Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 3, No. 3, 2023 

 

34 

Path tracking of Mr Fluid braking System based on PID 
control 

Peining Wen1, Yilin Liu2, Tong Liu2, Liqiang Zhou2, Qinyan Cheng2 

1 College of Automation Central South University, Changsha, Hunan 410083, China 
2 The College of electrical and mechanical engineering, Central South University, Changsha Hunan 410083, China 

 

Abstract: With the rise of intelligent vehicle and driverless technology and the continuous improvement of vehicle speed, 

people have higher and higher requirements for vehicle safety performance. Mr Brake can be integrated with vehicle electronic 

control system to improve the corresponding speed of the braking system to a large extent and shorten the braking distance of 

the vehicle. This paper introduces the design theory and analysis of Mr Fluid. The working principle is introduced, and the torque 

required by the front and rear wheels of the vehicle braking process is deduced by the vehicle dynamics model. The Mr Fluid 

braking model based on PID control is built by using Matlab-simulink software. By tracking the straight line without fluctuation, 

the straight line with fluctuation and the right Angle path, the simulation results show that By adding PID control model to the 

magnetorheological braking control, it can effectively reduce the overshoot in the fluctuating route during the braking process, 

and the corresponding speed is faster and the path tracking effect is better, which provides an innovative thinking method for the 

development of vehicle brake and the development of driverless technology. 

Keywords: Mr Fluid; PID; Matlab-simulink; Braking. 

 

1. Introduction 

In recent years, with the increasingly prominent energy and 

environmental problems, people's requirements for vehicles 

have been improved, and the development of the automobile 

industry has also put forward higher requirements for 

automobile braking system [1, 2]. At present, the mainstream 

braking system includes electronic hydraulic braking system 

and electronic mechanical braking system [3]. Mr Fluid 

braking system is a new linear control system available for 

electric vehicles. Using the advanced manufacturing 

technology of Mr Fluid and the special mechanical and 

magnetic properties of Mr Fluid, it can brake accurately and 

quickly, which puts forward an innovative idea and 

development direction for the development of vehicle brake 

[4, 5]. 

2. Design theory and analysis of Mr 
Fluid 

2.1. Characteristics of Mr Fluid effects 

The effect of Mr Fluid refers to that the viscosity of the 

fluid changes greatly under the action of the magnetic field. 

When the Mr Fluid reaches a critical value, the fluid no longer 

flows into a solid, and maintains its shape or has certain shear 

resistance. At the same time, it also has the unique yield 

phenomenon of solid.  

Under the action of magnetic field, the viscosity of Mr 

Fluid increases with the increase of magnetic field intensity. 

When the critical magnetic field intensity is reached, the flow 

stops and becomes solid. However, when the magnetic field 

intensity is weakened, the Mr Fluid can become a fluid and 

return to the original viscosity.  

Under the action of magnetic field, the Mr Fluid-solid 

liquid conversion is reversible and controllable, and requires 

less energy. 

2.2. Working principle of Mr Fluid 

When the magnetic fluid flows through the excitation coil 

with incoming current, the magnetic field passes through the 

magnetic fluid. Therefore, the magnetic fluid will produce 

needle-like bulge along the direction of the magnetic field. 

After the magnetic field passes for several hours, the magnetic 

fluid will change from the bulge state to the free-flowing state 

(Figure 1). 

 

Figure 1. Magnetic liquid-solid transformation 

According to its basic principle, Mr Fluids work in three 

modes: flow mode, shear mode and extrusion mode. 

For the flow mode, the Mfluid is located between two 

sufficiently large plates that remain relatively stationary. 

There is a pressure difference between the front and rear sides 

of the liquid, which enables the mfluid to flow between the 

plates and thus create a certain friction force between the 

plates. This mode is commonly used in current dampers and 

shock absorbers. For the shear mode, the magnetic fluid is 

located between two plates in relative motion, and the 

magnetic chain generated in the magnetic fluid is constantly 

broken under the shear of two plates in relative motion, which 

is manifested as shear stress on a macro level, hindering the 

relative motion of two plates. Currently, the devices that apply 

this working mode include clutch, brake, etc. For the 

extrusion mode, the Mr Fluid is squeezed by the proximity of 

the relative position between the two plates. It is usually 

applied to vibration dampers with small motion and large 



 

35 

damping force. 

2.3. Derivation of braking torque of Mr Fluid 

brake 

In the actual working process, Mr Fluid is complicated and 

affected by many small factors. In order to simplify the 

calculation of braking torque, our team makes the following 

assumptions about Mr Fluid brake:  

(1) The velocity of Mr Fluid is only related to the radius.  

(2) Although the Mr Fluid is affected by gravity and 

centrifugal force, it is still evenly distributed in the working 

gap of the brake.  

(3) Under the influence of time and temperature, the basic 

properties of Mr Fluid remain stable and do not change.  

(4) The volume of Mr Fluid is incompressible.  

(5) Ignore the influence of Mr Fluid injection channel on 

brake performance.  

For the disc Mr Fluid brake, the effective working face in 

the braking process is a torus (Figure 2). 

 

Figure 2. Disc brake torque analysis and calculation diagram 

The shear strain rate at any point on the working surface 

can be expressed as: 

γ =
𝑟𝜔

ℎ
                     (1) 

r represents the radius of rotation of the selected point. 

𝜔 represents the rotational angular velocity of the torus. 

h represents the working clearance of the Mr Fluid brake. A 

miniature torus with radial width is taken at radius r, and its 

area dA can be calculated as follows: 

dA = 2πrdr                 (2) 

In this case, the force dF generated by the miniature torus 

is:  

dF = τ × dA                                                     (3) 

The acting moment dT generated by the miniature torus is:  

dT = r × dF = 2πτr2dr                        (4) 

By integrating the braking torque generated by the 

miniature torus, the braking torque generated by the Mr Fluid 

brake on torus 𝑅1 to 𝑅2can be obtained as follows: 

T = ∫ 2πτ𝑟2𝑑𝑟
𝑅2

𝑅1
= 2π ∫ [𝜏𝑦(𝐵) + 𝛾𝜂]𝑟2𝑑𝑟 =

𝑅2

𝑅1
2π

3
𝜏𝑦(𝐵)(𝑅2

3 − 𝑅1
3) +

𝜋𝜂𝜔

2ℎ
(𝑅2

4 − 𝑅1
4)      (5) 

Considering that the above equation is only an analysis of 

a disc surface, while the Mr Fluid brake has two working 

faces, the total braking torque is: 

T =
2π

3
𝜏𝑦(𝐵)(𝑅2

3 − 𝑅1
3) +

𝜋𝜂𝜔

2ℎ
(𝑅2

4 − 𝑅1
4)         (6) 

The braking torque 𝑇𝑚 =
4π

3
𝜏𝑦(𝐵)(𝑅2

3 − 𝑅1
3)  is mainly 

controlled by the magnetic field intensity of Mr Fluid and 

belongs to the controllable part, while the braking torque 

𝑇𝑚 =
𝜋𝜂𝜔

2ℎ
(𝑅2

4 − 𝑅1
4) is determined by the vehicle speed, the 

working gap of Mr Fluid, the viscosity of Mr Fluid and the 

radius of brake disc, and belongs to the uncontrollable part of 

the braking torque. Therefore, in the design of Mr Fluid brake, 

the controllable part should be adjusted around. 

 

Figure 3. Automobile brake force diagram 

According to the vehicle dynamic model (Figure 3), the 

balance equation of front and rear wheel torque can be written: 

{
𝐹𝑍1𝐿 = 𝐺𝑏 + 𝑚

𝑑𝑣

𝑑𝑡
ℎ𝑔

𝐹𝑍2𝐿 = 𝐺𝑎 − 𝑚
𝑑𝑣

𝑑𝑡
ℎ𝑔

              (7) 

When the car is locked on the road surface with adhesion 

coefficient is, the sum of braking force generated by the 

vehicle is equal to the adhesion of the road surface to the 

wheel, and the braking force 𝐹𝜇1 and 𝐹𝜇2 generated by the 

front and rear wheels of the vehicle are equal to the respective 

adhesion of the road surface: 

{

𝜇𝐺 = 𝐹𝜇1 + 𝐹𝜇2

𝐹𝜇1 = 𝜇𝐹𝑍1

𝐹𝜇2 = 𝜇𝐹𝑍2

             (8) 

By combining the above two formulas, we can get: 

{
𝐹𝜇1 = 𝜇 (

𝑚𝑔𝑏

𝐿
+ 𝑚

𝑑𝑣

𝑑𝑡

ℎ𝑔

𝐿
)

𝐹𝜇2 = 𝜇 (
𝑚𝑔𝑎

𝐿
− 𝑚

𝑑𝑣

𝑑𝑡

ℎ𝑔

𝐿
)

                 (9) 

At this time, the distance between the wheel axle and the 

ground is 𝑅𝑟, so the maximum braking torque of the front, 

rear and axle of the vehicle is 𝑀1 and 𝑀2 are: 

{
𝑀1 = (

𝑚𝑔𝑏

𝐿
+ 𝑚

𝑑𝑣

𝑑𝑡

ℎ𝑔

𝐿
) 𝜇𝑅𝑟

𝑀1 = (
𝑚𝑔𝑎

𝐿
− 𝑚

𝑑𝑣

𝑑𝑡

ℎ𝑔

𝐿
) 𝜇𝑅𝑟

                        (10) 

3. Establishment of simulation model 
of Mr Fluid braking system 

3.1. PID braking system model 

Since the Mr Fluid brake itself does not have a braking 

system, the braking torque can be adjusted through its own 

control of the current in the electromagnetic coil, so as to 

realize the braking process. The PID model has a simple 

algorithm, good robustness and high reliability. Figure 4 

shows the adaptive variable damping controller model. 

 
Figure 4. PID control model 



 

36 

This paper simulates an electric vehicle, and its relevant 

parameters are shown in the following table 1. 

Table 1. Correlation parameter table 

Name 
Code 

name 
Unit 

Numerical 

value 

total mass m Kg 1710 

wheelbase L Mm 2730 

height of center of mass ℎ𝑔 Mm 0.8 

horizontal distance 

from center of mass to 

front axle 

a Mm 1530 

horizontal distance 

from center of mass to 

rear axle 

b Mm 1200 

front axle load 𝑚𝑓 Kg 1020 

rear axle load 𝑚𝑟 Kg 690 

wheel rolling radius 𝑅𝑟 Mm 345 

3.2. Straight-line path tracking 

3.2.1. Undulating line tracking 

 

Figure 5. Target speed and tracking speed 

 

Figure 6. Target path and tracking path 

 

 

Figure 7. Deviaton 

In the linear path without fluctuation, the linear path can be 

well tracked under the condition that the speed can be 

reasonably tracked in the control model without overshoot. 

The maximum control error is within 10cm, which can be 

well applied to the path tracking of the linear path without 

fluctuation (Figure 5, Figure 6 and Figure 7). 

3.2.2. Linear path tracking with fluctuation 

 

Figure 8. Target path and tracking path 

Considering that it is ideal for the vehicle to keep a straight 

line completely in the process of straight driving, this paper 

simulates the linear path to track the relevant fluctuations. 

Under the PID control model, I=0 and D=3 are maintained, 

and the Mr Fluid is controlled by adjusting P. According to 

the results, under the PID control model, It can well reduce 

the overshoot of the car in deceleration swing, and can better 

track the path with a straight line of fluctuation (Figure 8). 



 

37 

3.3. Limit right-angle path tracking 

 

Figure 9. Target velocity and tracking velocity 

 

Figure 10. Target path and tracking path 

In the path simulation of limit right-angle turn, PID model 

braking control is carried out on the car by giving different 

initial speeds to the car. The results show that the car can 

adjust its own speed adaptively to plan a path to the final point 

(Figure 9 and Figure 10). 

4. Summarize 

In summary, by introducing the design theory of Mr Fluid 

and correlation analysis, this paper introduces the working 

principle of Mr Fluid, deduces the braking torque of Mr Fluid 

brake, obtains the braking force torque of front and rear 

wheels through the vehicle dynamics model, and establishes 

a braking system model based on PID control to track the road 

surface without fluctuation and the road surface with 

fluctuation and the right-angle turning path. It can fully 

reduce the overshoot of the vehicle in the process of braking 

deceleration, and better track the target path, thus increasing 

the development space of the vehicle driverless. 

References 

[1] Zhang hequan Study on Structure optimization and 
performance of vehicle Magnetorheological Brake[C] 
Changchun University Of Technology 2018. 

[2] Songyu Structure research and performance simulation of Mr 
Fluid ABS brake[C] Northeast Forestry University 2011. 

[3] Ma liangxu Research and development of Magnetorheological 
fluid braking system for electric vehicle[D] Tsinghua 
University 2018. 

[4] Jin aili Speed tracking control of heavy duty train based on 
active disturbance rejection control [C] East China Jiaotong 
University 2021. 

[5] Chen wei Research on Active Disturbance Rejection control 
for two degrees of Freedom flexible mechanisms [C] Xidian 
University 2014. 

 

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