Comparative Analysis for the Inverse Kinematics of RedundantManipulators based on Repetitive Tracking Tasks

Jingguo Wang and Yangmin Li

Abstract— Considering the redundant degree of freedom(DoF)of open loop kinematic chain, the closed-loop inverse kinematicsalgorithm(CLIK) via pseudoinverse method is used to solve thecorresponding joint trajectories if the end-effector trajectory isgiven and the constraints of joint limits are applied to modify thenull space in order for some optimized solutions. The repetitivetrajectories are tracked in the simulations made on the 3-DoFplanar and 4-DoF spatial manipulators respectively and twooptimization methods with joint limits avoidance are considered.The simulation results verify the effectiveness of the proposedalgorithm and provide detailed comparative analysis amongdifferent methods and cases.

Index Terms— Redundant manipulator, closed-loop algorithm,joint limit avoidance, objective function, repetitive tracking task

I. INTRODUCTION

Since it is easy for a manipulator to control motion in thejoint space while the tasks are specified usually in the cartesianspace, therefore it is necessary to solve the inverse kinematicsproblem in order to find the corresponding set of joint anglesgiven the desired position and orientation of the end-effector.

For a redundant robot manipulator which has more DoFsthan required to achieve the desired position and orientationof the end-effector [1]-[16], the robot redundancy is to improveits flexibility and versatility and to implement collisions-freemotion in the robot workspace by using the redundant DoFs.A nonempty null space exists in a kinematically redundantrobot manipulator, which allows the user to analytically applythe redundancies of the system in a strategic manner thatwill improve performance, and its elements may be changedwithout affecting the validity of the solution because thepseudoinverse leaves the null-space untouched.

Many schemes and profound comparative analysis for theinverse kinematics of redundant manipulators are presentedand proposed in previous literature. The generalized inverseof Jacobian matrix, pseudoinverse[2] and inertia-weightedpseudoinverse[3] were widely applied for a redundant robot.The most popular method is the gradient projection methodfirst introduced in [4] to utilize the redundancy to avoidmechanical joint limits. To overcome the joint drift for open-chain robot manipulators, the well- known CLIK algorithmwas proposed in [5][6], which was developed to include

This work was supported by the Research Committee of University ofMacau under grant no. UL016/08-Y2/EME/LYM01/FST and Macao Scienceand Technology Development Fund under Grant no. 016/2008/A1.

Jingguo Wang is with Department of Electromechanical Engineering,Faculty of Science and Technology, University of Macau, Av. Padre TomasPereira, Taipa, Macao S.A.R., China ya77403@umac.mo

Yangmin Li is with Department of Electromechanical Engineering, Facultyof Science and Technology, University of Macau, Av. Padre Tomas Pereira,Taipa, Macao S.A.R., China YMLi@umac.mo

feedback for the end-effector’s position and orientation andthe algorithm can employ either the transpose of the in-verse Jacobian or the direct Jacobian. Another scheme basedon the constrained least-squares method[7] was proposed togenerate feasible output around singularities, which utilizeda generalized-inverse matrix of the Jacobian, known as thesingularity-robust pseudoinverse. A new dynamical closed-loop system for kinematic control, aiming at further improvingthe performance at and around the singularity, was proposedin[8] without deteriorating the performance at regular points.A simulation package was applied for dynamic simulationof n-DOF planar manipulators with revolute joints of n-Rmanipulator [9] .

Although related works had been made on the inversekinematics problem of redundant manipulators by follow-ing the given trajectory, no much attentions were paid onthe repeated-trajectory problem. This paper focuses on therepetitive trajectory-following issue of the 3-DoF and 4-DoFredundant manipulators and provides the detailed comparativeanalysis among different optimization methods. In sectionII, the kinematics analysis of the redundant manipulator ispresented. Then the improved CLIK algorithm is proposedto obtain the general inverse solution in section III. Theoptimization method with joint limit avoidance is given outin section IV and the related case studies on 3-DoF and 4-DoF manipulators are carried out in section V. Finally theconclusion and future work are given out in the last section.

II. KINEMATIC ANALYSIS OF REDUNDANTMANIPULATOR

The forward kinematics model for an n-DoF redundantmanipulator based on Denavit-Hartenberg transformation canbe formed as:

x = f (q) (1)

where x is an m-dimensional vector (x∈ Rm) representing boththe position and orientation of the end-effector in the taskspace, q is an n-dimensional vector (q ∈ Rn) representing thejoint position in the joint space (for redundant manipulators,n ≥ m) and f (q) denotes the forward kinematics of themanipulator.

The differential kinematics equation, in terms of either thegeometric or the analytical Jacobian, represents the differentialkinematics for the manipulator which has the form

x = Ja(q)q (2)

where x represents the task space velocity, q denotes thejoint velocity, Ja is the (m× n) manipulator Jacobian, which

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is a function of the joint configuration establishing a linearmapping between joint space velocities (q) and task spacevelocities (x).

For the redundant manipulator, the redundancy degree isr = n − m(r ≥ 1). Although redundancy degree plays animportant role in the kinematic control, due to the existenceof redundancy, joint velocities in (2) can not be obtaineddirectly by solving the differential kinematics equation sincethe inverse of non-square matrix J can not be available.

The nonempty null space (N(J)) exists for a kinematicallyredundant manipulator due to the excess of input space relativeto the manipulable space. The null space (N(J)) is the subspaceof Rn and its dimension is n−m′, where m′ is the rank ofthe matrix Ja. If Ja has full column rank, then the dimensionof N(J) is equal to the degree of redundancy. The jointvelocities belonging to N(J), referred to internal joint motionand denoted by qN , can be specified without affecting the taskspace velocities[16].

With reference to the geometric Jacobian, the particularinverse solution to (2) is obtained by using the pseudoinverseJ† of the matrix J and the solution can then be written as

qp = J†a (q)x (3)

The pseudoinverse J†a can be computed as(Rank(Ja) = m)

J†a = JT

a (JaJTa )−1 (4)

Although the pseudoinverse technique can be utilized to getthe least-square solution, there are some major problems withthe particular solution. As pointed out in [12], the solutiongiven by (3) does not guarantee generation of joint motionswhich avoid singular configurations and its norm will becomevery large near singular configurations. Another problem withthe pseudo-inverse approach is that the joint motions generatedby this approach do not preserve the repeatability and cyclicitycondition, i.e., a closed path in Cartesian space may not resultin a closed path in joint space [10].

Therefore, redundant DoFs will be necessarily consideredfor a kinematically redundant manipulator, and null space N(J)is available to set up systematic procedures for an effectivehandling of redundant DoFs. As (3) is the particular solutionto the inverse problem of differential kinematics equation (2),the homogeneous solution to the problem (Jaq = 0) in the nullspace can be obtained

qh = (I− J†a (q)Ja(q))q0 (5)

where the matrix (I− J†a (q)Ja(q)) is a projector of the joint

vector q0 onto N(J).By combining the the particular solution qp and homoge-

neous solution qh (2) together, the general inverse solution canbe written as

q = qp + qh (6)

The discrete-time version of first-order inverse kinematicsalgorithm can be written as

qk = qk−1 + qk4t (7)

where qk denotes the current joint value, qk−1 denotes the for-mer joint value in the last step and 4t is the time interval(4t =tk− tk−1).

Therefore, combining (3), (5), (6) and (7) together, theinverse kinematics problem can be solved by numerical in-tegration.

III. CLOSED-LOOP INVERSE KINEMATICSALGORITHM

Assume that a task space trajectory is given (xt), and thegoal of the inverse kinematics problem is to find a feasiblejoint space trajectory (qt) that reproduces the given trajectory.

Since open-loop reconstruction of joint variables throughnumerical integration (see in the above section) unavoidablyleads to solution drift and then to task space errors, in orderto overcome this drawback, a CLIK algorithm is utilizedwhich is based on the task space error between the desiredxd and current end-effector trajectory computed via the directkinematics based on Denavit-Hartenberg transformation.

To avoid the numerical drift, a error correction term is usedas the feedback into the algorithm,

xk = xd,k +Kek (8)

where xk represents the end-effector velocity to be achieved,xd,k represents the desired end-effector velocity(defined before-hand), K ia a suitable chosen positive definite matrix, and ekis the introduced position error.

It is obvious that the end-effector position error is given by

ek = xd,k− xk−1 (9)

where xd,k and xk−1 denote the desired and actual achievedend-effector positions.

Further, its derivative is

ek = xd,k− xk−1 (10)

Considering the detailed form of the desired end-effectortrajectory, both the position and orientation, the position error(9) and its derivative (10) can be rewritten as

ek =[

ep,keϕ,k

]=

[pd,k− p(qk−1)ϕd,k−ϕ(qk−1)

](11)

ek =[

ep,keϕ,k

]=

[pd,k− p(qk−1)ϕd,k− ϕ(qk−1)

](12)

where pd,k and ϕd,k denote the desired end-effector positionand orientation(Euler angles), p(qk−1) and ϕ(qk−1) are thecurrent ones, and ep,k and eϕ,k are the position and orientationerrors.

The parameters with notations of the time k, k− 1 hereis only to express clearly the relations between desired end-effector trajectory and actually solved trajectory, and thesenotations will be neglected in the next part for more gen-eral derivations. Also the definition of end-effector positionincludes the position and orientation, if no special declaration.

Combining (8), (9) and (10) together, it gives

xd − x+K(xd − x) = 0 (13)

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dx

dx

q q

K

x

e

Forward

Kinematics

Particular

Solution

Homogeneous

Solution

Fig. 1. The block scheme of improved CLIK algorithm with performancecriterion

e+Ke = 0 (14)

where K = diag{kpI3,kϕ I3} is a symmetric positive definitematrix.

It is not difficult to prove that (14) is exponentially stablewith the choice of K, implying that the error uniformlyconverges to zero.

Substituting (8) into (3), the joint velocity solution can bewritten as

q = J†a (q)(xd +K(xd − x)) (15)

The general joint velocity of a kinematically redundantmanipulator will have the form

q = J†a (q)(xd +K(xd − x))+(I− J†

a Ja(q))q0 (16)

IV. PERFORMANCE CRITERION OF JOINT LIMITAVOIDANCE

A. Redundancy resolution

In general, there are two tools that can be used to solve theinverse kinematics problem for redundant manipulators, one isredundancy resolution techniques and the other is performancecriteria. Redundancy plays an important role in the kinematiccontrol as redundant joints allow a manipulator to avoid jointlimits, singularities and obstacles, while following a desiredend-effector trajectory. A performance criterion is defined asa function of joint limits and its gradient is projected ontothe null space to obtain the self motion. Then modifying thenull-space allows for the generation of an infinite number ofsolutions to the original problem. The joint range availabilitycriterion was defined in [4] and another function reflecting theobjective of joint limit avoidance is presented in [10][11][12].Other performance criteria such as velocity limit avoidance,peak torque avoidance, obstacle avoidance, mathematical sin-gularity avoidance and other operational goal-base criteria areanalyzed in[13] [14].

As the redundancy resolution methods, pseudoinversemethod is applied in the above section. The homogeneous so-lution can be obtained by projecting an arbitrary n-dimensionalvector q0 to the null space of the Jacobian as shown in (5) and

(16), and a typical choice of the null space joint velocity vectoris

q0 = k0OΦ(q) = k0∂Φ(q)

∂q=

[∂Φ(q)

∂q1· · · ∂Φ(q)

∂qn

]T(17)

where k0 > 0, Φ(q) is a scalar objective function of the jointvariables and ∂Φ(q)

∂q is the vector function representing thegradient of Φ.

With this choice of the vector, the homogeneous solutionacts as a gradient optimization method which converges to alocal minimum of the cost function[16]. The cost function canbe selected to satisfy different objectives, such as joint limitavoidance in the following section.

B. Constraint-based criterion-joint limit avoidance

Utilization of redundancy for a particular task such as jointlimit avoidance is generally achieved through global or localoptimization of a performance criterion while satisfying thekinematic equations relating the end-effector trajectory to thejoint trajectory.

Every joint in a manipulator has its travel limits whichcannot be exceeded and any attempt to move a joint over itslimit can not only potentially damage the robot components,but also the workspace-boundary will make the Jacobianmatrix become rank deficient.

Joint limit avoidance is the most useful additional taskswhere joint variables of actual mechanisms are obviouslylimited by mechanical constraints. In actual implementations,if some joint variables computed by the inverse kinematicmodule exceed their limits, these joints will be fixed attheir extreme values which will restrict movements in certaindirections in task space.

C. Optimization function

In this section, under the configuration control approach,the criterion of joint limit avoidance will be formulated asa kinematic constraint function. The approach, defining thesecondary task as minimization of a desired cost function, willsolve the problem of joint limit avoidance and its performancewill be analyzed by using computer simulations in next part.

1) Optimization Method A: The first optimization method(OM) considered is the distance from mechanical joint limitsdefined in [4] as

Φ(q) =1n

n

∑i=1

(qi− qi

qiM−qim)2 (18)

where qiM(qim) denotes the maximum(minimum) limit for qiand qi the middle value of the joint range, thus redundancymay be exploited to keep the manipulator off joint limits.

2) Optimization Method B: As the quadratic form of equa-tion (18) is the most used function to be minimized, anotherfunction which reflects the objective of joint limit avoidancehas been proposed in[11] as:

Φ(q) = max(qi− qi

qiM−qim) = ‖ q− q

qM−qm‖p (19)

where p≥ 2.

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V. CASE STUDIES

In order to verify the algorithm discussed above and theeffectiveness of the OMs, two kinds of case studies are madewith computer simulations. The first one will be made on the3-DoF planar manipulator and the second is on the 4-DoFspatial manipulator. The desired trajectory is a closed curveand it will be followed twice in each simulation.

A. 3-DoF Manipulator for the repetitive planar trajectory

The parameters of the three-link planar manipulator aredefined as follows: link lengths (0.5m,0.4m,0.3m), jointslimits (−360deg ≤ q1 ≤ 360deg, −160deg ≤ q2 ≤ 160deg,−160deg ≤ q3 ≤ 160deg) and the initial configuration (180deg, −90 deg, −90 deg) corresponding to the initial positionp0 =

[ −0.2m 0.4m]T . The constant parameter in (17) is

k0 = 250.The end-effector of 3-DoF planar manipulator is only to

follow the motion trajectory, so one redundant DoF exists. Themotion trajectory is defined as:

pd(t) =[ −0.2cos(πt)

0.4+0.3sin(πt)

]0≤ t ≤ 4

Since the whole trajectory is in closed form, it will befollowed twice and it can be divided into two periods: the firstone is from the initial position(t = 0) to the end position(t = 2),and the second starts from the end position of the first periodsand ends at t = 4.

The simulation of free motion is carried out firstly withoutconsidering the redundancy in the null space and then theconstraints of joint limit avoidance with OM A and OM Bare taken into account.

No null space motion is considered in the first simulation asthe results are shown in Fig. 2(a), and it is easily found that thesecond joint and third joint pass over their lower joint limits,which is not possible in the real motion planning. Then thenull space motion is considered and the constraints of jointlimit avoidance with OM A and OM B are combined intothe algorithm as shown from Fig. 2(b) to Fig. 2(e). Fig. 2(b)shows the curves of joint variables as OM A is considered asthe objective function and the second and third joints do notexceed their joint limits in this case. After that, the OM B isutilized with p = 2, p = 4 and p = 6 respectively correspondingto Fig. 2(c), Fig. 2(d) and Fig. 2(e), which show all joints don’tviolate their joint limits in case of completing the task motion.In order to clearly compare the joint variables, the curves ofevery joint in four simulations are rearranged as shown in Fig.3. By comparing these figures in Fig.2 and Fig. 3 together,it is not difficult to find that the case of OM B with p = 6produces the best result.

As the desired trajectory is to be followed twice, the end-effector will return to its initial position at t = 2 and beginthe same tracking again. From the simulation results, it isfound that the joint variables don’t return their initial valueswhen no null space motion is considered as shown in Fig. 2(a).However, in the cases that OM A and OM B are considered,

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join

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Fig. 2. Five simulations of 3-DOF manipulator tracking the planar motiontrajectory

the joint variables will approximately be close to their initialvalues at t = 2 and the curves are basically symmetricalaccording to the central line t = 2. From Fig. 2 and Fig. 3,we can find that the cases of OM A and OM B with p = 6have better results and OM B with p = 6 is preferable forthat the third joint stays far away from its lower joint limit asshown in Fig. 3(c) in this case study.

B. 4-DoF Manipulator for the repetitive planar trajectory

The D-H parameters of the four-link planar manipulatorare defined as follows: where the lengths of four links are(a1 = 0.1m,a2 = 0.25m,a3 = 0.2m,a4 = 0.2m), a4 representsthe length from the last joint and the tip of the last link.

TABLE ITHE PARAMETERS OF JOINTS AND LINKS OF THE 4-DOF MANIPULATOR

(UNIT: DEG, M)

Link αi−1 ai−1 di θi1 0 0 0 q12 −90 a1 0 q23 0 a2 0 q34 0 a3 0 q4

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q3 join

t va

lue

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JLA with OM A

JLA with OM B p=4

JLA with OM B p=6

(c) q3

Fig. 3. Comparison of joint variables in four simulations of 3-DOFmanipulator tracking the planar motion trajectory

The motion trajectory is set to be in the XZ plane:

pd(t) =[

0.15(1− cos(πt/2))0.3914−0.15sin(πt/2)

]0≤ t ≤ 8

and obviously there are two redundant DoFs. The trajectory isin closed form and will be followed up twice, which is similarwith the first case study.

The initial configuration is (90 deg, −90 deg, −90 deg,45 deg) corresponding to the initial location of the end-effector on the XZ plane p0 =

[0 0.3914

]T . Note that theproper choice of initial configuration is important since manyconfigurations will flow into the kinematic singularities which

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Fig. 4. Three simulations of 4-DOF manipulator tracking the planar motiontrajectory

we want to avoid. Joints limits are(−360deg ≤ q1 ≤ 360deg,−120deg ≤ q2 ≤ 90deg, −100deg ≤ q3 ≤ 90deg, −160deg ≤q4 ≤ 160deg).

Similarly, the first simulation of free motion is carried outwithout considering the redundancy in the null space and thesimulation results are shown in Fig. 4(a), in this case, the thirdjoint variable violates the lower joint limit and the first andsecond joints variables are very close to the lower limit andupper limit respectively.

Then the constraints of joint limit avoidance with OM A andOM B are taken into account. Here, in order for comparison,two cases are shown, the case of OM A and the case ofOM B with p = 6 corresponding to Fig. 4(b) and Fig. 4(c)respectively. Fig. 4(b) shows the simulation results with OMA that the third joint does not violate the lower limit any moreand all the joint variables are located in the range of theirlimits, the same result as shown in Fig. 4(c). The variables ofthe third and fourth joints in three simulations are rearrangedas shown in Fig. 5. From Fig.4 and Fig. 5 together, we foundthat the joint variables can not return their initial values inthe free motion case, but they can return their initial valuesin the second period(4≤ t ≤ 8), since it is not difficult to findthat joint variables have the same value at t = 4 and t = 8.However, in the first period, the values at t = 4 is not equalto the initial values (t = 0) for two possible reasons, one isthe more redundant DoFs and the other is the choice of initialjoint values.

VI. CONCLUSION AND FUTURE WORK

A. Conclusion

This paper firstly presents the redundancy analysis of theredundant manipulator via the pseudoinverse approach. Theimproved CLIK algorithm is utilized in order to avoid the

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Fig. 5. Comparison of the third and fourth joint variables in three simulationsof 4-DOF manipulator tracking the planar motion trajectory

numerical drift of open-loop integration of joint variablesand the redundancy resolution is optimized by the constraintcriteria of joint limits.

Two optimization methods are applied and the correspond-ing simulations are made on both the 3-DoF and 4-DoF re-dundant manipulators tracking the repetitive planar trajectoriesrespectively. The results show that the joint variables will notviolate the joint limits if both OM A and OM B of jointlimit avoidance are considered whereas it will occur if notany optimization method is considered. In the case of freemotion, the joints can not repeat their variables although therepetitive trajectory is given. However, the joints can repeattheir variables partly in the cases of constrained motion andthe cases of OM A and OM B with p = 6 perform better thanothers.

B. Future work

In this paper, redundancy resolution is analyzed only at thevelocity level and it should be performed at the accelerationlevel in the next step. More performance criteria such as veloc-ity limit avoidance, peak torque avoidance, obstacle avoidanceand mathematical singularity avoidance will be added into

the algorithm and the experiments on the real redundantmanipulator (Powercube robots in the lab) will be carried outto verify the algorithm.

REFERENCES

[1] R. Hooper and D. Tesar, “Motion coordination based on multiple perfor-mance criteria with a hyper-redundant serial robot example,” IEEE Int.Symp. on Intelligent Control, Monterey, California, 1995, pp. 133–138.

[2] C. Carignan, “Trajectory Optimization for Kinematically RedundantArms,” Journal of Robotic Systems, Vol. 6(5), pp. 221–248, 1991.

[3] D. E. Whitney, “Resolved Motion Rate Control of Manipulator andHuman Prosthesis,” IEEE Trans. Man-Machine Systems, Vol. 10, pp. 47–53, 1969.

[4] A. Liegeois, “Automatic Supervisory Control of the Configuration andBehavior of Multibody Mechanisms,” IEEE Trans. Systems, Man, andCybernetics, Vol. 7(12), pp. 868–871, 1977.

[5] A. De Santis, B. Siciliano and L. Villani, “A unified fuzzy logic approachto trajectory planning and inverse kinematics for a fire fighting robotoperating in tunnels,” Intelligent Service Robotics, Vol. 1(1), pp. 41–49,2008.

[6] L. Sciavicco and B. Siciliano, “A solution algorithm to the inversekinematic problem for redundant manipulators,” IEEE J. of Robotics andAutomation, Vol. 4(4), pp. 303–310, 1988.

[7] C. W. Wampler, II, “Manipulator inverse kinematic solutions based onvector formulations and damped least-squares methods,” IEEE Trans. onSystems, Man, and Cybernetics, Vol. 16(1), pp. 93–101, 1986.

[8] D. N. Nenchev, Y. Tsumaki and M. Uchiyania, “Adjoint Jacobian closed-Loop kinematic control of robots,” IEEE Int. Conf. on Robotics andAutomation, Minnesota, USA, 1996, pp. 1235–1240.

[9] L. Zlajpah, “Simulation of n-R planar manipulators,” Simulation Practiceand Theory, Vol. 6(3), pp. 305–321, 1998.

[10] C. A. Klein and C. H. Hung,“Review of pseudoinverse control for usewith kinematically redundant manipulators,” IEEE Trans. on Systems,Man, and Cybernetics, vol. 13, pp. 245–250, 1983.

[11] C. A. Klein, “Use of redundancy in design of robotic systems,” RoboticsResearch: The Second Int. Symposium, ed. H. Hanafusa and H. Inoue,pp. 207–214, Cambridge, Mass.: MIT Press, 1984.

[12] A. A. Maciejewski and C. A. Klein, “The singular value decomposi-tion: Computation and application to robotics,” Int. Journal of RoboticsResearch, vol. 8(6), 1989.

[13] C. Kapoor, M. Cetin and D. Tesar, “Performance Based RedundancyResolution with Multiple Criteria,” ASME Design Engineering TechnicalConference, Atlanta, Georgia, 1998, pp. 1-6.

[14] C. Pholsiri and D. Tesar, “Task-based Decision Making and Control ofRobotic Manipulations,” Ph.D Dissertation, The University of Texas atAustin, 2004.

[15] J. De Schutter, T. De Laet, J. Rutgeerts, W. Decre, R. Smits, et al,“Constraint-based Task Specification and Estimation for Sensor-BasedRobot Systems in the Presence of Geometric Uncertainty,” The Int. J.Robotics Research, Vol. 26(5), pp. 433–455, 2007.

[16] R. V. Patel and F. Shadpey, Control of Redundant Robot Manipulators,Springer-Verlag, Berlin Heidelberg, 2005.

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