Pursuer Navigation Based on Proportional Navigation and Optimal Information Fusion

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Introduction
tVarious navigation and obstacle avoidance methods are the important issues. e proportional navigation is a method well known and widely applied in the aerospace community. In [1], the augmented IPN is deduced for interception. In [2], the authors present a new homing guidance law using wellknown BPN to perform an impact time constraint and impact angle constraint. Over and above the case of infinite maneuverability of the missile, the full condition that captures a nonmaneuvering target is deduced in [3]. Real-time navigation is given in [4] by integrating the backstepping method and neurodynamics model. In [5], the proportional navigation applied to missile guidance problems is tailored. In [6], the authors propose collision avoidance strategy for multiagent. A receding horizon control method for convergent navigation of the robot is given in [7], and this method includes a scientific procedure for the generation of potential control sequences. In [8], a modified cooperative proportional navigation is presented to avoid singularity, and the time-to-go control efficiency under the small leading angle is improved in this paper. e capturability of 3D PPN against the lower speed freely-maneuvering target for the homing phase is restudied in [9], extending the NOR method of the 2D PPN to 3D space. In the study of [10], pure proportional navigation (PPN) and a look angle-constrained guidance law consisting of PPN and look angle control are designed. In [11], a novel augmented proportional navigation (APN) is proposed for midrange autonomous rendezvous, and the midrange autonomous rendezvous can be absolutely implemented. e application of proportional navigation to the pursuer requires improvements. e presentation of this paper is different from the classical presentation.
e proportional navigation is proposed by using the flight path and heading angles of the pursuer. is presentation is more proper for the pursuer than the classical presentation, where proportional navigation is proposed in accordance with the lateral and vertical acceleration. en, the presentation of proportional navigation can be easily adapted to the collision avoidance mode since the proportional navigation is written as a function of the flight path and heading angles of the pursuer. It allows a rapid change in the path of the pursuer under proportional navigation.
In the study of [12], obstacle avoidance and navigation are addressed by using the model-based control method.
is method can be used for both online and offline. In [12], the proportional navigation is written based on the robotic steering angle. Moreover, the collision avoidance mode is implemented by using proportional navigation. Notwithstanding the method of [12] seems to be quite competent, and it sustains the following problems.
(1) e control law for the orientation angle can continue to be expanded (2) e proportional navigation was proposed by assuming no sensor noise (3) Information fusion is not combined with the proportional navigation to improve the tracking process In this paper, the work is mainly motivated by the study in [12]. e aim of this paper is to consider the solution of the pursuer tracking toward a target in the 3D space. Based on the geometric relationship of pursuer-target, this paper presents the polar kinematics models. In this paper, the proportional navigation is given in terms of the flight path and heading angles for the pursuer. Moreover, the proportional navigation can also be adapted to the collision avoidance mode. e method can be used for indoor and outdoor navigation as well, especially to reach goals that are at a long distance from the pursuer, and as a result, they are out of the range of view of the sensors (such as the camera), but their position is known to the pursuer. e proportional navigation of Belkhouche and Belkhouche [12] was proposed by assuming no sensor noise. For sensor noise, the filter method can be used to improve the tracking process. As studied in [13], a data-driven method combining the EKF and RBF neural network is given to estimate the internal temperature for the lithiumion battery. In [14], the particle filter is applied to predict the aging trajectory of the lithium-ion battery. Even though the algorithm of Belkhouche and Belkhouche [12] seems to be quite efficient, it suffers from that Kalman filter techniques are not used for dynamic state estimation. Various multisensor fusion methods have been studied to solve this problem. Under the optimal fusion criterion of Sun and Deng [15], the multisensor fusion decentralized Kalman filter is obtained. In [16], the authors propose the two-sensor information fusion steady-state Kalman filter. In [17,18], distributed optimal information fusion filter theory is presented under the classical Kalman filter. e device are argued in [19] in accordance with sensor data fusion methods, sensor design, and prototype setup. Based on multisensor fusion, a hybrid indoor localization system is given in [20]. In [21], the authors present the information fusion Kalman filter weighted by scalars. In [22], the functional equivalence of two optimal measurement fusion methods is proved under the steady-state Kalman filter. Information fusion weighted by diagonal matrices is proposed in [16][17][18]. As studied in [12], this method is not considered the negative influence of sensor noise. Based on the above control theory of information fusion, the control strategy of [12] can be further improved. In this paper, the proportional navigation combined with information fusion weighted by diagonal matrices are used to implement more reasonable tracking performance.
Control objective of this paper is to implement pursuer navigation and obstacle avoidance using a easy and valid model-based control law. It can be applied to both online and offline navigation and obstacle avoidance. is method consists of a family of methods for pursuer navigation under proportional navigation, where this paper applies the pursuer kinematics equations combined with the geometric rule. e challenge of this paper is how to design the control law for proportional navigation and implement obstacle avoidance. To deal with the challenge, this paper presents the polar kinematics models of pursuer-target. e control law of proportional navigation is given in terms of the flight path and heading angles for the pursuer. Under sensor noise, twosensor information fusion is applied to improve the control law. Moreover, the proportional navigation can implement collision avoidance by using point-to-point navigation. e contribution of this paper is mainly to present threedimensional proportional navigation to implement tracking the target, outperforming the pure proportional navigation (PPN) in terms of interception time. Under sensor noise, two-sensor information fusion together with proportional navigation can enhance the tracking precision. Moreover, obstacle avoidance is implemented by using point-to-point navigation combined with proportional navigation. e remainder of this paper is organized as follows. e dynamic model of pursuer-target is derived. en, the proportional navigation law is discussed. is paper designs the control method of obstacle avoidance. Two-sensor information fusion is used in this paper. Simulation results are given to formulate the availability of the obtained results, and then, some conclusions are drawn.

Dynamic Model
e geometry of the navigation is illustrated in Figure 1. In the 3D coordinate system, the linear velocity of the pursuer is v P . e flight path and heading angles are θ P and ϕ P , respectively. LOS for target-pursuer is TP. σ TP is the pitch angle of TP, and c TP is the yaw angle of TP. r TP is the relative distance pursuer-target.
Based on [23,24], the differential equations for r TP , σ TP , and c TP are 2 Complexity Since the target is motionless, one can have v T � 0, and thus, e robustness of the method is a critical issue. It is should be noted that this method belongs to a family of methods in terms of the kinematics equation and geometric rule. ese methods are famous for the robustness.

Three-Dimensional Proportional
Navigation Law is paper designs the proportional navigation law in conformity to the pursuer flight path and heading angles as follows: where G and E are the navigation constant with (G ≥ 1; E ≥ 1) and δ and μ are the deviation angles.
Combining equation (2) with equation (3), the differential equations for r TP , σ TP , and c TP are Results in relation to the pursuer that tracks an immovable point are given as follows.

eory
1. By using pure pursuit with (G � E � 1; δ � μ � 0), the pursuer can reach the target from any original condition. (4), it can be written as Since _ r TP < 0, r TP is decreasing and the pursuer can reach the target, with the final flight path angle θ P (t f ) � σ TP (t 0 ) and heading angle ϕ is completes the proof.   Complexity 3 Proof. On the basis of the first equation in the relative kinematics model, one can obtain _ r TP is a decreasing function when δ and μ ∈ [−π/2, π/2] or δ and μ ∈ [π/2, 3π/2].
is completes the proof.
□ eory 3. For G > 1 and E > 1, the pursuer navigating under equation (3) reaches the target for nearly all original states.

Complexity
Since the solutions for σ TP and c TP approach their asymptotically stable equilibrium positions, one can get , t 1 ≥ t 0 ; then, one can get _ r TP < 0 after t 1 . is completes the proof. At t 0 , the proportional navigation law is en, two cases are (i) Select (G, E) and (δ, μ) according to equation (11) on (G and E ≥ 1). (ii) Put into use heading regulation which deduces θ P and ϕ P from their original values to the values which satisfy equation (11) for (G, E) and (δ, μ). is method which gives more adaptability for the selection of (G, E) and (δ, μ).

Obstacle Avoidance
For simplicity and without loss of generality, obstacles are denoted by spheres S j . Spheres S j has d as a radius.
Points E 1 and E 2 are shown in Figure 2. r j 1 and r j 2 are the distances from the pursuer to E 1 and E 2 , respectively. e differential equations for r jk , σ jk , and c jk between the pursuer and the center of obstacle S j are _ r jk � −v P cos Gσ TP + δ − σ jk cos Ec TP + μ − c jk , With k � 1 and 2, one can identify whether the pursuer is oncoming or deviating from the obstacle based on equation (12). e pursuer is in a collision when where σ j 1 < σ j 2 and c j 1 < c j 2 . e avoidance course of the proportional navigation is when the pursuer is within a definite distance d 0 from the obstacle.
is section can provide free or obstacle directions, which is designed in consideration of the obstacles as follows: where K is the total amount of obstacles.
Point T 1 corresponds to a free direction. T 0 is the point where the pursuer starts deviating from a possible obstacle. When (r i − d) < d 0 , the pursuer is driven to an intermediary target that occurs in a free direction.
σ 0 TP 0 and c 0 TP 0 denote the pitch angle and yaw angle of pursuer-target measured at point T 0 at time t 0 1 , and σ 1 TP 0 and c 1 TP 0 the pitch angle and yaw angle pursuer-point T 1 measured at point T 0 at the same time. (σ 0 TP 0 , c 0 TP 0 ) and (σ 1 TP 0 , c 1 TP 0 ) are where (x T 1 , y T 1 , z T 1 ) are the coordinates of point T 1 . For the flatness of the path, then One can determine values of (G 1 , δ 1 ) and (E 1 , μ 1 ) that fulfill equation (17) and move the pursuer to point T 1 . If the pursuer suffers other obstacles, this strategy is repeated.

Two-Sensor Information Fusion
Two-sensor discrete-time system is where 6 Complexity Based on [27], the local optimal Kalman filter is us, the optimal Kalman filter is e optimal matrix of weight coefficients designed in [17,18] can be calculated as where the optimal weight coefficients are where P ii i and P ii 12 are the diagonal element of P i and P 12 . e error covariance matrix is e trace of the error covariance matrix for information fusion is trP 0 (τ|τ) � P 01 + P 02 + · · · + P 0n , where trP 0 ≤ trP i , i � 1 and 2. By using the Kalman estimator, fusion position of the target (x T Fu , y T Fu , z T Fu ) and the pursuer (x P Fu , y P Fu , z P Fu ) can be obtained. us,

Simulation Results
is section proposes several simulations, where tracking can be implemented under proportional navigation. In this section, distances velocities and time have been with units to achieve realistic results.
ere exist various potentialities for the selection of (x i P , y i P , z i P ). One takes (x i P , y i P , z i P ) � (101.9 m, − 101.5 m, 125.3 m); thus, θ i P 0 � 1.61 rad and ϕ i P 0 � 3.2 rad. One takes (x i P , y i P , z i P ) � (105.5 m, − 132.8 m, 161.7 m); thus, θ i P 0 � 1.33 rad and ϕ i P 0 � 3.26 rad. Some methods from control theory are applied to the intention of heading regulation. e air vehicle navigation applying this method is shown in Figure 4. e dashed lines illustrate the path of the air vehicle under the heading regulation phase. is method can give more adaptability for the selection of (G, δ) and (E, μ).   Example 3. At the appearance of spherical obstacles in a complicated environment, the point-to-point navigation method is applied to reach the aerial target and avoid the obstacles. e pursuer initiates from the initial position (0 m, 0 m, 0 m) and aims to reach the aerial target situated at (300 m, 300 m, 424.2 m). As illustrated in Figure 8, online deviation towards intermediary aerial targets T 1 , T 2 , T 3 , and T 4 is applied with the following different control parameters. Phase PT 1 : (G � 2, δ � π/4) and (E � 1.2, μ � 3π/40). Phase T 1 T 2 : (G � 2.1, δ � −10π/34) and (E � 6.8, μ � 10π/21). Phase T 2 T 3 : (G � 2.1, δ � −π/4) and (E � 4, μ � π/3). Phase T 3 T 4 : (G � 1.3, δ � −π/4) and (E � 2, μ � π/2). Phase T 4 T: (G � 2, δ � π/4) and (E � 1.2, μ � 3π/40). e path of the air vehicle is P ⟶ T 1 ⟶ T 2 ⟶ T 3 ⟶ T 4 ⟶ T. ese points can be selected so that the differences from the titular trajectory are small, which keeps smoothness of the trajectory. e air vehicle applies the point-to-point method to navigate towards the aerial target and avoid the obstacles.  (90 m, 90 m, 118.98 m). us, one can take (G � 2, δ � π/8) and (E � 2, μ � −π/8). In this case, two-sensor information fusion weighted by diagonal matrices together with proportional navigation is given to enhance the tracking precision. Figure 9 shows the filtered trajectory under proportional navigation. Enlargement in tracking result of Figure 9 is shown in Figure 10. e analysis of tracking performance indicates that the more higher the precision, the less the trace of the error covariance matrix. From Figure 11, one obtains trP i (k|k) > trP 0 (k|k). As a result, the trace of error the covariance matrix under information fusion is lower than the value of the single sensor. en, twosensor information fusion provides proportional navigation with more accurate target estimates.
is example shows     that the fusion result is better than that of the single sensor and the fusion method is effective.

Conclusion
is paper proposes a method for pursuer navigation under proportional navigation. e control strategy is primitive and depends on only the position of the target. For obstacle avoidance, this paper can avoid it by adjusting the control parameters. In the presence of sensor noise, the proportional navigation combined with information fusion weighted by diagonal matrices can achieve more reasonable interception performance. e method opens new directions for research, such as navigation using the proportional navigation under kinematics of pursuer and dynamics constraints and the influence of the control parameters, especially G and E.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.