CN113778117A - Multi-stage pseudo-spectrum method for intelligently selecting initial values for planning longitudinal optimal paths of airplanes - Google Patents

Multi-stage pseudo-spectrum method for intelligently selecting initial values for planning longitudinal optimal paths of airplanes Download PDF

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CN113778117A
CN113778117A CN202111039165.6A CN202111039165A CN113778117A CN 113778117 A CN113778117 A CN 113778117A CN 202111039165 A CN202111039165 A CN 202111039165A CN 113778117 A CN113778117 A CN 113778117A
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overload
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薛文超
何昕怡
陶呈纲
李导
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Academy of Mathematics and Systems Science of CAS
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Abstract

The invention provides an intelligent multi-stage pseudo-spectrum method for selecting initial values of the planning of the longitudinal optimal path of an airplane, and provides a control quantity design method based on multi-stage pseudo-spectrum conversion, such as normal overload, longitudinal overload and the like, wherein the normal overload and the longitudinal overload meet corresponding actual constraint conditions in each stage, so that the control quantity is guaranteed to be stably changed and actually realized in the whole stage; and learning to obtain the optimal track in a given initial and final state as an initial value of a pseudo-spectrum method by adopting a data set consisting of a neural network, the optimal track and boundary conditions. The method specifically comprises the following steps: firstly, establishing an airplane longitudinal track planning problem model containing nonlinear dynamics; the second step is that: a multi-stage pseudo-spectrometry that is actually operational based on the overload reference instruction; the third step: intelligently selecting an initial value through a neural network learning data set; the fourth step: obtaining an optimal solution by solving a nonlinear programming problem; the invention improves the convergence rate of the optimal solution calculation and realizes the high efficiency of the calculation.

Description

Multi-stage pseudo-spectrum method for intelligently selecting initial values for planning longitudinal optimal paths of airplanes
Technical Field
The invention belongs to the field of maneuvering path planning methods of aircrafts, and relates to an initial value selection method for optimal path calculation and an optimal control solving method under multi-stage constraint of control variables. The technology adopts a multi-stage pseudo-spectrum method to realize equivalent transformation from a continuous optimal control problem to a nonlinear programming problem, and realizes intelligent selection of initial values of iterative solution by using a BP neural network method. The technology is a solution for realizing the calculation of the longitudinal optimal track of the airplane with high efficiency and stability.
Background
The online trajectory planning is a necessary condition for realizing simulation of the trajectory of the aircraft, and the optimal control problem corresponding to the online trajectory planning often has singular arcs, and the numerical solution of the singular arcs can jump rapidly and frequently, so that the singular arcs are difficult to be used as reference instructions for actual operation. In addition, after the continuous optimal control problem is dispersed into a nonlinear programming problem, the iterative solution process is sensitive to initial values, and improper initial values can cause the calculation to be trapped in infeasible points. Therefore, how to obtain the optimal control quantity which accords with the actual operation and select a proper initial value track to enable the iteration process to avoid an infeasible area has great challenge. At present, the existing aircraft trajectory planning method is mainly an adaptive pseudo-spectrum method[1-3]And indirect method of conversion based on the maximum principle[4-5]Etc., there are limitations as follows:
1. the self-adaptive pseudo-spectrum method needs to consider curvature conditions at the same time, and does not consider the influence of grid time on control, so that algorithm errors are caused; an indirect method based on maximum principle transformation converts the original problem into a two-point boundary value problem, thereby increasing the complexity of the problem and the like;
2. for the problem of singular arcs, the optimal control quantity obtained by the self-adaptive pseudo spectrum is represented by high-frequency switching between an upper boundary and a lower boundary on the singular arcs, the number of discrete points on the singular arcs is increased, and the solving speed is slowed.
3. The numerical method for solving the nonlinear programming is sensitive to initial values, and improper selection of the initial values can cause problems to fall into non-optimal points or even infeasible points.
Therefore, the high efficiency and stability of calculation are difficult to guarantee when the existing method solves the problem of aircraft trajectory planning with singular arcs.
Aiming at the problem of online track planning of the longitudinal channel of the airplane, the invention adopts a multi-stage pseudo-spectrum method to avoid the generation of singular arcs caused by the affine nonlinearity of the airplane dynamics model to cause the corresponding optimal control problem, and provides a method for intelligently selecting an initial value by a neural network, thereby reducing the range of the initial value track and ensuring the high efficiency and stability of the online track planning calculation.
The invention content is as follows:
the technical problems solved by the invention are as follows: the method for designing the control quantity based on multi-stage pseudo-spectrum conversion, such as normal overload, longitudinal overload and the like, is provided, and the normal overload and the longitudinal overload meet corresponding actual constraint conditions in each stage, so that the control quantity is guaranteed to be stably changed and actually realized in the whole stage; and a data set consisting of a neural network, the optimal trajectory and boundary conditions is adopted, the optimal trajectory in a given initial and final state is obtained through learning and is used as an initial value of a pseudo-spectrum method, the convergence speed of optimal solution calculation is improved, and the high efficiency of calculation is realized.
The technical solution of the invention comprises the following 4 steps:
the first step is to establish an airplane longitudinal track planning problem model containing nonlinear dynamics
The trajectory planning goal is: the height Z, the speed V and the track inclination angle gamma reach given state values at the initial and the tail ends in the longitudinal plane, and the process time is shortest. The aircraft longitudinal dynamics model considered by the invention is as follows:
Figure BDA0003248424130000031
wherein t is time, Z (t) epsilon R is the height of the airplane in the longitudinal plane, X (t) epsilon R is the position of the airplane, V (t) is the speed of the airplane, gamma (t) epsilon R is the track inclination angle of the airplane, g is the gravity acceleration, n is the gravity accelerationz(t) is e.g. R and nx(t) is respectively the normal overload and the longitudinal overload of the airplane,
Figure BDA0003248424130000032
is the airplane control command corresponding to the command; the 5 th and 6 th expressions represent a first-order inertia element, and the parameter knz∈R,knxThe epsilon R is a first-order inertia link parameter of the normal overload instruction and the longitudinal overload instruction respectively, and can be selected according to different airplane characteristics. α ═ G (n)zM, g) is the aircraft angle of attack, is the normal overload nzThe aircraft mass m and the gravitational acceleration g, the specific functional form depends on different aircraft characteristics.
In model (1), (X (t), Z (t), V (t), γ (t), nz(t),nx(t)) is the state quantity,
Figure BDA0003248424130000033
for the control quantity, the optimization index is tf-t0.tfTo end time, t0Is the starting time. Each state control parameter also needs to satisfy the following boundary and process constraints:
and (3) boundary constraint:
Figure BDA0003248424130000034
and (3) process constraint:
Figure BDA0003248424130000035
the second step is that: multi-stage pseudo-spectrometry that is practically operable based on overload reference instructions
The normal overload control command and the longitudinal overload control command need to have practical operability, so the normal overload control command and the longitudinal overload control command are designed to be intermittent constant values, and the intermittent/switching times do not exceed T. The following gives a multi-stage pseudo-spectral method that is actually operable based on the overload reference instruction:
the following notations are first introduced:
Figure BDA0003248424130000041
f (-) is a state equation vector function.
Then models (1) - (3) are equivalently transformed into:
Figure BDA0003248424130000042
wherein t isi(i 1, 2.., T +1) represents the duration of the ith stage,
Figure BDA0003248424130000043
and
Figure BDA0003248424130000044
respectively, the start time and the end time of the ith stage.
The continuous model (4) was further transformed using multi-stage pseudo-spectroscopy as follows:
note N(i)The number of matches specified in the i stage is calculated as follows, wherein the value range of the matches is a positive integer set:
Figure BDA0003248424130000051
i stage distribution point
Figure BDA00032484241300000511
N(i)×N(i)Differential matrix D(i)The calculation formula of the ith row and the jth column element is as follows:
Figure BDA0003248424130000052
N(i)×(N(i)+1) differential matrix
Figure BDA0003248424130000053
The calculation of the following conversion relation is carried out:
Figure BDA0003248424130000054
Figure BDA0003248424130000055
as an approximation function X of the i-th stage state(i)(τ) at point
Figure BDA0003248424130000056
Value of (U)(i)The value of the control quantity of the ith stage is taken,
Figure BDA0003248424130000057
at points of approximation function for track inclination in stage T +1
Figure BDA0003248424130000058
Value of (a), Vi (T+1)At points that are an approximate function of aircraft velocity during phase T +1
Figure BDA0003248424130000059
The value of (a) is selected,
Figure BDA00032484241300000510
is the Nth of the R +1 stage differential matrix(T+1)The ith element of the row.
The state derivative equation for the first T stages is discretized as:
Figure BDA0003248424130000061
the state derivative equation for the T +1 stage is discretized as:
Figure BDA0003248424130000062
boundary conditions without derivatives
Figure BDA0003248424130000063
The dispersion is:
Figure BDA0003248424130000064
note the book
Figure BDA0003248424130000065
Then
Figure BDA0003248424130000066
The dispersion is:
φ′2＝0 (10)
cmin≤c(x,u)≤cmaxthe dispersion is:
Figure BDA0003248424130000067
through equations (5) - (11), the continuous optimal trajectory planning equivalence is transformed into the following discrete form:
min(t1+…+tT+1)
Figure BDA0003248424130000068
the third step: intelligent selection of initial values through neural network learning data sets
After the continuous problems are dispersed by using a multi-stage pseudo-spectrum method, a feedforward neural network is designed to select a proper initial value according to the adjustment parameters of a learning data set:
the learning data set is recorded as: a, which contains m boundary condition values without state derivative constraints
(Z0,Zf,V0,Vf,γ0)kAnd m optimal track samples satisfying corresponding boundary values
Figure BDA0003248424130000071
Designing a neural network with the following structure:
the neural network structure has s layers, and the input is a boundary condition without state derivative constraint
(Z0,Zf,V0,Vf,γ0)kOutput as the corresponding optimal trajectory
Figure BDA0003248424130000072
The iterative formula is as follows:
Figure BDA0003248424130000073
wherein
Figure BDA0003248424130000074
Representing the input from layer i-1 to layer i, M(i)The number of the layer i neurons is,
Figure BDA0003248424130000075
as a function of the activation of the i-th layer,
Figure BDA0003248424130000076
is the output of the i-th layer,
Figure BDA0003248424130000077
is a weight parameter of the i-1 st layer,
Figure BDA0003248424130000078
is the bias term for layer i-1.
Let θ be wi,biI 1.. s }, calculated by minimizing the following loss function[6]：
Figure BDA0003248424130000079
Calculating to obtain the track x under the condition of a given initial and final state by using a BP (Back propagation) neuron network learning method0Recording the initial value trace calculated by the pseudo-spectrum method as x0；
The fourth step: obtaining an optimal solution by solving a nonlinear programming problem
The initial value track x calculated by the third step0As an initial solution for the multi-stage discrete trajectory planning model (12), the optimal trajectory x is calculated by using an iterative algorithm solution of the nonlinear planning*Optimum trajectory x*Corresponding boundary conditions (Z)0,Zf,V0,Vf,γ0)kAnd adding the learning data set A as a learning data set of the next optimization problem.
The invention has the advantages that:
1. aiming at the optimal control problem of the existing singular arc, under the condition that the control input is physically constrained, a multistage pseudo spectrum which can be actually operated based on an overload instruction is designed, so that the multistage pseudo spectrum can be used as an important reference for operating an aircraft in the real life of a pilot.
2. The invention adopts a method of optimizing the initial value by the neural network, designs the neural network with a multilayer structure, trains the neural network by using the learning data set, and intelligently selects the initial value by the neural network so as to lead iteration to avoid infeasible areas, thereby avoiding the problem that the traditional problem is sensitive to the initial value.
3. Aiming at the optimal control problem containing state derivative constraint, the multi-stage pseudo spectrum is converted into a discrete problem, the multi-arc track is divided, the discrete approximation precision is increased, the constraint condition is simplified through a linearized state derivative condition, and the calculation time is reduced.
Drawings
Fig. 1 is a flowchart of an algorithm for selecting an initial value and a multi-stage pseudo spectrum based on a neural network according to the present invention.
Fig. 2 is a graph of the change in aircraft altitude.
Fig. 3 is a graph of the change in aircraft velocity.
Fig. 4 is a change curve of the normal overload control command and the longitudinal overload control command of the airplane.
Fig. 5 is a graph showing the variation of normal overload and longitudinal overload of an airplane.
The symbols are as follows:
t: time;
α ∈ R: angle of attack; beta epsilon R: a sideslip angle; v epsilon R: the speed of the aircraft relative to the air;
z belongs to R: the altitude of the aircraft;
x belongs to R: an aircraft position;
γ ∈ R: aircraft track inclination angle;
g: acceleration of gravity;
nze.g. R: normal overload;
nxe.g. R: longitudinal overload;
Figure BDA0003248424130000091
normal overload control instructions;
Figure BDA0003248424130000092
a longitudinal overload control instruction;
knze.g. R: normal overload instruction first-order inertia link parameters;
knxe.g. R: longitudinal overload instruction first-order inertia link parameters;
t0: starting time;
tf: an end time;
ti: the duration of the ith stage;
Figure BDA0003248424130000093
the starting time of the ith stage;
Figure BDA0003248424130000094
the termination time of the ith stage;
Figure BDA0003248424130000095
kth distribution point of ith stage
N(i)E.g. R: matching points in the ith stage;
PN(x) The method comprises the following steps An Legendre polynomial of order N;
Figure BDA0003248424130000101
the ith stage State at Point
Figure BDA0003248424130000102
Taking the value of (A);
Figure BDA0003248424130000103
control of phase i at point
Figure BDA0003248424130000104
Taking the value of (A);
Figure BDA0003248424130000105
a differential matrix of an i-th stage;
Figure BDA0003248424130000106
a differential matrix of the T +1 th stage;
Figure BDA0003248424130000107
nth of differential matrix of T +1 th stage(T+1)The ith element of the row;
Figure BDA0003248424130000108
taking the ith value of the track inclination angle of the T +1 th stage;
Vi (T+1)the ith value of the speed of the aircraft in the T +1 th stage belongs to the element R;
Y∈Rm×n: a learning data set;
Figure BDA0003248424130000109
an i-1 th layer to an i-th layer;
M(i)e.g. R: number of layer i neurons;
Figure BDA00032484241300001010
activation function of the ith layer;
Figure BDA00032484241300001011
the output of the ith layer;
Figure BDA00032484241300001012
a weight parameter of the i-1 st layer;
Figure BDA00032484241300001013
bias term for the i-1 th layer;
Detailed Description
In order to test the practicability of the method, a simulation experiment is carried out by taking the trajectory plan in a typical deceleration climbing scene as an example, and the calculation time of the problem under the same condition is compared with the calculation time of the GPOPS.
Simulation conditions are as follows:
the number of times of switching of the control command R is set to be 3, and the number of points in each stage is set to be 10, 10, 20 and 10. The initial and terminal conditions are:
Figure BDA0003248424130000111
first order parameter k of inertia elementnz，knxRespectively selecting 0.5 and 6;
the upper and lower limits of the normal overload and the longitudinal overload and the control command corresponding to the normal overload and the longitudinal overload are as follows:
Figure BDA0003248424130000112
the method comprises the following specific implementation steps:
1. and (4) discretizing the shortest time trajectory optimization problem (4) into a result (12) by using the steps (5) to (11).
J＝t1+t2+t3+t4
Figure BDA0003248424130000113
Figure BDA0003248424130000121
Figure BDA0003248424130000122
Figure BDA0003248424130000123
Figure BDA0003248424130000124
Figure BDA0003248424130000131
2. Designing a 6-layer neural network, wherein the number of points of each layer is respectively 4, 50, 50, 20 and 330. The activation functions corresponding to the middle layer are a sigmoid function, a purelin function, a sigmoid function and a hyperbolic tangent S-type transmission function. And inputting the known optimal track sample set and the corresponding boundary conditions into a neural network structure, and training to obtain neural network parameters.
3. Obtaining an initial trajectory x by inputting initial and terminal conditions into a neural network0And calculating by adopting a nonlinear programming iterative algorithm to obtain a corresponding optimal solution.
4. And adding the obtained optimal solution and the boundary condition into a learning data set as a new learning sample.
Fig. 2-5 show simulation results. As can be seen from the graphs in FIGS. 2 and 3, the algorithm provided by the invention can effectively enable the optimal track to reach the boundary condition, and the control instruction curve in FIG. 4 shows that the algorithm provided by the invention can effectively avoid singular arcs, is easy to realize and has actual reference value.
GPOPs algorithm inputs are models (1) - (3), and grid precision is 1e-7The maximum number of grid iterations is 15, the rest settings are the same as the algorithm of the invention, and the program running time and the optimal arrival time are shown in the following table:
algorithm of the invention GPOPs algorithm
Program calculating time 6.938124 seconds 198.178704 seconds
Optimal time of arrival 35.92104 seconds 34.00809 seconds
The above table shows that the algorithm of the invention simplifies the problem form and effectively shortens the program calculation time on the premise of losing the optimality of the minimum trajectory.
Reference documents:
[1] and an inner step of six-degree-of-freedom motion trajectory optimization research [ D ] based on an hp self-adaptive pseudo-spectral method, Nanjing aerospace university, 2020.
[2] Autonomous parking path planning [ J ] based on hp adaptive pseudo-spectral method, mechanical engineering journal 2020,56(04): 125-.
[3] The research on the closed-loop guidance of the ascending section of the hypersonic aircraft based on the hp-adaptive pseudo-spectral method [ J ] computer measurement and control, 2019,27(01): 136-charge 140+145.
[4]Shen H X，Casalino L，Luo Y Z.Global search capabilities of indirect methods for impulsive transfers[J].Journal of the Astronautical Sciences，2015，62(3):212-232
[5] Fourier yog, Roc, Runge, reentry vehicle trajectory optimization based on sequence gradient-repair algorithm [ J ]. ballistic declaration, 2012,24(03):36-42.
[6] Guohiting, log peak, structural optimization design based on BP neural network and genetic algorithm [ D ]. Nanjing university of aerospace, 2003.

Claims (4)

1. A multi-stage pseudo-spectral method for intelligently selecting initial values of aircraft longitudinal optimal path planning is characterized by comprising the following steps of:
establishing an airplane longitudinal track planning problem model containing nonlinear dynamics;
the trajectory planning goal is: the height Z, the speed V and the track inclination angle gamma reach given state values at the initial and the tail ends in a longitudinal plane, and the process time is shortest; the aircraft longitudinal dynamics model considered is as follows:
Figure FDA0003248424120000011
wherein t is time, Z (t) epsilon R is the height of the airplane in the longitudinal plane, X (t) epsilon R is the position of the airplane, V (t) is the speed of the airplane, gamma (t) epsilon R is the track inclination angle of the airplane, g is the gravity acceleration, n is the gravity accelerationz(t) is e.g. R and nx(t) is respectively the normal overload and the longitudinal overload of the airplane,
Figure FDA0003248424120000012
is the airplane control command corresponding to the command; the 5 th and 6 th expressions represent a first-order inertia element, and the parameter knz∈R,knxSelecting first-order inertial link parameters of the normal overload instruction and the longitudinal overload instruction according to different airplane characteristics by using the epsilon R respectively; α ═ G (n)zM, g) is the aircraft angle of attack, is the normal overload nzThe mass m and the gravitational acceleration g of the aircraft, the specific function form depending on different aircraft characteristics;
in model (1), (X (t), Z (t), V (t), γ (t), nz(t),nx(t)) is the state quantity,
Figure FDA0003248424120000013
for the control quantity, the optimization index is tf-t0，tfTo end time, t0Is the starting time; each state control parameter also needs to satisfy the following boundary and process constraints:
and (3) boundary constraint:
Figure FDA0003248424120000021
and (3) process constraint:
Figure FDA0003248424120000022
step two: a multi-stage pseudo-spectrometry that is actually operational based on the overload reference instruction;
the normal overload control instruction and the longitudinal overload control instruction need to have actual operability, so the normal overload control instruction and the longitudinal overload control instruction are designed to be intermittent constant values, and the intermittent/switching times do not exceed T; the following gives a multi-stage pseudo-spectral method that is actually operable based on the overload reference instruction:
the following notations are first introduced:
Figure FDA0003248424120000023
f (-) is a state equation vector function;
then models (1) - (3) are equivalently transformed into:
Figure FDA0003248424120000031
wherein t isi(i 1, 2.., T +1) represents the duration of the ith stage,
Figure FDA0003248424120000032
and
Figure FDA0003248424120000033
respectively the starting time and the ending time of the ith stage;
the continuous model (4) was further transformed using multi-stage pseudo-spectroscopy as follows:
note N(i)The number of matches specified in the i stage is calculated as follows, wherein the value range of the matches is a positive integer set:
Figure FDA0003248424120000034
i stage distribution point
Figure FDA0003248424120000035
N(i)×N(i)Differential matrix D(i)The calculation model of the ith row and jth column element is as follows:
Figure FDA0003248424120000036
N(i)×(N(i)+1) differential matrix
Figure FDA0003248424120000037
The calculation of the following conversion relation is carried out:
Figure FDA0003248424120000041
Figure FDA0003248424120000042
as an approximation function X of the i-th stage state(i)(τ) at point
Figure FDA0003248424120000043
Value of (U)(i)The value of the control quantity of the ith stage is taken,
Figure FDA0003248424120000044
at points of approximation function for track inclination in stage T +1
Figure FDA0003248424120000045
Value of (a), Vi (T+1)At points that are an approximate function of aircraft velocity during phase T +1
Figure FDA0003248424120000046
The value of (a) is selected,
Figure FDA0003248424120000047
is the Nth of the R +1 stage differential matrix(T+1)The ith element of the row;
the state derivative equation for the first T stages is discretized as:
Figure FDA0003248424120000048
the state derivative equation for the T +1 stage is discretized as:
Figure FDA0003248424120000049
boundary conditions without derivatives
Figure FDA00032484241200000410
The dispersion is:
Figure FDA00032484241200000411
note the book
Figure FDA00032484241200000412
Then
Figure FDA00032484241200000413
The dispersion is:
φ′2＝0 (10)
cmin≤c(x,u)≤cmaxthe dispersion is:
Figure FDA00032484241200000414
and (5) equivalently converting the continuous optimal trajectory planning into the following discrete form through models (5) to (11):
min(t1+…+tT+1)
Figure FDA0003248424120000051
step three: intelligent selection of initial values through neural network learning data sets
After the continuous problems are dispersed by using a multi-stage pseudo-spectrum method, a feedforward neural network is designed to select a proper initial value according to the adjustment parameters of a learning data set:
the learning data set is recorded as: a, which contains m boundary condition values (Z) without state derivative constraints0,Zf,V0,Vf,γ0)kAnd m optimal track samples satisfying corresponding boundary values
Figure FDA0003248424120000052
Figure FDA0003248424120000053
Designing a neural network with the following structure:
the neural network structure has s layers, and the input is a boundary condition without state derivative constraint
(Z0,Zf,V0,Vf,γ0)kOutput as the corresponding optimal trajectory
Figure FDA0003248424120000054
The iterative model is as follows:
Figure FDA0003248424120000055
wherein
Figure FDA0003248424120000061
Representing the input from layer i-1 to layer i, M(i)The number of the layer i neurons is,
Figure FDA0003248424120000062
as a function of the activation of the i-th layer,
Figure FDA0003248424120000063
is the output of the i-th layer,
Figure FDA0003248424120000064
is a weight parameter of the i-1 st layer,
Figure FDA0003248424120000065
is the bias term for layer i-1;
let θ be wi,bi1., s }, calculated by minimizing the loss function:
Figure FDA0003248424120000066
calculating to obtain the track x under the condition of a given initial and final state by using a BP (Back propagation) neuron network learning method0The initial value track calculated by the pseudo-spectrum method is recorded asx0；
Step four: obtaining an optimal solution by solving a nonlinear programming problem;
the initial value track x calculated by the third step0As an initial solution of the multi-stage discrete trajectory planning model (9), the optimal trajectory x is calculated by using an iterative algorithm of nonlinear planning to solve*Optimum trajectory x*Corresponding boundary conditions (Z)0,Zf,V0,Vf,γ0)kAnd adding the learning data set A as a learning data set of the next optimization problem.
2. The method of claim 1, wherein the method comprises the following steps: setting the switching frequency R of the control instruction as 3, and the point number of each stage as 10, 10, 20 and 10; the initial and terminal conditions are:
Figure FDA0003248424120000067
3. the method of claim 1, wherein the method comprises the following steps: first order parameter k of inertia elementnz，knx0.5 and 6 were chosen, respectively.
4. The method of claim 1, wherein the method comprises the following steps: the upper and lower limits of the normal overload and the longitudinal overload and the control command corresponding to the normal overload and the longitudinal overload are as follows:
Figure FDA0003248424120000071
CN202111039165.6A 2021-09-06 2021-09-06 Multi-stage pseudo-spectrum method for intelligently selecting initial values for planning longitudinal optimal paths of airplanes Active CN113778117B (en)

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