Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 8, No. 3, 2023 

 

139 

Research	on	UAV	Fixed-point	Delivery	Problem	Based	
on	Aerodynamics 

Lulu Wang* 

School of Mechanical and Electrical Engineering, Sanjiang University, Nanjing, 210012, China 
* Corresponding author 

 

Abstract: Aiming at the problem of UAV fixed-point delivery. In this paper, the air resistance model in the aerodynamic model 
is combined with Newton's second law to construct an efficient solution to the ball projection problem at low speed. Firstly, the 
motion of the ball is decomposed into two directions: horizontal and vertical. By simplifying the model, the horizontal flight 
distance of the ball under three wind directions is obtained, and then the delivery distance of the aircraft is obtained. Secondly, 
considering the function of UAV to perform the blasting task, the aircraft used a certain Angle of dive, and subject to the distance 
constraints, followed by the dynamic model. Finally, through numerical iteration, the dive Angle of the aircraft under the wind 
direction and the straight-line distance between the position of the aircraft when it is launched and the target were obtained. 
Under the influence of different wind direction and wind speed, the aerodynamic model that can calculate the position and dive 
Angle of the UAV when it is launched at fixed point is obtained according to the direction and size of the influencing factors. 

Keywords: Aerodynamic Model, Air Resistance Model, Newton Second Law. 
 

1. Introduction 
With the continuous development of science and 

technology, many material delivery tasks are gradually 
carried out by Uavs instead of humans. 

Uavs have also become important data acquisition 
equipment in anti-terrorism and disaster relief operations, and 
have shown unique potential and performance advantages[1] 
in fire protection, environmental investigation and earthquake 
rescue in deep forests. The most important function of 
unmanned aerial vehicles (Uavs) performing fixed-point 
delivery tasks in the air is nothing more than the delivery 
accuracy of stability in the execution of the task. The delivery 
accuracy not only depends on the operation and flight 
technology of the UAV, but also is related to the state and 
environment of the UAV during the execution of the task, 
such as the height and speed of the UAV when approaching 
the delivery point. The wind speed at the location of the UAV, 
the geographical environment around the delivery point and 
other factors are all decision variables for whether the UAV 
can hit the target stably and accurately. Its motion state is 
decomposed to analyze its speed in all directions, the dynamic 
model is established and the characteristics are analyzed, and 
the external disturbance and model structure uncertainty are 
considered to meet the performance requirements of the flight 
control system[2]. 

It is of great significance to further solve the flight time and 
distance of the ball for the UAV fixed-point delivery problem. 

2. Materials and Methods 
This paper adopted the May 2023 A topic mathematical 

contest in modeling 
(https://51mcm.cumt.edu.cn/d8/19/c14143a645145/page.htm) 
is analyzed and studied. 

3. Model Establishment and Solution 

3.1. Basic model of UAV flight attitude 
3.1.1. Establishment of model 

It is assumed that the aircraft flies normally and is not 
restricted by the wind during the whole process of delivery. 
The spherical materials it places are affected by the air 
resistance and the wind during the falling process, and the 
materials have velocities 𝑣   in three directions and 
displacements 𝐿  in each direction. 

According to the above assumptions, for this problem, only 
the conditions required for the spherical materials to fall to 
the specified position after throwing are analyzed. See Figure 
1 for details. 

 

 
Figure 1. Motion analysis diagram of spherical materials 

placed horizontally 
 
Under the conditions of three known wind directions and 

wind speeds, the additional speed of the wind will 
displacement the sphere in a corresponding direction, so the 
final delivery distance is synthesized by the displacement of 
the sphere in each direction. 

Firstly, the air resistance [3] of the sphere in the air is 
calculated. The air resistance consists of two parts: one is the 



 

140 

resistance 𝐹  of the windward side right in front of the sphere, 
and the other is the pull resistance caused by the vortex behind 
the sphere. In this problem, the influence of eddy current and 
turbulence on the motion of the small ball is ignored through 
problem simplification, and only the resistance in front of the 
ball is considered. See Figure 2 for details. 

 

 
Figure 2. The wind resistance of the sphere in the air 

 
Since the sphere is different from ordinary flat objects, the 

surface of the sphere is continuously changing, so in order to 
determine the resistance in front of it, the integral is used to 
calculate it. 

The first step is to decompose the first half of the sphere in 
Figure 2 into a combination of many rings. Set on each ring 
with central Angle 𝛼 and opening Angle 𝑑𝛼, and the area of 
the rings is 2𝜋𝑟 sin 𝛼 𝑑𝛼. 

According to the literature, when the air flow on the 
upwind surface encounters the ring with the center Angle 𝛼 
as the position, the velocity of the air flow is 𝑣 sin𝛼 . 
According to the kinetic energy theorem, the following 
formula can be obtained: 

 

 

2

2 2

2

222

0

2 2 32

0

1
sin

2 1
1 42 sin cos sin
2

sin cos

f

A r

F A v

F v A
Fv t r d v t v

F r v d







 


     

   

 

 
  

      









 (1) 

 
𝐹   is the air resistance of a spherical object of radius 𝑟 

when its velocity is 𝑣. 
Next, the influence of wind 𝑣 on the delivery process of 

spherical materials under different wind directions is 
analyzed separately: 
① In the case of downwind, that is, the direction of the 

wind speed is consistent with the direction of the aircraft 
projecting the sphere, because the wind speed is slow. In the 
above simplified model, the initial wind speed at the moment 
and the initial speed of the ball have been superimposed, and 
the resistance of the ball is mainly calculated by the relative 
speed of the ball and the air. 

Next, we just need to figure out 𝑡 how long it takes for the 
ball to reach the ground if it is in free fall and under the 
influence of wind resistance. 

According to the influence of wind speed on the sphere 
under different wind direction, the model is established 
respectively, and the projection distance in three cases is 
solved. 

0

2 2

2 222
1

2 2 22

1

0

, ( , )

0

21

2
1 , ( , )
2

, ( )

( )

x wind

f

f

x
x x

x y
f

y wind

x y

v v v Tailwind Headwind

mg ma F

F
a g

m
H

tH at a

L H L Tailwind HeadwindL v t a t

L H L L VerticalWind
F

a
m

v v
VerticalWind

v v

L v t

 
   


 
        
 

   
  
  







(2) 

3.1.2. Solution of the model 
The relationship between air resistance coefficient, gravity 

acceleration and velocity was determined by the cross-
sectional area of the ball, and then the dynamic model and 
Newton's second law were used to iterate. 

Finally, it is concluded that the landing time of the ball is 
𝑡 8.129 seconds. At this point, the movement of the ball 
stops, and the horizontal movement distance of the ball under 
the influence of the wind speed can be calculated by the 
determined dynamic model. 
①  In the case of downwind, the initial velocity of the 

projection of the sphere is known as: 

0 / 83.3 /
3.6x

v
v m s m s                (3) 

 
The wind speed 𝑣 5𝑚/𝑠  is known, so it is 

superimposed on the velocity of the sphere in the same 
direction: 

' 88.3 /x x wv v v m s               (4) 

 
The synthetic original velocity 𝑣,  and air resistance model 

are iterated under the constraint of time 𝑡 8.129𝑠. The final 
result is obtained 𝐿 571.3011𝑚. 

According to the distance synthesis calculation formula 
𝐿 𝐻 𝐿   when the wind is downwind, the delivery 
distance of the aircraft is obtained 𝐿 645.279𝑚. 
②  In the case of upwind, the initial velocity and the 

upwind wind speed of the sphere are known. The original 
velocity of the sphere is obtained as follows. 

 

'' 78.3 /x x wv v v m s                (5) 

 
The synthetic velocity 𝑣,,  and air resistance model are 

iterated under the constraints of time 𝑡 . The final result is 
obtained 𝐿 517.8954𝑚. 

According to the distance synthesis calculation formula 
𝐿 𝐻 𝐿   when the wind is upwind, the aircraft's 
delivery distance when the wind is upwind is obtained 𝐿
598.5112𝑚. 

③ In the case of vertical wind, the initial velocity and 
vertical wind speed of the sphere are known. The original 
velocity of the sphere is obtained as follows. 

 

''' 83.3 /

5 /
x

y

v m s

v m s


 

                    (6) 

 
The same can be obtained 𝐿𝑥 544.8736𝑚； 𝐿

40.645𝑚; . 



 

141 

According to the distance synthesis formula 𝐿
𝐻 𝐿 𝐿  in vertical wind, the landing distance of the 

aircraft in vertical wind is obtained 𝐿 623.3294𝑚. 

3.2. Basic model adjustment of UAV flight 
attitude considering blasting delivery 
function 

3.2.1. Establishment of model 
In order to achieve higher accuracy than the horizontal 

airdrop method, the dive airdrop method is proposed: firstly, 
the fixed-wing aircraft approaches the target from a large 
height, and then dives to it at a certain Angle while aiming. 
After releasing the projectile, the fixed-wing aircraft pulls up 
its fuselage and flies away, and the projectile falls to the 
target[5] in a nearly straight trajectory. 

It is assumed that the final launch position and dive Angle 
of the UAV are determined under the precondition that the 
explosives finally hit the target. In addition, the effects of 
gravity acceleration, wind speed and air resistance on the 
aircraft and explosives are ignored in the process of UAV 
diving. Only the effects[6][7] of gravity acceleration, wind 
speed and air resistance after the launch of explosives are 
considered. 

 

 
Figure 3. UAV bombing motion model diagram 

 
Analyze the conditions required for a spherical explosive 

to fall to a specified location after launch. See Figure 3 for 
details. 

In Question II, we continue to use the influence of the three 
wind directions in Question I. According to the air resistance 
model in problem 1, the horizontal delivery is changed to the 
target delivery at a certain inclination Angle 𝜃. At this time, 
the speed of the projectile 𝑣 is the sum of the speed of the 
UAV 𝑣  and the launching speed 𝑣 . 

To study the whole process from the time the explosive is 
launched to the time it hits the target, we need to decompose 
this velocity 𝑣 at a certain Angle 𝜃 into a horizontal velocity 
𝑣  and a vertical velocity 𝑣 . The relationship between the 
two and the total velocity 𝑣 is as follows: 

 

 
cos

sin
x

z

v v

v v





 

                 (7) 

 
The initial values of the movement state of the spherical 

explosive in all directions at the moment of launching are 
obtained. 

With the help of the air resistance model in Question 1, the 
following conclusions are drawn: 

 21

4fF v A                 (8) 

 
The aircraft starts to land from the cruising altitude and 

always maintains a constant speed, that is, it always maintains 
a force balance at this stage. Because the parameters of some 
factors that affect the lift and drag of the aircraft change 
constantly during the descent process, this paper fits the 
change curve and takes the optimal value to change it into a 
fixed value for building the model[8]. 

First of all, we assume that the plane has dived to the 
altitude 𝐻   when dropping the bomb, and the ball 
explosive reaches the hit point after a horizontal distance 
𝐿   after launching. Secondly, in order to determine the 
final forward distance of the explosive in the horizontal 
direction, it is necessary to obtain the time 𝑡  consumed 
by the explosive to fall on the ground in advance, and 
decompose the entire process of projecting the explosive into 
a vertical linear motion with variable acceleration and a 
horizontal linear motion with variable deceleration. These 
two motions are solved separately, and the distances of the 
two motions are finally synthesized. The specific relationship 
is as follows: 

 

22

2

2
1

1

' 2 22

''

'''

2

1

2

1

2

, ( )

, ( )

, (

f

f z z bow
down

bow z down down

level x down down

f

x x wind bow level

x x wind

y wind

x x

mg ma F

F v v aH
a g t

m a

H v t at

L v t a t

F
a

m

v v v Tailwind L H L

v v v Headwind

v v
VerticalWin

v v


 

       

  
  

 


    
 




''' 2
1

1
)

2level x down down y downd L v t a t v t





















   


 (9) 

 
With the final fall time of the spherical explosive, we can 

figure out how far the explosive will eventually travel in the 
horizontal direction. The horizontal motion of the spherical 
explosive has three different states of motion according to 
three wind directions. In the case of vertical wind, the wind 
speed is perpendicular to the horizontal direction of the 
explosive and provides a lateral offset speed. At this moment, 
the motion trajectory of the spherical explosive is changed 
from two-dimensional to three-dimensional. It is assumed 
that the horizontal displacement of the spherical explosive 
driven by the wind speed is not affected by the air resistance. 

3.2.2. Solution of the model 
Within the computational framework of the above model, 

the constraints and decision variables given in the title have 
been replaced into the model, and the lift coefficient and air 
density have been regarded as unchanged[9] at the moment, 
and the launch strategy [10]that the UAV needs to execute 
under the constraints has been obtained. 

The radius 𝑟 0.08𝑚  and mass 𝑚 5𝑘𝑔  of the 
spherical explosive are known. And the flying speed of the 
UAV is 𝑣 300𝑘𝑚/ℎ , the launching speed is 𝑣
600𝑘𝑚/ℎ, and the total speed is 𝑣 𝑣 𝑣 900𝑘𝑚/ℎ. 

It is known that the position of the aircraft before starting 



 

142 

the dive motion is the vertical distance 𝐻 800𝑚 
from the ground and the horizontal distance 𝐿
10000𝑚 from the target. And there are constraints: 

 

 

300

1000 3000

6 /

bow

wind

H m

m L m

v m s


  
 

           (10) 

 
To satisfy this constraint, the maximum descent height 

𝐻 𝐻 𝐻 500𝑚  of the UAV during the 
dive; The diving Angle of UAV in the process of diving is 
expressed as follows: 

 

 
max

2 22
max

arcsin
original

H

H L L
 

 
         (11) 

 
The flight time of the ball explosive in the air: 
 

 
22 2z z bow

down

v v aH
t

a

  
          (12) 

 
Given the time it takes for a ball explosive to hit the ground 

for the two dive angles, you only need to find the horizontal 
distance it travels for the three wind directions. 
① In the case of downwind, the horizontal velocity of the 

explosive is: 

 '
x wind xv v v                 (13) 

 
Therefore, according to the above model substitution 

operation: 

 ' 2
1

1

2level x down downL v t a t            (14) 

 
And finally obtain the distance between the UAV and the 

target point when the explosives are launched: 
 

 2 22
bow levelL H L                (15) 

 
In order to make the explosive hit the target, the launch 

position of the aircraft 1000𝑚 𝐿 3000𝑚  and 𝐻
300𝑚. Then the matlab numerical simulation solution is used 
to finally determine the UAV in the wind at the height from 

the ground 500.001138m , at the horizontal distance 
1000.726019𝑚  from the target, as the dive Angle 𝜃
26.548474° of the launch strategy. At this time, the straight-
line distance 𝐿 1118.6834𝑚  between the UAV and the 
target point meets the constraints. 
②In the case of headwind, the horizontal velocity of the 

explosive is: 

 ''
x x windv v v                (16) 

 
Substitute the model in ① as follows. 
 

 

' 2
1

2 22

1

2level x down down

bow level

L v t a t

L H L

  

  

           (17) 

 
Similarly, it can be obtained that the height 500.001138𝑚 

of the UAV from the ground and the distance 
1000.686864𝑚  from the target horizontal plane under the 
headwind condition are considered 26.549370°  as the 
launching strategy of the dive Angle. At this time, the straight-
line distance 𝐿 1118.6484𝑚  between the UAV and the 
target point meets the constraints. 
③ In the case of vertical wind, the horizontal speed of the 

explosive is: 

 

'''
x z

y wind

v v

v v

 
 

                 (18) 

 
According to the above model, the trajectory of the 

spherical explosive at this time is a three-dimensional graph. 
 

 
2

2' 22
1

2 22

1

2level x down down y down

bow level

L v t a t v t

L H L

       
 


 

 (19) 

 
Similarly, it can be obtained that the height of the UAV 

from the ground under the condition of vertical wind is 
500.001138𝑚, and the distance from the horizontal plane of 
the target is 1001.3134𝑚, which is the launching strategy of 
the dive Angle 𝜃 26.548922° . At this time, the straight-
line distance 𝐿 1119.2089𝑚  between the UAV and the 
target point meets the constraint conditions. The details are 
shown in Table 2. 

 
Table 2. UAV launch strategies for three wind directions 

 H L Dive Angle 𝜃 Linear distance L 
Tailwind 500.001138m 1000.726019𝑚 26.548474° 1118.6834𝑚 
Headwind 500.001138𝑚 1000.686864𝑚 26.549370° 1118.6484𝑚 

Vertical wind 500.001138𝑚 1001.3134𝑚 26.548922° 1119.2089𝑚 
 

4. Conclusion 
By combining the air resistance model in the aerodynamic 

model with Newton's second law, an efficient solution to the 
ball projection problem at low speed is constructed. 

When the aircraft is thrown horizontally, the sphere motion 
is decomposed into horizontal and vertical directions, and the 
velocity is 𝑣 , 𝑣 . The horizontal initial velocity of the ball is 

the cruise speed of the aircraft 83.3𝑚/𝑠 . In the vertical 
direction, the ball moves in a linear motion with variable 
acceleration under the action of air resistance and universal 
gravitation, and the air resistance increases with the falling 
speed of the ball. The landing time of the sphere is t=8.129 
seconds. After that, no matter the direction of the wind, the 
movement time of the ball is constant. By simplifying the 
model, the horizontal wind speed in the horizontal direction 



 

143 

is directly superimposed on the sphere, and the horizontal 
speed of the sphere under the three wind directions is 
determined as: 𝑣, 88.3𝑚/𝑠, 𝑣,, 83.3𝑚/𝑠, 𝑣,,, 78.3𝑚/
𝑠 . By solving the dynamic model, the horizontal flight 
distance of the sphere under the three wind directions is 
obtained: 571.3011𝑚 , 517.8954𝑚 , 544.8736𝑚 . The 
delivery distance of the aircraft is calculated by the Euclidean 
distance: 717.7907𝑚, 636.5007𝑚, 677.1457𝑚. 

And through numerical iteration, the dive angles of the 
aircraft under the three wind directions are respectively 
26.5485° , 26.5494°, 26.5489° , and the position of the 
aircraft when launching is (500,1000.726), (500,1000.687) 
and (500,1001.313), and the straight-line distance to the target 
is: 1118.6834𝑚，1118.6484𝑚，1119.2089𝑚. Considering 
the accuracy of the above model for solving the specified 
constraints, the model can be extended to obtain the UAV 
fixed-point delivery model suitable for complex and 
changeable situations. 

This model is extended as a core special case, which can be 
extended after the introduction of other factors, and has good 
compatibility and universality. 

References 
[1] Zang Qian. Research and Implementation of UAV precise 

Point Control Based on Multi-information Fusion [D]. 
Shenyang University of Aeronautics and Astronautics, 2022.  

[2] Ning Xin. Research on Control Method of Quad-rotor UAV 
[D]. Harbin Institute of Technology,2018. 

[3] Wu Qingchun, Qian Yangde. [3] WU Q C, QIAN Y D. 
Research on the cause and Quantitative Analysis of Air 
Resistance of moving Objects by Air resistance Coefficient 
measuring instrument [J]. College Physics, 2017, 4 (5) : 27-32.  

[4] Zhao Bingyan, Liu Weijun, Chen Zonghua. Wind speed cases 
swash cast ball flight trajectory analysis [J]. Journal of Yulin 
Normal University, 2020, 9 (3) : 36-42.  

[5] Gong Yongshou, Shen Yijun, Li Chunxu. Uav precision and 
toss life buoy scheme research [J]. Journal of Pearl River Water, 
2020 (22) : 38-40.  

[6] Wang Zhenxin, Su Wenbin, Shi Yanpeng. To explore the 
influence of the air resistance of sphere stant cast movement [J]. 
Science and Technology, the Wind, 2023, No. 520 (8) : 163-
165.  

[7] Liu Hao. Exploring the real motion process of falling objects 
from high altitude [J]. Hunan Middle School Physics, 2020, 
35(06): 31-32+49. 

[8] Huang Heji, Chen Shijie, Ma Junling et al. Research on 
Mathematical model of aircraft landing process [J]. Science 
Consulting (Science and Technology Management), 2022, 
No.781 (04):35-37. 

[9] Wang Xiaolei, Zhong Yuming. Analysis of drifting and 
Landing process Based on aerodynamic Parameters [J]. Journal 
of Civil Aviation,2022,6(01):73-76+84. 

[10] Si Shoukui, Sun Xijing, Mathematical Modeling Algorithm 
and Application, Beijing: National Defense Industry Press, 
2021.4. page416.