Large-scale stealth trajectory optimization based on hybrid A*-Gauss pseudospectral method

For airborne electronic countermeasures, a practical and feasible stealth trajectory with low observability significantly impacts mission success. However, for long-range and large-scale trajectory optimization problems, the significant increase in the state space size will affect the feasibility and optimality of the problem solution. This paper proposes a hybrid trajectory optimization method to address the above issues. First, the A* (A-star) algorithm and cubic B-spline curve fitting method are used to generate the corresponding waypoints in the predetermined grid map under the radar detection threat to satisfy the trajectory stealth effect at the macro level. Then, the optimal control model (OCP) with the shortest flight time is solved between the waypoints to obtain segmented trajectories. Finally, the above state-control variable sequences are further used to solve the optimal control problem of the micro-level stealth trajectory coupling the radar cross section (RCS) by Gauss pseudospectral method (GPM). The numerical simulation results validate the proposed hybrid optimization method.


Introduction
The primary purpose of stealth design is to reduce the intensity of the reflected or radiated signals of the aerial vehicle so that it is difficult to be detected, identified, and tracked under the radar perspective.This paper will conduct the corresponding research and discussion of indirectly reducing the characteristic signals, such as designing the stealth trajectory.
The stealth trajectory design problem essentially belongs to the class of optimization problems.It can be divided into two categories: traditional planning algorithm and intelligent planning algorithm, which are usually applied in different fields.Many scholars have adopted traditional trajectory optimization algorithms such as the optimal control model to design the stealth trajectory under radar threats, and most of them include the RCS into the variables of the radar detection model.Inanc T et al. [1] constructed the framework of low-observable trajectory generation by opponent multiple radar detection systems in two-dimensional space using a nonlinear trajectory generation package [2].Chen J et al. [3] adopted B-spline fitting method to construct the RCS model, which is applied in the proposed three-dimension low-observable trajectory optimization method.Considering it is difficult to solve most complex optimal control models analytically, the direct method has the features of appealing to complicated applications and promises versatility and robustness [4], including the famous Gauss pseudospectral method [5] widely applied in aerospace optimization problems.Scholars have also applied the pseudospectral method to the studied stealth trajectory problem in recent years and obtained good performance [3,6,7].Huang H et al. [6] used the Radau pseudospectral method to design the low-observable glide trajectory.Xu Q et al. [7] proposed the framework based on the pseudospectral method and the stepwise increasing constraints method to design the aircraft penetration trajectory.However, the traditional optimization scheme has an inevitable problem: the "combinatorial explosion" phenomenon will occur when the optimization problem scale becomes large.On the other hand, direct numerical methods such as GPM are easily affected by the initial guess values.If the initial guess values are inappropriate, the problem will easily fall into the local optimum or even cause the problem to be infeasible in a large computation time.
In some fields with more freedom of movement, such as robotics, another type of path planning algorithm exists based on search or probability, including the Dijkstra algorithm, A* algorithm, rapidly-exploring random tree (RRT) algorithm, and so on.The above algorithm can also be applied to stealth trajectory planning under the corresponding map environment model, as many researchers have performed extensive research [8][9][10][11][12][13].Zhao Z et al. [8] proposed a special A* algorithm considering flight limitation and radars' influence in the multi-radar network.Pharpatara P et al. [9] proposed a trajectory planning algorithm based on RRT* algorithm, and artificial potential fields are combined with the RRT* algorithm to accelerate the convergence speed.Zhang Z et al. [11] designed a route penetration strategy based on the improved A* algorithm for stealth trajectory replanning problems in 3D complex dynamic environment.Farid G et al. [13] implemented a modified heuristic-based A* algorithm that uses a truncation mechanism in order to avoid extra expanded nodes.Similarly, although such methods can adjust the algorithm's complexity in a large-scale environment by adjusting the level of detail of the map model, there are certain drawbacks.The trajectory obtained is a fold line connected by the trajectory points rather than the actual smooth trajectory under the constraints of the aerial vehicle itself.
The contributions of this work are as follows: a hybrid stealth trajectory optimization framework is proposed under the large-scale environment based on the A* algorithm and GPM.Benefiting from the ability of the A* algorithm to establish the cost and heuristic functions, it can guarantee the macrolevel stealth performance considering the flight time.Furtherly, it provides better initial guess values for GPM to solve the feasible and optimal micro-level stealth trajectory.Thus, the hybrid method will have a good performance on both solution's feasibility and optimality.

Aerial vehicle dynamic model
An aerial vehicle model with an active thrust, such as a missile, is considered in the paper.In order to simplify the analysis, a three-degree-of-freedom particle system in a Cartesian coordinate system is adopted while ignoring the rotation and curvature of the Earth.The parameters are defined in Figure 1.
where m is the mass of the vehicle, g is the gravitational acceleration, V is the velocity,  , V  , and V  are respectively trajectory inclination, trajectory declination, and velocity tilt angles,  and  are the attack angle and sideslip angle, T , X , Y and Z are the thrust, drag, lift and lateral forces, x , y and z are the position coordinate components of the coordinate axis.To further simplify the system model, this paper considers the case of 0 V   so that the state vector

Radar detection model A definite state x
 determines its radar detection threat.The metric in the radar threat environment can be described as the instantaneous probability of detection d P in the current state, and the radar detection probability can be expressed as Equation ( 2) based on the previous study [14], where is the slant distance between the radar and the aerial vehicle, max R is the max detection range of the radar,  is the instantaneous RCS, and the parameters 1 c and 2 c can be specified in advance as it depends on the specific performance and the type of the radar.
RCS is an important variable affecting radar detection probability, so the RCS model is very important.RCS is affected by the surface material and the line-of-sight (LOS) angle of the radar beam.Electromagnetic simulation software can easily establish the RCS characteristic model.This paper considers an axisymmetric physical model so that the RCS can be determined by a single angle variable LOS  , as shown in Figure 2. The Gaussian filtering (GF) and polynomial fitting (PF) can smooth the characteristic curve, which is conducive to the subsequent optimization.In order to calculate the LOS  in a specific state, the aerial vehicle body axial vector BA V  needs to be transferred from the body coordinate system to the ground coordinate system as Equation (3), , 0 0 0 , , 0 where T-E L , V-T L and B-V L are the components of the coordinate system transformation matrix L .
Therefore LOS  is equal to the vector angle between the body coordinate axial vector BA V  and the

LOS vector LOS
V  , as shown in Equation ( 4).Therefore, the attitude of the vehicle can ultimately affect the radar detection probability in Equation ( 2) by affecting the RCS.

Waypoints generation model
Most studies consider path planning and trajectory optimization to be the same concept.However, this paper considers path planning as "finding some waypoints (WPs) that satisfy certain conditions."The waypoints can be connected by straight lines to get the path with a broken line, which is not the final smooth trajectory curve with control variables.By adjusting the degree of map grid subdivision or the step size of waypoints, path planning can generate the waypoints with different sparsity and provide different levels of help for subsequent trajectory optimization.The proposed method adopts the above thought.The size of a large-scale space problem can be reduced by constructing a grid map and adopting the "first-order hold" to the original space.A* algorithm combines the advantages of the Dijkstra and the greedy best-first-search algorithms by establishing the cost function   g n and the heuristic function   h n .In particular, in the large-scale environment, the value of the height axis is small compared to the size of the ground plane.So the A* algorithm is adopted in a xO z two-dimensional plane of fixed height, which means that each node n is equal to a two-dimensional vector   T , z x .
In a constructed grid map under the threat of radar detection, the cost function is defined as the path cumulative detection probability, which can be expressed as shown in Equation (5), where p n is the parent node of the current node n .The heuristic function is defined as the Euclidean distance between the current node n and the goal node g n with a weight coefficient  , which can be expressed as Equation ( 6) and 0.2 To smooth the waypoints path so that it is easier to generate the initial guess values of the timestate-control variable sequence between waypoints through GPM, the final WPs 2 k P  are further obtained by B-spline curve fitting and sampling with an equal interval as shown in Equation (7), where u  is the sampling interval, A* i P are the control points generated by A* algorithm, and , i p N are the basic functions which are the polynomial functions with the order of p .

Constraints and objective function
In order to obtain the complete initial guess sequences, the optimal control model with the shortest flight time between WPs is adopted.The initial constraints of connectivity from the waypoint k P to 1 k P  are defined as Equation ( 8), where the subscript " 0 " and " f " denote the initial and terminal variables, respectively.It should be pointed that the feasibility constraints are specifically considered in this paper.It can be expressed as Equation ( 9) for each subproblem from waypoint k P to , , , where the  are the small enough angle variables, and , 1 V k k    is the trajectory inclination angle of the straight line from k P to 1 k P  .These proposed constraint designs can guarantee with a high probability that the terminal state of each segment is acceptable as the initial state for the next segment.In addition, the objective is the shortest-time performance index as , 0 , min , 0, , 1 After getting the complete initial guess sequences, the fully process stealth trajectory OCP can also be solved by GPM.Considering the continuity of the control variable, the virtual control variables are defined as the first-time derivative of the attack angle  and sideslip angle   are adopted.The initial, terminal, and path constraints can be expressed as Equation (10), the objective function can be expressed as

Framework of large-scale hybrid trajectory optimization method
By combining all the above theory models, the framework of the proposed hybrid optimization method of large-scale stealth trajectory is shown in Figure 3. Firstly, the A* algorithm and B-spline fitting method are adopted based on the constructed 2-D grid map, cost functions, and heuristic functions to generate the 2-D WPs.Secondly, the WPs will be used as the start and end points to solve the segmented minimum flight time OCP to yield the complete initial guess value sequences.Finally, the large-scale stealth trajectory OCP jointly constructed by the above model, constraints, and the objective function is solved through the GPM method and GPOPS software package [15] to get the final 3-D trajectory results with the state-control variables.The initial guess values provide feasibility and optimality.

Numerical simulation results
In order to verify the effectiveness and performance of the proposed scheme, a stealth trajectory optimization problem with the scale over 2000 km was simulated.The aerial vehicle's start and end  Figure 4 shows the simulation results of the initial guess values generation model for stealth trajectory GPM optimization.It is a process of path finding through solving the minimum flight time OCP between the waypoints.Therefore, the objective is to obtain a feasible path in a large-scale space that satisfies the stealth performance at the macro level based on WPs and generate a sequence of consecutive variables as initial guesses for further refining.9).The terminal states of the trajectory declination angle in each segmented OCP are designed to be conducive to reaching the next WP.On the other hand, the trajectory is not adjusted to change the RCS because the RCS is considered to be a constant in the A* algorithm.Figure 5 shows the results of solving the large-scale stealth trajectory OCP by GPM based on the initial guess values obtained above.After the path planning based on WPs, the feasibility of trajectory optimization is guaranteed with high probability.As shown in Figure 5(c), the trajectory is basically refined based on the initial guess trajectory, which considers the RCS characteristic model of Figure 2 to reduce the radar threat level.The fluctuation level of the trajectory parameters in Figure 5 is greater than that in Figure 4, which also shows that the aerial vehicle is constantly adjusting the attitude during flight to further improve the stealth performance at a more subtle level under the radar threat.However, it also results in increased flight time consumption as ,initial guess s 1177.4  Figure 6 shows the radar detection performance of the initial guess trajectory and the refined stealth trajectory to evaluate the stealth performance of the trajectory results.It can be seen that the radar detection probability of the initial guess trajectory increases as the distance to the radar decreases.Although the detection probability does not exceed 0.25 due to the WPs ensuring a long distance from the radar, the effect of RCS is not taken into account, which results in the long-term threat of radar detection from about 300 s to 1000 s .After the optimization by solving OCP, the stealth performance of trajectory shows that the cumulative detection probability

Conclusions
In order to improve the problem that stealth trajectory solving in large-scale space is challenging to obtain the optimal or even feasible solutions, a hybrid optimization scheme based on the A* algorithm and GPM is proposed in the paper.The A* algorithm-based WPs and initial guess sequences generation model is constructed, and GPM further solves the stealth trajectory optimization coupling RCS.The simulation results validated that the proposed scheme can provide feasible initial guesses for large-scale GPM iterations.The final refined trajectory results have effective stealth performance in a space with the scale over 2000 km .On the other hand, the proposed scheme is still an offline optimization method so that the future work will focus on the problem of online optimization in largescale scenarios.

Figure 1 .
Figure 1.Coordinate system and parameters definition.The dynamical equations of the system model are determined by Equation (1),

Figure 2 .
Figure 2. RCS data at 10 GHz of the aerial vehicle.

Figure 3 .
Figure 3. Framework of the proposed hybrid trajectory optimization method.
c  .The simulation results are shown in Figures 4-6.(a) Trajectory inclination angle (b) Trajectory declination angle (c) Initial guess trajectory and WPs

Figure 4 .Figure 5 .
Figure 4. Results of initial guess values generation based on A* algorithm.
Figure4shows the simulation results of the initial guess values generation model for stealth trajectory GPM optimization.It is a process of path finding through solving the minimum flight time OCP between the waypoints.Therefore, the objective is to obtain a feasible path in a large-scale space that satisfies the stealth performance at the macro level based on WPs and generate a sequence of consecutive variables as initial guesses for further refining.Figure 4(b) shows the constructed addition, it must be pointed out that the altitude axis in the resulting figure is not proportional to the ground axis, so the actual attitude fluctuation is not as large as shown in the figure.
due to iterative optimization of GPM based on the initial guesses.