Different-Level Redundancy-Resolution and Its Equivalent Relationship Analysis for
Different-Level Redundancy-Resolution and Its Equivalent Relationship Analysis for Robot Manipulators Using Gradient-Descent and
Zhang et al.’s Neural-Dynamic Methods
Binghuang Cai,Member,IEEE,and Yunong Zhang,Member,IEEE
Abstract—To solve the inverse kinematic problem of redundant robot manipulators,two redundancy-resolution schemes are in-vestigated:one is resolved at joint-velocity level,and the other is resolved at joint-acceleration level.Both schemes are reformulated as a quadratic programming(QP)problem.Two recurrent neural networks(RNNs)are then developed for the online solution of the resultant QP problem.The?rst RNN solver is based on the gradient-descent method and is termed as gradient neural net-work(GNN).The other solver is based on Zhang et al.’s neural-dynamic method and is termed as Zhang neural network(ZNN). The computer simulations performed on a three-link planar robot arm and the PUMA560manipulator demonstrate the ef?cacy of the two redundancy-resolution schemes and two RNN QP–solvers presented,as well as the superiority of the ZNN QP-solver compared to the GNN one.More importantly,the simulation results show that the solutions of the two presented schemes?t well with each other,i.e.,the two different-level redundancy-resolution schemes could be equivalent in some sense.The theo-retical analysis based on the gradient-descent method and Zhang et al.’s neural-dynamic method further substantiates the new?nd-ing about the different-level redundancy-resolution equivalence.
Index Terms—Equivalence,neural dynamics,quadratic pro-gramming(QP),redundancy resolution,robot arms.
I.I NTRODUCTION
I N RECENT YEARS,robotic researchers have focused on
solving a variety of tasks requiring sophisticated motion in complex or dangerous environments[1]–[6].Robots could thus have good prospects in numerous?elds of science and engineering,which motivates robot designers to improve the functionality and?exibility of robot manipulators[1],[3]–[6]. Redundant robot manipulators are therefore proposed,de-
Manuscript received December3,2009;revised July26,2010and September14,2010;accepted December28,2010.Date of publication January 30,2011;date of current version March30,2012.This work was supported by the National Natural Science Foundation of China under Grants61075121and 60935001,by the Program for New Century Excellent Talents in University under Grant NCET-07-0887,by the Yat-sen Innovative Talents Cultivation Program for Excellent Tutors,and by the Fundamental Research Funds for the Central Universities of China.
B.Cai was with the School of Information Science and Technology,Sun Yat-sen University,Guangzhou510006,China.He is now with the Centre for Autonomous Systems,University of Technology,Sydney,NSW2007,Australia (e-mail:bhcai@https://www.wendangku.net/doc/6715506120.html,).
Y.Zhang is with the School of Information Science and Technology,Sun Yat-sen University,Guangzhou510006,China(e-mail:ynzhang@https://www.wendangku.net/doc/6715506120.html,). Color versions of one or more of the?gures in this paper are available online at https://www.wendangku.net/doc/6715506120.html,.
Digital Object Identi?er10.1109/TIE.2011.2106092veloped,and widely investigated[1]–[3],[7]–[20].A robot
manipulator is said redundant when more degrees-of-freedom
(DOF)are available than the minimum number of DOF re-
quired to perform a given end-effector primary task[1],[2].
The potential ef?cacy of such a redundant manipulator is
determined by the DOF number(as well as the manipulator’s
structure and its control scheme)in some sense.A manipulator
with just enough DOF for a speci?c end-effector task(termed
as a nonredundant manipulator)may not have the ability to
achieve alternative goals when completing the primary task due
to the solution uniqueness and/or isolatedness required.On the
other hand,redundant manipulators have a wider operational
space and an extra DOF to meet a number of functional con-
straints because an in?nite number of joint con?gurations could
be feasible solutions to the problem[1],[2],[7],[8].Human
arms,elephant trunks,and snakes are examples of this kind of
redundant system[1],[2].
A fundamental issue in operating redundant manipulators
is the online redundancy-resolution(i.e.,inverse-kinematic or
motion-planning)problem[1],[2].The general description of
this robotic issue is that,given the desired or user-de?ned Carte-
sian trajectory r(t)∈R m for the manipulator’s end-effector to
track,we need to generate the corresponding joint trajectory
θ(t)∈R n online or in real time t.Note that m issue.Robotic research shows that the redundancy-resolution problem could be solved via the optimization of suitable per- formance indices[1],[2],[18],[21]or based on task space aug- mentation(e.g.,extended Jacobian method)[1],[2],[22].The conventional solution to such a redundancy-resolution method is the pseudoinverse-type formulation[1],[3],[10],[14],[18]– [29].Most recent research shows that the redundancy-resolution problem might be better solved via online optimization techniques[2],[7]–[13],[15]–[17],[30],[31]such as quadratic programming(QP)and recurrent neural networks(RNNs). Due to the parallel distributed computational nature,RNNs have been developed for real-time solution to redundancy- resolution problems[2],[7],[8],[10],[17].In particular,a neural network approach incorporating the pseudoinverse net- work and the linear programming neural network was applied to robot kinematic planning[9],[10].An improved two-layered primal-dual neural network(PDNN)was presented in[11]and [15]to minimize online the weighted joint velocity.To reduce 0278-0046/$26.00?2011IEEE network complexity and to increase computational ef?ciency, an early dual neural network(DNN)model was developed for the kinematic planning of redundant manipulators[16],[31].As the DNN may encounter a matrix-inversion bottleneck(which is time consuming and dif?cult even in hardware computation), a PDNN based on linear variational inequalities was developed for the online resolution of the manipulators’redundancy,with a simple piecewise-linear dynamics and without any matrix inversion[7],[8],[30],[31].Note that most RNNs mentioned earlier are developed based on the gradient-descent method [14],[32]–[38]. To take full advantage of the redundancy,various compu-tational schemes have thus been proposed and developed for robot manipulators in the past three decades.These schemes select an optimal joint-space solution from an in?nite number of possible solutions(or trajectories),given the primary end-effector task of tracking a desired workspace trajectory in terms of position and orientation.Such redundancy-resolution schemes are developed at different levels,such as the joint-velocity(termed as the?rst-order differential)level[1],[2], [7]–[14],[20],[22]–[27],[31]or the joint-acceleration(termed as the second-order differential)level[1]–[3],[7]–[9],[16]–[19],[21],[28]–[30].Is there any relationship among the redundancy-resolution schemes at different levels(e.g.,at joint-velocity and joint-acceleration levels)?Could the redundancy resolution at different levels be equivalent in some sense? In this paper,two redundancy-resolution schemes(one at the joint-velocity level and the other at the joint-acceleration level) are presented for redundant manipulators.Both two schemes are reformulated as a QP problem that is subject to an equality constraint.Two RNNs,namely,gradient neural network(GNN) and Zhang neural network(ZNN),are then developed for the online solution of the resultant QP problem.The computer simulations performed on a three-link planar robot arm and on a PUMA560manipulator demonstrate the ef?cacy of the two neural-QP-based redundancy-resolution schemes and the superiority of the ZNN QP-solver as compared to the GNN one. The simulation results also show the equivalent relationship be-tween the two presented different-level redundancy-resolution schemes,which is theoretically proved by the gradient-descent method[14],[32]–[38]and by the Zhang et al.’s neural-dynamic method[2],[38]–[41]. The remainder of this paper is organized in?ve sections. Section II presents the two different-level redundancy-resolution schemes and their QP formulations.In Section III, the GNN and ZNN solvers are developed for handling the resultant QP problem.Section IV provides the computer sim-ulation results based on a three-link planar robot arm and on a PUMA560manipulator.In Section V,the relationship between the two presented different-level schemes is further analyzed via two methods.Section VI concludes this paper with?nal remarks.The main contributions of this paper are as follows. 1)The two presented redundancy-resolution schemes are both reformulated as a quadratic program,which could be solved readily by two novel RNNs,i.e.,GNN and ZNN. Such neural-dynamic models are generated by the new methods for online QP problem solving. 2)The computer simulations are conducted based on the three-link robot arm and on the PUMA560manipulator. The simulation results,including the tracking examples of the elliptical and triangular paths,substantiate well the ef?cacy and equivalence of the two presented different-level redundancy-resolution schemes. 3)The equivalent relationship between the velocity-and acceleration-level redundancy resolutions is established using suitable performance indices,with the theoretical analysis provided via the gradient-descent method and Zhang et al.’s neural-dynamic method.To the best of the authors’knowledge,such a redundancy-resolution relationship between two different levels has not been investigated before by others,and there is very little(or even no)work published on the solution equivalence. II.P ROBLEM F ORMULATION The end-effector position/orientation vector r(t)∈R m in Cartesian space is related to the joint space vectorθ(t)∈R n through a forward-kinematic equation r(t)=f(θ(t)).(1) Since(1)is usually dif?cult(if not impossible)to solve due to the nonlinearity and redundancy of mapping f(·),such a problem is often solved at the velocity or acceleration level. Differentiating(1)with respect to time t yields the relationship between the Cartesian velocity˙r(t)and the joint-velocity˙θ(t) ˙r=J(θ)˙θ(2) where J(θ)=?f(θ)/?θ∈R m×n is the Jacobian matrix.For a given manipulator,the analytical expression of J(θ)with respect to argumentθcould be determined[12],[13],[16]. Differentiating(2)could further yield the relationship between the end-effector acceleration¨r(t)and the joint-acceleration ¨θ(t),i.e., ¨r a=J(θ)¨θ(3) where¨r a:=¨r?˙J(θ)˙θ∈R m and˙J(θ)is the time derivative of the Jacobian matrix J(θ).Equations(1)–(3)are all under-determined,admitting an in?nite number of solutions,because the manipulator is redundant(i.e.,m The conventional pseudoinverse-type solutions to(2)and (3)are normally formulated as one minimum-norm partic-ular solution plus a homogeneous solution,i.e.,˙θ=P˙r+ (I?P J)z for(2)and¨θ=P¨r a+(I?P J)z for(3),where P=J T(JJ T)?1∈R n×m denotes the pseudoinverse of J and z∈R n is an arbitrary vector selected as gradients of some performance indices.However,the pseudoinverse-type formulation may have some weaknesses(e.g.,high computa-tional complexity due to high calculation load of the matrix pseudoinverse)[2],[7].Note that the extended Jacobian method (which could be viewed as a special case of the pseudoinverse-type method)may also encounter high computational load due to the inversion of the extended Jacobian matrix.Moreover, some other optimization methods have been presented for the redundancy-resolution problem,e.g.,the global optimization technique using the calculus of variations,which results in globally stable optimal solutions[2],[16],[21].Since such a method requires a complete trajectory information in advance and is very computationally intensive,it is not well suited for real-time motion control,usually requiring a continuous trajectory modi?cation based on the sensory feedback[2],[16]. In this paper,we consider the quadratic program formulation for handling the redundancy-resolution problem.The following redundancy-resolution schemes(with one at the velocity level and the other at the acceleration level)are presented,inves-tigated,and formulated?nally as solvable QPs via effective RNNs,which could be applicable and ef?cient in real-time motion control of robot manipulators. A.Velocity-Level Redundancy-Resolution Scheme Using the simplest effective performance index ˙θ 22/2,the following so-called minimum-velocity-norm(MVN)scheme has been widely adopted by most researchers for the redundancy-resolution problem[1],[2],[7],[8]: minimize 1 2 ˙θ 22(4) subject to J(θ)˙θ=˙r(5) where · 2denotes the two-norm of a vector and(5)is exactly(2),which relates to the end-effector’s primary task.As performance index ˙θ 22/2=˙θT˙θ/2,the aforementioned MVN scheme(4)and(5)could be expressed as the time-varying QP that is subject to an equality constraint,as shown in the following: minimize 1 2x T W x+q T x(6) subject to Cx=d(7) where x:=˙θ∈R n,W:=I∈R n×n,C:=J∈R m×n,d:=˙r∈R m,and q:=0∈R n. B.Acceleration-Level Redundancy-Resolution Scheme We present the following acceleration-level redundancy-resolution scheme for the motion planning of redundant robots: minimize 1 2 ¨θ+λ˙θ 22(8) subject to J(θ)¨θ=¨r a(9) where design parameterλ>0should be large enough (e.g.,being20Hz).The minimization of ¨θ+λ˙θ 22/2 is equivalent to minimizing¨θT¨θ/2+λ˙θT¨θat the joint-acceleration level because ¨θ+λ˙θ 22/2=(¨θ+λ˙θ)T(¨θ+λ˙θ)/2=¨θT¨θ/2+λ˙θT¨θ+λ2˙θT˙θ/2.Thus,the aforemen-tioned acceleration-level scheme(8)and(9)could be formu-lated as minimize 1 2x T W x+q T x(10) subject to Cx=d(11)where,in the acceleration scheme,x:=¨θ∈R n,W:=I∈R n×n,C:=J∈R m×n,d:=¨r a∈R m,and q:=λ˙θ∈R n. From the aforementioned two sections,we can see that the QP formulations[i.e.,(6)and(7)and(10)and(11)]resulting from the presented velocity-and acceleration-level redundancy-resolution schemes[i.e.,(4)and(5)and(8)and(9)]are of the same mathematical form but with different de?nitions of de-cision vectors and coef?cient matrices/vectors.For description convenience,QP(10)and(11)is considered hereafter as the representation of such two QPs,which will be solved by two RNNs(i.e.,GNN and ZNN)in the ensuing section. III.N EURAL QP S OLVERS The formulation in the previous section shows that coef?-cient matrix C(i.e.,Jacobian matrix J,being of full row rank) and coef?cient vector d,together with their time derivatives, could be accurately estimated.Due to the positive de?niteness of matrix W=I,QP(10)and(11)is strictly convex,which guarantees the solution uniqueness of the problem[42].Hence, based on the preliminary results on equality-constrained opti-mization problems[9],[43],the Lagrangian function related to QP(10)and(11)could be de?ned as L(x,η)=x T W x/2+q T x+ηT(Cx?d) whereη∈R m denotes the Lagrange multiplier vector.Solving QP(10)and(11)could then be done by zeroing equations ?L(x,η) ?x =W x+q+C Tη=0 ?L(x,η) ?η =Cx?d=0 which could be further written as Qy=u(12) with Q:= W C T C0 ∈R(n+m)×(n+m) y:= x η ∈R n+m u:= ?q d ∈R n+m. Thus,the solution of QP(10)and(11)becomes the solution of linear system(12),i.e.,when(12)is solved,the optimal solution of QP(10)and(11)can be achieved from the?rst n elements of y. A.GNN Solver By following the gradient-descent method[14],[32]–[38] and the RNN design method[2],[9],[11]–[13],[15],[16],the GNN model could be established in solving QP(10)and(11), which has been reformulated as linear system(12).The design procedure of the GNN solver is shown as follows. First,a scalar-valued norm-based energy function (y)= Qy?u 22/2∈R Fig.1.Block diagrams of (a)GNN (13)and (b)ZNN (15)for QPs (6)and (7)and (10)and (11),as well as redundancy-resolution schemes (4)and (5)and (8)and (9). is de?ned such that its minimum point corresponds to the theoretical solution of linear system Qy =u [i.e.,(12)]. Next,an algorithm is designed to evolve along a descent direction of this energy function until the minimum point is reached.The typical descent direction is the negative of the gradient of energy function (y ),i.e., ? ? (y )=?? Qy ?u 22/2=?Q T (Qy ?u ).Finally,by using the aforementioned negative gradient to construct the neural network for linear system (12),we could have the following GNN model which solves QP (10)and (11): ˙y =?γQ T Φ(Qy ?u ) (13) where design parameter γ>0,being a set of reciprocals of the capacitance parameters used to scale the convergence rate,should be set as large as the hardware permits or should be set appropriately for simulative/experimental purposes [43].In addition,Φ(·):R n +m →R n +m denotes an activation func-tion processing array,in which each scalar-valued processing unit φ(·)should be a monotonically increasing odd activation function.For example,φ(·)could be linear activation function φ(y i )=y i or power-sigmoid activation function φ(y i )= y p i ,if |y i | 11+exp(?ξ)1?exp(?ξ) · 1?exp(?ξy i )1+exp(?ξy i ), otherwise with suitable ξ 2and p 3(being an odd integer).The power-sigmoid activation function with ξ=4and p =3is used in the ensuing simulation section.The block diagram of the GNN model (13)for QP (10)and (11)is shown in Fig.1(a). B.ZNN Solver To monitor the solving process of QP (10)and (11)via linear system (12),we could ?rst de?ne the following vector-valued error function (rather than the scalar-valued norm-based energy function used in the gradient-descent method): e (t )=Qy ?u ∈R n +m of which each element could be positive,negative,or lower unbounded.Then,to make each element e i (t )of error vector e (t )∈R n +m converge to zero,the following ZNN design formula can be adopted [2],[38]–[41]: de (t ) dt =?γΦ(e (t ))(14) where design parameter γ>0and activation function array Φ(·)are the same as the ones in (13). Expanding ZNN design formula (14)leads to the following ZNN model depicted in an implicit-dynamic equation: Q ˙y =?˙Qy ?γΦ(Qy ?u )+˙ u (15) with the block diagram shown in Fig.1(b).The convergence properties of ZNN (15)are shown in the following theorem.Theorem 1:Consider QP (10)and (11).If a monotonically increasing odd activation function array Φ(·)is used,state vector y (t )of ZNN (15),starting from any initial state y (0)∈R n +m ,could globally converge to the unique theoretical solu-tion of linear system (12),with its ?rst n elements constituting the optimal solution to QP (10)and (11).Moreover,if the linear activation function is used,the global exponential convergence can be achieved for (15)with rate γ.If the power-sigmoid ac-tivation function is used,superior convergence can be achieved for (15)compared to the linear activation function situation. Proof:The proof can be generalized from [2],[39],and [40]by taking into account the Lyapunov function candidate v =e T (t )e (t )/2and by using the Lyapunov theory [9],[44]. Remark 1:In the hardware implementation of the ZNN model,the differentiation errors about matrix Q and/or vector u ,as well as the dynamics implementation error (collectively termed as the model implementation error),appear most fre-quently.By following [45],if the implementation error is bounded,the computational error of ZNN model (15)is also up-per bounded,with its maximum steady-state error proportional to the reciprocal of design parameter γ.When design parameter γtends to be positive in?nity,the steady-state computational error can be decreased to zero.A detailed theoretical analysis could be found in [45].Thus,the presented ZNN QP-solver is robust for the inverse kinematic problem solving with respect to the bounded differentiation errors.C.Solver Comparisons The dynamic equations of GNN and ZNN for QP (10)and (11)are developed in (13)and (15),where the block diagrams are shown,respectively,in Fig.1(a)and (b).Fig.1shows the different structures of such two neural network models and also describes the operations for generating the system coef?cients Fig.2.Motion trajectories of the three-link planar robot arm when its end-effector tracks an elliptical path. (i.e.,Q and u)of the online neural network solvers.These operations could both be done via matrix/vector augmentation. The circuits realizing such neural-dynamic solvers consist of summators,multipliers,integrators,differentiator,matrix trans-poser,and weighted connections[2],[39],[40],[43],[46].Note that the gradient-type neural networks have been developed intrinsically for optimization problems with constant coef?-cients[2],[42].A much faster convergence rate of GNN is thus required for handling the time-varying situation,very stringent restrictions may then be needed for hardware realization,and the GNN solution precision may also be sacri?ced.On the other hand,ZNN is proposed originally for solving time-varying problems[2],[38]–[41].The main differences of GNN(13)and ZNN(15)for QP(10)and(11)may thus lie in the following facts. 1)The design of GNN(13)is based on the elimination of the scalar-valued norm-based energy function Qy?u 22, while the design of ZNN(15)is based on the elimina-tion of every element of the vector-valued error function e(t)=Qy?u. 2)GNN(13)is depicted in an explicit dynamics,i.e.,˙y= ···.In contrast,ZNN(15)is depicted in an implicit dynamics,i.e.,Q˙y=···,which coincides well with the systems in nature and in practice(e.g.,analog electronic circuits and mechanical systems,owing to Kirchhoff’s and Newton’s laws,respectively[2],[39],[40],[46]). 3)GNN(13)has not exploited the time derivative infor- mation of the coef?cient matrices and vectors.Thus,it may be less effective in solving time-varying QP(10)and (11).In contrast,ZNN(15)could methodologically and effectively exploit the time derivative information of the coef?cient matrices and vectors(e.g.,˙Q and˙u)during the real-time solving process of QP(10)and(11). IV.C OMPUTER S IMULATION S TUDIES In this section,the two presented different-level redundancy-resolution schemes[i.e.,(4)and(5)and(8)and(9)]are simulated based on a three-link planar robot arm and on the Unimation PUMA560manipulator.The end-effector of the three-link planar robot arm is expected to track an ellipse and a triangle,while the PUMA560end-effector is expected to track a straight-line path.Note that the simulation is conducted by MATLAB7.1,performed on a personal digital computer with a Pentium IV3.20GHz CPU,1-GB memory,and Windows XP Professional operating system. A.Elliptical-Path Tracking Task In this section,the simulation is performed based on the three-link planar robot arm,which has three DOF in the2-D operational space and workspace.Thus,the Jacobian matrix J(θ)is2×3in dimension,with one redundant degree avail-able.The Jacobian matrix could be obtained via a similar derivation for the?ve-link planar robot manipulator in[13]. The three-link manipulator’s end-effector is expected to track an elliptical path,with the major radius being0.4m and the minor radius being0.2m.Task duration T=10s,initial joint vectorθ(0)=[π/12,π/12,π/6]T rad,and design parameters λ=20Hz andγ=106.The simulation results are shown in Figs.2and3and Tables I and II. Fig.2shows the motion trajectories of the three-link planar robot arm operating in the2-D plane,which is synthesized by velocity-level scheme(4)and(5)and GNN QP-solver(13).The arrow in the?gure shows the motion direction.The motion trajectories of the other cases[i.e.,scheme(8)and(9)using GNN(13),scheme(4)and(5)using ZNN(15),and scheme (8)and(9)using ZNN(15)]are very similar to those shown in Fig.2and are thus omitted due to space limitation.As observed from the simulation results,the motion trajectories of the robot end-effector are suf?ciently close to the desired ellipse,with the maximum Cartesian positioning errors all less than1.1×10?4 m(as shown in Table I).This substantiates the ef?cacy of the presented different-level redundancy-resolution schemes and their resultant QP,as well as the GNN and ZNN QP-solvers. More speci?cally,Table I shows the maximum position- ing errors of the three-link planar robot arm’s end-effector. From the table,we can see that the maximum positioning error of the solution by the ZNN solver is smaller than the ones by the GNN solver.In addition,by using the ZNN solver,the maximum positioning errors are similar between the velocity- and acceleration-level schemes(i.e.,1.0×10?6m versus7×10?7m).By using the GNN solver,such maximum errors are quite different between the different-level schemes(i.e.,2.5×10?6m versus1.1×10?4m).This shows that the ZNN solver is more effective and accurate for inverse kinematic problem solving than the GNN solver,especially in the acceleration-level case. Table II shows the simulation time of the elliptical-path tracking task via velocity-level scheme(4)and(5)and acceleration-level scheme(8)and(9)using GNN solver(13) and ZNN solver(15).The simulation time corresponds to the running time of the computer used to complete the simulation of the whole system(including the robot,neural network, computing procedure display,and so on)for a speci?c robot task,which is different from the task duration(termed as the task execution time),because the task duration is the operation time of the manipulator completing a given end-effector task (e.g.,10s for the elliptical-path tracking task).In Table II, we can observe that the simulation time of the ZNN solver Fig.3.(a)Joint-angle pro?les (in radians)and (b)joint-velocity pro?les (in radians per second)of the three-link planar robot arm when its end-effector tracks an ellipse,where the solid curves with circle markers correspond to the joint pro?les synthesized by velocity-level scheme (4)and (5)and the dash-dotted curves with square markers correspond to the ones synthesized by acceleration-level scheme (8)and (9).TABLE I M AXIMUM P OSITIONING E RRORS (IN M ETERS )OF THE T HREE -L INK P LANAR R OBOT A RM ’S E ND -E FFECTOR W HEN T RACKING AN E LLIPTICAL P ATH VIA V ELOCITY -L EVEL S CHEME (4)AND (5)AND A CCELERATION -L EVEL S CHEME (8)AND (9)BY U SING GNN (13)AND ZNN (15) is also smaller than that of the GNN solver,especially in the acceleration-level resolution case.This again substantiates the superiority of the ZNN model for redundancy-resolution problem solving,especially at the joint-acceleration level. According to the reviewers’inspiring comments,we have tested the algorithmic time at each iteration of the QP solvers.The results show that the calculation time at each iteration of GNN model (13)is about 2.4×10?4s,while the one of ZNN model (15)is about 1.1×10?4s.Such computational intervals are suitable for real-time robot control.This means that the proposed method could be applicable to online inverse kine-matic problem solving in practice.As the presented schemes are expected to be ?nally implemented in hardware,with a parallel processing nature,it would hopefully be applicable in future practice for real-time redundancy-resolution computation. We have also used the pseudoinverse method to solve the inverse kinematic problem for the elliptical-path tracking task.The simulation results show that the solution by the pseudoin-verse method could follow the prescribed elliptical path,with the calculation time at each iteration being about 4.1×10?4s.By comparing the simulation results mentioned earlier,we can see that the QP-based redundancy-resolution schemes (espe-cially via the ZNN solver)are much more ef?cient and effective than the traditional pseudoinverse method. Fig.3shows the joint-angle and joint-velocity pro?les when the three-link robot’s end-effector tracks the elliptical path,which is synthesized by the two presented redundancy-resolution schemes using the ZNN solver.The pro?les gener-ated by the GNN solver are very similar to the ones in Fig.3and are thus omitted due to space limitation.As shown in Fig.3,the joint pro?les synthesized by velocity-level scheme (4)and (5)and acceleration-level scheme (8)and (9)?t very well with each other.This demonstrates that the solutions of the presented different-level redundancy-resolution schemes could be equivalent using suitable performance indices.The theoretical analysis on such an equivalent relationship would be given in Section V. B.Triangular-Path Tacking Task In this section,as another illustrative example,the three-link robot’s end-effector is expected to move along a desired isosceles-right-triangle path,with the side length being 0.8m.The motion duration T is 30s,and the initial joint variable θ(0)=[3π/4,?π/2,?π/4]T rad.Design parameters λand γare set in the same manner as in the elliptical example,i.e.,λ=20Hz and γ=106.Only ZNN (15)is considered in this triangular example,with the simulation results shown in Figs.4and 5. The motion trajectories of the three-link planar robot manip-ulator synthesized by acceleration-level redundancy-resolution scheme (8)and (9)are shown in Fig.4.The arrow appearing in the ?gure shows the motion direction.The motion trajectories synthesized by velocity-level scheme (4)and (5)are very similar to the ones in Fig.4and are thus omitted due to space limitation.In Fig.4and in the related simulation data,we can see that the motion trajectory of the robot end-effector is very close to the desired isosceles right triangle,with the maximum Cartesian positioning error less than 1×10?6m.This substan-tiates once more the ef?cacy of velocity-level scheme (4)and (5)and acceleration-level scheme (8)and (9),as well as their QP formulations (6)and (7)and (10)and (11)for the inverse-kinematic solution of the redundant manipulators. Fig.5shows the transient behaviors of the three-link planar robot’s joint angles and joint velocities for the triangular path tracking task.In the ?gure,we could see that the joint pro?les synthesized by the two presented different-level schemes ?t very well with each other.This once again veri?es that the redundancy resolution at velocity and acceleration levels could be equivalent for robot manipulators. TABLE II S IMULATION T IME (IN S ECONDS )OF THE E LLIPTICAL -P ATH T RACKING T ASK VIA V ELOCITY -L EVEL S CHEME (4)AND (5)AND A CCELERATION -L EVEL S CHEME (8)AND (9)BY U SING GNN S OLVER (13)AND ZNN S OLVER (15) Fig.4.Motion trajectories of the three-link planar robot arm when its end-effector tracks a triangular path. C.Straight-Line Path Tracking Task In this section,the simulation is performed based on the six-joint PUMA560manipulator.For testing purposes,we consider only the positioning of the end-effector,and the PUMA560manipulator then becomes a functionally redundant manipu-lator,with the associated Jacobian matrix J (θ)being 3×6in dimension.The PUMA560’s end-effector is expected to track a straight-line path,with the length being 1m.The motion duration T is 10s,and the initial joint variable θ(0)=[0,0,0,0,0,0]T rad.Design parameters λ=60Hz and γ=106.Only ZNN (15)is considered in this PUMA560example,with the simulation results shown in Figs.6and 7. The motion trajectories of the PUMA560manipulator syn-thesized by acceleration-level redundancy-resolution scheme (8)and (9)are shown in Fig.6.Similarly,the motion trajectories synthesized by velocity-level scheme (4)and (5)are omitted due to space limitation.In Fig.6and in the related simulation data,we can see that the motion trajectory of the robot end-effector is very close to the desired straight-line path,with the maximum Cartesian positioning error less than 1.1×10?6m.This substantiates once more the ef?cacy of velocity-level scheme (4)and (5)and acceleration-level scheme (8)and (9),as well as their QP formulations (6)and (7)and (10)and (11)for the redundancy resolution of the robot manipulators.Moreover,in Fig.7,we could see that the joint pro?les synthesized by the two presented different-level schemes ?t very well with each other.This once again veri?es that the redundancy resolution at the velocity and acceleration levels could be equivalent for robots. In summary,the aforementioned computer simulation and veri?cation results based on the examples of the three-link pla-nar robot arm’s elliptical and triangular paths tracking,as well as PUMA560’s straight-line path tracking,have demonstrated the ef?cacy of the presented velocity-and acceleration-level redundancy-resolution schemes,as well as the GNN and ZNN QP-solvers.The simulations also show the superiority of the ZNN model compared to the GNN model.More importantly,the simulation results demonstrate the equivalent relationship of the solutions synthesized by the two different-level schemes for solving the redundancy-resolution problem of the robot manipulators. As suggested by the respected reviewer,we would like to discuss some points about how to handle the mechanical con-straints in robotic systems via the proposed methods as follows.Note that mechanical constraints (especially,joint physical limits)exist in robotic systems,which should be considered into the control system of the practical robot system.As for the joint physical limits,the presented QP-based scheme formulation could easily handle such mechanical constraints as the bound constraints of the QP formulation [2],[7],[8].Currently,a small team of our research group works on the application of our QP-based redundancy-resolution method into a practical six-link robot,with the mechanical constraints considered.The pre-liminary results have also shown the feasibility of the method in practice.The systematical study on practical applications will be conducted,summarized,and presented as one of our future main research directions. V .E QUIVALENT R ELATIONSHIP A NALYSIS In this section,the neural network design methods (i.e.,gradient-descent method and Zhang et al.’s neural-dynamic method)used in Section III are employed for the relationship analysis between the two presented different-level redundancy-resolution schemes [i.e.,velocity-level scheme (4)and (5)and acceleration-level scheme (8)and (9)]. A.By the Gradient-Descent Method The gradient-descent method [14],[32]–[38]employed in Section III-A for QP problem solving is used here to analyze the equivalent relationship of the different-level redundancy resolution.The steps are listed as follows. 1)To minimize the performance index in velocity-level redundancy-resolution scheme (4)and (5),we can de-?ne the scalar-valued norm-based energy function as the performance index itself,i.e.,ε(˙θ)= ˙θ 22 /2.Evidently,such an energy function ε(˙θ) 0for any ˙θ ∈R n .2)To reach the minimum point of ˙θ 22 /2,a computational method could be designed to evolve along the negative Fig.5.(a)Joint-angle pro?les (in radians)and (b)joint-velocity pro?les (in radians per second)when the three-link planar robot arm’s end-effector tracks a triangular-path.The solid curves with circular markers correspond to the joint pro?les synthesized by velocity-level scheme (4)and (5),and the dash-dotted curves with square markers correspond to the ones synthesized by acceleration-level scheme (8)and (9). gradient of the energy function ε(˙θ ),i.e.,?(?ε(˙θ)/?˙θ)∈R n .We could thus have the following computational method which minimizes ε(˙θ)= ˙θ 22 /2:¨θ=?λ(?ε(˙θ)/?˙θ)=?λ ?( ˙θ 22/2)/?˙θ (16)where design parameter λ>0∈R ,being large enough, is used to scale the convergence rate of ε(˙θ ).3)Expanding gradient-descent formula (16)yields ¨θ =?λ˙θ(17) which could be immediately rewritten as ¨θ +λ˙θ=0.Evidently,ε(˙θ )could be guaranteed to converge exponen-tially to its minimum point with convergence rate λ. 4)As the end-effector motion-trajectory requirement has to be considered in the redundancy-resolution scheme,the dynamic equation ¨θ +λ˙θ=0could only be theoretically achieved.Thus,minimizing ¨θ+λ˙θ 22 /2appears to be more favorable for the motion planning of the redundant manipulators,instead of forcing ¨θ +λ˙θ=0directly and exactly as a constraint.In this sense,we could have the performance index minimized at the acceleration level ¨θ+λ˙θ 22 /2,which is depicted exactly in (8).Considering the end-effector trajectory requirement at the joint-acceleration level [i.e.,(9)],we could thus obtain acceleration-level redundancy-resolution scheme (8)and (9).From the aforementioned analysis and for the purposes of minimizing ˙θ 22 /2,we could say that the velocity-level redundancy resolution [depicted as (4)and (5)]is equivalent to the acceleration-level redundancy resolution [depicted as (8)and (9)]for the robot manipulators.B.By the Zhang et al.’s Neural-Dynamic Method For comparative purposes and for a more suf?cient analysis,we could introduce the Zhang et al.’s neural-dynamic method (mentioned in Section III-B)as an alternative way to prove the equivalence of the two presented different-level redundancy- Fig.6.Motion trajectories of the PUMA560manipulator when its end-effector tracks a straight-line path. resolution schemes [i.e.,(4)and (5)and (8)and (9)].According to such a neural-dynamic method,we could derive the equiva-lent relationship as follows. 1)To minimize the performance index ˙θ 22 /2in velocity-level scheme (4)and (5),instead of using the aforemen-tioned scalar-valued norm-based energy function ε(˙θ )shown in the preceding section via the gradient-descent method,we can de?ne alternatively the following vector-valued inde?nite unbounded error function in this section: E (t ):=˙θ (t )∈R n where,related to the preceding section,E (t )=0 achieves the minimum point of ε(˙θ )if and only if ˙θ=0.2)To make E (t )converge exponentially to zero,we could adopt the Zhang et al.’s neural-dynamic formula [2],[38]–[41]by simply setting ˙E (t )=?λE (t )(18) where λ>0∈R ,being large enough,denotes the rate of the exponential convergence of E to zero. Fig.7.(a)Joint-angle pro?les(in radians)and(b)joint-velocity pro?les(in radians per second)when the PUMA560manipulator’s end-effector tracks a straight-line path.The solid curves with circular markers correspond to the joint pro?les synthesized by velocity-level scheme(4)and(5),and the dash-dotted curves with square markers correspond to the ones synthesized by acceleration-level scheme(8)and(9). 3)Substituting E(t)=˙θ(t)into(18)yields¨θ(t)=?λ˙θ(t), which is the same as(17)and which could be rewritten as¨θ+λ˙θ=0,i.e.,via the Zhang et al.’s neural-dynamic method,we could achieve the same results as those via the gradient-descent method in Section V-A. 4)Similar to the?nal step of Section V-A,minimizing ¨θ+λ˙θ 22/2appears to be more favorable for the re-dundancy resolution,instead of forcing¨θ+λ˙θ=0.Con-sidering the end-effector trajectory requirement at the joint-acceleration level[i.e.,(9)],we could also obtain acceleration-level redundancy-resolution scheme(8)and (9).The analysis on the equivalence of the different- level redundancy resolution via the Zhang et al.’s neural-dynamic method is thus complete. In summary,via two different approaches,we have both shown that the two different-level schemes[i.e.,(4)and(5) and(8)and(9)]could be equivalent for redundancy-resolution problem solving.To achieve the equivalence,suitable perfor-mance indices[e.g.,(4)and(8)]have to be exploited for the redundancy resolution of the robot manipulators.From the analysis procedures,we can see that the Zhang et al.’s neural-dynamic method is much simpler than the gradient-descent method,especially in view of the comparison of key formulas (16)and(18). VI.C ONCLUSION Two different-level(velocity and acceleration levels) redundancy-resolution schemes and their equivalent relation-ship have been investigated in this paper.Both schemes have been reformulated and computed?nally as solvable quadratic programs.The gradient-descent method and the Zhang et al.’s neural-dynamic method have been employed for problem solv-ing and for equivalent-relationship analysis.The computer simulations based on the three-link planar robot arm and on the PUMA560manipulator have illustrated the ef?cacy and equivalence of the two present redundancy-resolution schemes, as well as the superiority of the ZNN QP-solver compared to the GNN QP-solver.Note that the equivalence of the different-level redundancy resolution may give us more insight into the motion planning and control of the redundant robot manipulators,as well as may impact future robotic research to some extent. 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Binghuang Cai(S’09–M’10)was born in Chenghai, Guangdong,China,in1981.He received the B.S. degree in electronic information engineering and the M.S.degree in signal and information processing from Shantou University,Shantou,China,in2004 and2007,respectively,and the Ph.D.degree in communication and information systems from Sun Yat-sen University,Guangzhou,China,in2010. He is currently a Research Fellow with the Centre for Autonomous Systems,University of Technol- ogy,Sydney,Australia.His research interests include neural networks,robotics,and intelligent information processing. Dr.Cai won the Best Paper Award of the2nd International Symposium on Systems and Control in Aeronautics and Astronautics(ISSCAA2008), Shenzhen,China,on December10–12,2008.He was also awarded the2nd Runner Up of the IEEE SMC Society Hong Kong Student Branch Chapter Student Paper Contest2009,Hong Kong,on September26, 2009. Yunong Zhang(S’02–M’03)was born in Xinyang, Henan,China,in1973.He received the B.S.de- gree from the Huazhong University of Science and Technology,Wuhan,China,in1996,the M.S.degree from the South China University of Technology, Guangzhou,China,in1999,and the Ph.D.degree from the Chinese University of Hong Kong,Shatin, Hong Kong,in2003. He is currently a Professor with the School of Information Science and Technology,Sun Yat-sen University(SYSU),Guangzhou.Before joining SYSU in2006,he has been with the National University of Ireland,Maynooth, Ireland,the University of Strathclyde,Glasgow,U.K.,and the National Univer-sity of Singapore,Singapore,Singapore,since2003.His main research interests include neural networks,robotics,and Gaussian processes.