数学建模 飞机管理模型

Air Traffic Control

Abstract

本文讨论了如何对新进入区域内的飞机是否会与区域内原有的飞机碰撞的问题，和如若碰撞，调整各架飞机方向角，使得飞机均能安全飞出正方形区域的问题。

针对判断飞行的飞机之间是否发生碰撞亦即可碰撞问题，把各架飞机的运动轨迹端点坐标用时间表示后，就可以写出第六架飞机与其它五架飞机的距离表达式，判断这个最小距离是否小于8km，如果小于，则碰撞，否则不碰撞。这一过程的实现可通过MATLAB编程动态模拟飞机在区域内的飞行过程。在时间轴上连续取样，最后得出第六架飞机和第五架相碰，碰撞时事件飞机的坐标位（），相碰时刻为？

针对检测到碰撞的存在并采取措施进行规避，即碰撞规避问题。需要对各架飞机的方向角进行调整，并且使得飞机方向角调整幅度最小。建立非线性规划模型，利用MATLAB求得第i架飞机)6,5,4,3,2,1

i调整的角度分别为：

(

关键字:飞机碰撞方向角最优解非线性规划

Contents

I. Introduction

Grow inside 160 kilometers of exact square districts in the about 10,000 meters high empty some side, usually how many the airplane make level flight.Position and speed vector of each airplane inside the district are recorded its data by the calculator, so that they carry on a flight management.When desire gets into the airplane of the district to arrive a district edge, after recording its data, immediately compute and judge whether meeting and airplane occurrence in the district collision.If will collide, then should compute how to adjust each(including is lately ingoing) direction Cape that the airplane flies.To avoid collision.Now suppose a condition as follows:

1)The standard that don't collide is more than 8 kilometers for the distance of arbitrarily two airplanes.

2)The airplane flies the range that the direction Cape adjusts to be higher than 30 degrees.

3)All airplane airspeeds are all per hours are 800 kilometers.

4)The airplane that gets into the district while arriving a district edge, with the distance of airplane inside the district in response to above 60 kilometers.

5)At most need to consider 6 airplanes.

6)Need not consider that the airplane leaves the condition of this empress in the district.

Please to the problem establishment mathematics model of the flight management that avoid collision.List to compute a step, carry on a calculation to the following data.(direction Cape the error margin isn't higher than 0.01 degrees)Requesting airplane to fly the range that the direction Cape adjusts is as far as possible small.

The coordinates that establishes the district's 4 tops (0,0),(160,0) , (160,160) , (0,160)

Airplane serial number Abscissa x Ordinate y

Direction

Cape

（degree）

1 150 140 243

2 85 85 236

3 150 155 220.5

4 14

5 50 159

5 130 150 230 New 0 0 52

Note:The direction Cape points to fly direction and X stalk just to of clip Cape. Try to carry on evaluation and expansion to your model according to the actual application background.

II. The Description of the Problem

2.1The analysis of the problem background

在一个确切的广场区，每架飞机的位置和速度矢量飞机到达区的边缘.其数据记录，计算器立即计算和判断飞机发生碰撞的区域.如果发生碰撞然后计算如何调整各方向开普敦的飞机飞以避免碰撞，.程序如下：

In an exact square district, position and speed vector of each airplane are recorded its data by the calculator.When the airplane arrives the edge of district, the calculator immediately computes and judge whether meeting and airplane occurrence in the district collision.If the occurrence collides and computes then how adjust each direction Cape that the airplane flies to avoid collision.Process such as figure a show:

access

leave

2.2 on the analysis to problem 1

由于模型假设飞机是在同一高度飞行，故可以认为飞机是在同一水平面上，这样只需讨论二维平面管理飞机不相碰撞的管理和调节问题。

首先要判断各架飞机之间是否会碰撞。可以在二维坐标上找出各个飞机的点坐标，以此确定飞机的相对位置。图1中虚线框代表整个正方形区域。左下角处为刚进入区域边缘的第6架。再可由题目中给出的飞机的坐标和飞行角度，借助MATLAB 编程及图像输出的方法确定初始运行轨道，由此可写出在t 时刻各个飞机

Square area

airplane Calculate and

determine

Adjustment flight path

的位置坐标（第六架飞机进入区域的时刻为初始时刻）。这样也可以算出在t 时刻第六架飞机与其他五架飞机的距离。如果存在某一时刻使得这五个距离中一个小于等于8公里，则就碰撞，需要进行具体角度调整措施的讨论，参考问题二。

Since the model is assumed the aircraft flying at the same altitude, the aircraft can be considered at the same level, so that only two-dimensional plane management discussion aircraft does not collide with the management and regulation problem.

We must first determine whether the collision between aircraft. Locate in the two-dimensional coordinates of each point of the coordinate plane, in order to determine the relative position of the aircraft. Dashed box in Figure 1 represents the entire square area. Just enter the area to the bottom left corner edge of the first six. Then be given the title of the coordinate plane and flying angle, and image output using MATLAB programming method to determine the initial orbit, which can be written at the time of each aircraft's position coordinates (sixth aircraft into the area as the initial moment time). This also can be calculated at the time of the sixth aircraft and other five aircraft in the distance. If there is a time makes it five in a distance of less than or equal 8 km, then in the collision, the need for the discussion of specific angle adjustment measures, refer to question two.

2.3 on the analysis to problem 2

在问题一的基础上，若新进入的飞机会与区域内的飞机碰撞，则需要调整各架飞机的方向角。可以假定各个飞机方向角的调整幅度为i θ(),...3,2,1=i ，各个飞机初始时刻坐标不变，飞行角度变为i i θα+，同样可以把调整后t 时刻各个飞机的位置表示出来。各个飞机的距离ij d 需满足大于8公里，并且对任意范围内的t 均成立。一定的约束条件下求出使得∑==6

12

i i f θ取最小值时的各个i θ的最优解。

In issue one, based on if the new aircraft will enter the area of the aircraft collision with, you need to adjust the angle of the direction of the aircraft. Individual aircraft can be assumed adjustment of the steering angle, the initial time coordinate

invariant individual aircraft, flying angle changes, the same can be adjusted each time the aircraft's position represented. The distance of each aircraft must meet more than eight kilometers, and are within the range for any establishment. Certain constraints such that the minimum value determined under the conditions of the individual's optimal solution.

III.model assumption and agreed

1．最多考虑同时6架飞机，不考虑飞机离开此区域后的情况。

2．假设每架飞机在正方形区域内飞行时没有任何干扰因素导致的故障。3．每架飞机在正方形区域内，飞机可作为质点来处理。

4．每架飞机调整的时间忽略，可调控，瞬时性。

5．每架飞机仅调整一次，且从飞机进入区域边缘的那一刻进行调整。

6．每架飞机均沿直线飞行，且所有飞机均在同一平面内飞行，飞行速度均为800km/h。

1.Up to consider while six aircraft, the aircraft leave the area without considering the situation after.

2.Assuming that each aircraft flying within the square area without any disturbance factors led to the failure.

3.Each aircraft in a square area, the aircraft can be handled as a particle.

4.Time to adjust each aircraft ignored, can be regulated, transient.

5.Adjusted only once per aircraft and from the aircraft into the edge of the region at the moment to adjust.

6.Each aircraft were flying along a straight line, and all aircraft are flying in the same plane, flight speed are 800km / h

IV. the sign explain

T第i架飞机和第j架飞机都在区的时间

ij

T第i架飞机飞出区域的时刻

i

T第j架飞机飞出区域的时刻

j

i θ

各飞机的最小调整角度

t 时刻

f 目标函数

ij T

The i and j aircraft aircraft are in the

district of the time

i T Regional aircraft flying out of the i-th

moment

j T

Regional aircraft flying out of the j-th

moment

i θ

Adjust the angle of the smallest aircraft

t

time f

The objective function

V.The establishment of model

5.1.1 The Foundation of Model 1

判断当一架欲进入区域的飞机到达区域边缘时是否会与区域内的飞机发生碰撞。

（1）不碰撞条件

()064)(2

≥-=t t f r ij ij （0 ≤ t ≤ij

T ）

其中：()222

)()

(y y x x r t

j

t i

t j t i

ij

t -+-=

（2） 两架飞机的距离（平方）

?????+=+=i

i

t i i i t i vt vt y y x x θθsin cos 00

（3） 时刻t 飞机的位置

},min{j i ij T T T =

不必考虑在区域外的碰撞 ,ij T 指的是第i 架飞机和第j 架飞机都在区的时间，

i T 为第i 架飞机飞出区域的时刻，j T 为第j 架飞机飞出区域的时刻。

??

?

???????=+=+>=1600)sin(1600)cos(:0min arg 0

0或或者，或i i i i i vt vt t T y x θθ 具体解释为：

整理：

()ij ij ij ij ij c z b z t f ++=2

其中：

2

sin

2j i ij

vt z θθ-=

]2

cos

)(2

sin

)]([20

00

j

i j

i

j

i j i ij y y x x b θθθθ+-++--=

64)()(20

20

--+-=y y x x j

i

j i ij c

5.1.2 模型求解：

???

???

????

???-≥-<≤≥≤<-≤<≤-≤-≤<--≥-<≤--≥≤<--≤-<<--≤<≤-=0

0000

00

0000

00000

0000tan ,223tan ,23,sin ,

tan ,23tan ,2,cos ,tan ,2tan ,20,sin ,tan ,223tan ,20,cos i i i i i i i i i i i i i

i i i i i i i i

i i i i i i i i i i i i i i i i i i i i x D y or x y if v y x y or x y D if v x x y D or x D y D if v x D x D y or x D y D if v x D T θπθπθπθπθθπθπθπθπθθπθπθπθθθπθπ

θπθθ

通过求解得到图1：

图1 飞机飞行线路图Ti为第i架飞机飞出区域的时刻，分别为：单位（s）

碰撞时刻的相关数据记录如下：

发生碰撞的事件飞机：5号与6号

碰撞时地(t =ptime)各架飞机间距（如下表）：

表1：碰撞时各架飞机间距

飞机编号

1 2 3 4 5 6 1 2 3

4

5

6

碰撞时事件飞机的坐标位置

所以新进入的飞机会与区域内的飞机碰撞，即需要调整各架飞机的方向角。

5.2.1 The Foundation of Model 2

如果会碰撞，计算如何调整各架(包括新进入的)飞机飞行方向角，以避免碰撞，并且各架飞机调整的角度尽可能小。

该问题是一个求最优解的问题，在避免飞机碰撞的前提下，求各飞机的最小调整角度i θ。设各飞机飞行方向的调整角度为约束变量，建立模型，目标函数为：

Min ∑==6

1i i f θ

由于i θ的值的区间为（-30°，30°），如果直接求和正负会相互抵消，故建立新的目标函数：

Min ∑==6

12

i i f θ

约束条件：

（1）按调整后的角度飞行，任意两架飞机在区域内的距离大于8公里，即

()()()[]()()()

[]

2

2

2

sin sin cos cos j j j i i i j j j i i i ij vt y vt y vt x vt x d θαθαθαθα++-+++++-++= 64> ()6,5,4,3,2,1,=

（2）实际要求飞机飞行最大调整角度i θ满足：-30°＜i θ＜30°。 故飞机飞行角度调整最优解模型如下： Min ∑==6

12i i f θ

()()()[]()()()[]

64

2sin sin cos cos 2

2

>+

=++-++++-++j

j j i i i

j

j j i i i ij d vt y vt y vt x vt x θαθαθαθα (1)

s.t. ()6,5,4,3,2,1,=

(2) -30°＜j θ＜30°

(3)

对上述不等式，可以采用数学方法和编程两种不同方法化简或求解。 数学方法化简： 对其求微分，

()()()ij ij ij ij ij ij

ij

vs vts y vc vtc x dt

d d +++=222

令其等于0，得

(

)

2

2ij

ij ij ij ij ij s c v s y c x t ++-

=

则可表示出

()()

()

2

222

2

22

2

m in ij

ij

ij

ij ij ij

ij ij

ij ij ij

ij s c c s x c y s c y s

x d +---=

64>

其中：

j i ij y x x -= j i ij y y y -=

()()j j i i ij c θαθα+-+=cos cos ()()j j i i ij s θαθα+-+=sin sin

这里采用编程化简的方法求解：这是一个非线性规划问题，借助MATLAB 的fimincon()函数编程计算出结果。

5.2.2 模型的求解

由matlab程序得出最优解，结果为

5.3 Improved Model

VI.The evaluation of model

6.1 Strength:

1.对题目中涉及到的众多影响因素进行了一般性分析，使得论文有说服力；

2.把飞机作为质点处理，因为飞机大小相对碰撞距离8公里较小，可忽略，这

样使得题目模型相对简单；

3.用MATLAB软件模拟出飞机飞行的线路，直观，一目了然。

1.Involved in the topic of the many factors affecting the general analysis, making the thesis convincing;

2.The aircraft as particle processing, because the aircraft size is small relative collision distance 8 km, can be ignored, which makes the topic model is relatively simple;

https://www.360docs.net/doc/028001249.html,ing MATLAB software to simulate the aircraft flight line, intuitive glance.

6.2 Weakness:

1.模型是在理想条件下进行的，不考虑天气、风向的影响，且设定飞行员接受

命令到执行的时间为零，飞行调整的角度为绝对精确，因此可能与实际情况有一定差距；

2.考虑飞机是在同一平面内飞行，未考虑飞机间的垂直距离

1.Model is carried out under ideal conditions, does not consider the weather, wind effects, and configured to accept commands to perform the pilot at time zero, adjust the angle of flight absolute precision and therefore may have a gap with the actual situation;

2. Consider the aircraft is flying in the same plane, without considering the vertical distance between aircraft

VII.Conclusions

7.1 Conclusions of the problem

7.2 Methods used in our models

VIII. References

[1]张兆宁，张晓燕，交叉航路碰撞风险研究，航空计算技术报，第37卷，第二期，2007年三月，第1～4页。

[2]姜启源、谢金星、叶俊，数学模型第三版, 北京；高等教育出版社，2003。

[3]黄忠裕，初等数学建模, 成都：四川大学出版社，2004.12。

[4]应爱玲、徐肖豪, 空域飞行侧向碰撞危险的Reich，模型方法研究中国民航学院学报，第20卷第四期，第6～10页。

[5]宫文娟，利用代数方法求解碰撞检测问题，山东大学硕士学位论文，绪论。