Design of Three-Dimensional Intelligent Guidance Law for Intercepting Highly Maneuvering Target

This paper investigates the three-dimensional guidance and control problem of missile intercepting highly maneuvering target, whose acceleration information is difficult to accurately predict. With the three-dimensional guidance model for intercepting single target established by using the principle of zeroing the rate of line-of-sight (LOS), a novel intelligence guidance law has been designed through backstepping sliding mode control method, radial basis function (RBF) neural network and adaptive control technique. Then, a Lyapunov-based stability analysis demonstrates that all the signals are bounded, and the LOS rates ultimately converge to a neighborhood of the origin. Following advantages are highlighted in this paper: (i) the target information is online estimated and compensated by the RBF neural network, which indicates that the proposed guidance law is easily put into practice only relying on the position information of target. (ii) an adaptive gain term is designed in the control system, which greatly reduces the inherent chattering of sliding mode method. At last, simulations are conducted, and results illustrate the effectiveness and superiority of the designed guidance law.


I. INTRODUCTION
Missile has gradually become the main offensive weapon in the modern battlefield owing to its high speed, long range and high accuracy [1], [2], [3]. Therefore, positively developing interception technology for highly maneuvering targets has great significance to national defense and security [4]. At present, the proportional guidance law is widely used in the field of missile interception because of its simple structure and easy implement in practice [5]. A bias proportional guidance law with the constraint of collision angle for lowspeed maneuvering target has been proposed in [6], and the effectiveness of the designed algorithm is verified through numerical simulation. In [7], the line of sight angle and line of sight angle velocity are considered as the system state variables, and a new extended proportional guidance law was designed based on Lyapunov stability theory. Further, a linearized three-dimensional proportional guidance law has The associate editor coordinating the review of this manuscript and approving it for publication was Amjad Ali. been developed in [8] with considering the influence of coupling terms in pitch and yaw direction. Although the above methods have good control effect in real engineering, they are only suitable for the interception of stationary or low-speed targets. For highly maneuvering targets, their acceleration information is hard to obtain directly, which causes that the aforementioned methods may exist the problem of unstable line of sight and excessive overload. Thus, the design of interception guidance law for highly maneuvering targets has become a popular research topic.
The sliding mode control (SMC) method has been extensively applied in the design of nonlinear guidance and control system due to its good robustness. In [9], a finite time guidance law was proposed by using fast terminal sliding mode control technology for maneuvering targets, which has a faster convergence rate than the traditional sliding mode algorithm. In [10], a super-twisting adaptive sliding mode guidance law was designed, and the target information in the guidance model is considered as the uncertainty, which is compensated through adaptive control technology. A high performance sliding mode guidance law was developed in [11], which theoretically achieves a zero miss distance at the final time. In [12], a robust guidance law was designed via terminal sliding mode (TSM) method, for the maneuvering targets in the presence of large initial heading errors. In [13], a continuous second-order sliding mode based impact angle guidance law has been proposed for intercepting moving targets with unknown acceleration bounds. In [14], an adaptive fast fixed-time convergent guidance law was presented through nonsingular terminal sliding mode for maneuvering targets with the impact angle constrains. In [15], a novel time-varying sliding mode guidance law for maneuvering targets with unknown target acceleration. Unfortunately, the shortcomings of the above methods are that the symbolic functions in switching term are discontinuous, which is easy to cause the problem of control system chattering. The simplest approach to reduce the chattering is to use the saturation function instead of the symbolic function in the control system, but for highly maneuvering targets, whose acceleration information may not be obtained precisely, the parameters of symbolic function are difficult to be determined. To overcome this problem, a observer-based sliding mode guidance law was presented in [16], and the expanded state observer was introduced to obtain the information of targets. In [17], a nonsingular sliding mode guidance law was developed by combining a linear extended state observer technique, and the target acceleration is estimated by observer employing current system states. A homogeneous high-order sliding mode guidance law based on an extended observer has been developed in [18], which improves guidance accuracy and system robustness by increasing the order of the observer. In [19], a finite-time dual-layer guidance law considering the second-order autopilot dynamics was designed, and an extended state observers are used to estimate the unknown target maneuver and acceleration derivative of the missile. with the development of artificial intelligence technology, neural networks [20], [21], [22] and deep reinforcement learning technique [23], [24], [25], [26], [27], [28] have also begun to be applied for the design of guidance laws to estimate the uncertain dynamics of maneuvering targets. Moreover, in order to reduce the computational cost of the neural networks, a non-fragile quantitative prescribed performance control method was developed in [29], a simplified finitetime fuzzy neural controller has been introduced in [30], an adaptive critic design-based fuzzy neural controller was proposed in [31].
Motivated by the above analysis, this paper proposed a robust intelligence guidance law through backstepping and sliding mode control techniques. An adaptive gain term is designed to reduce the control system chattering and improve system transient characteristics. Moreover, a neuro-estimator has been developed to obtain the acceleration information of maneuvering target.
The rest of this paper is organized as follows.Preliminary and Problem statement are presented in Section II. The robust intelligence guidance law is proposed in Section III.
The stability analysis of the control system is proposed in Section IV. Simulation results are presented to illustrate the effectiveness of the proposed guidance scheme in Section V. Finally, conclusions are discussed in Section VI.

II. PRELIMINARY AND PROBLEM STATEMENT A. NEURAL NETWORK
Radial basis function neural network (RBFNN) is extensively employed in the area of robot control because of its intrinsic capabilities in function approximations [32]. The typical RBFNN consists of input layer, hidden layer and output layer, whose structure is shown as Figure 1. For a arbitrary continuous function y(x) : ℜ p → ℜ q , there is a RBFNN such that [33] . . µ np ] T and λ n are the center vector and standard deviation,respectively. e w (x) = [e w1 (x), . . . , e wq (x)] T denotes the vector of approximation error, where ∥e w (x)∥ ≤ e M , e M > 0. Due to the fact that the exact value of the ideal weight matrix W * d cannot be directly obtained, an estimation of weight matrix W d is introduced to approximate the unknown function as y(x) = W d σ d (x), which is usually defined by using an adaptive rule.

B. PROBLEM STATEMENT
The dynamic interception model between missile and target in the three-dimensional space is proposed in this section. As shown in Figure 2, OX I Y I Z I denotes the inertial coordinate frame, MX M Y M Z M and TX T Y T Z T are VOLUME 11, 2023 velocity coordinate frames of missile and target, respectively. R denotes the relative distance between missile and target, V M denotes the velocity of missile, V T denotes the velocity of target. ε L is the elevation angle of the line-of-sight, β L is the azimuth angle of the line-of-sight. ε m and β m are the angles between velocity of the missile and line-of-sight coordinate frame. ε t and β t denote the angles between velocity of the target and line-of-sight coordinate frame. Based on the coordinate transformation, the kinematics equation between missile and target in three-dimensional space is defined as follows [34] where a ym and a zm denote the acceleration in pitch and yaw directions of the missile, respectively. a yt and a zt represent the acceleration in pitch and yaw directions of the target, respectively. To facilitate the design process of guidance law, taking the derivation ofε L andβ L , we get Further, (3) can be rewritten as where D = [d 1 , d 2 ] T is the unknown parameter matrix of moving target. u denotes the control input vector of guidance control system, which is defined as follows As a result, our control objective is to develop the guidance laws a zm and a ym to let the line-of-sight angular ratesε L anḋ β L converge to a small neighborhood of zero.

III. ROBUST GUIDANCE LAW DESIGN
The robust intelligence guidance law is developed by using backstepping method, sliding mode control approach and RBF neural network technique in this section. First of all, the following integral sliding surface S is chosen as whereε L (t 0 ) andβ L (t 0 ) denote the initial value ofε L andβ L . Then, taking the derivation of S, and we geṫ Further, the following exponential approach law can be constructed asṠ where k, ϑ and σ are positive constants, function sat(·) is introduced in [35]. Then, substituting (6), (7) and (8) into (4), and the robust sliding mode guidance law can be expressed as Here the unknown dynamics D(x) can be compensated by using RBF neural network, which is constructed as where x = [s 1 , s 2 ,ṡ 1 ,ṡ 2 ] T denotes the input vector of NN, W d = W * d − W d is the weight matrix error. Then, guidance law (9) can be rewritten as follows The corresponding adaptive law is defined aṡ where λ d and k d denote the positive constants. Then, to suppress the chattering of control system, the switching gain ϑ is estimated by RBF neural network, and we get Based on the gradient descent method, the RBF neural network weight adjustment algorithm can be designed as where Then, (14) can be rewritten as follows where λ ϑ ∈ (0, 1) denotes the learning rate parameter of RBF neural network. ∂S/∂u is the Jacobian information of the control object. Finally, the RBF neural network weight learning algorithm is expressed as where ℓ ϑ ∈ (0, 1) denotes the momentum factor. Similarly, Jacobian information of the control object ∂S/∂u can be online identified by RBF neural network, and we get where x I = [a zm , a ym , s 1 , s 2 ] T is the input vector of neural network. W I denotes the weight matrix, σ I (x I ) denotes the vector of activation function, y I is the output vector. Then, the following indicator function is defined as Based on the gradient descent method, and we get where λ I ∈ (0, 1) is the learning rate parameter, ℓ I ∈ (0, 1) is the momentum factor.
As a result, parameter ∂S/∂u is expressed as where m denotes the number of neurons. c Ij denotes the center vector parameter of neuron. b Ij is the base width of neural network. Figure 3 illustrates a block diagram of the proposed threedimensional intelligent guidance law. In the next section, the stability analysis of the guidance control system is proposed.

IV. STABILITY ANALYSIS
Theorem 1: Consider the three-dimensional intercepting mathematical model for maneuvering target denoted by (4). The robust guidance law (11), adaptive law (12) and weight update law of RBF neural network (14) guarantee that the line-of-sight angular ratesε L andβ L converge to a small neighborhood of zero in finite time.
Proof: Consider the following Lyapunov function for the overall guidance control system: Taking the derivation of (24) by combining equation (6), and we geṫ  Then, substituting guidance law (11) and adaptive law (12) into (25), we geṫ To ensure the stability of guidance and control system, the following conditions should be satisfied [36], [37] k ∥S∥ + ϑ ≥ δ + 1 4 k d ∥S∥ W 2 max (27) or Further, (26) and (27) are can be rewritten as (29) or In summary, sliding mode surface S can be converged to a small neighborhood of origin by adjusting the guidance law and neural network parameters.

V. SIMULATION EXAMPLES
In this section, numerical simulations are performed to verify the effectiveness of the proposed guidance law (NIGL) via MATLAB 2020a software. The dimension of target and missile are assumed to be approximated. In addition, to illustrate superiority of the proposed algorithm, proportional guidance law (PGL) [6] and terminal sliding mode guidance law (TSML) [16] are chosen as contrast methods. The environmental disturbance is set to Gaussian white noise with a variance of 0.07 2 . The maximum lateral and normal overloads that the missile actuator can provide are both 25g, where g = 9.8m/s 2 .
The initial distance between the missile and target is selected as 5000m. The velocity of missile is chosen as 800m/s, the velocity of target is selected as 500m/s, and the maneuver law of target is assumed to be a zt = a yt = 20cos(4t)m/s 2 . The initial values of the elevation angle ε L and azimuth angle β L are chosen as 5 deg. The initial values of ε m , β m , ε t and β t are set to 20 deg.
The simulation results are shown as Fig. 4 to Fig. 8. Fig.4 illustrates the intercepting trajectory of the three guidance laws, which indicates that all the guidance laws can guarantee  the successful interception for the moving target. As shown in Fig.5, the proposed guidance law in this paper has higher convergence accuracy compared with the other two guidance laws.
The overload curves for three guidance laws are provided in Fig. 6. Fig. 7 shows the approximation result of RBF neural network, which indicates that the neural network can effectively estimate and compensate the uncertain dynamics of the maneuvering target. Table 1 gives the interception time and miss distance of the three guidance laws, which indicates that the presented guidance method has shorter interception time and higher interception accuracy compared with the other two guidance laws. In addition, NIGL does not require the acceleration information of the target, which is more easily implemented in practice. Figure 8 shows the distribution of miss distance for 50 Monte Carlo shots, which verifies the effectiveness of the numerical simulation method.

VI. CONCLUSION AND FUTUREWORK
In this paper, a three-dimensional intelligent guidance law is proposed by using adaptive control technology, sliding mode control method and RBF neural network.The stability of the guidance and control system is proved based on the Lyapunov's stability theorem, and the superiority of the proposed guidance law in interception time and interception accuracy is verified by the comparative simulation. The futurework can be summarized as follows: (i) intelligent guidance law with saturation and dead-zone [38], [39], [40] will be investigated.
(ii) the improvement on computational cost and real-time performance will be discussed. (iii) integrated guidance and control technology will be considered in the future. (iV) an experimental platform will be established to verify the effectiveness of guidance and control algorithm for missiles.