation vector of the end-effector, which are definedas yaw, pitch, and roll relative to the robot base coordinateframe; are the joint angles. When

, the robot manipulator is redundant. In this paper, atask was defined as moving the end-effector from an initialto a goal by using a teaching pendant. The input pa-

rameter to the inverse kinematics of the manipulator is thechange in the position and orientation of the end-effector(i.e., ) set by the userthrough a teaching pendant, which is determined by the end-ef-fector velocity and sampling time. The output parameter ofthe inverse kinematics is the change in the configuration (i.e.,

caused by .After several steps, the end-effector can be moved from theinitial 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, wechoose the Moore–Penrose pseudoinverse to calculate joint ve-locities. Although there are many other criteria in solving thesingularity avoidance problem, such as dynamically consistentpseudoinverse [30], which may complicate the problem sinceit 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 thepseudo-inverse of the Jacobian .When rank is not the full rank and the determinant

of is zero, singularity occurs in this configuration. Thisconfiguration 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 areain which the end-effector locality loses the ability to move in anarbitrary 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 singularregion is calculated based on the estimation of the smallest sin-gular value of Jacobian matrix [31], [32]. The robot manipulatoris checked inside or outside the singular region by comparingthe estimation of smallest singular value to a threshold value.If the estimation of smallest singular value is smaller than thethreshold value, then the robot is singular. Otherwise, the robotis not singular. The estimation of smallest singular value can berealized 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 thresholdvalue for these criteria and much computation time is requiredto evaluate the criteria. In this paper, we use a different methodto detect singularity by checking whether the calculated jointvelocity exceed the maximal joint velocity. Low computationaltime is required to calculate the joint velocity and the maximaljoint 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 inthe end-effector trajectory occurs when the end-effector enters asingular region, we investigated the end-effector trajectory errorincluding 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 , andare the orthonormal unit vectors which specify the orientationof the end-effector. are the orthonormal vectorswhich specify the desired orientation of the end-effector. Notethat when calculating the maximal position error and orien-tation error, the absolute value of (4) and (5) should be used,respectively.



Fig. 2. Overview of singularity avoidance for a teaching playback robot ma-nipulator.

Because a robotic task is completed by a user through ateaching pendant, we investigate the users' feelings abouttheir robot control intentions, smoothness in the movementof the end-effector, efficiency with which a task can be com-pleted, sense of danger, and tension, from the viewpoint ofhuman-robot interaction [36], [37]. The analysis of joint trajec-tory and end-effector trajectory is an objectiveness analysis inthe evaluation of the singularity avoidance methods. The us-ability of a teaching playback robot manipulator system can beevaluated 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 manipulatorsystem. Fig. 2 shows an overview of singularity avoidancefor a teaching-playback robot manipulator. Because a singularregion is not established in advance by using a teaching pen-dant to control the robot, we check whether the end-effectoris about to a singular region in real time after the user entersthe command , but before moving the end-effector. If theend-effector is found to be entering a singular region afterthe user enters the command , the singularity avoidancealgorithm is applied. Otherwise, the robot will move based onan inverse kinematics solution. The inverse kinematics solutionis that configuration which requires the least time to movefrom the previous configuration [38]. When the end-effectorbypasses a singular region, its movement will differ from theuser. This reduces the manipulability. We therefore proposethree singularity avoidance methods from the perspectiveof 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 singularregion through the application of these three methods is shown

Fig. 3. Movement of end-effector when avoiding singular region with threesingularity avoidance methods. (a) Movement of end-effector when avoidingsingular region in NRSA. (b) Movement of end-effector when avoiding singularregion in RSA. (c) Movement of end-effector when avoiding singular region inPTPSA.

in Fig. 3. With NRSA, both the position and orientation ofthe end-effector are important and the position and orienta-tion errors need to be reduced with the same priority. Theend-effector passes through the singular region by following atrajectory crosses over that region. With RSA, the position ofthe end-effector is performed as a subtask with the first priorityand a singularity is avoided by using redundancy. The positionerror is reduced with first priority and the orientation error isreduced with second priority. With PTPSA, neither the positionnor the orientation of the end-effector are maintained. Rather,the joints are controlled to make the end-effector pass throughthe singular region.When the end-effector is in a singular region, a discontinuity

occurs because the singular components of the robot task spaceare removed and the control algorithms are calculated with adifferent task space specification [17]. Due to the ill-conditionedinverse kinematics and dynamics of a robot manipulator system,motion in the singular direction within the singular region is not



Fig. 4. Singularity avoidance. is the number of steps through which the robot joints move from the current configuration to the saved configuration when theycannot move from the current configuration to the saved configuration in one step. is the current end-effector position and orientation; is the next end-effectorposition and orientation; is the current configuration; is the next configuration; is the velocity limit for the th joint; is the configuration in whichthe robot joints move from the current configuration to the saved configuration. is the joint change limit of the th joint in sampling time; , is theangle 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 forwardkinematic function. Ceil() is the ceiling function which maps a real number to the smallest following integer. The difference between non-redundancy singularityavoidance 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 PTPSAare described in Sections III-A, III-B1, III-B2, and III-B3, re-spectively. The overall implementation of NRSA and RSA isshown in Fig. 4. The difference between NRSA and RSA ison “calculate joint velocity,” which is described in detail inSections III-B1 and III-B2, respectively.

A. Singularity Detection

If the end-effector were to enter a singular region, the jointvelocity would change considerably, exceeding the allowablemaximum. To prevent the end-effector from entering a singularregion, 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 thesampling time.In this study, we consider that the singular region is an area

that makes the exceed themaximal 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, itis 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 entera singular region. We define this position and orientation of the



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 thismethod, 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 formaking 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 thesame priority when determining the motion of the end-effectorto bypass a singular region. In this study, we implement DLSfor a teaching-playback robot manipulator system, and proposea method for eliminating the end-effector trajectory error whenthe 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 matrixinversion which produces unfavorable conditions close to thesingular 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] ormanipulability measure [39] is usually used to adjust the valuesof . In this study, we calculate the value of according to thesmallest singular value, as shown in (9) [32]

(9)

where is the smallest singular value; is a user definednumber; is the maximal value of the damping factor setby the user.Because the robot manipulator is controlled by a teaching

pendant, it is difficult for a user to eliminate the position andorientation error through the teaching pendant. We propose amethod to automatically eliminate the errors by moving theend-effector to a specified position and orientation. When a sin-gularity is detected, the current position and orientation of theend-effector (i.e., critical singular position and orientation) aresaved, as the “Save Data” shown in Fig. 4. After the end-effectorescapes the singular region, the robot control is returned to itsfull degree-of-freedom and the end-effector is returned to thespecified position and orientation by loading the saved data asthe “Load Data” shown in Fig. 4. A specified position and ori-entation is defined as a modified critical singular position andorientation that replaces the saved critical singular position inthe desired end-effector moving direction by the next positionin the desired end-effector moving direction when the end-ef-fector escapes the singular region, as shown in Fig. 5. When therobot joints cannot move from the current configuration to thespecified 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 velocitybe less than their limits. It is worthy to note that when the speci-fied position and orientation of the end-effector is in the singularregion, the singularity avoidance algorithm will be called againto make the end-effector pass through the singular region. WithNRSA, when a singularity is detected, the (8) is used to calcu-late the joint velocity, as the “calculate joint velocity” shown inFig. 4. The elimination of the position and orientation errors isnewly implemented for NRSA in this study, relative to NRSAin [28].2) Redundancy Singularity Avoidance: RSA is proposed for

redundant manipulators. This method is based on the concept ofsubtasks with order of priority for a robot manipulator with re-dundancy [40]. Normally, a complicated task given to the robotswith many degrees of freedom can be decomposed into severalsubtasks with order of priority. The subtask with the first priorityis first performed, and then for the second subtask as much aspossible, 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 theredundancy is used to avoid singularities. In this paper, we con-sider a 6R robot manipulator, and assume the first subtask as tomove the end-effector along a desired trajectory, while tryingto 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. Weimplement a potential approach which uses a potential functionto control the position of the end-effector to avoid singularitiesfor the redundant manipulator, because the potential approachis normally used to avoid singularity in the case of redundantmanipulators [40], [41]. The potential approach corresponds tothe 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 asingularity is detected, equation (10) is used to calculate the



joint velocity, as the “Calculate joint velocity” shown in Fig. 4.The elimination of the end-effector trajectory error is integratedinto the RSA as the “Save Data” and “Load Data” shown inFig. 4. The concept of error elimination used in the RSA is thesame 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 singularregion, the position and orientation errors can be eliminated bymoving the end-effector to the specified position and orientationbased on joint-interpolated motion. It is notable that the poten-tial approach can be modified for application to other redundantmanipulator 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 andthe redundancy is used to avoid a singularity, the orientation ofthe end-effector is considered as subtask with the second pri-ority. We therefore say that the orientation of the end-effectoris sacrificed by using RSA. While with NRSA, the position andorientation of the end-effector are performed with the same pri-ority. From the perspective of task constraints, we can considerthe subtask with the first priority as the first task constraint andthe subtask with the second priority as the second task con-straint. If the subtasks are with the same priority, they can beconsidered as the task constraints with the same priority and besolved simultaneously. Considering the example described forRSA, the position of the end-effector can be considered as thetask constraint with the first priority and the orientation of theend-effector can be considered as the task constraint with thesecond priority. While with NRSA, we can consider the posi-tion and orientation of the end-effector as the task constraintswith the same priority.3) Point-to-Point Singularity Avoidance: PTPSA is pro-

posed to avoid singularity based on the concept of making theend-effector pass through the singular region under joint-inter-polated control. As described in Section III-A, we can obtainthe critical singular position and orientation. If we can find asubposition and orientation in the desired end-effector movingdirection which is outside the singular region, then we canmake the end-effector pass through the singular region based onpoint-to-point jumping motion. When the end-effector passesthrough the singular region based on PTPSA, only the criticalsingular position and orientation, and the subposition and orien-tation are exactly identified. The trajectory of the end-effectorfrom the critical singular position and orientation to the sub-position and orientation is undefined due to the point-to-pointjumping motion. In this case, the position and orientation of theend-effector can be considered as unconstrained axes becausethe motion of robot manipulator is under the joint-interpolatedcontrol. Therefore, the position and orientation errors of theend-effector are not guaranteed when the end-effector passesthrough the singular region based on PTPSA. However, be-cause the subposition and orientation is exactly defined, theposition and orientation errors occur in passing through thesingular region can be eliminated. Before entering and afterescaping the singular region, the position and orientation of theend-effector can be considered as the task constraints. Whichaxis is unconstrained and can be sacrificed? This depends onthe task requirement and the used method.

Fig. 6. Singularity avoidance with PTPSA.

Fig. 6 shows the concept of PTPSA with an example whichassumes that the end-effector is moving in the Z-direction. Ifthe current position and orientation is the critical singularposition and orientation, singularity would be detected whenthe end-effector were to move to the next position and orien-tation given by the teaching pendant. To makethe end-effector pass through the singular region, we add a rea-sonable distance to the critical singular position in the desiredend-effector moving direction to specify the subposition andorientation. Therefore, the subposition and orientation canbe represented by , where is a 6-dimensionalvector. We calculate the configurations and correspondingto and , respectively, based on inverse kinematics. Then,we made the manipulator joint move from to in severalsteps based on joint-interpolated motion. Because and areexactly determined, the end-effector can bemoved 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 maydiffer. We compare the number of steps needed for each jointand choose the maximal one for all the joints to ensure that allthe joints arrive at at the same time. The PTPSA procedureis shown in Fig. 7. The kinematic control of the teaching-play-back robot manipulator system with PTPSA can be consideredas a hybrid of Cartesian trajectory control and joint-interpolatedcontrol. 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 themotion 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 orientationof the end-effector is outside the singular region. Therefore, wecan calculate several under the conditions that the end-effectormoves from a different initial to a different goal , and choosethe maximal to guarantee that the subposition and orientationof the end-effector are outside the singular region.



Fig. 7. PTPSA procedure. is the maximal number of steps throughwhich the robot joints move from to is the angle throughwhich the th joint moves; is the th joint angle.

IV. EXPERIMENTS

A. Experiment Conditions

We evaluated the performance of NRSA, RSA, and PTPSAthrough experimental case studies. These three methods werecompared with a method which does not consider singularityavoidance, 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, asshown in Fig. 8. The Denavit–Hartenberg parameters, jointlimit, and joint velocity are listed in Table I [42]. The wrist,shoulder, and elbow singularities are three typical singularitiesfor a 6R manipulator [43]. They are often used as case studiesin solving the problem of singularity avoidance [9], [17], [31],[43]. Because the elbow singularity will not occur for themanipulator considered in this study, given the limitation onthis joint (the limitation of joint 3), we only investigated theperformance of the proposed methods regarding wrist andshoulder singularity. Because the singular region is taken intoaccount 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 whenthe end-effector moves from the initial to the goal in the

Fig. 8. Teaching-playback robot manipulator system.

TABLE IDENAVIT-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 singularitytest.” In task II, the end-effector moves from the initial to thefirst goal and then to the second goal . Shoulder singularityoccurs when the end-effector moves from the first goal to thesecond goal in the direction.We referred to this task as the“shoulder singularity test.” Both tests were completed by usinga teaching pendant to control the motion of the end-effector.We used a heuristic approach to determine the parameter valuesfor PTPSA, NRSA, and RSA in the wrist singularity test andshoulder singularity test. The details of parameter setting wasdiscussed in [28]. For parameter setting in NRSA, was set to



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) Trajectoryof joint 5. (f) Trajectory of joint 6.

0.01; the damping factor was set to 0.01. For parametersetting in RSA, was set to 0.01; the positive constant was setto 1. For parameter setting in PTPSA, was set to 50 mm. Thevelocity of the end-effector was set 250 mm/s. The samplingtime 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 whohad no experience with controlling a robot manipulator wereasked to complete two tests by using a teaching pendant. First,the users were given three minutes to read a manual and thentwo minutes to practice the robot control. Each test consisted offour 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 byeach participant, the posture of robot manipulator that resultsin singularity may be different. We statistically analyzed theend-effector trajectory error and users' feelings derived by the24 participants.

B. Results and Discussion

In both wrist and shoulder singularity test, the end-effectoravoided the singular region and reached the goal by usingPTPSA, 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 canfind that NRSA, RSA, and PTPSA provided an effective meansof avoiding singularity. Because the difference between NSAand our proposed methods was obvious, we focused on thedifferences between NRSA, RSA, and PTPSA in this section.

The joint trajectories in the wrist and shoulder singularitytests derived by a participant are shown in Figs. 10 and 11, re-spectively. Due to the robot structure, the wrist singularity canmake the joint angle of joint 4 and 6 change largely. Consideringthe trajectory of joints 4 and 6 shown in Fig. 10, the trajectoryderived by PTPSA is smooth. There are fewer inflection in thetrajectory derived by PTPSA than that in the trajectory derivedby NRSA or RSA. The changed amplitude of joint angle derivedby PTPSA is smaller than that derived by NRSA or RSA. Due tothe robot structure, the shoulder singularity can make the jointangle of joints 1 and 6 change largely. From Fig. 11, we can findthat PTPSA can make the joints 1 and 6 rotate smoothly whenthe end-effector passes through the singular region. The jointtrajectory derived by PTPSA is smooth, this is because PTPSAmakes the end-effector pass through the singular region basedon 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, singularityoccurred when the end-effector was moving in the direc-tion; we therefore analyzed the end-effector position error forthe Y- and Z-axes, and orientation error. Fig. 12 shows the max-imal end-effector position and orientation error derived whenusing 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 PTPSAin the shoulder singularity test. The significant difference in themaximal position and orientation errors between NRSA, RSA,and PTPSA was tested by applying an analysis of variance andthe significant difference between each of the two methods wastested by using Tukey's honest significant difference (Tukey'sHSD) [45]. In the wrist singularity test, the maximal position



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 largerthan that derived with either NRSA or RSA. The maximal po-sition error for both the X- and Y-axes, as derived by RSA waslarger than that derived with NRSA. Considering the orientationerror, the maximal error derived with PTPSA was smaller thanthat derived with either NRSA or RSA. Themaximal orientationerror derived with NRSAwas larger than that derived with RSA.In the shoulder singularity test, the position error occurs mainlyalong the Y-axis. The maximal position error in the Y-axis asderived by PTPSA was smaller than that derived by both NRSAand RSA. Considering the position error for the Z-axis, the max-imal error as derived with PTPSA was larger than that derivedwith either NRSA or RSA. Considering the orientation error, themaximal error derived by RSA was large, but there were onlysmall 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. Thisis because, with PTPSA, the end-effector is controlled so as tomove 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 andorientation is the same as that in the subposition and orientation.The use of RSA resulted in an orientation error in both tests; thisis because the emphasis is on reducing the position error of theend-effector. With NRSA, both a position error and orientationerror occurred in the wrist singularity test, but only a small ori-entation error occurred in the shoulder singularity test. This isbecause both the end-effector position and orientation errors areattempted to be reduced with NRSA. The position error and ori-entation error are affected by both the singularity type and thesingularity avoidance methods.

C. Statistical Analysis of Users' Feelings

To analyze users' feelings regarding the performance of therobot manipulator with PTPSA, NRSA, and RSA, a ques-tionnaire was used to evaluate the users' impressions of theperformance of the robot manipulator using the three differentmethods. 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 movesmoothly? (Smoothness)

• Question 3: Was the test completed efficiently? (Effi-ciency)

• Question 4: Did you have any sense of danger when youwere 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. Toevaluate them quantitatively, we defined a point score for eachfeeling, as shown in Table II. The results of the experimentswere quantified by using the average and standard deviation forthe points.Fig. 14 shows the results obtained for users' feelings re-

garding control intention, smoothness, efficiency, sense ofdanger, and tension. The significant difference of results de-rived between NRSA, RSA, and PTPSA was tested by applyingan analysis of variance and the significant difference betweenevery 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 becausethe difference between the maximal end-effector position errorsas derived by these three methods is less than 2 mm, as shown in



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 alongY-axis. (c) Maximal end-effector orientation error.

Figs. 12 and 13. Given that the difference is so small, the usersfelt that the robot control intention under these three methodswas similar. Considering the smoothness of the end-effector,the users felt that the end-effector moved more smoothly withPTPSA than that with NRSA in the shoulder singularity test.This is because the maximal Y-axis position error for NRSAis larger than that for PTPSA, as shown in Fig. 13(a). Consid-ering the efficiency with which a test was completed, PTPSAperformed better than either NRSA or RSA. The efficiencyderived 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 regionunder joint-interpolated control, which made the users feel thatthe test was completed more quickly. Considering the users'feelings about danger and tension, the results derived by usingNRSA, RSA, and PTPSA were similar. This is because the

Fig. 13. Maximal end-effector position and orientation in shoulder singularitytest. Significant difference was indicated if . (a) Maximal end-effector position error along Y-axis. (b) Maximal end-effector position erroralong Z-axis. (c) Maximal end-effector orientation error.

TABLE IIPOINTS OF USER'S FEELING

end-effector can avoid the singular region with any of NRSA,RSA, and PTPSA, which gave the users similar impressionsabout danger and tension. The motion of the robot with theapplication of NSA gave the users a greater sense of dangerand tension than that with any of NRSA, RSA, and PTPSA.This is because, with NSA, the positions of the joints changeconsiderably 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



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 ofusers' 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 involvingshoulder singularity, however, PTPSA performed better thanNRSA 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 modificationof the Jacobian matrix. When the end-effector was requested toquickly pass through a singular region, PTPSA proved to be abetter 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 choosethe PTPSA for their task in terms of the end-effector trajectoryerror, moving smoothness of the end-effector, and efficiency incompleting a task. When a task can be decomposed into severalsubtasks with order of priority, then the RSA is a better choice.

V. CONCLUSION AND FUTURE WORK

In this study, we proposed a novel singularity avoidancemethod named PTPSA, whereby the end-effector passesthrough a singular region with point-to-point jumping motion,and compared the use of PTPSA, NRSA, and RSA for ateaching-playback robot manipulator system. These methodshave been successfully tested with wrist and shoulder singu-larity through experimental case studies. The results of theseexperiments showed that PTPSA, NRSA, and RSA can suc-cessfully avoid a singular region. PTPSA and NRSA resultedin large position but small orientation errors in the end-effectortrajectory. RSA mainly incurred orientation errors. A statisticalanalysis of the users' feelings showed that PTPSA performedbetter than NRSA, and RSA in terms of smoothness of themovement of the end-effector and the efficiency with whichthe test could be completed. The efficiency impression derivedwith PTPSA was improved on average by 24.93% and 20.70%relative to that derived with NRSA and RSA, respectively. Theperformance of PTPSA, NRSA, and RSA were all similar interms 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 asingular region. The obstacle avoidance can be considered asa task constraint. The relationship between singularity and jerkfor each trajectory will be studied.

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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 mechanicalengineering from Xiamen University, Xiamen,China, in 2008, and the Ph.D. degree in precisionmechanical engineering from the University ofTokyo, 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 systemoptimization, robot manipulator kinematics, and multirobot coordination.

Yoon Seong Yong received the B.S. degree inmechanical engineering from Technical Universityof Malaysia Malacca in 2009 and the M.S. degree inprecision mechanical engineering from the Univer-sity of Tokyo, Tokyo, Japan, in 2013. He is currentlyan 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 theUniversity of Tokyo, Tokyo, Japan, in 1999, 2001,and 2004, respectively.In 2002, he visited the University of Edinburgh

and researched a coevolutional design process. From2004 to 2008, he was a Project Assistant Researcherwith the Department of Precision Engineering,University of Tokyo. From 2008 to 2014, he wasan Assistant Professor at Tokyo Metropolitan Uni-versity. Currently, he is an Associate Professor with

Asahikawa Medical University. His research interests are multiple mobilerobot systems, transportation systems, environmental design for robot systemsand coevolutional design processes.



Tamio Arai received the B.S., M.S., and Ph.D. de-grees in precision mechanical engineering from theUniversity 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, asa Visiting Researcher. From 2000 to 2005, he wasDirector of Research Into Artifacts, Center for Engi-neering, University of Tokyo, in which he proposedservice engineering. From 1987 to 2012, he was aProfessor with the Department of Precision Mechan-

ical Engineering, University of Tokyo. He is currently a Professor with theShibaura 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 andthe Asian Society for Precision Engineering and Nanotechnology, and HonoraryPresident of the Japan Association for Automation Advancement.

Tsuyoshi Ueyama received the B.S., M.S., andPh.D. degrees in engineering from the NagoyaUniversity, Nagoya, Japan, in 1989, 1991, and 1994,respectively.He is currently an Engineer in Denso Wave Inc.,

Aichi, Japan. His main research interests are robotmanipulator system control.

Jun Ota received the B.S., M.S., and Ph.D. degreesin precisionmechanical 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 Associatewith the University of Tokyo. In 1996, he was anAssociate Professor. From April 2009, he was aProfessor at the Graduate School of Engineering,the University of Tokyo. From June 2009, he wasa Professor at Research Into Artifacts, Center for

Engineering (RACE), University of Tokyo. From 1996 to 1997, he was aVisiting Scholar at Stanford University. His research interests are multi-agentrobot systems, design support for large-scale production/material handlingsystems, 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 forExcellent Papers, FANAC FA and Robot Foundation Excellent Papers, BestPaper Award of the IEEE International Workshop on Robot and Human Interac-tive Communication, and Advanced Engineering Informatics Top Cited Article.

kinematic control with singularity avoidance for teaching-playback robot manipulator system

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