Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 13, No. 2, 2024 

 

297 

Motion	Path	Study	Based	on	Isometric	Solenoids	and	
Discrete	Simulation	
Chenhui Wang1, * 

1School of Meteorology and Oceanography, National University of Defense Technology, Changsha, China 
*Corresponding author: 1992029712@qq.com 

 

Abstract: This paper focuses on the position, speed change and collision detection problems during the traveling process of 
the bench dragon from both physical and mathematical perspectives. Firstly, based on the equidistance solenoid equation and the 
mechanical arm model, this paper iteratively recursively starts from the dragon head and sequentially solves the position of each 
section of the bench in the traveling process. Secondly, this paper combines the velocity analysis of the rigid rod and the nature 
of isometric solenoid, and also solves the position of each section of the bench in the traveling process sequentially in the form 
of iterative recursion. Finally, this paper discretizes the disk-in trajectory and carries out collision detection, so as to obtain the 
time, position, velocity and other information at the termination moment. The study shows that the model can effectively solve 
the complex optimization problem in the marching process of the dragon dance team. 

Keywords: Isometric Solenoids, Iterative Recursion, Rigid Body Rods, Discrete Simulation. 
 

1. Introduction 
The purpose of this paper is to study in depth the dynamics 

of the bench dragon during its traveling process from the 
physical and mathematical perspectives, so as to analyze the 
key factors such as the position, velocity change and collision 
detection during the traveling process of the bench dragon 
[1,2,3]. In this paper, firstly, based on the equidistant 
solenoidal equation and robotic arm model, the position of 
each section of the bench during the traveling process is 
solved by iterative recursive method [4,5]. Secondly, this 
paper combines the velocity analysis of rigid-body rod and 
the nature of isometric solenoid to further solve the position 
of each section of the bench during the traveling process. The 
introduction of the rigid-body rod model allows us to analyze 
the dynamic characteristics of the bench dragon during the 
traveling process more accurately, including the changes of 
velocity and acceleration [6,7]. Finally, in this paper, the disk-
in trajectory is discretized and collision detection is carried 
out to obtain the key information such as time, position and 
velocity at the termination moment. Through the discrete 
simulation technique, we can simulate various situations that 
may be encountered by the bench dragon in the actual 
traveling process, so as to optimize the performance scheme 
and avoid the occurrence of collision and other unforeseen 
situations [8,9]. 

2. Modeling of Bench Dragon Travel 
Process Based on Isometric 
Solenoidal Equation and Recursive 
Method  

Through field research, we collected data on a bench 
dragon, which consists of 223 sections of benches, of which 
the first section is the dragon's head, the next 221 sections are 
the dragon's body, and the last section is the dragon's tail. The 
length of the dragon's head is 341 cm, the length of the 
dragon's body and tail is 220 cm, and the width of all the 
benches is 30 cm. each bench has two holes, the diameter of 
which is 5.5 cm, and the centre of the hole is 27.5 cm away 

from the head of the nearest bench. the two adjacent benches 
are connected by a handle, and the two benches are connected 
by a handle, and the two benches are connected by a handle. 

According to the research, the dragon dance team coils in 
clockwise along an isometric solenoid with a pitch of 55cm, 
and the center of each handle is located on the solenoid. The 
traveling speed of the handles in front of the dragon head is 
always 1 m/s. After consulting the information, the thread 
belongs to the isometric thread and the general expression can 
be given by the following equation: 

 

  𝜌 𝜌 𝑏𝜃             (1) 
 
where, 𝜌0, 𝜃  are the parameters to be solved. After that, 

firstly, from the known parameter conditions, the pitch is 55 
cm, 𝜃 >0, the isometric solenoid with counterclockwise 
positive direction centered at point 𝑂 is established, and the 
parametric equation of the solenoid is: 

 

 𝜌 𝜃               (2) 

 
Since the subsequent modeling process requires the use of 

the arc length formula in isometric solenoids, the arc length 
formula is derived by line integration, and the derivation 
process is shown below, which takes a combination of 
differential and integral methods for the derivation: 

 

 𝑑𝑠 𝑎𝜃𝑑𝜃 𝑎𝑑𝜃 𝑎𝜃 𝑎 𝑑𝜃  (3) 
 

  √𝐴 𝐵𝑥 𝐶𝑥 𝑑𝑥

          (4) 

 
Integrating both sides simultaneously and applying the 

integral formula yields: 
 

 
𝑑𝑥

𝐴 𝐵𝑥 𝐶𝑥2

1

√𝐶
arcsinh

2𝐶𝑥 𝐵

4𝐴𝐶 𝐵2
      (5) 



 

298 

The final derived equation for the arc length of an isometric 
solenoid is as follows: 

 

𝐿 𝜃
⋅ ⋅

⋅
⋅ 𝜌 𝑏 2 ⋅ 𝜌 ⋅ 𝑏 ⋅ 𝜃 𝑏 ⋅ 𝜃

|𝑏|

2
⋅ arcsinh

𝜌0 𝑏⋅𝜃

𝑏
     (6) 

 
In order to obtain the speed as well as the position 

information of each node of the bench dragon under each 
moment, we regard each boarding as a robotic arm, and regard 
the handle as a joint, to construct a more intuitive robotic arm 
model, and the simulation model is shown in Figure 1 below. 

 

 
Figure 1. Simulation of robotic arm model 

 
For this model, each plateen is considered as a linkage, and 

each linkage has a length of 220 cm (dragon body) or 341 cm 
(dragon head) and a width of 30 cm. Each handle is 
represented as a joint, and all joints are confined to isometric 
solenoids in the two-dimensional plane, so the problem of 
solving the motion of the handles before and after all the 
boards is converted into a multi-joint robot arm problem. For 
such a robotic arm model, it is only necessary to solve the 
position and speed of the faucet joint, and then through the 
recursive formula of speed and position between the joints, 
the position and speed of each joint can be solved sequentially. 

Let the angle in the isometric solenoid corresponding to the 
faucet when it reaches a certain position be 𝜃, the arc length 
as a function of 𝜃 be 𝐿 𝜃 , and the total arc length be 𝐿0. 
For the faucet, its velocity is along the tangential direction of 
the solenoid, with a magnitude of 𝑣0 1m/s. Therefore, the 
arc length traveled by the faucet is equal to the total arc length 
minus the tangential velocity times time, so the parametric 
equation for arc length versus time can be derived as: 

 

 𝐿 𝜃  𝐿 𝑣 𝑡             (7) 
 

 𝜃 𝜃 𝑡                 (8) 
 

Since arc length, radian are functions of time, it is further 
possible to introduce 𝑥 𝑡   and 𝑦 𝑡  , which are obtained 
according to the conversion relationship between polar and 
Cartesian coordinates: 

 

 
𝑥 𝑡 𝜌 𝑡 cos 𝜃 𝑡
𝑦 𝑡 𝜌 𝑡 sin 𝜃 𝑡            (9) 

 
Let the position of the dragon head be 𝑥0 𝑡 , 𝑦0 𝑡 , the 

boarding length of the dragon head be 𝐿0 286 cm , the 
position of the dragon body of the i-th section i
1,2, … 222) be 𝑥𝑖 𝑡 , 𝑦𝑖 𝑡 , and the boarding length of the 

dragon body be 𝐿1 165 cm, so that the relationship of the 
adjacent joints can be obtained according to the definition of 
Euclidean distance: 

 
𝑥 𝑡 𝑥 𝑡 𝑦 𝑡 𝑦 𝑡 𝐿

𝑥 𝑡 𝑥 𝑡 𝑦 𝑡 𝑦 𝑡 𝐿 𝑖 1,2, … 222
 (10) 

 
The position of each joint as a function of time can be 

obtained by combining the equations of the circle centered on 
the joint, the equations of the parameters of the solenoid, and 
the equations of the polar coordinates, i.e., equations (2), (8), 
and (9), being careful to limit the length of the arcs or else 
more than one solution may be found. Initially, the faucet is 
located at point A on the 16-th turn of the spiral, and this 
condition is used to find the required position information. 

 

 
Figure 2. Joint Velocity Schematic 

 
Figure 2 shows the two joints before and after boarding, the 

back handle of the 𝑖-th boarding is also the front handle of the 
(𝑖 1)-th boarding, 𝜃  represents the polar angle of the back 
handle of the 𝑖-th boarding, 𝜃  represents the polar angle 
of the back handle of the (𝑖 1)st boarding, 𝛽 represents the 
difference of the polar angles, 𝑇 , 𝑅  represent the tangential 
and radial vectors, and the direction of the velocity on the 
spiral �⃗�   is in the same direction as �⃗� . 

In this paper, the angle between each velocity and the rod 
is calculated, and the velocities at each joint are solved 
recursively. Checking the literature, it can be obtained that for 
any point 𝑃 on the solenoid, its tangential vector consists of 
the parameters of the solenoid equation, and let the tangential 
vector at the point be �⃗� 𝑇 , 𝑇 T ⃗_𝑖=(𝑇 , 𝑇  ), then the 
tangential vector is given by the following equation: 

 

 
𝑇 𝑏 ⋅ cos 𝜃 𝜌 𝑏 ⋅ 𝜃 sin 𝜃

𝑇 𝑏 ⋅ sin 𝜃 𝜌 𝑏 ⋅ 𝜃 cos 𝜃
 (11) 

 
For the 𝑖-th joint, the vector pointing to the next joint is 

𝑅 𝑥 𝑥 , 𝑦 𝑦 , so for finding the angle 𝛼 , it is 
simpler to find the angle of the two vectors i.e. 

 

 𝛼 arccos 
⃗ ⋅ ⃗

⃗ ⃗             (12) 

 



 

299 

The link between the forward and backward velocities is 
given by the recursive modeling of the velocities of this joint: 

 

 𝑣 ⋅ 𝛼 𝑣 ⋅ 𝛼           (13) 
 
Therefore, the 1m/s of the dragon's head can be used to 

solve the speed of the back part of the handle by iterative 
recursion. 

The horizontal and vertical coordinate positions of the 
dragon dance team at each moment are given in Table 1 below, 
and the speed of the dragon dance team at each moment is 
given in Table 2 below. 

 
 

Table 1. Horizontal and vertical coordinates of the position of the dragon dance team at each moment in time 
 0 s 60 s 120 s 180 s 240 s 300 s 

Mixer x (m) 8.800000 5.662178 4.453844 2.327034 2.351795 4.699969 
Mixer y (m) 0 -5.902779 6.042917 6.355608 5.463400 1.644806 

Section 1 of the Dragon's 
Body x (m) 

 7.372824 1.886354 4.763584 4.655571 3.063900 

Section 1 of the Dragon's 
Body y (m) 

 3.611935 7.300518 4.859957 3.769426 3.989175 

Section 51 of the Dragon's 
Body x (m) 

  5.169406 3.536561 5.813751 -6.189657 

Section 51 of the Dragon's 
Body y (m) 

  6.677874 6.947749 4.069224 1.215279 

Section 101 of the Dragon's 
Body x (m) 

   1.170586 5.162728 -5.776372 

Section 101 of the Dragon's 
Body y (m) 

   8.594989 6.178448 4.571165 

Section 151 of the Dragon's 
Body x (m) 

     7.427815 

Section 151 of the Dragon's 
Body y (m) 

     3.681309 

Section 201 of the Dragon's 
Body x (m) 

      

Section 201 of the Dragon's 
Body y (m) 

      

Dragon's Tail (back) x (m)       
Dragon's Tail (back) y (m) \ \ \ \ λ  

 
 

Table 2. The speed of the dragon dance team at each moment 
 0 s 60 s 120 s 180 s 240 s 300 s 

Mixer (m/s) 1 1 1 1 1 1 
Section 1 of the Dragon's Body (m/s)  0.999950 0.999934 0.999893 0.999827 0.999675 

Section 51 of the Dragon's Body (m/s)   0.999506 0.999296 0.998910 0.998014 
Section 101 of the Dragon's Body 

(m/s) 
   0.998961 0.998421 0.997258 

Section 151 of the Dragon's Body 
(m/s) 

     0.996799 

Section 20 of the Dragon's Body (m/s)       
Dragon's Tail (back) (m/s)       

 

3. A Discrete Simulation-based 
Collision Conflict Detection Model 
for Bench Dragons 

During the disking-in process along the isometric solenoid, 
the boarden dragon may collide with different sections of the 
boarden due to the small radius of the dragon's head path. In 
order to avoid the collision, it can be determined first that no 
collision occurs in the first 300s of the benched dragon, so 
this paper starts to find the collision point after 300 s. The 
collision detection model is developed to determine the 
boundary collision condition. This paper establishes a 
collision detection model to determine the boundary collision 
conditions, and adopts the method of gradually narrowing the 

time and node search range in the process of solving the 
collision moments, which reduces the time complexity of the 
program and improves the solving accuracy. 

First of all, it needs to be clear that the collision in the 
question is the collision between the dragon head and the 
dragon body, and it occurs in the inner circle. Because the 
radius is smaller in the inner circle, the arc of the dragon head 
under the same length (chord length) is larger, the angle 
across is larger, and it is more likely to collide with the dragon 
body segments, and the schematic diagram of the collision 
nodes is given in Figure 3 below. 
 



 

300 

 
Figure 3. Schematic diagram of collision nodes 

 
As can be seen from the figure, on the inside of the coil, the 

curvature of the arc is large, the red dragon swings through a 
large angle, and is more likely to collide with the blue dragon 
body. On the other hand, on the outside of the coil, the 
curvature of the arc is small, and the angle of rotation of the 
dragon's body is small, so there will be no collision. Secondly, 
the collision cannot be an edge-to-edge collision, but a top 
corner-to-edge collision. Because the two handles are located 
on the same thread, the distance between the two threads is 55 
cm, and the distance between the center axes of the board is 
2 ∗ 15 30 cm, which is smaller than the distance between 
the threads, so it is impossible for the board to have an edge-
to-edge collision. We start looking for the collision point after 
300 s. The first step is to determine the location of the faucet. 
First, we determine the location of the faucet. According to 
the model established in the previous chapter, the location of 
the faucet corresponds to the time, so the termination moment 
of the disk-in can be determined by the location of the faucet 
collision. 

Discretize the trajectory of the dragon head after 300s, and 
gradually increase the degree of discretization of the inner 
circle. The angle of the inner circle is uniformly discretized 
into 1000 copies, and the front handles of the faucet are placed 
at the discretized points in turn, so that the geometric 
positions of the faucet and the dragon body are determined, 
and then the collision detection is carried out. 

Let the positions of the two handles be 𝑥1, 𝑦1 , 𝑥2, 𝑦2  

respectively, and the two auxiliary direction vectors are �⃗�1, �⃗�2 
as shown in the figure, and we utilize the property of vector 
addition and subtraction to solve for the position of the top 
angle, which is derived as follows: 

The two handles form a vector �⃗� 𝑥 𝑥 , 𝑦 𝑦  
and the unit vector in that direction is given in the following 
equation: 

 

 𝑒
,

          (14) 

 
Similarly, the unit vector perpendicular to it is given by the 

following equation: 
 

 𝑒
,

         (15) 

The distance of the handle from the front of the dragon is 
27.5 cm, and the direction is 𝑒  ; half of the width of the 
boarder is 15 cm, and the direction is 𝑒 , so we can get the 
vector 𝑏 of the top angle as follows: 

 
 𝑏 �⃗� 27.5𝑒 15𝑒           (16) 

 
After obtaining the coordinates of the top corners, it is 

sufficient to use the four top corners of the dragon head to 
detect the collision with the dragon body of the innermost and 
penultimate circles without using the outer dragon body for 
detection. 

Since the innermost and penultimate circles have more loci 
after discretization and it takes longer computation time to 
detect the collision one by one, we do a round of screening 
first, taking the distance 𝑠𝑖𝑗 from the 𝑖-th vertex corner of the 
dragon's headboard to the side of the 𝑗-th boardboard as the 
basket selection criterion, and set a smaller threshold m, and 
perform the second round of collision modeling only when 
𝑠𝑖𝑗 𝑚. 

𝑠𝑖𝑗 is given by the equation of the distance from a point to 
a straight line, which passes through the top corner 
𝑥 , 𝑦 , 𝑥 , 𝑦  points, then the equation of the straight line 

can be expressed in the coordinates of the two points. Let the 
equation of the line for 𝑦 𝑘𝑥 𝑏, the coordinates of the top 
corner of 𝑥 , 𝑦 , then 

 

 𝑠
| |

            (17) 

 
In this paper, a geometric method is used to construct a 

collision conflict detection model. For any point in the plane 
of the dragon body boarding, respectively, connecting its four 
vertices with the dragon body boarding, constituting four 
triangles, if the sum of the area of these four triangles is equal 
to the boarding rectangle area, the point is located in the 
boarding rectangle, that is, the collision occurs; if the sum of 
the area of the triangle is greater than the boarding rectangle 
area, it is located outside the rectangle, that is, the collision 
has not occurred. 

The area of a triangle is given by the coordinates of three 
points, which are (in any order) 𝑥 , 𝑦 , 𝑥 , 𝑦 , 𝑥 , 𝑦 : 

 

 𝑆 | 𝑥 𝑥 𝑦 𝑦 𝑥 𝑥 𝑦 𝑦 | (18) 

 
Using this model, this collision problem is converted into a 

coordinate solving problem. From the above analysis, we can 
see that we only need to determine the collision of the 
innermost circle and the penultimate circle of the boarden. 
The termination moment before the collision that we solved 
for is 416.4578 s. Table 3 gives the specific positions and 
velocities of the benches at this time. 

 
Table 3. Position and velocity of the bench at the moment of termination of the collision 
 horizontal coordinate x (m) vertical coordinate y (m) velocity (m/s) 

Mixer 1.889317 -1.03866 1.000000 
Section 1 of the Dragon's Body 1.420626- 1.781327 0.988705 

Section 51 of the Dragon's Body 4.494389 3.263509 0.924827 
Section 101 of the Dragon's Body -5.552990 -5.02482 0.920103 
Section 151 of the Dragon's Body 8.233446 0.616415 0.918579 
Section 201 of the Dragon's Body 8.056790 2.695702 0.916375 

Dragon's Tail 8.7005811233970 1233970 0.914978 



 

301 

 

4. Conclusions 
In this paper, we propose a mathematical model based on 

isometric solenoids and discrete simulation by studying the 
position, velocity change and collision detection problems 
during the traveling process of the bench dragon. First, we 
calculate the position of each section of the bench during the 
traveling process of the bench dragon by iterative recursion 
using the equations of isometric solenoids and the robotic arm 
model. This method not only takes into account the physical 
structure of the bench dragon, but also makes full use of the 
mathematical properties of isometric solenoids, so that the 
bench dragon can maintain uniform speed and smooth 
dynamics during the traveling process. Then, we further 
optimize the position solution process of the bench dragon by 
combining the velocity analysis of rigid body rod. By 
introducing the rigid-body rod model, we are able to more 
accurately analyze the velocity and acceleration changes of 
the bench dragon during the traveling process, which provides 
more specific operational guidance for the dragon dancers. In 
addition, we also consider the collision problem that the 
bench dragon may encounter during the marching process, 
and through discrete simulation technology, we discretize the 
trajectory of the disk entry and perform collision detection, so 
as to obtain key information such as the time, position and 
speed of the termination moment. This not only improves the 
safety of the bench dragon performance, but also provides an 
important reference for the optimization of the performance. 

In summary, the model proposed in this paper can 
effectively solve the optimization problem in the process of 
bench dragon marching and provide scientific theoretical 
support for the smooth progress of bench dragon performance. 
Through the application of mathematical modeling and 
simulation technology, we can not only improve the artistic 
effect of the bench dragon performance, but also ensure the 
safety and ornamental performance. In the future, we will 
continue to study the dynamics of the bench dragon and 
explore more ways to optimize the bench dragon performance, 

so as to make greater contributions to the inheritance and 
development of Chinese traditional folk culture. 

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