A Novel Trajectory Planning Method for Parafoil Airdrop System Based on Geometric Segmentation Strategy

: Reasonable trajectory planning is the precondition for the parafoil airdrop system to achieve autonomous accurate homing, and safe landing. To successfully realize the self-homing of the parafoil airdrop system, a new trajectory optimization design scheme is proposed in this paper. The scheme is based on the parafoil’s unique ﬂight and control characteristics and adopts a segmented homing design. The current common trajectory design method faces a problem, whereby straight-line ﬂight distance before landing is limited by the radius of the height-reducing area. The core feature of the proposed design scheme is its avoidance of this problem, thereby ensuring landing accuracy and safety. Firstly, the different starting states of the parafoil airdrop system and the landing requirements were comprehensively considered, and the homing trajectory reasonably segmented. Based on the requirements of energy control, stable ﬂight, and landing accuracy, the optimal objective function of the trajectory was established, and the trajectory parameters, calculation methods, and constraints were given. Secondly, the cuckoo search algorithm was applied to optimize the objective function to obtain the ﬁnal home trajectory. Finally, the trajectory planning under different airdrop conditions was simulated and veriﬁed. The results showed that the planned trajectories could reach the target point accurately and meet the ﬂight direction requirements, proving the proposed scheme’s correctness and feasibility.


Introduction
The parafoil is controllable. Compared with the traditional circular parachute airdrop system, the airdrop system composed of the parafoil can achieve a precise airdrop of fixedpoint targets under the controller's action. The parafoil airdrop system has a wide range of application values, such as the airdrop of relief materials in a disaster environment, the efficient delivery of military materials, such as weapons and ammunition in combat areas, and the accurate recovery of spacecraft [1]. For example, in June 2021, China successfully recovered the Chang San Yi rocket booster through its recovery system. Similar to other unmanned aerial vehicles, the parafoil airdrop system also needs to track the planned trajectory when it realizes the autonomous homing operation [2][3][4][5]. Therefore, scientifically planning a high-quality homing trajectory is the premise for the parafoil airdrop system to achieve reliable autonomous flight. It is also an essential guarantee for the final realization of an accurate and safe airdrop [6][7][8][9]. The parafoil airdrop system is a soft-wing system, and its flight is easily disturbed by the surrounding environment, such as terrain and wind [10]. It is a nonlinear system and has many restrictions on its control. Therefore, various conditions and constraints must be considered for its homing trajectory planning [11][12][13]. The research on the autonomous homing trajectory of the parafoil airdrop system mainly focused on two aspects.

•
Based on the geometric segmentation strategy, a new autonomous homing scheme for the parafoil airdrop system was designed. The straight-line flight segment before landing was designed on the tangent of the circle with reduced height, and its length can be freely controlled. Compared with the target point being developed at the center of the circle with reduced altitude, it avoids the problem that the length of the straight flight segment before landing is subject to the circling radius and provides a guarantee for the smooth implementation of a bird landing, in terms of landing accuracy and safety; • The objective function of trajectory optimization is established, and the calculation method and constraints of trajectory parameters are introduced in detail. The realization process of the new method of trajectory planning is described with examples; • Different initial conditions for airdrops are set. The trajectory planned by the new scheme was simulated and verified, and a comparative analysis and discussion with the traditional scheme are carried out, which proves the effectiveness of the trajectory planning scheme designed in this paper.

Construction of Point-Mass Model of Airdrop System
The parafoil autonomous airdrop system generally consists of the parafoil, the loadcarrying object, the object to be airdropped, and the controller [25,26]. The parafoil airdrop system with an autonomous homing function does not have a power device. Its homing is realized by controlling the left and right pull-down parafoil ropes at the rear edge of the parafoil during its descent. The controller controls the servo mechanism that performs the pull-down action of the parafoil rope. When the parachute rope's pull-down amplitude on the parafoil's left side is more significant than that on the right side, the airdrop system performs a left turn flight. Instead, the airdrop system performs a right-turn flight. The airdrop system performs gliding flight when the left and right pulling amplitudes are equal. Through reasonable flight control by the controller, the parafoil airdrop system can achieve autonomous flight and airdrop materials to the designated destination.
Without external interference, the airdrop system drops steadily under the action of gravity and aerodynamic force. In order to ensure overall stability and safety, the moment of inertia is ignored, and the pull-down amplitude of one side of the parafoil rope must be kept within a small range. Therefore, the following settings are made in this paper:

•
The horizontal speed v s and vertical descent speed v z of parafoil airdrop system are constant; • Response without delay.
Based on the geodetic coordinate system, the point-mass motion equation of parafoil airdrop system is established, as shown in Equation (1), which is used for trajectory planning in the homing process of the system: In Equation (1), (x, y, z) is the position of the system, u is the control quantity, ψ is the turning angle, and w x , w y is the transverse wind speed of the space where the system is located. The wind is usually greatly affected by the airdrop environment, and the parafoil airdrop system has relatively high requirements for the airdrop environment. Therefore, generally, only the impact of the average wind field on the parafoil flight is considered. For the convenience of research, the impact of the average wind field is usually converted into the trajectory tracking error, and only the effect of the wind direction is considered. This paper also follows this treatment method. That is w x , w y = (0, 0). The point-mass model of can be further simplified, as shown in Equation (2): (2)

Design of Homing Trajectory
The parafoil airdrop system itself does not have a power device. By controlling the two pull-down parachute ropes on the left and right of the trailing edge, the parafoil can home autonomously. The entire segmented homing process can generally be divided into three parts: centripetal, energy control, and landing, as shown in Figure 1. considered. For the convenience of research, the impact of the average wind field is usually converted into the trajectory tracking error, and only the effect of the wind direction is considered. This paper also follows this treatment method. That is , = (0, 0). The point-mass model of can be further simplified, as shown in Equation (2): (2)

Design of Homing Trajectory
The parafoil airdrop system itself does not have a power device. By controlling the two pull-down parachute ropes on the left and right of the trailing edge, the parafoil can home autonomously. The entire segmented homing process can generally be divided into three parts: centripetal, energy control, and landing, as shown in Figure 1. Centripetal flight is used to reduce the distance difference between the airdrop system and the target point. Capability control flight is used to eliminate the high degree of redundancy existing in the airdrop system. Landing flight is mainly used to adjust the flight direction, which is prepared for the windward landing of the airdrop system.

Trajectory Design Scheme
Based on the classical segmented trajectory design, this paper designed a new trajectory planning scheme, which not only considers the energy loss generated in the homing process, homing accuracy, and the overall flight safety of the system, but also takes into account the convenience of the bird landing operation. The specific scheme is shown in Figure 2. Centripetal flight is used to reduce the distance difference between the airdrop system and the target point. Capability control flight is used to eliminate the high degree of redundancy existing in the airdrop system. Landing flight is mainly used to adjust the flight direction, which is prepared for the windward landing of the airdrop system.

Trajectory Design Scheme
Based on the classical segmented trajectory design, this paper designed a new trajectory planning scheme, which not only considers the energy loss generated in the homing process, homing accuracy, and the overall flight safety of the system, but also takes into account the convenience of the bird landing operation. The specific scheme is shown in Figure 2.
In Figure 2, x and y are the two coordinate axes on the horizontal plane in the wind coordinate system. Axis z is perpendicular to plane x, y and intersects the plane x, y at point F. The origin F of the wind coordinate system is set as the target point, and the wind direction is consistent with the x-axis direction. Point A is the starting point of the airdrop system after parachute opening. After turning through arc AB, the parafoil airdrop system flies to the circling area near the target point, and point D is the entry point. Circle O 3 is the center of the circle in the circling area, and the parafoil airdrop system makes a turning flight around this point with R ep as the radius, so as to reduce the flight altitude, and θ ep is the angle between the line connecting the two points D and O 3 and the positive direction of the x-axis. EF is the upwind flight phase before entering the bird's landing, and it is tangent to the circle O 3 . In Figure 2, and are the two coordinate axes on the horizontal plane in the wind coordinate system. Axis is perpendicular to plane , and intersects the plane , at point F. The origin F of the wind coordinate system is set as the target point, and the wind direction is consistent with the -axis direction. Point A is the starting point of the airdrop system after parachute opening. After turning through arc AB, the parafoil airdrop system flies to the circling area near the target point, and point D is the entry point. Circle O is the center of the circle in the circling area, and the parafoil airdrop system makes a turning flight around this point with as the radius, so as to reduce the flight altitude, and is the angle between the line connecting the two points D and O and the positive direction of the -axis. EF is the upwind flight phase before entering the bird's landing, and it is tangent to the circle O .
In order to reduce the control frequency of the controller and reduce the energy loss generated as much as possible, the trajectory in Figure 2 is designed with a single-side pull-down control method. When the parafoil system needs to adjust the flight direction, in order to achieve the purpose of fast and safe adjustment, it turns with the minimum turning radius. As shown in Figure 1, the two arcs AB and CD are the flight direction adjustment sections, both of which are carried out with the minimum turning radius. The value range of the central angle corresponding to the two arcs is 0 ≤ , ≤ π, so as to reduce the control energy consumption as much as possible. The key to segment optimization is to determine the position of entry point D, which can be transformed into the calculation of the values of the two parameters and . In order to make the whole planned trajectory optimal, the objective function shown in Equation (3) is established to obtain the optimal values of parameters and .
In Equation (3), is the landing accuracy objective function, which represents the deviation between the landing point and the target point, where represents the length of the arc AB and CD, • represents the length of the arc DE, ⃑ and ⃑ represents the length of line segments BC and EF, respectively, • represents the horizontal flight distance corresponding to the initial height of the parafoil system, and is the glide ratio of the parafoil airdrop system.

Calculation and Constraints of Trajectory Parameters
Let ( , , ) be the coordinates of the initial point, represents the turning flight direction of the parafoil system, = −1 is the clockwise flight, = 1 is the counterclockwise flight, = ⃑ , is the initial heading angle, and the center In order to reduce the control frequency of the controller and reduce the energy loss generated as much as possible, the trajectory in Figure 2 is designed with a single-side pull-down control method. When the parafoil system needs to adjust the flight direction, in order to achieve the purpose of fast and safe adjustment, it turns with the minimum turning radius. As shown in Figure 1, the two arcs AB and CD are the flight direction adjustment sections, both of which are carried out with the minimum turning radius. The value range of the central angle corresponding to the two arcs is 0 ≤ β 1 , β 2 ≤ π, so as to reduce the control energy consumption as much as possible. The key to segment optimization is to determine the position of entry point D, which can be transformed into the calculation of the values of the two parameters R ep and θ ep . In order to make the whole planned trajectory optimal, the objective function shown in Equation (3) is established to obtain the optimal values of parameters R ep and θ ep .
In Equation (3), J is the landing accuracy objective function, which represents the deviation between the landing point and the target point, where R min ·(β 1 + β 2 ) represents the length of the arc AB and CD, R ep ·β 3 represents the length of the arc DE, BC and EF represents the length of line segments BC and EF, respectively, f ·z 0 represents the horizontal flight distance corresponding to the initial height of the parafoil system, and f is the glide ratio of the parafoil airdrop system.

Calculation and Constraints of Trajectory Parameters
Let (x 0 , y 0 , z 0 ) be the coordinates of the initial point, dir represents the turning flight direction of the parafoil system, dir = −1 is the clockwise flight, dir = 1 is the counterclockwise flight, L EF = EF , α 0 is the initial heading angle, and the center positions of β 1 and β 2 are represented by O 1 and O 2 , respectively, then the vector L O 1 O 2 from O 1 to O 2 can be calculated by Equation (4): The included angle δ O 1 O 2 between L O 1 O 2 and the positive direction of the x-axis can be calculated by Equation (5): The included angles β 1 , β 2 and β 3 of the circular arc in Figure 1 can be calculated by Equations (6) and (7): The parameter α 2 in Equation (6) can be obtained by Equation (8): In order to save the control energy as much as possible and improve the safety and stability of the airdrop system during flight, the value ranges of the agreed parameters R ep and θ ep are limited by Formula (9): (9) In Formula (9), R 1 is the minimum turning radius allowed by the parafoil airdrop system when flying in the hovering high-flying area, corresponding to the maximum unilateral pull-down range of the parachute rope, and R 2 is the maximum turning radius.

Trajectory Optimization
The cuckoo search (CS) algorithm was proposed in 2009 [27]. It is a bionic algorithm generated by imitating the cuckoo's breeding and random flight behavior. The search process does not depend on the gradient and has the advantages of fewer parameters, high algorithm execution efficiency, and easy implementation. It has attracted the attention of many researchers. On this basis, many improved algorithms have been proposed to improve the overall performance of the algorithm [28][29][30], and, based on the CS algorithm, combined with the piecewise trajectory design scheme proposed above, this paper searched for the optimal solution for the objective function, shown in Equation (3), within the given interval of parameters R ep and θ ep , and determined the trajectory. The specific implementation steps are shown in Figure 3.
Step 1: Initialize parameters, such as population size N, the maximum number G m of iterations, the probability P a of the host discovering cuckoo eggs, the starting point coordinates (x 0 , y 0 , z 0 ), the initial direction α 0 , the radius R ep and the angle range θ ep of the entry point, and other parameters. According to the value range shown in Formula (9), a two-dimensional initial population composed of R ep and θ ep is randomly generated, and its fitness value is calculated using the objective function shown in Equation (3).
within the given interval of parameters and , and determined the trajectory. The specific implementation steps are shown in Figure 3. Step 1: Initialize parameters, such as population size , the maximum number of iterations, the probability of the host discovering cuckoo eggs, the starting point coordinates ( , , ), the initial direction , the radius and the angle range of the entry point, and other parameters. According to the value range shown in Formula (9), a two-dimensional initial population composed of and is randomly generated, and its fitness value is calculated using the objective function shown in Equation (3).
Step 2: Generate a new levy flight solution from Equation (10) where , represents the -th solution in the -th generation, ( )~ / / , ~(0, σ ), ~(0, 1), and the parameter can be calculated by Equation (11): Step 2: Generate a new levy flight solution from Equation (10) where X g,i represents the i-th solution in the g-th generation, N(0, 1), and the parameter σ can be calculated by Equation (11): Step 3: Calculate the fitness value J of solution A, and update the solution set according to the following rules: if J(X g+1,i ) > J X g,i , then X g+1,i ⇔ X g,i .
Step 4: Update the solution set with the probability P a of the host finding the cuckoo egg, as shown in Equation (12), where H(·) is the Heaviside step function, and r, ε ∼ U(0, 1): Step 5: Calculate the fitness value of the solution set updated by Equation (12), and update the solution set using the rules in Step 3, then, find and save the optimal solution in the new solution set. Judge whether the search end condition is satisfied; if not, skip to step 2 for loop execution. Otherwise, end the loop and find the global optimal solution.
Step 6: Substitute the obtained optimal parameter R ep , θ ep into Equations (4)-(8) to calculate the parameters of each segment, and calculate the corresponding homing trajectory according to the initial conditions and the point-mass model of the parafoil airdrop system.

Parameter Setting
In order to prove the feasibility of the segmented trajectory planning scheme proposed in this paper, the trajectory optimization results were simulated and verified based on the point-mass model of the parafoil airdrop system and the cuckoo optimization algorithm. The relevant parameter settings in the trajectory optimization scheme and the cuckoo optimization algorithm are shown in Table 1. In Table 1, the glide ratio f of the parafoil airdrop system was 3, its horizontal flight speed v s was 13.8 m/s, the vertical descent speed v z was 4.6 m/s, and the minimum turning radius R min was set as 100 m. The minimum R 1 and maximum R 2 of the radius R ep of spiral height elimination were set as 100 m and 500 m, respectively. The headwind flight distance EF before bird landing was set as 100 m. Let the number of nests N in the cuckoo optimization algorithm be 100. That is, the number of solutions was 100, the maximum number of iterations G m was 200, the probability P a of the host discovering cuckoo eggs was 0.25, the step scaling factor α was 1, and β was 1.5.

Result Analysis
The cuckoo optimization algorithm was used to solve the objective function shown in Equation (3), to obtain the optimal solutions of R ep and θ ep , and, then, to determine the optimal trajectory. In order to ensure the comprehensiveness of the verification, Table 2 shows three different airdrop situations, and the simulation analysis of their trajectory planning was carried out, respectively. The simulation results are shown in the following figures. When the starting point of the parafoil airdrop system was at (800 m, −650 m, 1000 m), the optimal solution of the objective function was obtained through the cuckoo search algorithm, where R ep = 272.3363 rad, θ ep = −3.1416 rad, and the objective function value was 0 at this time. Figure 4 shows the convergence curve when the objective function was optimized. It can be seen that the curve converged when the search reached 16 generations, and the convergence speed was faster. When the starting point of the parafoil airdrop system was at (800 m, −650 m, 1000 m), the optimal solution of the objective function was obtained through the cuckoo search algorithm, where = 272.3363 rad, = −3.1416 rad, and the objective function value was 0 at this time. Figure 4 shows the convergence curve when the objective function was optimized. It can be seen that the curve converged when the search reached 16 generations, and the convergence speed was faster.                 In state 2, the starting point of the parafoil airdrop system was located at (800, 650, 1000), which was located at the upper right of the target point in the horizontal trajectory, and the airdrop height was consistent with state 1. The search algorithm was used to search for the optimal solution of the objective function, and the optimal entry point was located at = 348.7353 m , = 3.0147 rad. It can be seen from the planned trajectory curve that, similar to state 1, the whole trajectory was still composed of turning towards the target point, flying straight to the energy control area, circling and height elimination, and upwind alignment. At this time, was 1.8173 rad and was 4.6877 rad.  In state 2, the starting point of the parafoil airdrop system was located at (800, 650, 1000), which was located at the upper right of the target point in the horizontal trajectory, and the airdrop height was consistent with state 1. The search algorithm was used to search for the optimal solution of the objective function, and the optimal entry point was located at R ep = 348.7353 m, θ ep = 3.0147 rad. It can be seen from the planned trajectory curve that, similar to state 1, the whole trajectory was still composed of turning towards the target point, flying straight to the energy control area, circling and height elimination, and upwind alignment. At this time, β 1 was 1.8173 rad and β 3 was 4.6877 rad.            Figures 5, 8 and 11. It can be seen from the control curve that the control quantity corresponding to the homing trajectory planned by the segmental trajectory planning scheme proposed in this paper was a piecewise function, having a value less than zero, and the control quantity remained constant in the corresponding time period. This control was relatively simple in practical application, and the flight of the parafoil airdrop system was also relatively stable. Figures 14-16 are the control curves under three different airdrop situations, respectively, corresponding to the trajectories planned in stages, as shown in Figures 5,8 and 11. It can be seen from the control curve that the control quantity corresponding to the homing trajectory planned by the segmental trajectory planning scheme proposed in this paper was a piecewise function, having a value less than zero, and the control quantity remained constant in the corresponding time period. This control was relatively simple in practical application, and the flight of the parafoil airdrop system was also relatively stable.    . It can be seen from the control curve that the control quantity corresponding to the homing trajectory planned by the segmental trajectory planning scheme proposed in this paper was a piecewise function, having a value less than zero, and the control quantity remained constant in the corresponding time period. This control was relatively simple in practical application, and the flight of the parafoil airdrop system was also relatively stable.    Table 3 shows the landing accuracy of the three trajectories at the target point. It can be seen that the lateral error of the landing was small, and the landing direction was opposite to the preset wind direction, which met the design requirements. In addition, a large number of simulation experiments showed that the optimization of the parameters of the trajectory optimization algorithm could be converged within 20 generations, showing good robustness.

Results of State 3
Comparing the segmented trajectory under the third airdrop cases with the first two, it can be seen that, due to the high position of the airdrop point in case 3 and the distance from the target point being farther than in cases 1 and 2, more time for circling and height elimination was required in the energy control area. On the contrary, if the airdrop altitude was too low and the distance from the target point far, the system would not have enough time to achieve the target landing. Therefore, for the unpowered parafoil airdrop system, its airdrop position needs to be reasonably selected.  Table 3 shows the landing accuracy of the three trajectories at the target point. It can be seen that the lateral error of the landing was small, and the landing direction was opposite to the preset wind direction, which met the design requirements. In addition, a large number of simulation experiments showed that the optimization of the parameters of the trajectory optimization algorithm could be converged within 20 generations, showing good robustness. Comparing the segmented trajectory under the third airdrop cases with the first two, it can be seen that, due to the high position of the airdrop point in case 3 and the distance from the target point being farther than in cases 1 and 2, more time for circling and height elimination was required in the energy control area. On the contrary, if the airdrop altitude was too low and the distance from the target point far, the system would not have enough time to achieve the target landing. Therefore, for the unpowered parafoil airdrop system, its airdrop position needs to be reasonably selected.

Discussions
Under the same conditions, when considering the landing accuracy of the target point, the accuracy of the trajectory planned by the method in this paper is basically the same order of magnitude compared with the existing classical segmented trajectory method. For example, when the coordinates of the starting point were at (800, 800, 2000) and the initial flight direction = − /3 rad, the landing direction of the segmented trajectory planning strategy proposed in this paper was 3.14, and the landing error was 0.6685 m, which was about 0.3742 m higher than the accuracy of 1.0427 m in Ref. [23]. Compared with the traditional classical segmented trajectory, the length of the glide trajectory before landing in the new scheme can be set freely according to the actual flight situation, which overcomes the defect of the length of the glide segment being affected by the radius of the parafoil circling flight circle, and has, therefore, higher landing accuracy and safety. Therefore, the segmented trajectory optimization strategy of the parafoil airdrop system given in this paper is effective and feasible.

Discussion
Under the same conditions, when considering the landing accuracy of the target point, the accuracy of the trajectory planned by the method in this paper is basically the same order of magnitude compared with the existing classical segmented trajectory method. For example, when the coordinates of the starting point were at (800, 800, 2000) and the initial flight direction α 0 = −π/3 rad, the landing direction of the segmented trajectory planning strategy proposed in this paper was 3.14, and the landing error was 0.6685 m, which was about 0.3742 m higher than the accuracy of 1.0427 m in Ref. [23].
Compared with the traditional classical segmented trajectory, the length of the glide trajectory before landing in the new scheme can be set freely according to the actual flight situation, which overcomes the defect of the length of the glide segment being affected by the radius of the parafoil circling flight circle, and has, therefore, higher landing accuracy and safety. Therefore, the segmented trajectory optimization strategy of the parafoil airdrop system given in this paper is effective and feasible.
It should be pointed out that the trajectory planning scheme designed in this paper is the basic optimal design method for the autonomous homing trajectory of the parafoil. The trajectory planning problem with constraints, such as obstacles and faults in the homing process, is not within the scope of this paper.

Conclusions
In the design scheme of the segmented homing of the parafoil airdrop system, the landing target point, or the position of executing the bird landing, is usually set at the center of the circling flight area. This method has the problem that the length of the straight flight segment before landing is affected by the helix radius. When the radius calculated by the optimization parameters is small, the gliding flight segment of the parafoil is shortened, which brings difficulties to the bird landing operation, or even failure to perform the bird landing successfully, resulting in damage to the airdropped material. In this paper, the segmentation scheme is improved. The straight flight segment before landing is designed at the tangent position of the spiral height elimination circle, and its length can be freely controlled. Compared with the target point designed at the center position of the spiral height elimination circle, it avoids the problem of the length of the straight flight segment before landing being subject to the spiral radius and provides a guarantee for bird landing in terms of landing accuracy and safety.
Based on the verification needs of the new trajectory scheme, this paper established the point mass model of the parafoil airdrop system. On the basis of analyzing the flight characteristics of the parafoil airdrop system, the segmentation idea and design method of the new scheme were introduced, and the calculation method of each segment trajectory analyzed. The calculation formula of the objective function and the constraint range of parameters R ep and θ ep were given. The application method and steps of the parameter optimization of the cuckoo optimization algorithm in the new scheme were introduced in detail. In order to fully verify the feasibility of the new scheme, the trajectory under different airdrop conditions was simulated in this paper. The results showed that the correct trajectory could be successfully obtained under different initial conditions. Compared with the existing design schemes, the results show that the landing direction and accuracy of the new scheme can achieve ideal results, and the landing glide segment can be freely set and is no longer limited by the helix radius length, which proves the correctness and feasibility of the new trajectory design scheme. Moreover, the new scheme has the characteristics of simple control, high safety and easy realization in engineering. It is suitable for the trajectory planning of parafoil fixed-point airdrop systems before autonomous homing and also provides a new reference for the landing flight of parafoil airdrop systems. In future research work, we will consider more complex environmental constraints, such as obstacles, enemy areas, etc., so that the research results in this paper can be applied more widely.