Abstract
This paper presents the flight penetration path planning algorithm in a complex environment with Bogie or Bandit (BB) threats for stealth unmanned aerial vehicle (UAV). The emergence of rigorous air defense radar net necessitates efficient flight path planning and replanning for stealth UAV concerning survivability and penetration ability. We propose the improved A-Star algorithm based on the multiple step search approach to deal with this uprising problem. The objective is to achieve rapid penetration path planning for stealth UAV in a complex environment. Firstly, the combination of single-base radar, dual-base radar, and BB threats is adopted to different threat scenarios which are closer to the real combat environment. Besides, the multistep search strategy, the prediction technique, and path planning algorithm are developed for stealth UAV to deal with BB threats and achieve the penetration path replanning in complex scenarios. Moreover, the attitude angle information is integrated into the flight path which can meet real flight requirements for stealth UAV. The theoretical analysis and numerical results prove the validity of our method.
1. Introduction
1.1. Background and Motivation
In recent years, the emergence of new military equipment and systems, the stealth unmanned aerial vehicles (UAV) are gaining much attention due to the air-defense radar net is increasingly in the field of close and low altitude combat. Meanwhile, the survivability and combat effectiveness of stealth UAV will have a great influence on modern air warfare [1, 2]. The nose, fuselage, and tail of stealth UAV have relatively small radar cross-section (RCS), which aims to reduce the detection probability of radar system and improve the penetration ability and survivability of the stealth UAV [3]. However, previous works have mainly concentrated on the path planning in the static environment, there are many theoretical challenges and practical problems for stealth UAV on the problem of penetration path planning, such as the constraint of attitude angle and control of track point, the efficiency of path planning algorithm, and real-time performance. Besides, with regard to the Bogie or Bandit (BB) threats in the real combat environment, it is difficult for stealth UAV to achieve penetration path planning which is rapidly and safely. Therefore, the kinematics model and path planning algorithm for stealth UAV should be studied in a complex environment.
1.2. Literature Review
More recently, a lot of research has been done in the area of penetration path planning for aircraft, and so many algorithms had been proposed. In Reference [4], a new concept on the aircraft path planning for the variation of radar cross-section (RCS) was firstly presented, which indicates that the stealth UAV can significantly reduce the RCS by adjusting the attitude angle. In Reference [5], a research framework of low detectability tactical trajectory planning was proposed in the presence of multiple radar-detected threat environments; the solution of the optimal trajectory is transformed into the optimal control problem with the minimum radar detection probability. A penetration trajectory optimization method considering the influence of aircraft radar RCS was proposed in Reference [6]. In Reference [7] and Reference [8], a hybrid heuristic adaptive pseudospectrum method was proposed to address the low-detectability trajectory planning problem. In contrast, these methods usually adopt a simplified kinematic model, which aims to obtain a rough reference track, and the constraints and control conditions of radar attitude are rarely considered in the reference trajectories. In Reference [9], a path planning method for stealth UAV based on the RCS ellipsoid model was proposed to find the optimal trajectory during flight. Inversely, this model requires a large amount of computation in addressing the problem of path planning. In Reference [10], a trajectory planning method based on dynamic game theory was presented to analyze the relationship between radar and UAV. In Reference [11], a real-time air battle trajectory optimization and game model for aircraft based on a rolling time-domain control strategy were proposed in a complex environment. However, two kinds of shortages exist in the research. One was that only can be applied to the static environment with known radar position, and it rarely achieves the path planning for stealth UAV in the dynamic environment. The other was that the calculation process is relatively complicated, and the real-time performance of the path planning method is poor. Erlandsson proposed a path planning model, and the expected path cost of air mission was discussed in a hostile environment; the particle swarm optimization algorithm (PSO) was adopted to address the optimization path planning problem [12, 13]. Meanwhile, many scholars are focus on analyzing a series of optimization algorithms to address the penetration path planning problem for unmanned aerial vehicle, such as A-Star algorithm [14], genetic algorithms (GA) [15], sparse A-Star algorithm (SAS) [16], rapidly exploring random-tree algorithm (RRT) [17, 18], particle swarm optimization algorithm (PSO) [19, 20], black hole algorithm [21], and other combinatorial optimization algorithms [22]. However, these optimization algorithms are only applied to address the stealth path planning problem in the static combat environment, the single-base radar is usually adopted in the combat scenario that the position of each radar is known, and the number is relatively small. Furthermore, the efficiency of the algorithm and the safety of the path are rarely optimal.
1.3. Contribution
In this paper, we focus on analyzing the penetration path planning for stealth UAV based on an improved A-Star algorithm, which aims to achieve rapid penetration path planning in the dynamic combat environment with BB threats. The novelty of this method is summarized as follows: firstly, the main idea of the model-based predictive control (MPC) and learning real-time A-Star algorithm (LRTA) is integrated into the path to devise the improved A-Star algorithm. Additionally, for the BB threats in a complex environment, the improved A-Star algorithm is applied for the penetration path replanning with fixed altitude to improve the survivability and combat effectiveness of stealth UAV. Moreover, numerical simulation results are applied to demonstrate the validity of the improved A-Star algorithm in the presence of the radar net with BB threats.
1.4. Paper Organization
The paper is organized as follows. Section 2 describes the mathematical model of stealth UAV. Section 3 discusses the detection probability calculation of multiple radar net, which is close to the real combat environment. Section 4 presents the improvement of the path planning algorithm, includes improved A-Star algorithm, LRTA-Star algorithm, and D-Star algorithm. Our numerical results and evaluation are presented in Section 5. Section 6 states our conclusions and future work.
2. The System Modeling
2.1. Kinematics
It is assumed that the stealth UAV can be defined as a particle with attitude information and ignores the influence of wind and other external conditions [1]. We are focus on studying the control of the flight path. The stealth UAV moves in a horizontal plane at a constant altitude . Therefore, the kinematics model can be given by where and are the Cartesian coordinates of the aircraft, is the heading angle, is the constant speed, is the input signal, and is the acceleration normal to the flight path vector.
Specify radar locations in the plane (on the ground) by the coordinates . The range from the th radar to the stealth UAV is given by
Additionally, let be the azimuth, aspect, and roll angles, respectively, measured to the th radar, is the acceleration of gravity.
2.2. Dynamic RCS Features
Radar cross-section (RCS) is a measure of ability for UAV to reflect electromagnetic radiation emitted by radar [23]; the RCS of the nose, fuselage, and tail of the stealth UAV is smaller than conventional UAV, and the RCS value will be changed due to different radar bands and radar polarization modes. It plays an important role in the penetration path of stealth UAV in combat. For the path planning problem, RCS is usually defined as a constant, but it is not reasonable in practical application. By convention [24], it is modeled as a function of the aspect angle and roll angle as viewed from the radar. Therefore, the dynamic RCS model is adopted to an ellipsoid [1], which is given by where represents the RCS value of UAV, represents aspect angle, represents roll angle, is relatively small frontal RCS, is a larger beam aspect RCS, and is relatively large RCS when viewed from above or below.
3. Radar Detecting Probability
When the acquired target information meets the common track criteria, the radar system confirms the detection of the target and transmits the acquired target information to the information fusion center. The detection probability of target in a single-base radar, dual-base radar, and a radar net system is depicted as follows.
3.1. Single-Base Radar
For the single-base radar, the detection probability of the UAV is only related to the distance from the UAV to the radar center. In a period, the radar instantaneous detection probability can be expressed as follows: where is the instantaneous detection probability of the single base radar, is given by Eq.(2), and and are the performance parameters of specific radar, respectively.
3.2. Dual-Base Radar
The dual-base radar system is considered to be effective systems to counter the four major threats due to their technical characteristics of transmitter-receiver separation and passive reception. For the performance of the radar system, it has the advantages that single-base radar can not match, such as a larger detection range and higher accuracy. Dual-base radars are usually deployed in modern air defense systems; therefore, we focus on analyzing the dual-base radar system.
The transmitter is also employed as a receiver for dual-base radar systems, which are T/R-R systems. A transmitter can work synchronously with other splitter receivers in the dual-base radar system. The receiver synchronizes with the transmitter in time, phase, and space. The configuration of the dual-base radar system is depicted in Figure 1.
where represents the position of the target, represents the transmitter, represents the receiver, represents the distance between the transmitter and receiver, represents the distance between the target and transmitter, represents the distance between the target and receiver, and represents the dual-base angle.
Additionally, is defined as the detection range of the bistatic radar, that is . where and . is a constant when the radar parameters are given. Hence, the maximum detection range of the bistatic radar can be obtained by where represents the peak power of the transmitter, represents the antenna gain, represents the system loss, represents Boltzmann constant, is the noise temperature of the receiver, is the signal to noise ratio of the receiver, and represents the wavelength. Similarly, the detection probability of the bistatic radar under exponential distribution is given by
3.3. Detection Probability of Networked Radar System
For a complete netted radar system, the radar net can improve the detection probability of UAV, significantly. The detection probability of a networked radar system mainly refers to the target detection probability calculated by the information fusion center. The rank K fusion rule is widely adopted in modern networked radar systems. Therefore, we adopt the rank K fusion rule to analyze the detection probability of networked radar [25, 26].
It is assumed that the number of radars in the netted radar system is , which includes single-base radar and bistatic radar. When the number of radar detected in the system exceeds the detection threshold according to the rank K fusion rule, that means, the target has been detected by the radar system, and the approximate value of the optimal detection threshold can be depicted as follows.
The radar makes a local judgment based on the detection of the stealth UAV, and the judgment results are either 0 or 1, which depends on whether the local threshold detection target exists or not. and are binary assumptions, where represents target does not exist and represents target exists. Therefore, the decision value of the th radar ) can be expressed as follows:
Additionally, the local decision result is passed to the information fusion center of the radar system to form a global decision matrix , that is, . And the radar information fusion rule for the network is denoted as ; the decision rule for rank K fusion is given by
Meanwhile, the total detection probability of the network radar system on the UAV is given by where is a set of local decision vectors that make the fusion center judge “1”, is a set of local decision vectors that make the fusion center judge “0”, and is the discovery probability of the first radar in the radar net.
4. Path Planning Algorithm
A-star algorithm is an effective search method to solve the shortest path in a static road network, which is widely used to settle the path planning problems of many agents. The closer the estimated distance is to the actual value, the faster the final search speed is [27–29]. This paper discusses the improvement method of the A-Star algorithm to address the penetration path planning problem for stealth UAV.
4.1. The A-Star Algorithm
The main idea of the standard A-Star algorithm is as follows. Firstly, select the appropriate heuristic function, estimate the generation value of the extensible search points in the search area, comprehensively. Moreover, compare the different cost values of each point. Moreover, consider the operation time and distance cost of the track point search. Finally, find an optimal path. In the A-Star algorithm, the operation of the OPEN list and CLOSE list are usually performed to achieve the storage and update of track points.
However, the standard A-Star algorithm that is applied to find the optimal path for stealth UAV has many disadvantages: (a) the route obtained by the A-star algorithm only has the position of stealth UAV, and it can not reflect the characteristics of the dynamic RCS and attitude angle information. (b) Unknown path cost estimation is rarely calculated in the process of path planning accurately, and the computation time is too long to find a globally optimal path. (c) the performance of real-time is poor, and the algorithm can not deal with BB threats. Therefore, we are focus on improving the performance of the A-Star algorithm.
4.2. The D-Star Algorithm
The D-Star algorithm is developed by the A-Star algorithm and the Dijkstra algorithm [30, 31], which is suitable for addressing the path planning problem in unknown environments. The main idea of the algorithm is to search the reverse path from the destination to the start, and the heuristic function expression of the D-Star algorithm is given by where represents the actual cost of the path from the destination to state , and is the estimated cost of the path from the state to the starting point.
The main steps of the D-Star algorithm are given as follows. Step 1: the value of all states are set to NEW, and are set to infinity for all states, is set to zero, and the destination is added to the OPEN list. Step 2: keep performing the search until the location of which is removed from the OPEN list, if the CLOSE list contains the tag value of the state , the complete path sequence will be obtained. In contrast, path planning has failed in the threat scenario. Step 3: if the path exists, the state can be employed to point to the destination by the back pointer. Besides, if the changes, the current path cost will be immediately recalculated. Step 4: insert the affected state which near the new threat into the OPEN list, and then return to Step 2. The pseudocodes of the D-Star algorithm are depicted as follows Algorithm 1.
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4.3. Improved A-Star Algorithm
We are focus on analyzing the learning real-time A-Star algorithm (LRTA-Star) which satisfies the requirements of real-time planning in a dynamic environment [32, 33]. In contrast, the path obtained by adopting the LRTA-Star algorithm is composed of a series of the tortuous track; it is difficult to achieve accurate track tracking control due to the limited maneuverability. Besides, in the flight process of stealth UAV, LRTA-Star algorithm is easy to fall into a local dead loop, which leads to the failure of path searching. Therefore, further combined with the idea of model-based predictive control (MPC) [34, 35], an improved A-Star algorithm with the multistep optimal search is proposed in this section.
MPC is an optimization control method, which is mainly composed of model prediction, feedback correction, and rolling optimization. The principle of the MPC system is depicted in Figure 2.
The closed-loop output prediction is given by where is the actual control quantity on the system at moment , is the reference track softened by the input filter, is the predicted output value of the model, is the closed-loop output prediction, and is the error correction coefficient.
According to Eq. (13), the optimal control law for multistep prediction is given by where is the length of the prediction domain; is the length of the control domain; is the output prediction error, and is the weighting coefficient of the control variable.
In the process of a multistep optimization search, the track cost of each predicted track is considered as the cumulative value of the costs of predicted track nodes. Therefore, the objective function of UAV flight path planning can be obtained according to the optimal control law of the MPC system. The cost of the predicted flight path in steps from the th node is given by where is the radar threat cost at the th prediction point on the current predicted track segment, which can be obtained by Eq. (11). is the weight of threat cost; is the distance cost weighted matrix, is the control sequence, and is the distance cost between the coordinate of destination and the coordinate of th track point and .
The stealth UAV is limited by its maneuverability in the process of a multistep search, where is sought by the current track point and heading angle, and the cost function for the node has a minimum value. Therefore, add the parameter to the heuristic function of the original algorithm, which aims to simplify the operation, reduce the search time and ensure the optimality of path planning, and the new heuristic function expression in the improved A-Star algorithm is given by
Where is the ratio of the path cost of the current state to the path cost of an unknown region, is the current node, is the adjacent extended node, and is the cost from the current node to the adjacent node, and . The pseudocode of the improved A-Star algorithm is depicted as follows Algorithm 2, and the flow chart of the improved A-Star algorithm is depicted in Figure 3.
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5. Numerical Results
Numerical simulations of penetration path planning are performed by employing the improved A-Star algorithm, LRTA-Star algorithm, and D-Star algorithm in different threat scenarios, which aims to verify the effectiveness of the improved A-Star algorithm. The simulation experiments are conducted with MATLAB2020Ra software and a Windows 10 system. The parameters of flight are depicted in Table 1, the value is determined by the number of radar , and it will change when there is a BB threat in the threat scenario. Additionally, it is considered as the probability state of radar high detection when exceeds 0.4, and is performed to evaluate the safety degree of the track segment. The RCS data of a certain type of stealth UAV is depicted in Figure 4.
5.1. The First Threat Scenario
The threat region has a range of ; the geographical location and types of threat sources are presented in Table 2. The coordinate of the starting point is (5,5) km, and the coordinate of the target point is (90,90) km. The numerical results of penetration path planning which is performed by different algorithms in the first scenario are depicted in Figure 5. The variation of heading angle and roll angle which is applied by the improved A-Star algorithm in the first scenario is depicted in Figure 6, the detection probability of netted radar in the first threat scenario is depicted in Figure 7, and the statistical result of flight in the first scenario is presented in Table 3.
(a) Improved A-Star algorithm
(b) LRTA-Star algorithm
(c) D-Star algorithm
(a) The variation of heading angle
(b) The variation of roll angle
(a) Original planning
(b) Replanning
Figure 5 demonstrates the penetration paths for stealth UAV which is performed by employing three different algorithms in a scenario. From Figure 7 and Table 3, we can infer that the stealth UAV can achieve penetration path planning by employing in the first scenario in which a static or dynamic environment. However, compared with the other two algorithms, the paths obtained by employing the improved A-Star algorithm has a smaller value of and takes less time. Figure 6 demonstrates that the improved A-Star algorithm can accurately adjust the attitude angle and reduce the value of RCS on the paths, and this algorithm achieves the rapid and safe path penetration planning for stealth UAV. Besides, the paths of the improved A-Star algorithm is shorter than the LRTA-Star algorithm, and the improved A-Star algorithm has higher path planning efficiency and safety, which further proves the effectiveness of the improved A-Star algorithm.
5.2. The Second Threat Scenario
The threat region has a range of ; the geographical location and types of threat sources are presented in Table 4. The coordinate of the starting point is (5,5) km, and the coordinate of the target point is (90,90) km. The numerical results of penetration path planning which is performed by different algorithms in the second scenario are depicted in Figure 8. The variation of heading angle and roll angle which is performed by an improved A-Star algorithm in the second scenario is depicted in Figure 9, the detection probability of netted radar in the second threat scenario is depicted in Figure 10, and the statistical result of flight in the second scenario is presented in Table 5.
(a) Improved A-Star algorithm
(b) LRTA-Star algorithm
(c) D-Star algorithm
(a) The variation of heading angle
(b) The variation of roll angle
(a) Original planning
(b) Replanning
Figure 8 describes the penetration paths for stealth UAV which is performed by applying three different algorithms in a scenario with more complex threat sources, and from Figure 10 and Table 5. For the original path planning, the penetration paths which are safe by employing all three algorithms in the scenario. However, for the path replanning in the presence of BB threats, the path replanning has failed in the second threat scenario, because the of paths by adopting the LRTA-Star algorithm and D-star algorithm are higher than 0.5. Inversely, stealth UAV can still achieve the path replanning in the second scenario with two BB threats by using an improved A-Star algorithm. Moreover, the improved A-Star algorithm takes less time and has higher path planning efficiency than the other two algorithms whether in the original path planning or replanning, and it further proves the validity of the algorithm. Figure 9 indicates that when the improved A-Star algorithm is performed to penetration path planning, the variation of heading angle and roll angle are within the preset range, which can satisfy the real flight requirements.
5.3. The Third Threat Scenario
The threat region has a range of ; the geographical location and types of threat sources are presented in Table 6. The coordinate of the starting point is (5,5) km, and the coordinate of the target point is (90,90) km. The numerical results of penetration path planning which is performed by different algorithms in the third scenario are depicted in Figure 11. The variation of heading angle and roll angle which is performed by an improved A-Star algorithm in the third scenario is depicted in Figure 12, the detection probability of netted radar in the third threat scenario is depicted in Figure 13, and the statistical result of flight in the third scenario is presented in Table 7.
(a) Improved A-Star algorithm
(b) LRTA-Star algorithm
(c) D-Star algorithm
(a) The variation of heading angle
(b) The variation of roll angle
(a) Original planning
(b) Replanning
Figure 11 describes the penetration paths for stealth UAV which are performed by applying three different algorithms in a scenario with a large number of threat sources and BB threats; from Figure 13 and Table 7, we can see that the paths by adopting the LRTA-Star algorithm and D-star algorithm are higher than 0.5 no matter in the original path planning or path replanning. Obviously, stealth UAV can rarely achieve the penetration path planning which is safely and rapidly by employing these two algorithms. However, the of paths by adopting improved A-Star algorithms are always lower than 0.5; therefore, stealth UAV can still achieve penetration path planning and replanning by using the improved A-Star algorithm in dynamic threat scenarios. Furthermore, paths of the improved A-Star algorithm have higher efficiency and safety, which further proves the validity of the improved A-Star algorithm in the scenario with high threat density. Figure 12 indicates that stealth UAV can meet constraints of attitude angle in the field of close and low altitude combat.
6. Conclusions
This paper presented a new solution for path replanning for stealth unmanned aerial vehicle in a complex radar net environment with BB threats.
We are focus on analyzing the kinematics model of stealth UAV, threat source in penetration environment, and path planning algorithm. Further combined with model-based predictive control (MPC) and the LRTA-Star algorithm, an improved A-Star algorithm is proposed to achieve penetration path planning and replanning for stealth UAV. Meanwhile, the attitude angle information of stealth UAV is added to the algorithm, which demonstrates the variation characteristics of dynamic RCS. Further combined with the kinematics analysis of stealth UAV and the detection performance analysis of radar net, the original paths, and the replanning paths can satisfy the real flight requirements.
Compared with the other two algorithms, the stealth UAV can rapidly achieve the penetration path planning by employing the improved A-Star algorithm in a different complex environment with BB threats, improve the survivability of stealth UAV, and the efficiency of penetration path planning.
The threat scenarios are composed of single-base radar, dual-base radar, and BB threats, which are closer to the real combat environment. The model and the improved A-Star algorithm proposed in this paper can quickly generate better penetration paths in the combat area under a dynamic environment, exhibiting certain military application value.
In future work, we are focus on the real-time rapid penetration path planning for stealth UAV account for a three-dimensional complex dynamic environment with the terrain and unknown motion targets in the field of close and low altitude combat.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under grant 61663032, Aeronautical Science Foundation of China under grant 2016ZC56003, and Innovation Special Fund of Nanchang Hangkong University for Graduate Student under grant YC2019026.