A Varying-Parameter Recurrent Neural Network Combined With Penalty Function for Solving Constrained Multi-Criteria Optimization Scheme for Redundant Robot Manipulators

To effectively solve the multi-objective motion planning problem for redundant robot manipulators, a penalty neural multi-criteria optimization (PNMCO) scheme is proposed and investigated. The scheme includes two parts: a constrained multi-criteria optimization (CMCO) subsystem, and a varying-parameter recurrent neural network combined with penalty function (VP-RNN-PF) subsystem. Specifically, the CMCO subsystem is made up of velocity two norm, repetitive motion, and infinity norm. With these criteria, it can achieve energy minimization, repetitive motion, and avoidance of speed peaks. In addition, the CMCO subsystem is then transformed into a standard quadratic programming (QP) problem, and the VP-RNN-PF subsystem is applied to solve the QP problem. Results of computer simulations based on the JACO2 robot manipulator demonstrate that the proposed PNMCO scheme is effective and feasible to plan the multi-objective motion tasks. Comparison experiments of two complex paths tracking between VP-RNN-PF and the traditional neural networks (e.g., simplified linear-variational-inequality-based primal-dual neural network, S-LVI-PDNN) shows that the proposed scheme as well as the neural network is more accurate and more efficient for solving multi-objective motion planning problem.


I. INTRODUCTION
Nowadays, robot technology has received widespread attention and robot manipulators have been widely applied in various fields, such as medical surgery [1], [2], domestic service [3] industrial production [4], [5]. If a robot manipulator has more DOFs than the minimum number of DOFs to complete the primary task of an end-effector, it is termed a redundant robot manipulator. The advantage of a redundant robot manipulator is that it has additional DOFs to complete other secondary subtasks in addition to the primary tasks of an end-effector. Some common secondary subtasks are joint The associate editor coordinating the review of this manuscript and approving it for publication was Saeid Nahavandi . limits avoidance [6], [7], repetitive motion [8], [9], energy minimum [10], [11], et al.
Since the number of joint variables for a redundant robot manipulator is larger than the number of robot kinematic equations, the solutions to the inverse kinematics problem are more than one. The traditional solution to the inverse kinematics problem was based on the pseudo-inverse method [12]. However, this method would take a lot of time to calculate the inverse of matrices. In addition, this method also cannot consider inequality problems (such as physical limit constraints) [13]. If these constraints are not considered, the moving joints may touch their physical limits, which may cause task failure, even robot damage. In recent years, the inverse kinematics problem is converted into an optimiza-tion problem, Cheng et al. transformed the redundant inverse kinematics problem into a quadratic programming (QP) problem, and considered the physical limits as the inequality constraint [14].
To solve the joint-angular deviation problem of redundant robot manipulators, optimization approaches were preferred by an increasing number of scholars, and many optimization criteria were designed in the past years. For instance, infinity-norm velocity minimization (INVM) [15], infinity-norm acceleration minimization (INAM) [16], minimum acceleration norm (MAN) [17], minimum kinetic energy (MKE) [18], minimum weighted velocity norm (MWVN) [19], repetitive motion planning (RMP) [8], [9], and minimum-velocity-norm (MVN) [11]. However, in the practical application of redundant robot manipulators, singlecriterion optimization schemes cannot meet the demand. In order to solve the multi-criteria optimization problem in practical applications, some bi-criteria minimization schemes were presented [20], [21]. Zhang et al. presented a bi-criteria scheme that combined the MVN and INVM [20]. Guo and Zhang et al. studied bi-criteria minimization schemes at torque, velocity, and acceleration levels, such as the MAN-MKE bi-criteria minimization scheme [21] and MVN-MAN bi-criteria minimization scheme [22]. Guo and Zhang also unified and compared the bi-criteria torque, velocity, and acceleration schemes [23]. Tri-criteria minimization schemes were presented and exploited in recent years [24], [25]. These multi-criteria schemes are reformulated as QP problems and then these QP problems are solved.
The solutions to QP problems are divided into two types, i.e., numerical methods [26], [27] and neural networks [28]- [36]. Compared with numerical methods, neural network methods can process information in parallel and have higher computational efficiency [28]. Scholars designed many neural network methods in the past years. Leung et al. presented a new gradient-based neural network (GNN) method to solve QP problems [29]. However, This method could not track the theoretical solution of QP problems well [30]. Zhang et al. presented a zeroing neural network (ZNN) [31], ZNN could track the theoretical solution of QP problems. However, it tracked the theoretical optimal solution only when the time approaches to infinity [30]. Miao et al. and Zhang et al. [33] respectively presented a finite time ZNN (FT-ZNN) and a varying parameter RNN (VP-RNN) to improve convergence speed of ZNN [32], [33]. However, these solutions usually fail to consider the joint physical limits.
To solve the optimization schemes of redundant robot manipulators with joint physical limit constraints, a dual neural network (DNN) was presented to solve the corresponding QP problem with inequality constraints [34], but the structure of this DNN was too simple. A linear variational inequality-based primal-dual neural network (LVI-PDNN) was presented [35]. To avoid solving the inverse of the matrix, reduce the computational complexity, a simplified LVI-PDNN (S-LVI-PDNN) was presented and applied [36].
The multi-objective motion tasks of redundant robot manipulators with joint physical limits was usually solved by S-LVI-PDNN [8], [16]- [25]. However, the rate of S-LVI-PDNN could not always reach exponential convergence [37]. Based on the research of VP-RNN [38], a penalty neural multi-criteria optimization (PNMCO) scheme is proposed. The scheme can effectively and feasible plan the multiobjective motion tasks of redundant robot manipulators.
This paper consists of five sections. In Section II, a PNMCO scheme based on QP form for redundant robot manipulators is constructed. For comparisons, we give the traditional S-LVI-PDNN. In Section III, computer simulation results demonstrate the feasibility of the PNMCO scheme. Meanwhile, comparison experiments between VP-RNN-PF and S-LVI-PDNN verify that the proposed schemes more accurate and more efficient for solving the multi-objective motion planning problem. Section IV draws the conclusion of this paper. Specifically, the main contributions of this paper are summarized below.
• A novel PNMCO scheme is proposed to solve the multiobjective motion planning problem for redundant robot manipulators. The proposed PNMCO scheme includes two subsystems, i.e., VP-RNN-PF and CMCO.
• The PNMCO scheme considers both the joint velocity and joint angle limits, since this scheme is solved at the velocity level, the joint velocity and joint angle limits combine as the piecewise function is proposed in this paper.
• Computer simulations by two different path tracking task results demonstrate the effectiveness and feasibility of the proposed PNMCO scheme to solve multiobjective motion planning problem for the JACO 2 robot manipulator.
• Comparison experiments between the VP-RNN-PF and traditional S-LVI-PDNN indicate that the proposed PNMCO scheme is more accurate and more effective in solving the multi-objective motion planning problem for redundant robot manipulators.

II. CONSTRUCTION OF PNMCO SCHEME AND S-LVI-PDNN SOLVER
The forward kinematics of the redundant robot manipulators is to obtain the end effector r path through the joint angle θ change, and the equation is considered as Due to the f (·) is a differentiable nonlinear mapping function, the equation (2.1) is difficult to solve, then the above equation is rewritten to solve at the velocity level [8]- [25] J (θ)θ =ṙ (2.2) whereθ ∈ R n andṙ ∈ R m denote joint velocity vector and end-effector velocity vector, respectively. The J (θ) ∈ R m×n is Jacobian matrix of the robot manipulators. The n and m denote the DOFs of manipulators and the dimension of endeffector Cartesian space, respectively. In this paper, for the VOLUME 9, 2021 six-DOF JACO 2 redundant robot manipulator used, the n = 6, m = 3.

A. CMCO SUBSYSTEM FORMULATION
The PNMCO scheme combines three different optimization criteria (i.e., MVN, RMP, and INVM), which can be expressed as where · ∞ denotes the infinite norm, · 2 denotes the two-norm. η and δ are weighting parameters for assigning different optimization criteria (i.e.,MVN, RMP and INVM), In practical applications, redundant robot manipulators are always subject to physical constraints, considering the joint velocity limits and joint angle limits of redundant robot manipulators, the PNMCO scheme can be formulated as where θ ± denote the physical limits for joint angle vector anḋ θ ± denote the physical limits for joint velocity vector. According to the mathematical formula, the MVN and RMP in equation (2.5) can be rewritten as where I ∈ R n×n is an identity matrix, and = 2τ .
In order to solve the schemes (2.5)-(2.8) at the velocity level, constraints (2.7)-(2.8) need be unified as the constraint of velocity level, the joint limits (2.7) are rewritten as [10], [34] υ > 0, the function of this parameter is to scales the feasible regions of joint-velocity vectorθ in equation (2.13). The equation (2.13) can guarantee joint velocity accelerate to zero when the joint-angle approaches its limits. Therefore, the joint limits (2.7) and joint velocity limits (2.8) combined into a variable joint physical constraint as (2.14) The PNMCO scheme of redundant robot manipulators is finally rewritten as a standard QP form is an identity matrix. is an infinite constant. The 0 and represent the constraint range of k.

B. VP-RNN-PF SOLVER
Firstly, the inequality constraint (2.18) is converted into a penalty term and which added to the objective function (2.16) of the QP problem [39]. The penalty function is designed as where p is the penalty factor, σ is a positive design parameter, Therefore, the QP problem (2.16)-(2.18) is transformed into a QP problem without inequality constraints as Secondly, for the solution of the above QP problem (2.20)-(2.21), we can construct a Lagrange function as where λ(t) ∈ R m is a vector of Lagrange multiplier. In order to solve the above equation (2.22), the equation (2.22) differentiates x(t) and λ(t) respectively, and then make them equal to 0, that is Equation (2.23) is reformulate to get the following matrix equation Through the above operations, solving QP problem (2.16)-(2.18) is transformed into solving matrix equation (2.24).
Thirdly, to find the optimal solution of (2.24), an error function is defined as To solve equation (2.24), the above equation (2.25) needs approach to zero, design method based on neural dynamic [28], [31], [33], [38], [40], the derivative of the ϕ(t) to time t is designed as where γ > 0, the γ e t is used to adjust the convergence rate, (·) : R n+m+1 → R n+m+1 denotes activation-function processing-array, every unit φ(·) of (·) is a monotonicallyincreasing odd activation function. Three activation functions are used in the simulation experiments of this paper.
The linear activation function as The sinh activation function as φ(ϕ(t)) = e (ϕ(t)) − e (−ϕ(t)) 2 (2.28) The power-sigmoid activation function as Finally, substituting equation (2.25) into equation (2.26), we can get obtain implicit-dynamic equation, then reconstruct this equation, we can obtain the following equation, that is The time complexity of the proposed neural network is O(n 3 ) when using the linear activation function. The structure of VP-RNN-PF is shown in Fig.1. Theorem 1: The state variable y(t) = [x T (t), λ T (t)] T ∈ R n+m+1 of VP-RNN-PF (2.30), starting from any initial state y(0) globally converges to the unique theoretical solution VOLUME 9, 2021 y * (t), i.e., lim t→+∞ (y(t) − y * (t)) = 0. We provide the following proof.

III. COMPUTER SIMULATIONS
In this section, to illustrate the feasibility and effectiveness of the PNMCO scheme, computer simulations of the fourleaf-clover path and Chinese character ''dai'' path are performed on the Kinova JACO 2 robot manipulator. Besides, computer simulation results of VP-RNN-PF and traditional S-LVI-PDNN are compared. Joint physical limits of JACO 2 robot manipulator are shown in Table 1.  Firstly, Fig. 2 shows the end-effector of JACO 2 redundant robot manipulator tracks the path of four-leafclover. As shown in Fig. 2(a) and (b), the end-effector can complete the path tracking task well. As shown in Fig. 2(c), the joint angle θ is always within the physical limits when executing the task. As shown in Fig. 2(a) and (c), After completing a task cycle, the joints return to the initial position. The second column of Table 3 shows the detailed joint angle drifts, which deviation reaches 10 −4 rad levels. From Fig. 2(d), when executing the path tracking task, theθ 1 andθ 4 reach the limits ±1.5 rad/s, but never exceed them, which can effectively avoid high velocity and damage to the robot manipulator. In short, this four-leaf-clover path tracking simulation results verify the feasibility and accuracy of the proposed PNMCO scheme.
Secondly, VP-RNN-PF (2.30) and S-LVI-PDNN (2.35) are applied to solve the QP problem (2.16)-(2.18) of the CMCO scheme, and the robot manipulator is controlled to track the four-leaf-clover path for comparisons. In simulations, the CMCO schemes with two different weight distribution are adopted, which are the CMCO scheme (η = 0.2 and  Table 2 and  Table 3, respectively. θ i (5) − θ i (0) denote the drifts of the ith joint between the final state and initial state of the joint angle. θ(5) − θ(0) 2 denotes the two norm of joint drifts. As shown in Table 2 and Table 3 Table 2 and  Table 3 when the end-effector tracks the four-leaf-clover path. All runtime of VP-RNN-PF with linear, sinh, and powersigmoid activation functions are very tiny. Among them, the VP-RNN-PF with linear activation function is the fastest, The runtime of the CMCO scheme (η = 0.2 and δ = 0.4) and CMCO scheme (η = 0.1 and δ = 0.8 ) are 2.1566 s and 2.0490 s, respectively. However, the runtime of S-LVI-PDNN is more than twice that of VP-RNN-PF with linear activation function. The runtime of the CMCO scheme (η = 0.2 and δ = 0.4 ) solved by S-LVI-PDNN is 5.1153 s. The runtime is over the task execution time T = 5 s, which means it cannot effectively solve the CMCO scheme. Fourthly, compare the position errors of the CMCO scheme (η = 0.2 and δ = 0.4 ) synthesized by VP-RNN-PF(three different activation functions) with S-LVI-PDNN during the four-leaf-clover path tracking task. we can see that there is not much difference between VP-RNN-PF with linear activation function Fig.3(a) and VP-RNN-PF with power-sigmoid activation function Fig.3(c) and S-LVI-PDNN Fig.3(d), but the VP-RNN-PF with sinh activation function Fig.3(b) is less than the S-LVI-PDNN. Specifically, the position error of VP-RNN-PF with sinh activation function ε is less than 5.0 × 10 −5 m. The position error with S-LVI-PDNN ε is less than 10 × 10 −5 m. Fig.4 shows the position errors of VOLUME 9, 2021  In summary, this four-leaf-clover path tracking simulation verifies that VP-RNN-PF can effectively and accurately solve the CMCO scheme.  Firstly, Fig.5 shows the end-effector of the JACO 2 redundant robot manipulator tracks Chinese character ''dai'' path. The path tracking task is completed well, which can be seen in Fig.5(a) and (b). After completing a task cycle, the joint angle drifts are very small, which can be regarded as the joints of the manipulator fully return to the initial position. Table 4 shows the specific data of joint angle drifts of the CMCO scheme (η = 0.1 and δ = 0.8 ) synthesized by VP-RNN-PF.  In summary, the above two path tracking task results illustrate that the VP-RNN-PF can effectively solve the CMCO scheme. In addition, compared with the traditional S-LVI-PDNN, the proposed PNMCO scheme with different activation functions is more effective and more accurate.

IV. CONCLUSION
In this paper, a PNMCO scheme has been proposed and investigated to solve the multi-objective motion planning problem for redundant robot manipulators. The scheme consists of CMCO and VP-RNN-PF subsystems. The CMCO subsystem can achieve energy minimization, repetitive motion, and avoidance of speed peaks. In addition, the CMCO subsystem considers a novel variable joint physical constraint, and then the CMCO subsystem is reformulated as a QP problem. Finally, the QP problem has been solved by the proposed VP-RNN-PF. Comparison experiments have validated that the proposed VP-RNN-PF is more accurate and more efficient compared with the traditional S-LVI-PDNN for solving the multi-objective motion planning problem. The future research direction will be the VP-RNN-PF solver investigated to solve multi-criteria schemes at different levels.