A composite planning method considering avoidance of multiple no-fly zones

Aiming at the problem that multiple no-fly zones need to be avoided in the reentry process of aircraft, a composite programming method combining conjugate gradient and dynamic programming is proposed. The method divides the reentry flight trajectory into two parts: the longitudinal plane and the transverse plane. The longitudinal plane uses the conjugate gradient method to guide the aircraft to meet the minimum mass of the thermal protection system during the whole flight. The horizontal plane mainly uses the dynamic programming method to obtain a reasonable obstacle avoidance route and inversely calculates the change law of the heading angle. The simulation results show that the three-dimensional reentry trajectory can reasonably solve the problem of avoiding multiple no-fly zones and meeting the aerodynamic constraints of the flight process.


Introduction
As a key component of the spacecraft mission, re-entry technology plays a vital role in guiding the spacecraft to return to the Earth or other celestial bodies.The key points of the re-entry mission include constraining the dynamic pressure, heating rate, and overload of the spacecraft during highspeed re-entry flight while ensuring accurate landing to avoid the no-fly zones.Given the many variables in the flight process, it is a complex task to determine a re-entry trajectory that satisfies constraint conditions and reduces the mass of the thermal protection system.
Aiming at this problem, domestic and foreign scholars have put forward a lot of opinions.Liao et al. proposed a closed-loop optimal feedback guidance method for the re-entry phase, which is designed by combining online trajectory optimization and optimal feedback control [1].In [2], the no-fly zone constraints are transformed, and the A* path planning method is used to solve this avoidance problem.In [3], the Dubins curve path planning method and the proportional navigation method are combined to track the virtual target in real time to realize the avoidance guidance of the aircraft to the no-fly zones.In [4], based on the energy-based motion model, the analytical prediction-correction guidance of range and altitude is carried out, aiming to repair the range altitude error caused by evasion.
In this paper, aiming at the re-entry flight process, considering the aerodynamic constraints, control constraints, and path constraints introduced by avoiding no-fly zones [5][6], a composite planning algorithm combining the conjugate gradient method and dynamic programming method is proposed to solve the no-fly zones avoidance problems.At the same time, the mass of the thermal protection system is reduced.The rationality and reliability of this method are verified by simulation.

Dynamic model of re-entry vehicle
The whole flight is in an unpowered gliding mode [7].Considering the arc and rotation of the Earth, the dynamic equations are given

Performance index and constraints of re-entry problem
Aiming at a reentry flight trajectory optimization problem, it is necessary to meet the requirements of the optimization performance index under the comprehensive consideration of various constraint conditions [8], so that the aircraft can achieve the reentry purpose reasonably and reliably, and realize the effect of real-time optimization during the flight.To reduce the cost of thermal protection system and other overall design schemes [9], the performance indexes of trajectory optimization during reentry flight are as follows (1) starting point and terminal constraints: (2) process constraint [10]: (3) control variables constraints: The reasonable range of angle of attack can make the aircraft fully play its ability under the premise of satisfying various process constraints, considering factors such as heat protection, aerodynamics, attitude control, and navigation [11].The constraint range of the bank angle determines the ability of the aircraft to change course during reentry.

Design of no-fly zones
To facilitate the calculation and analysis, this paper regards the no-fly zones as infinitely high cylinders, and the aircraft can only be bypassed from both sides of these cylinders [12].Considering the Earth's arc surface, it is assumed that ,

Design of longitudinal trajectory
Due to the longitudinal and lateral separation guidance method, when designing the longitudinal guidance logic, the trajectory deviation angle value is zero and the bank angle value is constant.The change of the angle of attack is used to meet the aerodynamic constraints and terminal constraints of the reentry process.

Optimization algorithm of longitudinal flight trajectory
We define the state quantity equation as The conjugate gradient method is used to obtain more accurate optimal control variables and optimal trajectories.

Simulation of longitudinal flight trajectory
The above optimization method is used to calculate all the flight trajectories of the aircraft under different initial reentry conditions, and the flight state parameters [ , , , , , ] V h

    and control parameters [ , ]
  of the aircraft at different times can be obtained.Based on the aerodynamic characteristics data of the aircraft and the corresponding atmospheric characteristics, the flight trajectory calculation when the flight simulation reentry condition are At this time, the drop point parameter is 3978 , 820 . It can be seen from Figures 1-2 that the flight trajectory obtained by this method changes reasonably and the state quantity changes smoothly.In the whole reentry process, the trajectory inclination angle is basically less than zero, which satisfies the equilibrium gliding condition [13].

Design of transverse trajectory
Based on the longitudinal trajectory design, a reasonable angle of attack change process is determined to control the flight trajectory to reach the target area under the condition of meeting the aerodynamic constraints.The design of the lateral trajectory mainly completes the task of avoiding the no-fly zone and determines the change law of the bank angle.

Planning algorithm of lateral flight trajectory
Based on the time and range of the longitudinal plane trajectory, the transverse plane discrete state equation is defined as ( 1) [ ( ), ( ), ( ), ], 0, 1, 2,..., ] (2) control variables constraints: 30 30 no-fly zone constraints: The cost function: The iterative process: , ( ), ( ), ] [ ( ) [ ( 1), ( 1), ( 1), 1]       The simulation results show that the above lateral guidance logic can guide the aircraft to complete the no-fly zone avoidance task.The three-dimensional reentry trajectory diagram verifies the effectiveness of the combination of longitudinal and lateral guidance methods as a whole.Figures 7-8 show that the aerodynamic heating rate, overload, and dynamic pressure of the whole flight do not IOP Publishing doi:10.1088/1742-6596/2764/1/0120546 exceed the constraints, indicating that the aircraft meets the reentry conditions and can achieve the reentry purpose well.

Conclusion
According to the characteristics of reentry gliding flight, this paper studies the longitudinal and lateral guidance design of no-fly zone avoidance.Firstly, the performance index of the longitudinal plane flight trajectory is optimized by the conjugate gradient method, and the amplitude of the angle of attack is ensured to meet the constraints.Secondly, based on the time and range of the longitudinal plane trajectory, the state quantity of the transverse plane is discretized, and the flight trajectory conforming to the no-fly zone constraint is designed by the dynamic programming method.Finally, the numerical value of the trajectory deviation angle is calculated in reverse, and the three-dimensional reentry trajectory diagram is drawn.The simulation diagram shows that the three-dimensional reentry trajectory can avoid multiple no-fly zones, which verifies the rationality of the composite planning algorithm.
are the longitude and latitude of the real-time position of the aircraft; , z z   are the latitude and longitude of the location of a no-fly zone; , e z R R are the radius of the radius of the Earth and the no-fly zone.The following constraints should be calculated

4 Figure 1 .
Figure 1.Distance x in the longitude direction.Figure 2. Attack angle  changes over time.

Figure 2 .
Figure 1.Distance x in the longitude direction.Figure 2. Attack angle  changes over time.

4 . 2 .
Simulation of flight trajectory Combined with the longitudinal flight trajectory simulation data, we let  0 ,  480.1 ,  3978 .The simulation trajectory is shown in Figures 3-8: Figure 3 is the lateral trajectory diagram of avoiding the no-fly zones; Figure 4 is the trajectory deflection angle change diagram; Figure 5 is the variation law of heeling angle; Figure 6 is a three-dimensional reentry IOP Publishing doi:10.1088/1742-6596/2764/1/0120545 trajectory.Among them, the heading angle is obtained by reversely solving the reentry dynamic equation based on the lateral trajectory.

Figure 4 .
Figure 4. Heading angle  changes over time.

Figure 8 .
Overload n changes over time.