Two DTZNN Models of O(τ4) Pattern for Online Solving Dynamic System of Linear Equations: Application to Manipulator Motion Generation

In this paper, firstly, a high accuracy one-step-ahead numerical differentiation formula with <inline-formula> <tex-math notation="LaTeX">$O(\tau ^{4})$ </tex-math></inline-formula> pattern is proposed for discretization. Meanwhile, a high precision first-order derivative formula of the backward difference rule with <inline-formula> <tex-math notation="LaTeX">$O(\tau ^{4})$ </tex-math></inline-formula> pattern error is given to approximate the derivative information. Then, two high accuracy discrete-time zeroing-type models (HADTZTM) with <inline-formula> <tex-math notation="LaTeX">$O(\tau ^{4})$ </tex-math></inline-formula> pattern, i.e., HADTZTM with derivative information known (HADTZTM-K) and HADTZTM with derivative information unknown (HADTZTM-U), are developed, analyzed and investigated for online solving the dynamic system of linear equations (DSLEs). In addition, the 0-stability, consistency, and convergence of the HADTZTM-K and HADTZTM-U are verified for DSLEs. From a theoretical/numerical viewpoint, the classical models are revisited and analyzed for online solving DSLEs. Ultimately, simulation experiment including an application to the path-tracking of the four-link planar manipulator is conducted to demonstrate the efficiency and superiority of the HADTZTM-K and HADTZTM-U, where the HADTZTM-U overcomes the difficulty of derivative information unknown in practical applications.


I. INTRODUCTION
The problem of solving linear equations has been widely researched by many scholars in the scientific and engineering fields as a fundamental mathematical one, including research on rapidity and accuracy of solutions under overdetermined, under-determined and normal conditions. Therefore, there are many applications for solving linear equation problems, for instance, robotics [1], [2], machine learning [3], optimization [4] etc. Many traditional methods such as Newton-Raphson iteration [5], gradient-related methods [3] and algorithms based on homotopic perturbation [6], [7], The associate editor coordinating the review of this manuscript and approving it for publication was Yan-Jun Liu. by reason of these proposed algorithms do not utilize time-varying derivative information in the design process. Therefore, these approaches are theoretically devoted to solve the static (or time-invariant) linear equations. Note that there are a large number of practical problems which should be formulated as dynamic (or time-varying) system of linear equations (DSLEs) [8]. That is to say, static solvers cannot exhibit satisfactory performance when applied to solve dynamic linear equations. Although a few algorithms have been developed to solve time-varying linear equations [14], the existing ones still have large errors in the application of DSLEs with high time-dependence such as robots [9], localization [10], and signal processing [11], etc. Therefore, research on the stability, accuracy and rapidity of algorithms still have a wide range of significance. In this paper, two high accuracy neural-network-based discrete-time models are designed to solve DSLEs with time-varying coefficients. The theoretical analysis and example simulations demonstrate the superior performance of the models in real-world applications.
In recent years, neural networks have been widely applied in various fields, both in science and engineering [12], [13], [13]- [19]. As a significant application, the kinematic control of robot redundant is frequently solved by the neural network method [20]. Aiming at eliminating the joint drift phenomenon of robot manipulator, Xie et al. have exploited the gradient descent method to construct the dynamic neural network, which would compel joint angle error to rapidly converged to zero [21]. This method has been extended to the research of the manipulator with unknown structure and achieved remarkable results [22]. In addition, according to the game theory, the manipulability optimization method of manipulators has been proposed in [23], which is assisted by a specially designed recurrent neural network with the great noise resistance. In practical applications, considering the convenience of algorithm implementation in hardware, Ferreira et al. have creatively proposed a classic gradient-based neural network (GNN) with parallel processing capabilities to solving the DSLEs [3]. However, although GNN has the potential capacity to handle problems in parallel, it just be theoretically and essentially designed for solving the static problems. Therefore, Zhang et al. proposed a new zeroing neural network (ZNN) based on recurrent neural network for online solving the mathematical problem in the engineering and science [24]. Based on the strong parallel processing ability and exponential convergence advantages of ZNN models in scientific applications, Zhang et al. have utilized several different functions to design different continuous-time zeroing-type neural network models (CTZTM) for solving DSLEs with time-varying coefficients [25], [26]. But there is no doubt that the similar continuous models can hardly be directly realized on digital computers and applied in real-world applications. Therefore, in order to facilitate the applications on hardware and make up for the shortcomings of the lack of accuracy of existing models, two discrete-time ZNN based models are designed for online solving DSLEs in this paper.
In this paper, a high accuracy first-order derivative approximation formula is firstly proposed and used for discretization. Sequentially, a novel high accuracy discrete-time zeroingtype neural network model with known derivative information (HADTZTM-K) is designed, analyzed and investigated for online solving DSLEs. Meanwhile, the 0-stability, consistency and convergence of the HADTZTM-K are theoretically proved for the first time. Two numerical examples and an illustrative application to a four-link planar redundant manipulator verify that the HADTZTM-K has a high accuracy with the order of residual error O(τ 4 ) pattern. However, for some real-world applications, the derivative information of HADTZTM-K will be difficult to know or obtain [27]. To overcome this disadvantage, the backward-difference rule with O(τ 4 ) pattern precision will be used to approximate the derivative information in the HADTZTM-K [28], and then a novel high accuracy discrete-time zeroing-type model with unknown derivative information (HADTZTM-U) is developed for online solving DSLEs in practical applications. The theoretical analysis and numerical simulation results demonstrate the efficiency and superiority of the proposed HADTZTM-K and HADTZTM-U for online solving DSLEs.
The remainder of this paper is organized as follows. In Section II, the problem formulation of DSLEs and a new Taylor-type difference rule are introduced and analyzed to design the models of HADTZTM-K and HADTZTM-U as a foundation. Section III develops, investigates and theoretically proves the superiority of HADTZTM-K and HADTZTM-U for online solving DSLEs. In Section IV, three models, including the discrete-time Newton-Raphsontype model (DTNTM), the discrete-time Euler-type neural network model (DTETM), and the discrete-time zeroingtype model (DTZTM) are presented for comparison with the HADTZTM-K and HADTZTM-U. To demonstrate the superior performance of the proposed models, two numerical examples and a four-link planar manipulator example are illustrated in Section V. In the end, Section VI concludes the future works of this paper. The main contributions of this paper are as follows.
1) This paper proposes two discrete-time zeroing type models, which overcome the shortcomings of traditional solvers that can only solve static problems, and facilitate hardware implementation. 2) Theoretical analyses and results demonstrate that the proposed HADTZTM-K and HADTZTM-U models with the higher accuracy of O(τ 4 ) pattern are superior compared with DTNTM, DTETM and DTZTM for online solving DSLEs. 3) Two numerical simulation results, together with an illustrative application to a four-link planar manipulator, which substantiate the efficacy and superiority of the proposed HADTZTM-K and HADTZTM-U for online solving DSLEs in real time.

II. PROBLEM FORMULATION AND PRELIMINARIES
To build the foundation for further investigation, the problem formulation of DSLEs is mathematically presented in this section. Meanwhile, a high accuracy first-order derivative approximation with the O(τ 3 ) truncation error pattern is given as the basis for the HADTZTM-K and HADTZTM-U.

A. PROBLEM FORMULATION
Considering the mathematically problem formulation of the continuous time-varying DSLEs: where the coefficient matrix A(t) ∈ R m×n is defined as a measurable, full column or row rank matrix with time-varying coefficients, B(t) ∈ R m denotes a time-varying vector, and VOLUME 8, 2020 X (t) ∈ R n signifies the unknown time-varying solutions of (1) to be obtained. As a basis for the design of discrete-time models, the corresponding discrete-time form of the DSLEs (1) deserves to be mentioned as follows: where matrices A k+1 and B k+1 represent the sample value of A(t) and B(t) at sampling time t = (k + 1)τ , respectively; τ denotes the sampling interval; X k+1 will be obtained in the [kτ, (k + 1)τ ). Therefore, utilizing the previous or real-time data, the time-varying solution X k+1 of linear equations will be updated online in each computational interval.

B. HIGH ACCURACY TAYLOR-TYPE DIFFERENCE RULE
To obtain a higher accuracy discretization formula in comparison with the Euler-type and Newton-Raphson-type models, a novel effective O(τ 3 ) formula is generalized for first-order derivative estimation. Furthermore, to compensate the lack of accuracy of existing models, the higher precision first-order derivative approximation formula with O(τ 3 ) pattern will be utilized for discretization. According the general first-order derivative approximation formula in [29] where f k+i = f (t k+i ) = f ((k + i)τ ); c j = R, (j = 1, 2, . . . n) and ζ = 0 ∈ R are the coefficients; O(τ m ) is the truncation error. The left-hand side first-order derivativė f k will be discretized into six parts, which include f k+1 , f k , which has a truncation error of O(τ 3 ). Proof: The detailed certification process is shown in Appendix A.
C. CONTINUOUS-TIME ZEROING-TYPE NEURAL NETWORK MODEL As a preliminary foundation for the establishment of HADTZTM-K and HADTZTM-U, in [25], [26], considering the over-determined, under-determined and normal conditions of DSLEs (2), the CTZTM is designed by using zeroing formula and error functions. Combined with the above high accuracy first-order derivative approximation formula (4), a discrete-time zeroing-type model is obtained to solve DSLEs in real time. According to the design process in [17], [26], the design steps of the CTZTM model is as follows.
Step 1: Define the vector-formed error function as the zero finding function which is Step 2: Choose the zeroing dynamics design formula (ZDF) and activation function as follows: Here, the linear activation function F(e(t)) = e(t) will be utilized to simplify the ZDF. Therefore, the ZDF can bė Step 3: Combining the equation (5) with (6), the CTZTM (7) can be obtained to online solving the DSLEs (2). (7) where the parameter γ that can be selected in accordance with the performance of the hardware, which is designed to ensure high convergence rate of the CTZTM (7). On the premise of (1), the A Hereto, the problem formulation and preliminaries have been introduced for the design of HADTZTM-K and HADTZTM-U.

III. DISCRETE-TIME ZEROING-TYPE MODELS AND THEORETICAL ANALYSES
In this section, it is seriously considered that the case where the derivative information is known and unknown, and the lack of accuracy of the existing model. Then, two novel high accuracy discrete-time zeroing-type neural network models with derivative information known and unknown, termed as HADTZTM-K and HADTZTM-U, are developed, analyzed and investigated for online solving DSLEs (2).

A. HADTZTM-K AND HADTZTM-U
Utilizing the Taylor-type difference rule to discretize the CTZTM (7) model, the HADTZTM-K and HADTZTM-U will be designed for online solving DSLEs (2).
Firstly, based on high accuracy first-order derivative approximation formula (4), the equation (4) can be described as follows.
which has a truncation error of O(τ 3 ).
Assuming that the derivative information is known in CTZTM (7), then, the formula (8) will be employed to discretize the CTZTM (7) model. As a result of discretization, the HADTZTM-K is (9) where the parameter h = γ τ > 0 ∈ R signifies the step size, which needs to be set in accordance with the performance of HADTZTM-K (9) in the simulation process. However, theȦ k andḂ k of the model (9) may be difficult to be acquired directly in real-world applications. Thus, in [28], to overcome the situation of the unknown derivative information, the firstorder derivative formula (10) of the backward-difference rule with O(τ 4 ) pattern truncation error: will be applied to approximate the derivative information of the model (9). Thereby, the HADTZTM-U can be described as below: where step size h = γ τ > 0 ∈ R, and τ denotes the sampling gap. So far, two high accuracy discrete-time zeroing-type neural network models have been designed for online solving the DSLEs. Compared with the traditional continuous-time zeroing-type model, the proposed HADTZTM-K (9) and HADTZTM-U (11) can be widely applied to the digital hardware. Meanwhile, two novel discrete-time models, which not only simultaneously combine higher precision and better real-time performance than the existing discrete-time models, but also overcome the challenging problem that the derivative information is difficult to be obtained in real-world applications.

B. THEORETICAL ANALYSIS AND RESULTS
Based on several definitions described below, the proposed HADTZTM-K (9) and HADTZTM-U (11) are certified and analyzed for the 0-stability, consistency, and convergence as follows.
i=0 β i F k+i can be proved to be 0-stability by calculating and checking the roots of characteristic polynomial P N (δ) = N i=0 α i δ i , if the roots of P N (δ) = 0 lie in/on the unit disk (i.e., |δ| ≤ 1 and |δ| = 1). Definition 2: An N step model can be testified to be consistent with order m , if the model has a truncation error of O(τ m ) when finding a smoothly exact solution, where m > 0. Definition 3: An N step model is convergent, i.e., X (t−t 0 )/τ → X * (t), ∀t ∈ [t 0 , t f ], as τ → 0, if and only if the model is consistent and 0-stability.
According to the above definitions, the theorems about the HADTZTM-K (9) and HADTZTM-U (11) models are proved.
Theorem 2: The HADTZTM-K (9) is 0-stable. Proof: Based on the Definition 1, the characteristic polynomial of HADTZT-K (9) can be described as where the five roots can be calculated as follows: where all the characteristic roots are located in/on the unit disk, and i signifies the imaginary unit. Therefore, according to the Definition 1, the proof about 0-stability of HADTZTM-K (9) is accomplished. Theorem 3: The HADTZTM-K (9) is consistent and convergent, which converges with the 4-order of the O(τ 4 ) pattern residual error for ∀t k ∈ [t 0 , t f ].
Proof: According to the Definition 2, Definition 3 and Theorem 1, equation (8) can be rewritten aṡ Utilizing the equation (12) to discretize the CTZTM (7), the HADTZTM-K (9) with the O(τ 4 ) pattern residual error can be developed as Comparing equation (13) and equation (9), it can be observed that the HADTZTM-K (9) can be obtained if the O(τ 4 ) pattern truncation error of (13) is dropped. Therefore, based on the Definition 2, HADTZTM-K (9) is consistent with 4-order of the O(τ 4 ) pattern truncation error of (13). Then, according to Definition 3, the HADTZTM-K (9) can be certified to be consistent and convergent, which converges with the 4-order of O(τ 4 ) truncation error for all t k ∈ [t 0 , t f ].
Theorem 4: HADTZTM-K (9) has O(τ 4 ) order of steady-state residual error lim k→∞ ||A k X k − B k || F in solving the problem of DSLEs (2), where the || · || F denotes the Frobenius norm of a matrix.
Proof: Let X * k ∈ R n be the exact solution of (2), based on the above Definitions, the solution of the HADTZTM-K (9) can be discribed as k being the exact solution of (2), thus, A k X * k = B k . Sequentially, the above results can be further given by Therefore, the steady-state residual error of HADTZTM-K (9) is Hereto, the proof is completed.
Similarly, due to Definition 1, Definition 2 and Definition 3, the theoretical analyses of the HADTZTM-U (11) are developed and investigated as follows.
Theorem 5: The HADTZTM-U (11) is 0-stable. Proof: In light of the Definition 1, the characteristic polynomial of HADTZTM-U (11) is listed below where the five roots can be seen as follows, δ 1 = 0.3833, Proof: According to the previous Definitions, employing the equation (12) to discretize the CTZTM model (7) and utilizing the equation (10) to estimate the unknown derivative information. Therefore, the HADTZTM-U (11) with O(τ 4 ) pattern residual error can be given by The formula (14) can be further rewritten as where the term 2A  (15), the HADTZTM-U (11) will be obtained to solving the problem (2). Based on the Definition 3, the proof about consistency and convergence of the HADTZTM-U (11) are completed.
To sum up, the 0-stability, consistency and convergence of the HADTZTM-K (9) and HADTZTM-U (11) have been investigated for online solving DSLEs (2). From the perspective of theoretical analysis, the proposed two novel models not only show higher accuracy, but also possess outstanding real-time performance when solving the DSLEs. In order to demonstrate the superior performance of the above two models in the numerical simulations, three comparative models are developed, analyzed and investigated in the following sections.

IV. DTZTM, DTETM AND DTNTM
In this section, according to [14], the theoretical analysis and results of the DTZTM, DTETM and DTNTM models are presented for contrasting with the models of HADTZTM-K (9) and HADTZTM-U (11) for online solving DSLEs (2).
which has an O(τ 2 ) pattern truncation error, thus, the DTZTM with error terms can be given by Getting rid of the O(τ 3 ) term on the right of (17), the DTZTM can be generalized as (18) where the parameter h = γ τ is the step size and τ is the sampling gap. Proof: According to [14] and Definitions 1, the roots of characteristic polynomial P 3 (δ) = δ 3 − 1.5δ 2 + δ − 0.5 of the (18) are located in the unit circle, where δ 1 = 0.25 + 0.6614i, δ 2 = 0.25 − 0.6614i and δ 3 = 1. Based on the Definitions 2-3, it can be summarized as the model of DTZTM (18) is 0-stable, consistent and convergent to the 3order of O(τ 3 ) pattern residual error for all t k ∈ [t 0 , t f ].

B. DTETM AND DTNTM
Owing to [28], the Euler-type first-order derivative approximation formula with O(τ ) pattern will be selected to discretize the CTZTM (7), theṅ thus, the DTETM can be expressed as where h = τ γ is the step size. Note that the DTNTM can be obtained if the parameter h = 1 and the derivative informationȦ k andḂ k are dropped from (19), which is given by Contrastively, it can be summarized as the DTNTM (20) is a particular case of the DTETM (19). Here, it is worth declaring that the DTZTM (18), DTETM (19) and DTNTM (20) are presented for comparing with the HADTZTM-K (9) and HADTZTM-U (11). Due to the page limit, it is unnecessary to analyze the theory of the three comparative models in detail. Therefore, for the presented DTETM (19) and DTNTM (20), the detailed proof of the following theorems 8-9 can be found in [14].
Theorem 8: The DTETM (19) is 0-stable, consistent and convergent, which has a residual error of O(τ 2 ) and converges with the 2-order of O(τ 2 ) pattern residual error when t k ∈ [t 0 , t f ].
Theorem 9: The DTNTM (20) is 0-stable, consistent and convergent, which the residual error is O(τ ) pattern for all t k ∈ [t 0 , t f ].
In conclusion, DTZTM (18), DTETM (19) and DTNTM (20) have different solution precisions which are O(τ 3 ), O(τ 2 ) and O(τ ), respectively, in solving discrete-time DSLEs. Obviously, the existing highest accuracy of the O(τ 3 ) pattern cannot guarantee the high accuracy in practical applications. In this and the previous section, the 0-stability, consistency, and convergence of the proposed five models have been analyzed for online solving DSLEs (2). However, it is more necessary to verify the performance of the proposed models in real-world applications. Therefore, in the next section, the previous models will be utilized to online solving the DSLEs involved in Example 1-3. For the reader's convenience and comparison, the aforementioned five models are listed in Table 1.

V. NUMERICAL SIMULATIONS AND APPLICATION
In this section, two numerical simulations and an illustrative application to a four-link planar manipulator are provided to further verify the excellent performance of the proposed HADTZTM-K (9) and HADTZTM-U (11) for online solving the DSLEs. Meanwhile, the presented DTZTM (18), DTETM (19) and DTNTM (20) will also be applied to compare with the proposed HADTZTM-K (9) and HADTZTM-U (11).
It is worth pointing out that the initial value of the model is an important part in the simulation process. For the proposed models, HADTZTM-K (9) and HADTZTM-U (11) require five initial states to start the iteration, while the DTZTM (18) need three initial states. In order to choose a more reasonable initial value to compensate for the impact of initial value setting on model performance, based on the initial state X 0 , the DTETM (19) will be utilized to generate four states (X 1 , X 2 , X 3 , X 4 ) as the initial states for HADTZTM-K (9), HADTZTM-U (11) and DTZTM (18). The four initial states can be obtained as follows Hence, X 0 , X 1 , X 2 will be the the initial states of DTZTM (18), and the HADTZTM-K (9) and HADTZTM-U (11) will iterate from five initial states (X 0 , X 1 , X 2 , X 3 , X 4 ). In addition, by reason of the derivative information of the two numerical simulations can be obtained by MATLAB software. Therefore, HADTZTM-K will be adopted to solving the two numerical simulations involved through Example 1 to 2 compared with DTZTM (18), DTETM (19) and DTNTM (20). Then, HADTZTM-U (11) will be employed to deal with the path-tracking problem of the four-link planar manipulator involved in Example 3.

A. NUMERICAL EXAMPLE 1
In this example, considering the linear equations under normal conditions (m = n), the A k and B k can be given by The aboved four models (9), (18), (19) and (20) will be applied to solve the DSLEs involved in Example 1, where the parameter h = 0.5 and the sampling gap is set to τ = 0.01s, 0.001s and 0.0001s. Then, starting from five randomly generated initial state X 0 ∈ [−1.2, 1.2] 3 , the corresponding numerical results are shown in Figure. 1. Assuming the theoretical solution is unknown, as can be seen from the Figure. 1 that the four discrete-time models are iterated from five different initial state, and the corresponding solution will converge together in a certain time. However, under the premise that the theoretical solution is unknown, the results in Figure. 1 cannot be directly considered as theoretical solutions. Therefore, combining with the steady-state residual errors of the models in Figure. 2, it can be summed up as the above four In order to demonstrate the performance of the model in solving the DSLEs, the residual errors ||e(kτ )|| F = ||A k X k − B k || F is utilized to reflect the performance of the model. The residual error of the models can be seen in Figure. 2. In addition, the maximal steady-state residual errors (MSSREs) of the above models are listed in Table 2 for further comparison. As shown in Figure. 2(b), the HADTZTM-K (9) achieves the highest convergence order of 10 −8 pattern, which is higher than those of DTZTM (18), DTETM (19) and DTNTM (20) (10 −6 , 10 −4 and 10 −2 ). Comparing the Figure. 2(a) with Figure. 2(b) and Figure.   residual error is order of 10 −5 , 10 −8 , 10 −11 as the sampling gap τ decreases from τ = 0.01s, to 0.001s and 0.0001s. As it can be seen in Figure. 1(b), the results are converge to the theoretical solution in the form of an O(τ −8 ) pattern residual error. 4) The steady-state residual error of the proposed HADTZTM-K (9) changes in the pattern of O(τ 4 ) as the sampling gap τ changes from τ = 0.01s, to 0.001s and 0.0001s. Specifically, Figure. 1(a) reveals that the results of HADTZTM-K (9) will converge to the In summary, the computational performance of HADTZTM-K (9) is superior to DTZTM (18), DTETM (19) and DTNTM (20). Table 2 reveals that the MSSREs of the aboved four models decrease with the parameter h increasing from h = 0.1 to 0.6. Moreover, as the sampling gap changes by 10 times, the steady-state residual error of the proposed HADTZTM-K (9) not only can be changed in an O(τ 4 ) pattern, but also the average computing time per updating 10 −5 is less than sampling gap of τ = 10 −2 s, τ = 10 −3 s and τ = 10 −4 s. In other words, HADTZTM-K (9), a higher accuracy solver, can be utilized to solving the DSLEs in real-time.

B. NUMERICAL EXAMPLE 2
In this example, the following under-determined DSLEs with m = 8 and n = 9 are designed to text the proposed HADTZTM-K (9) as: a 2n (t k ) . . .
where the a mn (t k ) and b m (t k ) are , m is odd sin(t k ), m is even.
In this example, the models of HADTZTM-K (9), DTZTM (18), DTETM (19) and DTNTM (20) will be selected to solve the above DSLEs. Starting from five randomly generated initial state X 0 ∈ [−1.  Figure. 3. Assuming the theoretical solution to be unknown, as we can see from the Figure. 3 that HADTZTM-K (9) starts from five initial states, and the corresponding solutions will also coincide together in a certain period of time. Similarly, the steady-state error ||e(kτ )|| F = ||A k X k − B k || F in Figure. 4 will be calculated to illustrate the solution of the HADTZTM-K (9) can converge to the theoretical solution. Due to space limitations, the numerical simulation results of  the DTZTM (18), DTETM (19) and DTNTM (20) to online solving the time-varying DSLEs involved in Example 2 are not shown in this section.
According to the Figure. 3, Figure. 4 and Table 3, the performance of the above four models can be summarized as follows.
1) For the presented DTNTM (20), with h = 1 and τ = 0.01s, 0.001s, 0.0001s, its steady-state residual error changes in the pattern of O(τ ). Looking at Table 3 as the sampling gap τ decreases from τ = 0.01s, to 0.001s and 0.0001s. In addition, the numerical simulation result in Figure. 3 will converge to the theoretical solution in the form of an O(τ −11 ) pattern residual error. To sum up, the computational performance of the HADTZTM-K (9) is superior to the model of DTZTM (18), DTETM (19) and DTNTM (20). In addition, it can be seen from Table 3 that the MSSREs of the DTZTM (18), DTETM (19) and DTNTM (20) will decrease with the parameter h changing from h = 0.1 to h = 0.6. However, what is different from Example 1 is that the steady-state residual error of the HADTZTM-K (9) will not decrease gradually with the parameter h increasing from h = 0.1 to h = 0.6 for online solving the DSLEs involved in Example 2. Note VOLUME 8, 2020 that, only in the case of h = 0.2 and h = 0.3, the MSSREs of the HADTZTM-K (9) changes in the pattern of O(τ 4 ) as the sampling gap τ changes by 10 times. That means choosing an appropriate h is an important step when the HADTZTM-K (9) is used to solve under-determined DSLEs. Therefore, it can be concluded from multiple numerical simulations that the HADTZTM-K (9) has outstanding performance at h = 0.3. Furthermore, the average computing time per updating 10 −5 is less than sampling gap of τ = 10 −2 s, τ = 10 −3 s and τ = 10 −4 s. In conclusion, the HADTZTM-K (9) can be utilized to solving the under-determined DSLEs with higher-accuracy in real-time.

C. APPLICATION ON PLANAR MANIPULATOR
In this example, the proposed HADTZTM-K (9) and HADTZTM-U (11) are applied to a four-link planar manipulator for path-tracking. In fact, the path-tracking problem of a planar manipulator can be transformed into solving the DSLEs online. From the [14], [30], it can be obtained as follows: where the q(t) andq(t) denote the joint-angle vector and velocity vector, and J (q(t)) = ∂ (q(t))/∂q(t) ∈ R 2×4 is the Jacobian matrix of the four-link planar manipulator, where the (·) is the forward kinematics function. The term oḟ r d (t) + κ(r d (t) − (q(t))) signifies the end-effector position velocity vector with position-error feedback, where the r d (t) andṙ d (t) represent the desired end-effector position vector and velocity vector, and the parameter k ∈ R + is the position-error feedback gain. Meanwhile, the r(t) can be calculated by utilizing the formula (q(t)) = r(t). In this application, the four-link planar manipulator is applied to track a three-leaf-clover path. The task will start from the initial joint-angle vector q(0) = [0, π/2, 0, π/2] T rad and complete the clover trajectory tracking during T = 10s. In addition, the parameters can be selected as follows: step size is h = 0.3, sampling gap is τ = 0.001s and position-error feedback gain is k = 15. As shown in Figure. 5, specifically, Figure. 5(a) shows the motion trajectories synthesized by the proposed HADTZTM-K (9). It can be seen from Figure. 5(b) and Figure. 5(c) that the joint-angle vector q and velocity vectorq synthesized by HADTZTM-K (9) are stable and smooth. Due to the space limitation, it is worth mentioning that the motion trajectories, joint-angle vector q and velocity vectorq of the HADTZTM-U (11), DTZTM (18) are similar to the Figure. 5(a)-5(c), therefore, it will be omitted.
In addition, from a qualitative perspective, see Figure. 5(d)-5(f), the position-error vector of the HADTZTM-K (9) and HADTZTM-U (11) are smaller than the DTZTM (18) with h = 0.3. Note that the HADTZTM-K (9) has excellent performance when solving the under-determined DSLEs involved in Example 2 at h = 0.3, therefore, the parameters can be chosen as h = 0.3 and τ = 0.001s to complete the three-leaf-clover path tracking of the  four-link planar manipulator. However, it cannot be directly confirmed that that the HADTZTM-K (9) is superior to other models under different h. Therefore, from a quantitative perspective, it can be summarized from Table 4 as follows: 1) The position-error of the HADTZTM-K (9) will decrease with the parameter h increasing from 0.1 to 0.3. Specifically, the HADTZTM-K (9) has outstanding performance compared to the DTZTM (18) and DTETM (19) at h = 0.3, where the maximal position-error e x = 2.0163×10 −5 , e y = 2.2032×10 −5 of the HADTZTM-K (9) is 2.87 times smaller than the one synthesized by DTZTM (18). 2) The position-error of the DTZTM (18) will decrease with the parameter h increasing from 0.1 to 1. Specifically, the end-effector has the minimum position-error e x = 2.5483 × 10 −5 , e y = 2.8921 × 10 −5 when the DTZTM (18) with h = 1 is applied to path tracking in a four-link planar manipulator. However, the minimum position-error of DTZTM (18) is still 1.2-1.3 times larger than the minimum position-error of HADTZTM-K (9). 3) After numerical simulations, we found that the parameter h of HADTZTM-K (9) must be less than 0.3. When h > 0.3, the task of the four-link planar manipulator path tracking will fail. In addition, the parameter h of DTZTM (18) must be less than 1. 4) When the derivative informationȦ k andḂ k of HADTZTM-K (9) is unknown, we utilize equation (10) to approximate the termsȦ k andḂ k , from Table 4 and Figure 6 we can obtain that the efficiency and superiority of the HADTZTM-U (11) in the path-tracking of the four-link planar manipulator. The position-error of the HADTZTM-U (11) will decrease with the parameter h increasing from 0.1 to 0.3. Note that the parameter h of HADTZTM-U (11) must be less than 0.3, the task of the four-link planar manipulator path tracking will also fail at h > 0.3.

VI. CONCLUSION
In this paper, firstly, it is subtlety designed the high accuracy Taylor-type difference rule (4) and the first-order derivative of the backward difference rule with O(τ 4 ) pattern residual error. Then, HADTZTM-K (9) and HADTZTM-U (11) are proposed, analyzed and investigated for online solving the DSLEs (2) in comparison with DTZTM (18), DTETM (19) and DTNTM (20). In addition, two numerical examples have illustrated that the HADTZTM-K (9) is superior to the three existing comparative models, and then an illustrative application to a four-link planar manipulator for path-tracking has also demonstrated that HADTZTM-K (9) has the minimum position-error. It meant that, when the derivative information is known, HADTZTM-K (9) has high solution accuracy, stability and real-time performance in both numerical simulations and real-world applications. Considering the derivative information may be difficult to obtain in practical applications, HADTZTM-U (11) has also been applied to the path-tracking task of the four-link planar manipulator. The numerical results have proved that HADTZTM-U (11) still has excellent performance when dealing with the case where the derivative information is unknown. For the future direction, the performance of the proposed models in a noisy environment could be discussed. Therefore, it will be further improved that the accuracy and anti-interference performance of the models should be considered and analyzed for the DSLEs. In addition, it is worth mentioning that the proposed two high accuracy discrete-time models will also be utilized to online solving zero finding problems [31]- [33], rehabilitation robots [34], trajectory planning [35], [36] and switched nonlinear systems [37]- [40] etc.