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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/ACCESS.2020.3002608, IEEE Access

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Recurrent Neural Network for StateAdjustment of Redundant ManipulatorsLONG JIN1,2, (Member, IEEE), LI HE2, ZHIGUAN HUANG1, AND JUN WANG31Guangdong Provincial Engineering Technology Research Center for Sports Assistive Device, Guangzhou Sport University, Guangzhou 510000, China2School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China3Gansu Institute of Food Inspection and Research, Lanzhou 730000, China

Corresponding author: Long Jin (e-mail: [email protected])

This work was supported in part by the National Natural Science Foundation of China (No. 61703189), in part by Guangzhou SportUniversity Innovation and Strengthen Project (No. 5200080589), in part by Ministry of Education Industry-Academic CooperationCollaborative Education Program of China (No. 201901007048) and Baidu Online Network Technology (Beijing) Co., Ltd, in part by theTeam Project of Natural Science Foundation of Qinghai Province, China (No. 2020-ZJ-903), in part by the Key Laboratory of IoT ofQinghai (No. 2020-ZJ-Y16), in part by the Fundamental Research Funds for the Central Universities (No. lzujbky-2019-89), in part by theNational Key Research and Development Program of China (No. 2017YFE0118900), and in part by the Natural Science Foundation ofGansu Province, China (No. 18JR3RA264).

ABSTRACT To make joints of a redundant manipulator moving automatically to a target state, a state-adjustment (SA) scheme is studied and modified in this paper. Specifically, owing to the problem of non-zeroinitial joint velocity in the SA scheme leading to a potential hazard to redundant manipulators, a modifiedstate-adjustment (MSA) scheme is obtained on the basis of the SA scheme. The MSA scheme achieves thestate adjustment by minimizing the differences between the joint angles and the target values. For solvingthe MSA scheme, a recurrent neural network (RNN) model is derived, of which the critical component isto iterate over the joint angles and joint velocities. The MSA scheme solved by the RNN model enables theredundant manipulator to adjust to the target state automatically while ensuring that the initial joint velocityis zero. Beyond that, several comparative simulations demonstrate the availability and accuracy of the MSAscheme solved by the RNN model in controlling the state adjustment of redundant manipulators.

INDEX TERMS State-adjustment scheme, recurrent neural network, redundant manipulators.

I. INTRODUCTION

RECENTLY, with the gradual maturity of robot tech-nology, manipulators have become a common tool in

industrial production and other fields [1]–[3]. A manipulatorwith more degrees of freedom than required to completetasks is called a redundant manipulator. Compared with non-redundant manipulators, redundant manipulators are moreflexible, which provides the possibility of controlling thepose and avoiding obstacles when executing tasks [4]–[6].Therefore, in recent years, redundant manipulators have re-ceived considerable attention, and studies on the motioncontrol and applications of redundant manipulators haveemerged in an endless stream [7]–[11]. For example, Xieet al. exploit data-driven technology to fulfill the flexibleperformance of redundant manipulators and present a high-performance repetitive motion scheme [10]. Besides, thecontrol approach is extended to the acceleration level withthe quantitative relationship between the joint space error andCartesian space error explored [11].

The redundancy solution scheme is the method to obtainthe optimal joint angles referring to the optimization in-dex when the target trajectory of the end-effector is known[12], [13]. The optimization technique, especially quadraticprogramming, is a common technique in the redundancysolution problem. Generally speaking, when performing aredundancy solution, the redundancy solution problem wouldbe transformed into an optimization problem and solvedby numerical algorithms (e.g., Newton iteration, gradient-related methods, neural networks) [14]. The Newton iterationand its related improvement methods can make the systemconverge to the theoretical solution, but some of them areonly applicable to the solution of static problems [15]. Forthe problem of redundancy resolution which needs to besolved dynamically, these methods bring about the time-delay problem due to the absence of leveraging the timederivatives of dynamic parameters. Neural networks, withuniversal approximation property [16], fault-tolerance [17],parallelism [18], and excellent learning capabilities [19], are

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L. Jin et al.: RNN for State Adjustment of Redundant Manipulators

widely applied to address all kinds of complicated problems[20]. Recurrent neural networks (RNNs) have natural ad-vantages in solving real-time problems because they containhidden layers that can store data from the past, which is con-ducive to subsequent computations [21]. To date, numerousRNNs have been exploited to solve dynamic optimizationproblems converted from robotics [22]–[25]. For instance,in [24], a neural algorithm with an estimation on Hessianmatrix inversion, as a special kind of RNN, is presentedto solve the dynamic unconstrained optimization problem.Besides, a discrete-time neural dynamics model is furtherinvestigated in [25] for handling complex-valued quadraticprogramming problems with equality constraints and appliedto robot motion generation.

State adjustment refers to the process of moving themanipulator from the current state to the target state [26]without the displacement of the end-effector. Most of theresearch on the redundant manipulator focuses on motionplanning such as the manipulability optimization methodstated in [27] during the task execution of the redundantmanipulator, but there are few achievements on the problemof state adjustment [28]–[30]. In fact, state adjustment isimperative in certain situations [31], [32]. For instance, sometasks require the initial pose of the manipulator to be in aspecific state, and thus the manipulator needs to be adjustedto the specific state for performing subsequent operations[33], [34]. Moreover, during the execution of repetitive mo-tion, the joint drift phenomenon happens occasionally, whichmeans that the joints of the manipulator fail to return to theiroriginal states for a completed task cycle [35], [36]. Consid-ering the problems described above, a state-adjustment (SA)scheme based on the perspective of kinematics is studiedand modified, which allows joint angles to converge to targetvalues with high precision automatically. Owing to the SAscheme failing to guarantee zero initial joint velocity, thispaper further investigates a modified state-adjustment (MSA)scheme that is exactly right to handle the problem of non-zeroinitial joint velocity, and then an RNN is proposed to solvethis MSA scheme. Eventually, the validity and accuracy ofthe redundant manipulator synthesized by the MSA schemesolved by the RNN model for moving from the initial state tothe target state are illustrated by simulations.

The remainder of the paper is divided into the followingsections. In Section II, the SA scheme for redundant ma-nipulators is firstly discussed, and then an MSA scheme isexplored as an improvement. As described in Section III, anRNN model is derived to solve the MSA scheme, and itsconvergence is presented as well. In Section IV, comparativesimulations are carried out to examine the effectiveness of theMSA scheme solved by the RNN. Section V concludes thispaper. Before ending the current section, the main contribu-tions of the paper are generalized as follows:

1) The MSA scheme researched in this paper is able tomake the redundant manipulator move from one state tothe target state via decreasing the difference between thejoint angle and the target state without the displacement

of the end-effector. Considering the physical realizabili-ty, the MSA scheme exploits and modifies the constraintof joint velocity to obtain the zero initial joint velocity.

2) An RNN is proposed to solve the MSA scheme atthe velocity level with the proof on its global conver-gence displayed. The simulations based on Cartesianthree-dimensional space under different parameter set-ting conditions verify the validity and accuracy of theMSA scheme solved by the RNN model. Besides, theinfluence of parameters on the convergence speed andaccuracy is explored through comparison simulation.

II. SA SCHEME AND MSA SCHEMEIn this section, an MSA scheme with zero initial joint velocityis studied to adjust the pose of the redundant manipulator tothe target state based on the SA scheme.

A. SA SCHEMEBefore the research on the SA scheme, the essential kinemat-ic equations of the redundant manipulator is explained. As-sume that the number of joints of the redundant manipulatoris m and the joint angle is σ(t) = [σ1, σ2, · · · , σm]T ∈ Rmwith the superscript T denoting the transpose operator. Bydefining f(·) as the mapping between the joint angles andposition of the end-effector of the redundant manipulator, theposition pe ∈ Rk is expressed as pe = f(σ(t)), and thevelocity of the end-effector pe is written as

pe = Jσ(t), (1)

where J = ∂f(σ(t))/∂σ(t) ∈ Rk×m, and σ(t) is the jointvelocity. In addition, suppose that the initial state or currentstate of the redundant manipulator is σ(0) ∈ Rm, and thetarget state is σs ∈ Rm. The ultimate goal of the designed SAscheme is σ(t) = σs, that is, each joint angle of the redundantmanipulator should change into the target value.

According to the above explanations, the SA scheme isconstructed as

min1

2σT(t)σ(t) + %T(t)σ(t)

s.t. Jσ(t) = pe

σ− ≤ σ(t) ≤ σ+

σ− ≤ σ(t) ≤ σ+,

(2)

where %(t) = r(σ(t) − σs) ∈ Rm; r > 0; σ(t) = dσ(t)/dtis the real-time joint velocity; σ∓ and σ∓ are the lowerand upper bound limits of the joint angle and joint velocityseparately. Note that, the position of the end-effector remainsunchanged during the state adjustment task, resulting inpe = 0. In SA scheme (2), the objective function can realizethe pose adjustment of the redundant manipulator, and itsconstraint conditions can guarantee that σ(t) and σ(t) arekept within the constraint range.

The following remark focuses on explaining the intrinsicconsistency between the minimized objective function andthe expected result σ(t) = σs.

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Remark 1: In the SA scheme (2), the redundant robotmanipulator can automatically approach the target state byminimizing σT(t)σ(t)/2 + %T(t)σ(t).

The derivation of the minimization index σT(t)σ(t)/2 +%T(t)σ(t) is explained in detail next. First, make the follow-ing definitions:

e(t) = σ(t)− σs, (3)

ande(t) = −re(t), (4)

where e(t) ∈ Rm denotes the error between the joint angleand the target state, and e(t) = de(t)/dt is its derivative. Thesolution to differential equation (4) is

e(t) = e(0) exp(−rt), (5)

where e(0) = σ(0)−σs is the initial angle error. As observedfrom (5), e(t) converges exponentially to zero and r canadjust the convergence rate. Thanks to the convergence ofe(t), the final state of the redundant manipulator can beconsistent with the target state if there is sufficient adjustmenttime. Secondly, the combination of (3) and (4) brings about

σ(t) + r(σ(t)− σs) = 0. (6)

The above analysis can be summarized that, the joint anglecan exponentially converge to σs as long as (6) is satisfied.Hence, the state adjustment of a redundant manipulator canbe carried out by minimizing σ(t) + r(σ(t) − σs). Due tothe hidden demands in the redundant manipulator’s task,it is ‖σ(t) + r(σ(t) − σs)‖22 more appropriate in physicaloperations rather than σ(t) + r(σ(t) − σs), where ‖ · ‖2is the operator to obtain the two-norm of a vector. Expand‖σ(t) + r(σ(t)−σs)‖22 and define %(t) = r(σ(t)−σs) to get

‖σ(t) + r(σ(t)− σs)‖22=σT(t)σ(t) + 2%T(t)σ(t) + %T(t)%(t).

(7)

In terms of the velocity level, %T(t)%(t) is a constant. Even-tually, the minimum index of the SA scheme (2) is describedas σT(t)σ(t)/2 + %T(t)σ(t), with ignoring the performanceof %T(t)%(t).

B. MSA SCHEMEIn the task of solving real-time joint velocities, it is a compli-cated thing to constrain joint angles, and thus, two differentlevels of joint physical constraints in SA scheme (2) shouldbe written as a comprehensive constraint. The new bounds ofthe joint velocity are redefined as ψ∓, and the new upper andlower limits of the jth joint velocity are expressed as

ψ+j =minσ+

j , λ(σ+j − σj)

ψ−j =maxσ−j , λ(σ−j − σj),(8)

where σ+j and σ+

j are the upper limits of the jth joint velocityand angle; σ−j and σ−j are the lower limits of jth jointvelocity and angle; λ > 0 is the scale factor. In (8), oncethe joint angle is too large, the inherent upper limit of jointvelocity would be replaced by λ(σ+

j − σj), which drives

the joint angle to change into the constraint range. Similarly,when the joint angle is too small, λ(σ−j − σj) would replacethe inherent lower limit of joint velocity, avoiding the jointvelocities out of the constraint range. In this way, the rangeof joint velocities can be controlled while the range of jointangles can be constrained. Note that, in this subsection andsubsequent derivations, these time-varying variables are sim-plified such as σ for σ(t).

Aided with the preparatory work above, SA scheme (2) issimplified as

min1

2ωTω + %Tω

s.t. Jω = pe

ψ− ≤ ω ≤ ψ+,

(9)

where ω = σ ∈ Rm. Formally, SA scheme (2) is simplifiedas a quadratic optimization scheme at the velocity level,which only contains a range constraint and is easier to solve,in contrast to SA scheme (2).

The SA scheme (2) generally leads to a phenomenon thatthe initial joint velocity is not zero with the influence ofω(0) = −r(σ(0) − σs). In practice, the joint velocity ofthe redundant manipulator maintains zero at the beginningof the movement, and then gradually increases. If there is alarge joint velocity at the initial state, the damage to the re-dundant manipulator is self-evident. Therefore, the followingimprovements are added to get the MSA scheme:

min1

2ωTω + %Tω

s.t. Jω = pe

Ψ− ≤ ω ≤ Ψ+,

(10)

where Ψ− and Ψ+ are the modified lower and upper boundsof the joint velocity. Specifically, Ψ− and Ψ+ are defined as

Ψ− = sin(πt/2T )ψ−

Ψ+ = sin(πt/2T )ψ+,(11)

where T is the stipulated state-adjustment time. Comparedwith (8), a time-varying sine function is added to the con-straint as a multiplier, which enables the limits of jointvelocity to expand gradually during the motion, makes thejoint velocity to increase from zero and avoids the problem ofexcessive initial joint velocity. Hence, through this method,MSA scheme (10) possesses the merit of zero initial jointvelocity and becomes more safe and reliable than SA scheme(2).

III. SOLUTIONTo obtain joint velocity data that can control the motion ofredundant manipulator, developing a corresponding solverto solve MSA scheme (10) is indispensable. The followingcontent is the derivation procedure of the effective RNNmodel solver.

First, based on MSA scheme (10), a Lagrangian functionis defined as F (ω) = ωTω/2 + %Tω + γTJω, with γ ∈ Rkdenoting the Lagrange-multiplier vector. As known from the

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FIGURE 1: Simulation results of the redundant manipulator synthesized by SA scheme (2) solved by RNN model (19) withT = 3 s and r = 4. (a) The initial state and final state. (b) Motion trajectory. (c) Position of the end-effector. (d) Profiles ofposition errors. (e) Profiles of joint angles. (f) Profiles of joint velocities.

Karush-Kuhn-Tucker condition [37], the optimal solution toMSA scheme (10) satisfies

NΨ(ω) +∂F

∂ω3 0,

Jω = 0,(12)

where NΨ(ω) is the normal cone of set Ψ at ω and satisfies

(η − ω)T(∂F

∂ω) ≥ 0, ∀η ∈ Ψ. (13)

Preform the following operations to get

‖η − (ω − ∂F

∂ω)‖22 − ‖ω − (ω − ∂F

∂ω)‖22

=‖ω − η‖22 + 2(η − ω)T(∂F

∂ω) ≥ 0.

(14)

Then, (12) is reformulated into

ω = ΓΨ(ω − ∂F

∂ω) = ΓΨ(−%− JTγ), (15)

where the projection function ΓΨ(·) is defined as

ΓΨ(a) = arg min‖h− a‖22, ∀h ∈ Ψ (16)

with 0 ∈ Ψ and is equivalent to

ΓΨ(a) = b, with ‖h−a‖22−‖b−a‖22 ≥ 0, b ∈ Ψ, ∀h ∈ Ψ.(17)

Finally, to solve equation (15), an RNN is exploited andwritten as a differential equation:

εω = −ω + ΓΨ(−%− JTγ),

εγ = Jω,(18)

where ε > 0 is the proportional coefficient.Facilitated by the same derivation method, the correspond-

ing RNN exploited for the SA scheme (2) is described asfollows:

εω = −ω + Γψ(−%− JTγ),

εγ = Jω,(19)

where the definition of nonlinear projection function Γψ(·) isthe same as (16), except that Ψ is replaced by ψ. Note thatRNN model (19) is different from RNN (18) although theyare similar except for the individual parameter. Besides, theRNN model (18) is applied to solve MSA scheme (10), andthe other is used for solving SA scheme (2).

The following theorem illustrates the convergence of RNNmodel (18) in solving MSA scheme (10).

Theorem 1: The output of proposed RNN model (18) forsolving MSA scheme (10) converges to its optimal solution.

Proof: First, by defining ρ = [ω, γ]T ∈ Ω with Ω =[ω, γ]T ∈ Rk+m|ω ∈ Ψ, γ ∈ Rk, (18) is rewritten as

ερ = −ρ+ ΓΩ(ρ−G(ρ)), (20)

where the function G(ρ) is defined as

G(ρ) =

[ω + %+ JTγ−Jω

]∈ Rm+k, (21)

and its gradient is

∇G =

[I JT

−J 0

]∈ Rk+m×k+m, (22)

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FIGURE 2: Simulation results of the redundant manipulator synthesized by MSA scheme (10) solved by RNN model (18) withT = 3 s and r = 4. (a) The initial state and final state. (b) Motion trajectory. (c) Position of the end-effector. (d) Profiles ofposition errors. (e) Profiles of joint angles. (f) Profiles of joint velocities.

with I standing for an identity matrix, and ∇ being thesymbol for the gradient. Then, the monotonicity of G(ω) isinferred from

(ρ1 − ρ2)T(G(ρ1)−G(ρ2))

=(ρ1 − ρ2)T∇G(ς)(ρ1 − ρ2)

=(ρ1 − ρ2)T(∇G(ς) +∇TG(ς))(ρ1 − ρ2) ≥ 0,

(23)

where ρ1 and ρ2 are two arbitrary vectors in set Ω, and ς ∈ Ωis a vector between ρ1 and ρ2. Facilitated by these derivationsand lemmas in [38], RNN model (18) is Lyapunov stable andits equilibrium point ρ′ conforms to the following variationalinequality:

(ρ− ρ′)TG(ρ′) ≥ 0, ρ ∈ Ω. (24)

According to the properties of variational inequalities, (24)can also be converted into the following equation:

−ρ′ + ΓΩ(ρ′ −G(ρ′)) = 0. (25)

The above projection equation has the same solution of (20),which indicates that the output of RNN model (18) convergesto the optimal solution of MSA scheme (10). The proof iscomplete.

IV. SIMULATIONIn current part, for verifying the effectiveness and reliabilityof the MSA scheme (10) solved by RNN model (18), a seriesof simulations are carried out.

A. SIMULATION SETTING

Simulations are based on the redundant manipulator KUKAwith seven degrees of freedom, and the joint physical limitsare as follows: the upper and lower limits of the first, third,fifth and seventh joint angles are 17/18π and −17/18π radseparately; the upper and lower limits for the second, fourth,and sixth joint angles are 2/3π and−2/3π rad separately; theupper and lower limits of each joint velocity are 1.5 and−1.5rad/s, respectively. Suppose that the initial states of eachjoint angle are 0.981028, −0.980346, 0.283290, 0.803084,−1.557347, −2.059395, and 1.070796 rad, and the targetstates are π/4, −π/4, π/6, π/3, −π/3, −π/2, and π/2rad. Moreover, the scale factor λ is 20 and the proportioncoefficient ε is 0.001. Simulations consist of two parts: thefirst part is the simulation of SA scheme (2) solved by RNNmodel (19); the second part includes the simulations of theMSA scheme (10) solved by RNN model (18) under differentparameter settings.

B. SIMULATION RESULTS

The simulation of the redundant manipulator synthesized bySA scheme (2) without zero initial joint velocity is carriedout with results shown in Fig. 1. More specifically, the initialstate and final state are revealed in Fig. 1(a). The motiontrajectory during the adjustment can be seen in Fig. 1(b),in which motion trajectories are sparse near the initial stateand become dense near the final state. Owing to the samesampling interval between two adjacent trajectories, this phe-

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FIGURE 3: Joint angles and velocities of the redundant manipulator synthesized by MSA scheme (10) solved by RNN model(18) with T = 3 s and r = 8. (a) Profiles of joint angles. (b) Profiles of joint velocities.

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FIGURE 4: Joint angles and velocities of the redundant manipulator synthesized by MSA scheme (10) solved by RNN model(18) with T = 5 s and r = 4. (a) Profiles of joint angles. (b) Profiles of joint velocities.

nomenon suggests that the redundant manipulator possessesa large joint velocity at the initial moment and a low jointvelocity at the end. Besides, from the position and positionerrors of the end-effector displayed in Fig. 1(c) and (d), itcan be generalized that the end-effector of the manipulatordoes not change significantly during the state adjustment.Moreover, as observed in Fig. 1(e) and (f), joint angles canconverge from the initial values to the target ones and thejoint velocities decrease to zero from a very large one, whichgoes against the reality that the joint velocities are zero at thestart of the task. In addition, the joint velocities of the sixthand seventh remain unchanged at 1.5 rad/s for a short time atthe beginning, of which the reason is that the two initial jointvelocities (σ6(0) ≈ 1.95 > 1.5 rad/s and σ7(0) ≈ 2.00 >1.5 rad/s) calculated by σ(0) = −r(σ(0) − σs) in Remark1 exceed their physical constraints and are adjusted to theboundary values 1.5 rad/s.

The rest content is the simulations of the MSA scheme(10) for the state adjustment of the redundant manipulator,which involves three cases with the corresponding parametersettings for each case as follows: case 1) T = 3 s and r = 4;case 2) T = 3 s and r = 8; case 3) T = 5 s and r = 4.The convergence rate and accuracy of the MSA scheme (10)solved by the proposed RNN model (18) are further explored

by comparing the simulations under these three cases. Incase 1), the initial state, final state, and motion trajectory ofthe redundant manipulator synthesized by MSA scheme (10)solved by RNN model (18) with T = 3 s and r = 4 are shownin Fig. 2(a) and (b). Evidently, motion trajectories are densewhen approaching the initial state and the final state, andsparse in the middle part, which indicates that the redundantmanipulator moves slowly at the beginning and end of thestate adjustment, and moves fast during the intermediateprocess. Furthermore, by observing the position and positionerrors of the end-effector in Fig. 2(c) and (d), it is foundthat there is no obvious displacement at the end-effectorof the manipulator, which is consistent with the theoreticalanalysis. Moreover, profiles of joint angles and velocities aredisplayed in Fig. 2(e) and (f), in which joint angles convergeto the target values in 2 s, and joint velocities increase fromzero. Compared with the results of SA scheme (2) in Fig.1(f), MSA scheme (10) can indeed deal with the problemof non-zero initial joint velocity such that it becomes moreapplicable. In addition, to observe the results of the stateadjustment more accurately, joint-angle errors are revealedin Table 1, in which the absolute values of joint errors undercase 1) are all less than 4.3503× 10−5 rad.

For further analysis, simulation of the redundant manipula-

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TABLE 1: Angle Error by Applying RNN (18) under Case 1

σ(0) σs σ(T ) |σ(T )− σs|+0.981028 +π/4 +0.785411 1.289461× 10−5

−0.980346 −π/4 −0.785409 1.177066× 10−5

+0.283290 +π/6 +0.523593 4.891301× 10−6

+0.803084 +π/3 +1.047182 1.499467× 10−5

−1.557347 −π/3 −1.047241 4.350270× 10−5

−2.059395 −π/2 −1.570835 3.934844× 10−5

+1.070796 +π/2 +1.570755 4.115238× 10−5

tor synthesized by MSA scheme (10) solved by RNN model(18) under case 2) is conducted and the corresponding resultsare revealed in Fig. 3. In the simulation, only joint angles andvelocities are analyzed, because other simulation results arealmost the same as those under case 1) in Fig. 2. By observingthe joint angles in Fig. 3(a), it is found that joint anglesconverge to the target values in 1.5 s, which indicates thatthe convergence rate of joint angles is improved when r isincreased. Furthermore, joint velocities in Fig. 3(b) graduallyincrease from zero at the beginning and drop to zero around1.5 s. To explore the change of accuracy with the increase ofr, the errors of joint angles are calculated and shown in Table2, in which the maximum of the absolute value of the error isless than 2.35×10−7 rad. In contrast to errors in Table 1, theaccuracy is almost a hundred times better, which implies thatthe accuracy can be improved by increasing r.


σ(0) σs σ(T ) |σ(T )− σs|+0.981028 +π/4 +0.785398 5.007557× 10−8

−0.980346 −π/4 −0.785398 9.679112× 10−8

+0.283290 +π/6 +0.523598 3.814844× 10−8

+0.803084 +π/3 +1.047197 2.349258× 10−7

−1.557347 −π/3 −1.047197 9.339705× 10−9

−2.059395 −π/2 −1.570796 1.783016× 10−7

+1.070796 +π/2 +1.570796 1.185032× 10−8

As for case 3), joint angles and velocities of the redundantmanipulator synthesized by MSA scheme (10) solved byRNN model (18) are shown in Fig. 4. Increasing the task timeto 5 s, joint angles in Fig. 4(a) still converge to the target valuein 2 s, and joint velocities in Fig. 4(b) drop to zero around2 s, which indicates that the increase of task time does notaffect the convergence rate of joint angles. For exploring theimpact of task time on the accuracy, the errors of joint anglesare calculated and revealed in Table 3, in which the absolutevalues of the error are all almost a hundred times lower thanthose obtained under case 1). It can be summarized fromsimulation results under case 3) that changing the task time Timproves the accuracy but does not change the convergencerate.

In conclusion, MSA scheme (10) solved by RNN model(18) can solve the problem of non-zero initial joint velocityduring the state adjustment, which can change the conver-gence rate by modifying r, and achieve the desired accuracyby adjusting r and T .


σ(0) σs σ(T ) |σ(T )− σs|+0.981028 +π/4 +0.785398 4.607877× 10−8

−0.980346 −π/4 −0.785398 9.687483× 10−8

+0.283290 +π/6 +0.523598 3.839616× 10−8

+0.803084 +π/3 +1.047197 2.433062× 10−7

−1.557347 −π/3 −1.047197 1.861330× 10−8

−2.059395 −π/2 −1.570796 2.174961× 10−7

+1.070796 +π/2 +1.570796 3.950501× 10−8

V. CONCLUSIONIn this paper, for controlling state adjustment of redundantmanipulators, based on the modification of the SA scheme,an MSA scheme has been discussed, which can deal withthe non-zero initial joint velocity. To be specific, the MSAscheme concentrates on minimizing the deviations betweenjoint angles and target values, and its constraints include thejoint angle constraints and joint velocity constraints. Besides,a sine function multiplier has been added to constrain theinitial joint velocity on the basis of the joint limits, whichis different from the SA scheme. Further, RNN models havebeen proposed to solve the SA scheme and the MSA schemeseparately. Eventually, the comparative simulation resultshave elucidated that the MSA scheme with the assistance ofthe proposed RNN model can remedy the problem of the non-zero initial joint velocity and improve the accuracy greatly bymodifying related parameters.

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LONG JIN (M’17) received the B.E. degree inautomation and the PhD degree in informationand communication engineering from Sun Yat-sen University, Guangzhou, China, in 2011 and2016, respectively. He is currently a full professorwith the School of Information Science and En-gineering, Lanzhou University, Lanzhou, China.Before joining Lanzhou University in 2017, hewas a Postdoctoral Fellow with the Department ofComputing, The Hong Kong Polytechnic Univer-

sity, Hung Hom, Kowloon, Hong Kong. His main research interests includeneural networks, robotics, and intelligent information processing. He is amember of the IEEE.

LI HE received the B.E. degree from Wuhan U-niversity, Wuhan, China, in 2017. She is current-ly pursuing her M.E. degree in communicationand information systems at School of Informa-tion Science and Engineering, Lanzhou Univer-sity, Lanzhou, China. Her main research inter-ests include robotics, neural networks, intelligentinformation processing, artificial intelligence andoptimization theory.

ZHIGUAN HUANG was born in Changning, Hu-nan, China, in 1980. He received the Ph.D. degreein human kinesiology from South China NormalUniversity, Guangzhou, China, in 2009. He is cur-rently an Associate Professor with the Engineer-ing Research Center for Sport Assistive Device,Guangzhou Sport University. He has publishedaround 30 papers in various journals and confer-ences. His current research interests include neuralnetworks, sport biomechanics, and sport engineer-

ing.

JUN WANG received the B.S. and M.S. degreesin inorganic chemistry from Northwest NormalUniversity, Lanzhou, China, in 2007 and 2010, re-spectively. She currently works at Gansu Instituteof Food Inspection and Research. Her researchinterests include chemistry and signal processing.

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