Towards non-linearly activated ZNN model for constrained manipulator trajectory tracking

As a powerful method for time-varying problems solving, the zeroing neural network (ZNN) is widely applied in many practical applications that can be modeled as time-varying linear matrix equations (TVLME). Generally, existing ZNN models solve these TVLME problems in the ideal no noise situation without inequality constraints, but the TVLME with noises and inequality constraints are rarely considered. Therefore, a non-linear activation function is designed, and based on the non-linear activation function, a non-linearly activated ZNN (NAZNN) model is proposed for solving constrained TVLME (CTVLME) problems. The convergence and robustness of the proposed NAZNN model are verified theoretically, and simulation results further demonstrate the effectiveness and superiority of the NAZNN model in dealing with CTVLME and the constrained robot manipulator trajectory tracking problems. In addition, the wheeled robot trajectory tracking fault problems with physical constraints are also analyzed theoretically, and the proposed NAZNN model is also applied to the manipulator trajectory tracking fault problem, and the experimental results prove that the NAZNN model also deal with the manipulator trajectory tracking fault problem effectively.


Introduction
With the acceleration of industrialization, manipulator has been widely used in industrial production. However, it is important to realize accurate control of robotic manipulators because as there are various disturbances and constraints in the production environment. In general, the behavioural control of a robot manipulator can be modelled as a TVLME problem [1][2][3]. When designing behavioural controllers for robot manipulators, there are necessary to consider robot dynamics [4,5], torque saturation [6] and obstacle avoidance [7,8] of the problem. For example, wheeled robots are non-linear systems and their controllers do not consider sliding, so it is difficult for these controllers to have good control when wheeled robots work with some complex environments, such as wet and icy, uneven surfaces. Moreover, the problems of dynamic coupling, dynamic limitations caused by the environments, and delay problems of the controller are also should be considered, and they complicate the control process of manipulator trajectory tracking. Therefore, researchers have proposed the PID control [9,10], feedback control [11], finitetime control [12][13][14], fuzzy control [15][16][17][18] and neural network control [19][20][21][22][23][24][25] to solve the above problems.
In recent years, with the continuous improvement and development of deep learning [26][27][28][29][30], neural networks [31][32][33][34][35][36] have become an efficient solution for various time-varying problems. For example, Jin et al. build the interference-tolerant fast convergence zeroing neural network (ITFCZNN) model [37] based on a new activation function, which exhibit excellent timevarying Robustness and convergence. One of the things that must be mentioned is that the activation function heavily influences the performance of the model. Therefore, the researchers have proposed a corresponding novel activation function in order to achieve the desired experimental results. For example [38], proposed zero-tuned neural networks (ZTNN) to solve the Stein matrix equation based on several new activation functions. In [39], it is proposed that the PSAF-based ZNN model is applied to the secondary programming problem. A ZNN model based on the activation function (HSAF) is designed in [40] to solve the time-varying square root of the matrix. In [41,42], it is proposed to apply the SBPAF-based ZNN model to solve the time-varying Sylvester equation. In [43,44], a ZNN model with adjustable convergence rate is designed based on the new activation function and applied to the control process of a robotic arm. However, most practical problems can be modelled as timevarying non-linear systems of equations with constraints. In contrast, the ZNN model described above only considers the unconstrained ideal environment and may not work as well as expected in practical applications, especially for the control of robot behaviour.
The physical constraints of industrial robots must be taken into account in the actual control process, otherwise specific tasks cannot be performed and even the hardware of the industrial robot is damaged. Considering the physical constraints of industrial robots are less studied. In [45], the motion process of an industrial robot is modelled as a multilayer time-varying problem in order to solve the problem of joint angle constraints in the control process [46]. proposed a physical limit constrained minimum velocity parametric coordination scheme to solve the constraints of a wheeled robot. In [47], the analytical solution of the robot controller is solved based on a parametric approach, which in turn yields the joint angle range of the industrial robot. However, the above methods only consider the angle overrun fault of the robotic manipulator, and they ignore other faults, such as speed overrun, noises, and the robotic manipulator does not stop in the case of angle overrun, which is possible in practical situations.
It is clear that the usage of neural networks to solve constrained manipulator trajectory tracking is still far from being investigated, and the potential of neural networks in this area remains to be exploited. Therefore, this paper develops the NAZNN for manipulator trajectory tracking with joint angle and joint velocity constraints in noisy environments.
The remainder of the paper is organized as follows. The modeling process and theoretical proof of the NAZNN model are analyzed in Section 2. Examples of the NAZNN model for solving CTVLME problems are presented in Section 3. Modeling of a wheeled robot is provided in Section 4, and simulation experiments of the NAZNN model for failure case of manipulator trajectory tracking with physical constraints are provided in Section 4. Finally, The conclusions of the paper are given in Section 5.

CTVLMA problem description and NAZNN model 2.1 The CTVLMA problem
In mathematics, a constrained time-varying linear matrix equation (CTVLMA) can be expressed as where A(t) ∈ R m×n and B(t) ∈ R p×n are smoothed full rank matrices, and m < n. C(t) ∈ R m and D(t) ∈ R p are smooth vectors. And X(t) ∈ R n is the unknown vector to be solved. The time-varying solution For CTVLMA (1), the following equation is obtained by Here v .2 (t) is also a unknown vector, and the superscript .2 denotes the square operation of each element of v(t) ∈ R p . Defining the logarithmic matrix Thus, Equation 2 can be represented in a matrix form below.
Then, we define V(t) where the upper label T denotes permutation. Then, the following equation is obtained based on the above equivalent substitution.

NAZNN model for CTVLMA problem solving
In ordered to find the solution of Eq. 4, the NAZNN is designed according to the following steps.
Firstly, denote the following error function E(t).
Then, the time derivatives of E(t) are deduced below.
where λ > 0 is the convergence factor, and F(•) is an activation function. In order to construct our model for solving Eq. 4, the following theorem 1 is introduced in advance. Lemma 1: The time derivatives of v .2 (t) can be written as where _ v(t) is the time derivatives of v(t). Proof: Let _ R(t) represent the temporal derivatives of R(t), and we can obtain the following equation. Then, Then, Eq. 8 can be expressed in the following form. v The proof is completed. Based on Eq. 5, Eq. 6 and Lemma 1, the following NAZNN model is obtained.
Let us set N(t) ] T , and Eq. 11 can be transformed to the following simplified NAZNN (12).
The NAZNN model with noise Y(t) can be expressed in the following form.
where λ > 0 and F(•) are the same definitions above. Actually, the performances of the ZNN models are closely related with the activation functions F(•), and the existing activation functions are listed in Table 1. Additionally, f(•) denotes the element of F(•).
In order to enhance the performances of the NAZNN (13), the following non-linear activation function (14) is designed.

Convergence and robustness analysis of the NAZNN model 2.3.1 Convergence analysis
The following Lemma 2 is provided in advance for the convergence analysis of the proposed NAZNN model. Lemma 2: [48,49] Considering a non-linear dynamic system as follows.
where f(•) is a continuously non-linear function. Assumed that there is a continuous function L(b) satisfying both of the following conditions.
Then, the above dynamic system (15) is fixed time stable, and the upper bound of its stable time is The following Theorem 1 and 2 guarantee the convergence and robustness of the proposed NAZNN model (12), respectively.
Theorem 1: If the NAZNN model (12) is not polluted by noise, and Eq. 1 is smooth at all times. For any initial system state X(t), the state solution of model (12) converges to the theoretical solution of Eq. 1 at a predetermined time t s .
Proof: If the error function E(t) in Eq. 5 converges to 0, the neural state solution of NAZNN (12) will be equal to the theoretical solution of Eq. 1, and the evolutionary formulation (6) assures the convergence of the error function E(t). Furthermore, Eq. 6 consists of n 2 independent subsystems. Therefore, we only need to show these subsystems are stable at a fixed time.
where the scalars _ e ij (t) and e ij (t) mean the elements of the ith row and jth column of _ E(t) and E(t) respectively. The flowing Lyapunov function is adopted for the convergence validation.
Derive for h(t): Then, substituting the non-linear activation function (14) into the above formula yields Based on Lemma 2, the upper bound of convergence time t ij for the subsystem in row i and column j of E(t) is obtained.
On basis of the above analysis, it can be drawn the conclusion that the state solution of the ZNN model (12) converges to the theory solutions of Eq. 1 in a predetermined time.

Robustness analysis
The robustness of NAZNN model (13) in noisy environment is discussed in this subsection. In order not to lose generality, the noise in Eq.13 is chosen to be Y(t) = 0.1exp(0.5t), and the following Theorem 2 guarantees the robustness of NAZNN model (13).
Theorem 2: Assume the theoretical solution of Eq. 1 exists, and each element of Y(t) satisfies the conditions |y ij (t)| ≤ δ|e ij (t)| and 0 < δ < λk 3 e/z. The NAZNN model (13) has the following fixed convergence time t s with noise Y(t).
Proof: Similar with the analysis of Theorem 1, Eq. 6 also contains n 2 mutually independent subsystems with noise Y(t).
The following Lyapunov function is selected for the stable validation of E(t).
The derived of h(t) is obtained below.
According to the conditions |y ij (t)| ≤ δ|e ij (t)|, we can obtain the following inequalities.
In order to simplify the approach, setting up the following equation Then, Let The derivative of (29) is provided in Eq. 30.
According to the above formula, it is clear that when 0 < ϖ < 1, the derivative of f(ϖ) is less than 0, and f(ϖ) decreases monotonically; when ϖ > 1, f(ϖ) > 0 and the derivative of f(ϖ) rises monotonically. Obviously, on basis of the above monotonic analysis, ϖ = 1 is equivalent to z being close to 0 and f(ϖ) achieving a minimum, which indicates f(t) reaches its minimum value.
Since δ ≤ λk 3 e/z, δ is less than the minimum value of f(t), and u 1 (t) is always less than 0. In summary, inequality (26) can be further simplified to the following form.
Then, based on Lemma 2, the convergence time of the ijth element of E(t) is obtained.
The maximal convergence time t s of the ZNN model (13) is Frontiers in Physics frontiersin.org From the above analyses, we can draw the conclusion that the proposed NAZNN model has the fixed convergence time t s in noiseless and noisy environments.

Numerical simulations for CTVLME solving
In this section, an illustrative example of the NAZNN model for solving CTVLME (1) is introduced to demonstrate its efficiency and accuracy.
Consider CTVLME (1) with the following coefficients. where Here, we set all the parameters m, p, q, z, k 1 , k 2 , k 3 , k 4 in the nonlinear activation function (14) to 5, and the state solutions generated by the proposed NAZNN model are shown in Figure 1.
According to Figure 1, it can be observed that the black curves quickly coincide with the red curves, which indicate that the state solutions generated by the NAZNN model converge to the theoretical solutions of Eq. 1 in a very brief period of time.
In addition, the residual errors of the NAZNN model (12) and the IZNN model activated by LAF, PSAF, HSAF and SBPAF for solving the above same equation are compared in Figure 2. Figure 2A shows only the residual error of the NAZNN model (12) converges rapidly to 0, while the IZNN model activated by the activation functions converges slowly.
Furthermore, it can be observed that the convergence speed becomes faster as the value of m increases in Figure 2B. This phenomenon indicates that the time required for the NAZNN model (12) to solve inequality (1) becomes shorter as m increases. Finally, from Figure 2C, we can see that the convergence time of the NAZNN model becomes smaller as the convergence coefficient λ increases, which indicates that the convergence speed of the NAZNN model is also closely related to its convergence coefficient. Therefore, we can set these parameter values according to the accuracy requirements.
It is worth mentioning that the system environment in reality is complex and variable. Therefore, the NAZNN model (13) is used to solve inequality (1) with the different noises to highlight its robustness, and the simulation result are shown in Figure 3.
As can be seen from Figure 3A   As can be seen in Figure 4A-D, the residual errors of the NAZNN model (13) drops to 0 in a very short time in the presence of four noises. However, the residual errors of the IZNN model fluctuate greatly or converge very slowly, or even fail to converge to 0 due to noises.

Applications on wheeled robots
In this section, the NAZNN model is applied to solve the wheeled robot trajectory tracking problems [50,54].

Modeling of the physical constrained manipulator
In this subsection, the modeling of a movable six-joint three-wheel manipulator is introduced and its 3D model is shown in Figure 5A.
The equations of motion are analyzed according to the wheeled robot mobile device in Figure 5B, and the parameters are shown below.
W 0 : the midpoint of the drive axis, expressed in the world coordinate system as (x 0 , y 0 , z 0 ). W s : the position of the manipulator, expressed in the world coordinate system (x s , y s , z s ). d: distance from W 0 to W s , d = 0.1m. b: distance from the midpoint of the drive axis to the left and right drive wheels, b = 0.32m. r: radius of each driving wheel, r = 0.1025m. θ: The heading angle of the mobile device, expressed as the angle of the mobile device from the X-axis to the symmetry axis; its time derivative is the heading speed. P: the mobile device rotates around the point P. R: the distance of point P from the left driving wheel. ω: the rotation speed of the mobile device around point P; and ω _ θ.
_ β 1 and _ β r : rotation speed of the left and right wheels. The next step is to establish robot model of the mobile device. Assume that each link of the wheeled robot is rigid and there is no relative sliding between the mobile device and the robot arm. Based on Figure 5B, the kinematic equations for the mobile device are established as follows, and the detailed derivation can be found in Supplementary Appendix S1A.
Rewrite Eq. 36 into matrix form Frontiers in Physics frontiersin.org where _ x s and _ y s denote the x-axis velocity component and y-axis velocity of the point W s , respectively. Figure 5A shows a 3D model of a wheeled robot. The origin of the moving platform coordinate system (x s , y s , z s ) is defined at the point w s in Figure 5B, while x s is the positive direction of the moving device. Let (x s , y s , z s ) indicates the reference points in the coordinate system of the moving platform, and for computational convenience we set z s to 0. Similarly, the coordinate system of the ith joint of the six-joint robot is defined as (X i , Y i , Z i ), as shown in Figure 5C (i = 1, 2, . . . , 6). The articulation of each joint with the previous joint is defined as the origin of each joint's coordinate system, and the direction along this joint is the Z-axis of this joint's coordinate system, which conforms to the left-hand rule to find the X-axis and Y-axis. The coordinate system of the end-effector (X 6 , Y 6 , Z 6 ) position vector 6 W end belongs to R m , and the homogeneity condition shows that 6 W end [ 6 W end T , 1] T belongs to R m+1 . (The superscript T denotes the transpose of the vector.) Based on the platform coordinate system (x s , y s , z s ), we can represent the end-effector vector as follows.
where s 1 T, 1 2 T, 2 3 T, 3 4 T, 4 5 T, 5 6 T denote the flush transformation matrices. The position coordinate vector of the end-effector in world coordinates can be expressed as where δ z ∈ R n indicates the position relative to the world coordinate system, and z s T cos θ −sin θ 0 x s sin θ cos θ 0 y s 0 According to the above analysis, the coordinates of the endeffector in the world coordinate system can be expressed as (x z , y z , z z ), and the kinematic relations of the end-effector can be obtained as follows: where δ s is a three-dimensional vector, and the detailed expressions and calculation procedure for δ s are shown in Supplementary Appendix S1B. Besides, Eq. 40 can be obtained by the time derivative of the speed level equation.
It can be reformulated as: In three-dimensional space, the position of the end-effector is calculated with n = 3 (n is the number of variables), and the position and direction of the endeffector is calculated with n = 6. To control the wheeled robot, we set n = 8 (including two rotation variables and six joint variables). _ θ 1 and _ θ r denote the rotational speed of the left and right drive wheels, and _ γ i (i = 1, 2, . . . , 6) denotes the joint velocity vector of the robotic arm. The combined velocity vector _ ϕ is a derivation of the combined angular vector ϕ with respect to t. In addition, J z ∈ R m×n denotes a generalized Jacobi matrix and J z ∈ R m×n . The detailed expressions and derivations are given in Supplementary Appendix S1C. The physical constraints of the robot are shown in Table 2. Conventional control approaches have difficulty in handling trajectory tracking with constraints, and physical constraints are rarely considered before. However, almost all controllers have physical constraints (angular constraints, speed constraints, etc.). Therefore, it is very realistic and necessary to consider the failure problem of trajectory tracking of wheeled robots subject to constraints. The physical constraints of wheeled robot trajectory tracking considered in this work are listed below.
where v > 0 is the feedback gain, _ δ zd and δ zd denote the position vector and velocity of the desired end position of the tracking path. ϕ − and ϕ + denote the lower and upper limits of the joint angle ϕ, _ ϕ − and _ ϕ + denote the lower and upper limits of the joint velocity _ ϕ, respectively. For the range of motion of the wheeled robot, we theoretically set the upper and lower limits of ϕ 1 and ϕ 2 to converge to infinity.
To deal with the problem at the joint velocities level, the range of active wheel rotation angles of the wheeled robot and the range of robot joint angles are transformed into the corresponding velocities as follows.
where the coefficient η > 0 is used to calculate the range of joint velocities, and the coefficient has η special points. The value of η may lead to a sudden deceleration of the joint velocity when the wheeled robot approaches the joint limit, in numerical terms, By calculating and setting the next simulation experiment in η = 4; the constraints of angle and velocity can be combined into constraint ψ − ≤ _ ϕ ≤ ψ + , where the κ th element of ψ − and ψ + can be expressed as In summary, the velocity control of wheeled robot trajectory tracking with physical constraints can be formulated as where q ∈ R m denotes the combined velocity vector _ ϕ, J = J z , and the equation restriction represents the linear relationship between the velocities. Besides, _ δ _ δ zd + v(δ zd − δ z ) ∈ R m , where v > 0 is the feedback gain. _ δ zd and δ zd denote the position vector and velocity at the desired end position of the tracking path. ψ denotes the range of active wheel rotation angles of the wheeled robot and the range of robot joint angles transformed into the corresponding velocities. ψ + and ψ − represent the upper and lower bounds of the transformed

NAZNN model for trajectory tracking control of wheeled robots
Equation 46 with the above conditions can be transformed as Convert (47) to matrix form by referring to the CTVLMA calculation process.
, y(t) and o(t) are process quantities generated by the calculation process with reference to CTVLMA. Let Then, define u(t) [ q y(t) o(t) ] T , the following matrix vector form of (45) is obtained The NAZNN model for trajectory tracking control of wheeled robots with physical constraints is obtained.

Simulation results on physically constrained wheeled robots
In this section, the above NAZNN model (50) and other related ZNN models are all used to solve the same wheeled robotic arm trajectory tracking problem with physical constraints for the purpose of comparison.

NAZNN model simulation results
In this subsection, the proposed NAZNN model is applied to control the wheeled robot to perform trajectory tracking, and the corresponding simulation results are represented in Figure 6. As observed in Figure 6, the wheeled robot controlled by the NAZNN model completes the tracking task effectively.
Additionally, the joint angle variation of the robot for spiral trajectory tracking is represented in Figure 7, and the joint velocity of the robot for the tracking process is represented in Figure 8.
It can be observed in Figures 7, 8 that the joint angles and joint speeds of the NAZNN model-controlled wheeled robotic arm are within the physical limits until the end of the trajectory tracking, which indicates that the NAZNN model-controlled wheeled robotic arm successfully completes the tracking task.

IZNN models simulation results
In this subsection, the IZNN models controlled robots are also used for the same tracking task for the purpose of comparison, and the corresponding simulation results are represented in Figures 9-12. Figure 9 shows the difference between the IZNN-1 and IZNN-2 models. Both the IZNN-1 and IZNN-2 models are activated by the linear activation function. Because the parameter size has a great impact on the models, the actual experimental results are also different.
As observed in Figure 9, the IZNN-1 model controlled robot fails the tracking task due to robot joint angle _ λ 1 exceeds limitation; besides, the IZNN-2 model controlled robot also fails the tracking task due to robot joint speed _ λ 5 exceeds limitation. Figure 9A shows that joint angle 1 of the IZNN-1 model exceeds the upper limit at 3.6 s, and it lead to the trajectory tracking task fail; Figure 9B shows that joint velocity 5 of the IZNN-2 model falls below the lower limit at 3.6 s, and it lead to the trajectory tracking task pause.  The NAZNN model-controlled wheeled robots trajectory tracking with 6 joint angle changes.
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FIGURE 8
The NAZNN model-controlled wheeled robot successfully finishes the trajectory tracking with 6 joint velocities change.  The wheeled robot controlled by the IZNN-3 model for the same tracking task is presented in Figures 10-12. As observed in Figure 10, the wheeled robot controlled by the IZNN-3 model also fails the tracking task. Additionally, the joint angle variation of the robot controlled by the IZNN-3 model for spiral trajectory tracking is represented in Figure 11, and the joint velocity of the robot FIGURE 11 The IZNN-3 model-controlled wheeled robot fails trajectory tracking with joint angles exceed limits.

FIGURE 12
The IZNN-3 model-controlled wheeled robot fails trajectory tracking with Joint velocities exceed limits.
Frontiers in Physics frontiersin.org 12 controlled by the IZNN-3 model for the tracking process is represented in Figure 12.
As observed in Figure 10, Figures 11, 12, the tracking task is interrupted, with the joint speed ( _ λ 3 ) exceeds the physical limit (3 rad/s) and the joint angle (λ 3 ) exceeds the physical limit (exceeded the lower limit -π/6). Speed 3 ( _ λ 3 ) in Figure 10 exceeds the limit at 3.3 s and angle 3 (λ 3 ) in Figure 11 exceeds the limit at 3.3 s, which stop the tracking task. The above analysis indicates that the wheeled robotic arm controlled by the IZNN-3 model does not complete the tracking task due to exceediton of the joint speed and joint angle constrains.
In summary, the proposed NAZNN model has better control performances compared to other models for physically constrained wheeled robot trajectory tracking, and it completes the physically constrained trajectory tracking task smoothly and accurately without exceeding the joint angle and joint speed. The detailed comparative results of the models are presented in Table 3.

Conclusion
In this paper, a NAZNN model is proposed and effective applied to solve CTVLME problems. It is theoretically demonstrated that the NAZNN model can obtain the exact solutions of CTVLME problems. The validity and superiority of the NAZNN model is further verified by two numerical examples. Besides, the proposed NAZNN model is also applied to the failure problem of trajectory tracking of wheeled robots with physical constraints. Finally, the NAZNN model is used to control the wheeled robot to complete trajectory tracking under restricted conditions to prove the feasibility of this control method.

Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.