Robotics and Computer-Integrated Manufacturing 27 (2011) 860–869

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Robotics and Computer-Integrated Manufacturing

0736-58

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/rcim

Remedy scheme and theoretical analysis of joint-angle drift phenomenon forredundant robot manipulators

Yunong Zhang �, Dongsheng Guo, Binghuang Cai, Ke Chen

School of Information Science and Technology, Sun Yat-sen University, Guangzhou 510006, China

a r t i c l e i n f o

Article history:

Received 31 December 2009

Received in revised form

23 January 2011

Accepted 7 February 2011Available online 26 February 2011

Keywords:

Repetitive motion planning

Quadratic programming

Kinematics

Redundant robots

Dynamics

45/$ - see front matter & 2011 Elsevier Ltd. A

016/j.rcim.2011.02.001

esponding author. Tel.: +86 13060687155; fa

ail addresses: ynzhang@ieee.org, zhynong@m

a b s t r a c t

A quadratic programming (QP)-based method, as a remedy for joint angle drifts, is employed for

redundant robot manipulators with physical constraints (e.g., joint-angle limits and joint-velocity

limits) considered. By using the QP-based redundancy-resolution scheme, real-time repetitive motion

planning (RMP) can be achieved in a drift-free manner. Theoretical analyses based on gradient-descent

and neural-dynamic methods are also conducted. Based on analyses, the efficacy of the presented

QP-based RMP scheme for redundant manipulators is successfully explained. To demonstrate the

effectiveness of the RMP scheme, different kinds of redundant robot manipulators, such as PA10,

PUMA560, and a six-link planar robot arm, are tested in order to perform circular and straight line end-

effector trajectories by using computer simulations. Both theoretical analysis and computer simulation

results have demonstrated the efficacy of the QP-based RMP scheme.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

A robot manipulator is redundant when more degrees offreedom (DOF) are observed in relation to the minimum numberof DOF required to perform a given end-effector primary task [1].The human arm, elephant trunk, and a snake are examples ofredundant systems [2]. The potential efficacy of such a redundantmanipulator is determined to some extent by the DOF number, aswell as the structure of the manipulator and its control scheme.A manipulator with sufficient DOF for a specific end-effector task(also known as non-redundant manipulator) may not have theability to achieve alternative goals when completing the task dueto the uniqueness of the solution. In contrast, if the manipulatordoes not have sufficient DOF, the end-effector motion might notbe performed accurately; at times, they could not even be fulfiled.These findings have motivated robot-practitioners (including theauthors of this paper [3–9]) to improve further the functionality,flexibility and efficacy of robot manipulators. In effect, redundantmechanisms [3–25] were considered. Regardless of the type ofredundancy (e.g., kinematical or functional), a variety of tasksrequiring sophisticated motion in complex environment can beconducted by such redundant robots. In particular, this includesworking in hazardous or rough-and-tumble environment [17],carrying heavy objects or radioactive materials [17,18], andexploring unpredictable regions (e.g., outer-space or blue-water

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x: +86 20 39943353.

ail.sysu.edu.cn (Y. Zhang).

exploration) [19,21]. Subsequently, redundant robot manipulatorshave been widely investigated [3–38].

One fundamental issue in operating such redundant robotsystems is the online redundancy-resolution problem, whichincludes the inverse-kinematic and inverse-dynamic sub-issues [3]. The general description of the robotic issue is that,given the desired or user-defined Cartesian trajectory rðtÞARm forthe manipulator’s end-effector to track, we need to generate thecorresponding joint trajectory yðtÞARn in real time t. Note thatmon, and thus there exist multiple solutions to this redundancy-resolution issue. In contrast, the general description of theforward-kinematic problem [3] is that, given the joint variablevector yðtÞ, the location of the end-effector, r(t), can then bedetermined uniquely and directly. By properly resolving theredundancy, the robots can avoid joint physical limits [4,7] andenvironmental obstacles [1,5,6,39], apart from optimizing varioussecondary criteria [10–14]. However, for a redundant manipulatorperforming a user-specified end-effector task, the redundancy ofjoint motion (i.e., the existence of multiple or even an infinitenumber of solutions) complicates the manipulator motion plan-ning and control problem considerably, including the inherentkinematic and dynamic nonlinearities.

To take full advantage of the redundancy, various computationalschemes, including approaches based on quadratic programming(QP), have thus been proposed, developed and investigated. Theconventional solution to such a redundancy-resolution problem (e.g.,inverse-kinematic problem) mostly takes the pseudoinverse-basedformulation (i.e., a minimum-norm particular solution plus a homo-geneous solution) [15,20,26–37]. Majority of current researchers

Y. Zhang et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 860–869 861

have applied the pseudoinverse technique to formulate and resolvethe redundancy by considering different optimization criteria, suchas least-square joint velocities [33], singularity avoidance [34],obstacle avoidance [35], and task priority control [36]. However,among those techniques, the physical constraints such as joint limitsand joint-velocity limits are usually neglected. This method may leadto a saturation and considerable tracking errors, or may even causepossible physical damage. Furthermore, the aforementioned pseu-doinverse techniques (also known as generalized inverse techniques)may exhibit the algorithmic singularities for some inverse-kinematicproblems; e.g., when the Jacobian matrix becomes rank-deficient.Thus, several modified generalized inverse techniques, such as theextended Jacobian technique [15,30,31] and integrable pseudoin-verse technique [32], have been proposed to avoid algorithmicsingularities. Research during the last 15 years [3–11,13,38] showsthat the redundancy-resolution problem might be solved in a morefavorable manner by using online optimization techniques (e.g., QPneural-dynamic approach).

In general, there are several extrinsic and inherent factors such asend-effector motion requirement, joint physical limits, and optimi-zation of secondary criteria, which can greatly affect the motionplanning of robot manipulators, some allowing kinematic controls tobecome non-repetitive. If the redundancy-resolution schemes(e.g., the conventional pseudoinverse-based schemes) are unsuitablefor a number of particular end-effector tasks, the final configurationof the robot manipulator may not coincide well with the initialconfiguration (i.e., the motion planning of the robot manipulators isnot repetitive). The so-called joint-angle drift phenomenon (alsoreferred to as the repeatability problem) implies that, when therobot end-effector tracks a closed path in its workspace, the jointvariables may not return to their initial values after completing theend-effector task. In other words, the trajectories obtained inthe joint space may not be linked together. Problem may arise fromthe robot manipulator’s unpredictable behavior, and in turn, lessefficiency in readjusting the manipulator’s configuration with self-motion at every cycle [40,41]. This paper presents and investigatesthe repetitive motion planning (RMP) of redundant manipulatorsbased on neural-optimization techniques [3–9]; its aim is toeliminate the joint-angle drift phenomenon [20,27–32,38].

To make the kinematic control repeatable (also referred toas repetitive, cyclic, or conservative), a QP-based RMP scheme(i.e., joint-angle drift remedy scheme) will be presented in thenext section for robot manipulators’ drift-free redundancy-reso-lution. In addition, the presented scheme takes into account theavoidance of joint physical limits (i.e., the limits of joint anglesand joint velocities). Moreover, the effectiveness of the drift-freeperformance index (i.e., QP performance index) is analyzed in twoways. Section 3 illustrates the computer simulation results of thecircular and straight line paths synthesized by the presented QP-based RMP scheme, wherein three different types of robot arms(i.e., PA10, PUMA560, and a six-link planar robot) can all achieverepetitive motions. Section 4 concludes this paper. The maincontributions of this study are the following.

(1)

This paper presents and further investigates the RMP perfor-mance index. It also summarizes the resultant RMP scheme (alsoknown as drift-free redundancy-resolution scheme) formulatedas a quadratic program with joint-angle limits and joint-velocitylimits both considered.

(2)

This paper provides illustrative cases of circular and straightline paths. The QP-based RMP scheme is applied successfullyto different kinds of redundant robot manipulators, includingkinematically and functionally redundant manipulators, aswell as a planar robot arm. Results from simulation based onthe different kinds of redundant robot manipulators demon-strate the efficacy of the RMP scheme.

(3)

The effectiveness of the RMP performance index is analyzedfor the first time and substantiated formally in a journal paperusing the gradient-descent method and a new method pro-posed recently for the neural-dynamic design.

2. Joint-angle drift remedy scheme

The functional relation between the end-effector position-and-orientation vector rðtÞARm and joint variable vector yðtÞARn ofredundant robot manipulators can be written readily as follows:

r¼ f ðyÞ, ð1Þ

where f ð�Þ : Rn-Rm is a continuous nonlinear forward-kinematicmapping (evidently, not a one-to-one mapping) with knownstructure and parameters for a given manipulator. By differentiat-ing (1) with respect to time t, we obtain the point-wise linearrelation between the end-effector Cartesian velocity _r and joint-velocity _y as follows:

JðyÞ _y ¼ _r , ð2Þ

where JðyÞARm�n is the Jacobian matrix defined as JðyÞ ¼ @f ðyÞ=@y.In redundant robot manipulators (i.e., mon), Eqs. (1) and (2) areboth underdetermined, as evidenced by the infinite number offeasible solutions.

The conventional pseudoinverse-based solution to the inverse-kinematic problem (i.e., r is given to solve y) is normallyformulated as the summation of a minimum-norm particularsolution and a homogeneous solution:

_y ¼ P _rþðI�PJÞz, ð3Þ

where P¼ JT ðJJTÞ�1ARn�m denotes the pseudoinverse of J; I is an

appropriately dimensioned identity matrix; and zARn is anarbitrary vector selected by using some performancecriteria [20,27–32].

2.1. Repetitive performance index

The pseudoinverse-based solution, such as (3), may not berepetitive [27–32,37]. In other words, a closed path of the end-effector may not yield the closed trajectories in the joint space.Such joint-angle drift is not applicable for cyclic motion planningand control. The manipulator’s configuration can be readjustedwith manipulator self-motion processes (i.e., without changingthe end-effector’s position-and-orientation) [40,41], but thiswould lead to less efficiency. To make the inverse-kinematicsolution repetitive, the minimization of the joint displacementbetween current and initial states should be exploited [8]. In theformulation, the performance index used is

12ð_yþcÞT ð _yþcÞ with c¼ lðy�yð0ÞÞ, ð4Þ

where yð0Þ is the initial state of the joint variable vector. Thedesign parameter, l40, is used to scale the magnitude of themanipulator response to the joint displacement. Design para-meter l should be set as large as the robot system would permit,or selected appropriately for experimental and/or simulativepurposes. Theoretical analysis for the effectiveness of perfor-mance index on RMP of robots has never attained considerableattention in literature up to the present. Motivated by severalsuccessful computer simulation results of the RMP scheme ondifferent robot manipulators, we analyzed the repetitive perfor-mance index (4) based on two different approaches, namely,the gradient-descent method [42,43] and the neural-dynamicmethod [3,44,45].

Y. Zhang et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 860–869862

2.1.1. Gradient descent method

Used as the first approach for analysis, the well-knowngradient-descent method [42,43] is used to explain the designprinciple and effectiveness of the drift-free performance index (4).Specifically, to achieve the quadratic performance index (4), weemploy the following steps:

�

First, to eliminate the joint-angle drift phenomenon [i.e., toachieve yðt¼ TÞ ¼ yð0Þ, where T denotes the final time of themotion task], it is natural to define the following scalar-valuednorm-based displacement-function that is to be minimizedover the whole range [0,T]:

e¼ Jy�yð0ÞJ22=2, ð5Þ

where JyJ2 :¼ffiffiffiffiffiffiffiffiyTy

pdenotes the two norm of joint vector y.

Evidently, a minimum point of displacement function,e¼ Jy�yð0ÞJ2

2=2, is obtained with e¼ 0 if and only ify�yð0Þ ¼ 0 (i.e., the drift-free situation).

� Second, a computational scheme is designed to evolve along a

descent direction of displacement-function eðyÞ until theminimum point is reached. A typical descent direction is thenegative gradient of eðyÞ, in particular, �ð@eðyÞ=@yÞ. Thus, wehave the following computational scheme aimed at minimiz-ing eðyÞ:_yðtÞ ¼�l@eðyÞ=@y¼�lðyðtÞ�yð0ÞÞ, ð6Þ

where scalar l40AR is used to adjust the e-convergence rate.Evidently, the displacement-function e converges to 0 expo-nentially with rate l. In addition, Eq. (6) can be rewritten as_yðtÞþlðyðtÞ�yð0ÞÞ ¼ 0.

� Finally, inasmuch as the other factors like the end-effector

motion requirement and joint physical limits have to beconsidered in the RMP scheme, the dynamic equation_yðtÞþlðyðtÞ�yð0ÞÞ ¼ 0 can only be achieved theoretically. Thus,minimizing J _yðtÞþlðyðtÞ�yð0ÞÞJ2

2=2 appears to be more feasiblefor the repetitive motion planning of redundant manipulators,rather than forcing _yðtÞþlðyðtÞ�yð0ÞÞ ¼ 0 directly in its exactform. Expanding J _yðtÞþlðyðtÞ�yð0ÞÞJ2

2=2 yields

ð _yþlðyðtÞ�yð0ÞÞÞT ð _yþlðyðtÞ�yð0ÞÞÞ=2: ð7Þ

Thereafter, c :¼ lðyðtÞ�yð0ÞÞ. The performance index (7)becomes exactly (4); that is, ð _yþcÞT ð _yþcÞ=2, which has beenproven effective for the drift-free motion planning of redun-dant robots.

2.1.2. Neural-dynamic method

For comparative purposes, we introduced a different approachin order to analyze the effectiveness of the drift-free performanceindex (4). This approach (also known as the neural-dynamicmethod) has been proposed in the online solution of time-varyingmatrix and/or vector problems [3,44,45]. In addition, this methodprovides for the drift-free quadratic performance index (4) enum-erated as follows.

�

First, to eliminate the joint-angle drift phenomenon, instead ofusing a scalar-valued norm-based displacement-function ein Section 2.1.1 for the gradient-descent method, we definealternatively the following vector-valued displacement-func-tion in this subsection, which is to be eliminated over thewhole range [0,T]:

EðtÞ :¼ yðtÞ�yð0ÞARn:

Similar to those presented in Section 2.1.1, E(t)¼0 if and only ify�yð0Þ ¼ 0.

�

Second, in order for E(t) to converge exponentially to 0, wesimply set:

_EðtÞ ¼�lEðtÞ, ð8Þ

where l40AR denotes the exponential convergence rate of E.The solution to Eq. (8) is evidently EðtÞ ¼ expð�ltÞEð0Þ. Thisalso suggests that within the time-period of 4=l s, jEjðtÞj wouldbe less than 1.85% of jEjð0Þj, 8jAf1,2, . . . ,ng [3].

� Third, substituting EðtÞ ¼ yðtÞ�yð0Þ into (8) yields _yðtÞ ¼�lðyðtÞ�yð0ÞÞ, which is the same as in Eq. (6). The same resultscan be achieved more easily using the neural-dynamic methodcompared with the gradient-descent method presentedin Section 2.1.1. Similarly, the final step of Section 2.1.1(gradient-descent method) suggests that the quadratic perfor-mance index (4), ð _yþcÞT ð _yþcÞ=2, could theoretically eliminatethe joint-angle drift phenomenon of redundant manipulators.The analysis is thus completed.

In summary, by using two different approaches, the designmethod and the theoretical effectiveness of drift-free perfor-mance index ð _yþcÞT ð _yþcÞ=2 used in the repetitive motion plan-ning of redundant robot manipulators have been demonstrated.

2.2. Joint physical bounds

Inasmuch as almost all robot manipulators are physicallyconstrained by their joint limits and/or joint-velocity limits, itseems more realistic and useful to consider the followingbounded RMP-scheme formulation:

minimize 12ð_yþcÞT ð _yþcÞ with c¼ lðy�yð0ÞÞ

subject to JðyÞ _y ¼ _r , ð9Þ

y�ryryþ , ð10Þ

_y�r _yr _y

þ, ð11Þ

where the RMP performance index is the same as that presentedin Section 2.1, and the equality constraint of Eq. (9) is the same as inEq. (2), which describes the end-effector primary task of thepath. More importantly, the avoidance of joint-angle limits y7

and joint-velocity limits _y7

are considered, with superscripts +

and � denoting the upper and lower limits of a joint variable vector,respectively.

As the redundancy-resolution problem is resolved at the joint-velocity level, limited joint range ½y�,yþ � of (10) has to beconverted into an expression based on joint-velocity _y [3,4]. Thenew bound constraint can be written as:

mðy��yÞr _yrmðyþ�yÞ, ð12Þ

where the large values of parameter m40 may cause quick jointdeceleration when the robot approaches its limits. Normally, m isdetermined in order for the converted feasible region of _y [i.e., byjoint limits (10)] to be not smaller than the original region madeby the joint-velocity limits (11) [4,7]. In mathematics, m should be

more than or equal to 2max1r irnf_yþ

i =ðyþ

i �y�

i Þ,�_y�

i =ðyþ

i �y�

i Þg.

In addition, design parameter m is set accordingly in the ensuingcomputer simulation for the robot arms of PA10, PUMA560 and asix-link planar robot.

Combining the bound-constraints (11) and (12) yields a uni-fied dynamic bound constraint, x�r _yrxþ , where the ith ele-ments of x� and xþ are defined, respectively, as follows [3,4,8]:

x�i ¼maxf _y�

i ,mðy�i �yiÞg, xþi ¼minf _yþ

i ,mðyþi �yiÞg:

Y. Zhang et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 860–869 863

2.3. Quadratic programming

The RMP scheme is resolved at the joint-velocity level and thedecision variable vector is joint-velocity _y. Subsequently, thefunction c [i.e., c¼ lðy�yð0ÞÞ in Eq. (4)] is viewed as a constantin the performance index. Moreover, cT c=2 is positive and viewedas a constant (with respect to _y), which is thus set aside from theperformance index. Therefore, based on Eqs. (9)–(11) and pre-vious analysis, the physically constrained drift-free redundancy-resolution scheme (i.e., joint-drift remedy scheme) of robotmanipulators can be reformulated into the following quadraticprogram:

minimize _yT _y=2þcT _y ð13Þ

subject to J _y ¼ _r , ð14Þ

x�r _yrxþ , ð15Þ

where coefficients c, J, _r and x7 are defined the same as before.For emphasis (with details omitted due to space limitation), the

presented QP-based RMP scheme (13)–(15) can be computedonline by using popular mathematical programming routinessuch as MATLAB ‘‘QUADPROG’’ with syntax being _y ¼QUADPROGðI,c,½ �,½ �,J, _r ,x�,xþ Þ [46]. Furthermore, considered as efficient paral-lel-processing alternatives to real-time computation of quadraticprograms, recurrent neural networks [e.g., the primal-dual neuralnetwork based on linear variational inequalities (LVI)] [7–9] can beexploited to remedy the presented scheme (13)–(15). Lastly, thecomparison between the proposed and investigated QP-based RMPscheme and the pseudoinverse-based techniques can be presentedas follows.

Remark 1. Unlike the conventional pseudoinverse-based techni-ques [20,27–32,37], the presented QP-based RMP scheme mayplan repetitive motion of redundant manipulators in an explicitinverse-free manner, in order to avoid expensive Oðn3Þ computa-tion. Moreover, physical constraints [i.e., in the form of boundconstraint (15)] can be readily inputted by the presentedQP-based RMP scheme (13)–(15), whereas pseudoinverse-basedtechniques do not consider physical constraints; this is also truefor algorithmic singularities [20].

−0.6−0.4

−0.20

0.2

−0.2

−0.1

0

0.1

0.2−0.5

0

0.5

1

−1

−0

0

1

Fig. 1. PA10 end-effector moves along a circular path with joint physical limits con

(b) joint-angle transients.

Remark 2. Based on the extended Jacobian technique [30,31], ifJacobian matrix J is a full rank matrix with dimension m�n, thenany matrix P for which JP¼ I can be obtained by adding appro-priately selected n�m constraint-rows to J such that the n�n

extended Jacobian matrix obtained becomes nonsingular. How-ever, such n�m constraints may not take full advantage ofredundancy (e.g., they may not permit inequality-based obstacleavoidance [3,5,6,38]), and may artificially limit the end-effectorworkspace. In addition, the extended Jacobian technique alsolimits the total number of the physical constraints (i.e., n�m),which may have less practical significance in the field of engi-neering. In contrast, by using the QP-based RMP scheme (13)–(15)which incorporates the avoidance of joint physical limits (15) as asubtask of redundancy-resolution, the proposed framework canalso handle other subtasks, such as obstacle avoidance [5,6],formulated in terms of constraints and/or performance indices.

3. Application to different manipulators

In this section, the QP-based RMP scheme (13)–(15) is simulatedbased on three kinds of robot manipulators: PA10, PUMA560, and asix-link planar robot. In addition, the different desired end-effectormotion trajectories including circular and straight line paths aretested for illustrative and comparative purposes. Note that, theQP-based RMP scheme (13)–(15) is solved in the simulations byusing the LVI-based primal-dual neural network [7–9] of which thedynamic equation is described as

_y ¼ gðIþMT ÞfPO½y�ðMyþqÞ��yg, ð16Þ

where g40, used to scale the network convergence, corresponds tothe reciprocal of a capacitance parameter in the possible hardwareimplementation of the network; it should also be set as large as thehardware could permit, or selected appropriately for experimental/simulative purposes [7–9]. POð�Þ is a piecewise-linear projectionoperator. In addition, primal-dual decision variable vector y, coeffi-cient vector q, and matrix M are defined, respectively, as

y¼_yx

" #ARnþm, q¼

c

�_r

� �ARnþm, M¼

I �JT

J 0

" #ARðnþmÞ�ðnþmÞ,

where xARm denotes the dual decision vector for equalityconstraint (14).

0 1 2 3 4 5 6 7 8 9 10.5

−1

.5

0

.5

1

.5

2

sidered but without drift-free criterion (i.e., l¼ 0): (a) motion trajectories and

−0.6−0.4

−0.20

0.2

−0.2−0.1

00.1

0.2−0.5

0

0.5

1

0 1 2 3 4 5 6 7 8 9 10−5

0

5

10

0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Fig. 2. PA10 end-effector moves along a circular path with both joint physical limits and drift-free criterion considered (i.e., l¼ 4): (a) motion trajectories, (b) end-effector

positioning errors, (c) joint-angle transients, and (d) joint-velocity transients.

Table 1Joint displacement of PA10 tracking a circle without considering the drift-free

criterion.

# Joint Initial yð0Þ in radians Final yð10Þ in radians yð0Þ�yð10Þ

1 y1 +0.0000000000 +0.4825307152 �0.4825307152

2 y2 �0.7853981633 �0.7883070086 +0.0029088452

3 y3 +0.0000000000 �0.0287131516 +0.0287131516

4 y4 +1.5707963267 +1.5705310337 +0.0002652930

5 y5 +0.0000000000 +0.0079733652 �0.0079733652

6 y6 �0.7853981633 �0.8049657550 +0.0195675916

7 y7 +0.0000000000 +0.0000000000 +0.0000000000

Table 2Joint displacement of PA10 tracking a circle in consideration of the drift-free

criterion.

# Joint Initial yð0Þ in radians Final yð10Þ in radians yð0Þ�yð10Þ

1 y1 +0.0000000000 �0.0000250981 +2.50�10�5

2 y2 �0.7853981633 �0.7853998801 +0.17�10�5

3 y3 +0.0000000000 +0.0000016960 �0.16�10�5

4 y4 +1.5707963267 +1.5707964280 �0.01�10�5

5 y5 +0.0000000000 +0.0000009783 �0.09�10�5

6 y6 �0.7853981633 �0.7853838037 �1.43�10�5

7 y7 +0.0000000000 +0.0000000000 +0.00�10�5

Y. Zhang et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 860–869864

The first three simulations are based on PA10 robot manip-ulator [3,4].

(1)

In Figs. 1 and 2 and Tables 1 and 2, the motion trajectory ofthe PA10 end-effector is expected to be a circle with radius of0.2 m and with the revolute angle around the X-axis beingp=6 rad. The motion duration is 10 s, and initial joint stateyð0Þ ¼ ½0,�p=4,0,p=2,0,�p=4,0�T rad.

(2)

In Fig. 3, the motion trajectory of the PA10 end-effector isexpected to be a circle with radius 0.2 m; this is performed inmultiple cycles (i.e., five cycles, T¼50 s). The revolute anglearound the X-axis and the initial joint state yð0Þ are the sameas previous.

(3)

In Fig. 4, the end-effector is expected to move forward andbackward along a straight line segment path (2.5 m). Anglesof this straight line, making with the XY, YZ and ZX planes, are

−0.6−0.4

−0.20

0.2

−0.2−0.1

00.1

0.2−0.5

0

0.5

1

0 5 10 15 20 25 30 35 40 45 50−4

−2

0

2

4

6

8

10

12

0 5 10 15 20 25 30 35 40 45 50−1.5

−1

−0.5

0

0.5

1

1.5

2

0 5 10 15 20 25 30 35 40 45 50−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Fig. 3. PA10 end-effector moves along a circle for multiple cycles with both physical limits and drift-free criterion considered: (a) motion trajectories, (b) end-effector

positioning errors, (c) joint-angle transients, and (d) joint-velocity transients.

Y. Zhang et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 860–869 865

p=4, p=6 and p=6 rad, respectively. The initial joint state isyð0Þ ¼ ½p=4,p=4,p=4,p=4, p=4,p=4,0�T rad. Task duration is 7 s.

Fig. 1 illustrates the redundancy-resolution simulation resultswith joint physical limits considered, but not the drift-free

criterion (i.e., by forcing the drift-free coefficient l¼ 0, the

performance index is _yT _y=2). As shown in Fig. 1(a), the end-

effector of the PA10 robot arm moves along a circle in a three-dimensional workspace, which is sufficiently close to the desiredcircular path. However, for robots working with a repetitivemotion, a serious weakness may exist in the sense that the finalstate of the PA10 manipulator does not coincide with its initialstate, as evidenced in Table 1. In addition, Fig. 1(b) shows that thefinal configuration does not match the initial configuration,especially y1. This solution implies a problem on joint-angle drift.For comparison and illustration, the redundancy-resolutionresults are shown in Fig. 2 where both joint physical limits anddrift-free criterion considered. In Fig. 2, which is also synthesizedby scheme for Eqs. (13)–(15) with l¼ 4, all joint trajectories areclosed, and the final state of the PA10 manipulator coincides verywell with initial state yð0Þ, albeit a tiny positioning error of the

end-effector (i.e., the maximum positioning error is less than1.0�10�6 m). In this paper, the variables eX, eY and eZ [Fig. 2(b)]denote X-, Y-, and Z-axis components, respectively, of positioningerror e with respect to the base frame. Tables 1 and 2 also showthe joint displacements of the PA10 robot manipulator aftertracking a circular path. Evidently, the solution generated by theQP-based RMP scheme [i.e., by using Eqs. (13)–(15)], whichconsider the drift-free criterion, is repetitive; that is, it solveswell the joint-angle drift problem.

For further investigation, the PA10 robot manipulator issimulated to track the aforementioned circle for multiple cycles(i.e., with motion duration T¼50 s). The simulation results areillustrated in Fig. 3, with both joint physical limits and drift-freecriterion considered (i.e., l¼ 4). As shown in Fig. 3, the end-effector trajectory is sufficiently close to the desired circular pathwith a maximum positioning error of less than 1.2�10�6 m. Thepositioning error shown in Fig. 3(b) has been kept minimal duringthe motion. In addition, Fig. 3 and corresponding simulation datashow that the joint-angle drift phenomenon could be remedied,such that the final state of the PA10 manipulator [i.e., yð50Þ]coincides very well with initial state yð0Þ. This demonstrates theefficacy of the QP-based RMP scheme (13)–(15). Fig. 4 illustrates

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Fig. 4. PA10 end-effector moves along a straight line path: (a) motion trajectories with l¼ 0, (b) joint-angle transients with l¼ 0, (c) motion trajectories with l¼ 4,

and (d) joint-angle transients with l¼ 4.

Y. Zhang et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 860–869866

the simulation results when the presented QP-based RMP schemeis applied to a straight line path-following case. Fig. 4(a) and(b) shows the existence of the joint-angle drift phenomenon(i.e., the final state of PA10 does not coincide well with its initialstate). Fig. 4(c) and (d), on the other hand, shows that the joint-angle drift problem is solved well using the scheme (13)–(15)with the drift-free criterion considered, hence verifying theefficacy of the QP-based RMP scheme.

In summary, Figs. 1–4 and Tables 1 and 2 prove that the drift-free criterion plays an important role in the repetitive motionplanning of redundant robot manipulators. Simply put, by usingthe QP-based RMP scheme (13)–(15) with the drift-free criterionconsidered, the joint-angle drift phenomenon disappears and therepetitive performance is achieved.

The second set of computer simulation examples is performedbased on a 6-DOF PUMA560 robot arm [3,7]. The PA10 robot armis kinematically redundant, whereas the PUMA560 robot arm isfunctionally redundant if only its end-effector positioning inconsidered. As shown in Fig. 5(a), the motion trajectory of thePUMA560 end-effector is expected to be a circle, with the robotstarting from initial state yð0Þ ¼ ½0,�p=4,0,p=2,�p=4,0�T rad.In Fig. 5(b), the end-effector is expected to move forward andbackward along a straight line segment, with the robot starting

from initial joint state of yð0Þ ¼ ½0,�p=4,0,2p=3,�p=4,0�T rad.The simulation results illustrated in Fig. 5 have substantiatedthe inverse-kinematic solution synthesized by the presentedQP-based RMP scheme (13)–(15), which is repetitive in the sensethat the initial and final states of the PUMA560 robot fit very wellwith each other for both circular and straight line simulations.In other words, the presented redundancy-resolution scheme(13)–(15) can solve the joint-angle drift problem. In addition, asevidenced by other simulation data, all joint variables (i.e., y and_y) were kept well within their mechanical ranges, with theinclusion of bound constraint (15).

The final simulation example is performed based on a six-linkplanar robot arm. The aforementioned PA10 and PUMA560 robotmanipulators both operate in three-dimensional space; in con-trast, the six-link planar robot manipulator in this example worksonly horizontally and with a point obstacle [3,5,6]. The drift-freeredundancy-resolution scheme (13)–(15), firstly without consid-ering the point obstacle, is applied to the six-link planar robotarm with the corresponding simulation results illustratedin Fig. 6. As seen from Fig. 6(a), the aim of RMP has been achievedsuccessfully (i.e., with no joint-angle drift phenomenon).However, by looking into the intermediate results, we can seefrom Fig. 6 that the minimum link-obstacle distance d had been

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Fig. 5. PUMA560 end-effector moves along circular and straight line paths with joint physical limits and drift-free criterion considered: (a) circular-path following

example and (b) straight line path-following example.

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Fig. 6. The end-effector of a six-link planar robot tracks a circular path with the drift-free criterion considered but without considering a point obstacle: (a) motion

trajectories and (b) minimum link-obstacle distance (colliding when dr0:05 m).

Y. Zhang et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 860–869 867

smaller than the inner safety threshold (e.g., 0.05 m). For such aclose distance, it might be considered as a collision in practice,which is dangerous for both manipulator and obstacle. Accordingto the authors’ previous work [3,5,6], the obstacle-avoidancerequirement can be represented by a dynamically updatedinequality constraint, A _yrb. By considering such an inequalityconstraint, the RMP scheme could thus be termed as an obstacle-avoidance RMP scheme. Fig. 7 illustrates the simulation resultssynthesized by such an obstacle-avoidance RMP scheme, where‘‘traj.’’ is the abbreviation of the word ‘‘trajectory’’. As shownin Fig. 7(b), the minimum link-obstacle distance is always safelygreater than the inner safety threshold 0.05 m. In addition, thejoint-angle drift phenomenon is remedied by using such ascheme, as seen from Fig. 7(a) and (c). These substantiate thatthe schemes [including the original RMP scheme (13)–(15)] aresuccessful repetitive motion planning schemes. For more detailsabout the obstacle-avoidance feature (e.g., the aforementioneddynamically updated inequality constraint and the meaning ofthe inner safety threshold), please refer to [3,5,6].

In summary, both circular-path example and straight line pathexample based on different kinds of robots have demonstrated the

efficacy of the QP-based RMP scheme (13)–(15) for joint-constrainedrobot manipulators. More specifically, the presented scheme cansolve the joint-angle drift problem that may occur in the motionplanning of redundant manipulators. Before ending this section, it isworth giving the following remark about design parameter l.

Remark 3. From the theoretical analysis in Section 2.1, we can seethat the unit of l is Hz (or 1/s). So, physical factors, such as hardwarerealizability, may have effect on the limit of l. By choosing differentvalues of l in the simulative tests (but omitted due to spacelimitation), the authors have observed that the repetitive perfor-mance could be achieved, but the end-effector positioning errorsmight increase when the value of l is increased (e.g., from 4 to 160).So, generally speaking, according to the authors’ experience, therequirements on an acceptable end-effector positioning error(e.g., less than 10�4 m), simulative/computational execution time(e.g., within a few minutes) and solution realizability (e.g., withoutchattering phenomenon [47]) determine the upper bound of thevalue of design parameter l. The comparative choosing and study ofdifferent values of design parameter l (including the determinationof its limit) could be a future research direction.

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Fig. 7. The end-effector of a six-link planar robot tracks a circular path with both drift-free criterion and point obstacle considered: (a) motion trajectories, (b) minimum

link-obstacle distance, and (c) joint-angle transients.

Y. Zhang et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 860–869868

4. Conclusions

In this paper, we have investigated a remedy scheme toeliminate the joint-angle drift problem of redundant manipula-tors. We have summarized various effective computer simulationresults based on different robot manipulators for drift-free pur-poses. In addition, we have analyzed the effectiveness of a drift-free performance index by using both the gradient-descentmethod and the neural-dynamic method. To conclude, the repe-titive motion planning of robot manipulators can be achieved byusing the QP-based drift-free redundancy-resolution scheme(13)–(15).

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant Nos. 61075121 and 60935001) andby the Fundamental Research Funds for the Central Universities ofChina. In addition, the authors sincerely thank the editors andanonymous reviewers for their constructive and inspiring com-ments and suggestions, which have really helped improve thequality of the paper very much.

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