










































  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 

    

 
 
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. 

SLIDING MODE-BASED 
TRACTION CONTROL  

FOR AN AUTONOMOUS  
MARTIAN ROVER 

JULIAN  BOWNE,  
DR. ANNALISA SCACCHIOLI  (FACULTY ADVISOR) 

 
 
 
 

✵ ABSTRACT 
 A traction control system was developed for 
an autonomous Martian rover using a sliding mode 
controller. The main inspiration for this project was 
NASA’s Mars rover, Curiosity, which suffered severe 
wheel damage due to the lack of an effective traction 
control system. A control system was sought out to 
effectively prevent wheel damage, slippage, and soil 
failure for a Martian rover. It was initially hypothe-
sized that a sliding mode controller would be most 
effective to control the vehicle’s traction. A Simulink 
model was created with a deformable soil-rigid tire 
mathematical model in order to simulate the traction 
control system. The sliding mode controller was 
tested to be more robust and stable compared to a 
proportional-integral-derivative (PID) controller for 
the rover. The results elaborate the possible applica-
tions for this project, which spans across commercial 
and military rovers, rescue robots, and planetary rov-
ers in the private and global space industry.  
 

1 INTRODUCTION 
Traction control has been a growing con-

cern for the robotics and automotive industry. For 
example, traction control can be applied to hybrid 
cars, rescue robots, and planetary rovers [2,8,12]. The 
main inspiration for this research was the severe 
damage of the wheels of the Mars Rover, Curiosity, 
that resulted from an incomplete understanding of 

the Martian terrain and a lack of a suitable traction 
control system[1]. NASA later produced an algorithm 
to reduce further damage to the wheels and to con-
tinue its exploration of Mars, but this method was not 
in the scope of difficulty for this project[14]. Instead, a 
similar approach to Kazuya Yoshida's article about 
Slip-Based Traction Control was followed[17].  

The goal of this research is to create a simu-
lation to test a model-based traction control system 
for a Martian Rover, like Curiosity. This project con-
sists of two main components: mathematical model-
ing of the vehicle-terrain interaction and the control-
ler, which generates a voltage applied to the wheel’s 
electric motor. This voltage forces the slip ratio to ap-
proach a desired value. In other words, the controller 
accelerates the vehicle efficiently in the deformable 
Martian terrain. 

In Yoshida's research, a slip-based model 
was developed using the terramechanics, which is 
the study of soil properties, of a rigid wheel on loose-
deformable soil[17]. The terrain model used in this ar-
ticle was the deformable soil-rigid tire model, which 
was then adopted into the current project[15]. This 

FIGURE 1: One of Curiosity’s Damaged Wheels.  
DESCRIPTION: The holes present on the middle tire, on the 
right side of the image, was caused by sharp rocks found 
in Gale crater. Due to the lack of a traction control system, 
the damage to the tire grew rapidly. 



  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

model consists of several equations of terramechan-
ics; specifically the interaction of wheeled vehicles 
on various surfaces. The equations involved model 
soil that is deformed by the vehicle’s wheels, as well 
as the resistive and driving force and torque gener-
ated by the soil. These relations allow for a more ac-
curate simulation for the soil and vehicle interaction. 
This model was also chosen based on the compu-
ting restrictions of the computer used. More ad-
vanced options like finite element models require 

computers with higher level of performance. 
Another key component is the controller. In 

Yoshida's research, a proportional-integral-deriva-
tive controller (PID) was used to control the slip-
page[17]. After thorough investigation, a faster and 
more robust method was sought out. A faster 
method specifically meant a quicker settling time 
(time to reach 2% of the desired slip ratio value), and 
more robust method meant a controller that func-
tions better with more uncertainties. The Sliding 
Mode controller was used for its robustness and ef-
fectiveness[5]. The more advanced controller is a 
nonlinear control method that uses a discontinuous 
control signal. It is a type of variable structure con-
trol, meaning it alters the dynamics of a nonlinear 
system by applying a switching control. 

Mathworks’ softwares Matlab and Simulink 
were integral tools used for this research[9]. Matlab is 
a programming tool with its own language that ex-
presses matrix math more directly. Simulink is a pro-
gram for modeling, simulating, and analyzing dy-
namical systems. Its main interface is a block dia-
gramming tool that allows the user to model and 
simulate a system made up of various intercon-
nected equations. They were ideal for this project 
because Simulink and Matlab are frequently used for 
professional vehicle based-control system design in 
the field. Simulink is user-friendly for mathematical 
modeling because of its block-based coding. This 
means that instead of writing lines of code, the con-
trol system can be designed with simple “blocks” 
that represent mathematical operations and tasks. 
Matlab was useful for inputting initial conditions and 
extracting results for plots. Both of these softwares 
were used together to simulate and extract results. 

 

2 MODELING 
A negative feedback system occurs when 

the output of a system is fed back into itself in order 
to influence its output. These system configurations 
are required for most control methods. In order to 
model a traction control system with block diagrams 
in Simulink, three basic components of a negative 
feedback loop are needed:  

1. The plant is the mathematical model of the 
dynamic equations of the problem. 

2. The actuator is the controlled motor.  

3. The controller is inputted with the error of 
the feedback loop (the desired slip ratio mi-
nus the plant’s modeled slip ratio) and out-
puts the voltage applied to the actuator. 

To begin understanding the problem of slip-based 
traction control on deformable soil, a basic under-
standing of a vehicle control system was established 
through the exploration of a simple Anti-lock Brak-
ing System (ABS). Two examples were used as a 
starting point: the first being an example from Math-
works' website and the second being a rudimentary 
Simulink block diagram[4,10]. The second example in-
corporates the quarter car model for the vehicle's 
motion equations. 

 

EQUATION SET 1: Vehicle Model and Slip Ratio. 



  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

The first set of equations modeled were the 
vehicle's motion equations. The quarter car model 
was used, which isolates each individual wheel to 
carry the load of a quarter of the car's weight. This 
allows the problem to be simplified to controlling a 
single wheel’s slip. The derived equations are New-
ton's Second Law for a quarter mass vehicle and 
Newton's Second Law for Circular Motion for the 
wheel. The input variables in the quarter car equa-
tions are the driving force, resistive force, motor 
torque, and soil torque (reaction torque generated 
by the soil). The acceleration taken by the two equa-
tions are integrated to calculate the velocity of the 
vehicle and the angular velocity of the wheel. These 
values along with the wheel’s radius is used to calcu-
late the slip ratio. 

In order to develop a relation for the slip ra-
tio and the forces and torques generated by the soil 
onto the wheel, a soil-vehicle mathematical model 
must be utilized. A simple model that was initially 
used for testing the controller was the Pacejka’s 
Magic Tire Formula[11]. It is a trigonometric function 
(made up of sine and arctangent) with generally four 
terrain specific coefficients. The more complex 
model that was used for the final results was the de-
formable soil-rigid tire model. It is composed of 

three integrals to calculate the two forces and one 
torque. The integrated functions are the compo-
nents of shear and normal stress produced by the 
soil. The Martian soil is represented by certain soil 
coefficients in the equations to calculate the shear 
and normal stress as functions of slip. 

FIGURE 2: Vehicle Model.  
DESCRIPTION: This diagram represents a simplified model, which focuses on a single wheel’s dynamics. 𝑀𝑀 is the mass of the 
car, 𝑟𝑟 is the radius of the wheel, 𝑣𝑣𝑥𝑥 is the vehicle’s velocity, 𝜔𝜔 is the angular velocity of the wheel, 𝐹𝐹𝑑𝑑 is the driving force, and 
𝑇𝑇𝑚𝑚 is the motor torque. 

TABLE 1: Vehicle Model and Slip Ratio Variables. 



  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 

FIGURE 3: Deformable Soil-Rigid Tire Model.  
DESCRIPTION: The diagram represents the tire interaction with de-formable soil. The main focus of the diagram is the shear 
and normal stresses represented by 𝜏𝜏 and 𝜎𝜎, respectively. The stress is experienced by the tire, and two resultant forces are 
generated, the driving force 𝐹𝐹𝑑𝑑 and the resistive force 𝐹𝐹𝑅𝑅. 

EQUATION SET 2: Terramechanics Equations. TABLE 2: Terramechanics Variables. 
 



  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 

    

 

Parameters used in this project for the equations mentioned earlier came from two sources. These in-
clude Yoshida’s article for soil properties and NASA’s Curiosity specifications for the rover’s dimensions and 
mass[13,17]. Below is a table of the parameters: 
 
 

In order to utilize the deformable soil-rigid tire model, the integrals to calculate the forces and torque 
must be solved. Several methods were considered for finding the solutions. First, an analytical method was in-
vestigated. However, using Matlab's Symbolic Toolbox (a solver that can be used for integrals) only resulted in a 
non-simplified output. Since the equations only can be solved numerically, Taylor series approximations were 
used. Adding higher order derivatives to the approximations makes it more accurate, which is shown in the 
graph below. The 5th order approximations were utilized in Simulink to complete the mathematical model. This 
approximation was sufficient since higher slip ratio values (closer to one) do not occur in the simulation.  

The slip ratio was determined to be the state variable, which means it is a variable that describes the state 
of the system. The friction or driving force, which is maximized for the ideal performance of the simulated vehicle, 
is a function of this slip ratio. The voltage is the control variable which is then inputted into a DC motor to generate 
a corresponding motor torque.  

A model of a DC motor was required for the actuator of the simulation. After investigation, the Armature 
Circuit was a good starting point to model an ideal motor[3]. The simple Armature Circuit consists of a voltage 
source, resistor, inductor, and rotor with a fixed magnetic field. For the DC motor, Newton’s Second Law and 
Kirchhoff’s Voltage Law were derived. Also, proportional relations were determined for the torque and electro-
motive force. By utilizing these equations, an input of voltage produces a corresponding motor torque. 

TABLE 3: Parameters for the Overall System. 



  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 
List of Relevant Terms 

NEGATIVE FEEDBACK LOOP – A system where the output is 
fed back into itself in order to influence its future  
output. 

PLANT – The mathematical model that has an input 
and output (without a feedback loop) 

ACTUATOR – The component of a system which is  
responsible for moving or controlling a device or 
machine. 

CONTROLLER – The component of a system that uses 
the system’s state to control its output.  

QUARTER CAR MODEL – A model that simplifies a vehicle 
into an isolated wheel and a quarter of the total 
mass. 

NEWTON’S SECOND LAW – The acceleration is directly 
proportional to the magnitude of the net force, and 
inversely proportional to the mass. 

SLIP RATIO – The value expressed in a ratio that  
represents the slipping behavior of a vehicle’s wheel. 

DRIVING FORCE – The force, also called the friction force, 
is the force that causes the vehicle to accelerate  
forward. 

PACEJKA’S MAGIC TIRE FORMULA – A simple formula that 
uses the slip ratio as an input and outputs the driving 
force. Different plugins of coefficients can model  
different terrains. 

DEFORMABLE SOIL-RIGID TIRE MODEL – A set of equations 
that model a complex deformable soil like the  
Martian terrain. 

TAYLOR SERIES APPROXIMATION – A way of approximating 
complex functions, using values of its derivative. 
Higher order expansions (more derivatives) allow for 
a more accurate approximation. 

SHEAR STRESS – The force applied to an object in the  
direction parallel to the cross-sectional area. 

NORMAL STRESS – The force applied to an object in the 
direction normal to the cross-sectional area. 

ARMATURE CIRCUIT – An electrical circuit that involves a 
rotating part or in the case of a DC motor, an input 
current that interacts with a fixed magnetic field to 
produce a torque. 

KIRCHHOFF'S VOLTAGE LAW – A form of conservation of  
energy for a closed electrical circuit. The law states 
that the sum of the voltage around a circuit must 
equal zero.

FIGURE 4: Taylor Series Approximation for Driving Force.  
DESCRIPTION: This graph shows the increase in the accuracy 
of the Taylor Series Approximation as higher order deriva-
tives are added. The approximations are compared to the 
numerical curve, which is colored blue. 

FIGURE 5: DC Motor Armature Circuit.  
DESCRIPTION: This diagram shows the electrical circuit on 
the left, and the rotor and electric field on the right. The 
circuit contains a voltage source, a resistor, an inductor, 
and a rotor. 



  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 

    

 

3 THE CONTROLLER 
The purpose of the controller is to output a 

motor voltage signal to force the slip ratio to ap-
proach the desired value. The PID controller was first 
experimented with and later used to compare final 
results. It consists of a proportional gain, an integral 
gain, and a derivative gain. These multiplicative co-
efficients allow the user to tune the controller for dif-
ferent dynamical systems. 

The sliding mode controller is a more ad-
vanced method than the PID. The controller, essen-
tially, has a phase where the state variables are 
forced to approach a function called the sliding sur-
face. Once on the surface, the sliding phase begins 
by trying to bring this sliding function to zero (as well 
as the error of the desired and modeled slip ratios). 

FIGURE 7: Simulink Model.  
DESCRIPTION: The three main components (left to right) are the controller, labeled SMC for sliding mode controller, the actuator, 
which is the DC motor, and the plant, which is the deformable soil-rigid tire model. The SMC is inputted with three tuning 
coefficients labeled K, lambda, and psi as well as the error and its rate of change. It then outputs a voltage applied to the 
actuator. The actuator then outputs the motor torque, which is inputted into the plant. Based on the vehicle-terrain model, the 
plant outputs the slip ratio which is fed back into the system to calculate the error. This completes the negative feedback loop 
that simulates a traction control system for a Martian rover. 

FIGURE 6: Sliding Mode Behavior.  
DESCRIPTION: First, the sliding mode controller forces the 
system onto the sliding surface, denoted by the reaching 
phase. Then, it oscillates about this surface until the error 
is zero, which is called the sliding phase. 



  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

The mathematics behind the sliding mode is divided 
into the sliding surface design and the control input 
design. For the sliding surface design, the function is 
made up of the error and a certain number of its de-
rivatives. For the control input design, the task is to 
steer the sliding surface function towards zero (in 
other words, to reduce the error). The control input 
for this project is the voltage applied to the electric 
motor powering the wheel. In the context of the 
problem, FIGURE 6 variables (x1 and x2) are the wheel’s 
angular velocity and the vehicle’s velocity, respec-
tively. These variables are the state variables of the 
system because they are used to calculate the slip 
ratio. Similar to the PID controller, the sliding mode 
controller has coefficients for tuning. These coeffi-
cients are involved in the equations for the sliding 
surface function and the control input function.  

After modeling the necessary mechanisms 
for controlling slip, the models must be integrated 
into Simulink for creating the simulations. To create 
the negative feedback system, different blocks for 
mathematical computation are incorporated to cre-
ate a block diagram version of the equations for the 
model and controller. Once all the coefficients and 
initial conditions have been inputted, the simulation 
can commence. 
 

 
List of Relevant Terms 

PID CONTROLLER – Proportional-integral-derivative  
controller. It uses a balance of the three to minimize 
the error of the control system.  

SLIDING MODE CONTROLLER – A nonlinear control method 
that alters the dynamics of a system with a discontin-
uous control signal that forces the system to “slide” 
along a function that minimizes the error. 

SLIDING SURFACE – The bounded function in which  
the sliding mode controller forces the system onto. 

CONTROL INPUT – The manipulated discontinuous  
signal that forces the system onto the sliding surface. 

TUNING – The way a control method is customized  
for the appropriate scenario. 

STATE VARIABLES – Variables that describe the  
mathematical state of the dynamical system. 

4 RESULTS  

The results for the preliminary Pacejka 
Model simulation showed the working PID controller 
and different traction control responses for each of 
the terrains: dry asphalt, wet asphalt, concrete, snow, 
and ice. The desired slip ratio was different for the 
terrains. The value was determined by maximizing 
the driving force using the Pacejka’s Magic Tire  
Formula. When looking at the plots, it is evident that 
the PID controller overshoots the desired slip ratio. 
Additionally, the oscillations in the response are not 
minimized. When comparing the different re-
sponses, the PID’s response for ice had more oscilla-
tions at a higher frequency. This shows the lack of  
robustness for the PID controller. The rise and  
settling time for the plots were nearly identical for 
the five terrains. These preliminary results were use-
ful to see the behavior of the PID for the simulated 
wheel and vehicle. In addition, these results already 
show the limitations of the PID controller for a  
simpler terrain model represented by the overshoot 
and oscillations. 

The main results of this project illustrate how 
a sliding mode-based traction control system works 
on a simplified Martian terrain using a deformable 
soil-rigid tire model. The sliding mode controller 
outperformed the PID controller in stabilizing the 
slip ratio to a value of 0.16. Looking at the step  
responses, the sliding mode controller did not over-
shoot the desired value, while the PID did. In addi-
tion, the response of the sliding mode controller was 
critically damped, which was ideal. The initial  
response for the sliding mode controller was also 
faster, which is desirable for a traction control  
system. This is evident in the faster settling time, 
which was reduced by around a factor of 7  
(decreased from 0.7 to 0.1 seconds). Additionally, 
the rise time was reduced by around a factor of 3. 
Looking at the tracking error plots, the error for the 
sliding mode controller response shows a single 
peak with a very small width. The same peak is  
evident in the PID response, but it is wider. This 
means the width of the initial error pulse is signifi-
cantly reduced. This further supports that the sliding 
mode controller has better performance for minimiz-
ing the error. To finalize, the sliding mode controller 
is more practical for a physical rover because of its 
faster and more efficient response. 



  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 

 
 

  
FIGURE 8: Preliminary PID with Pacejka’s Magic Tire Formula 
for Dry Asphalt.  
DESCRIPTION: The plot shows the slip ratio being controlled 
to achieve the desired value for dry asphalt. 

FIGURE 9: Preliminary PID with Pacejka’s Magic Tire Formula 
for Wet Asphalt.  
DESCRIPTION: The plot shows the slip ratio being controlled 
to achieve the desired value for wet asphalt. 

FIGURE 10: Preliminary PID with Pacejka’s Magic Tire Formula  
for Dry Concrete.  
DESCRIPTION: The plot shows the slip ratio being controlled to 
achieve the desired value for dry concrete. 

 
FIGURE 11: Preliminary PID with Pacejka’s Magic Tire Formula for Ice.  
DESCRIPTION: The plot shows the slip ratio being controlled to achieve 
the desired value for ice. 

 
FIGURE 12: Preliminary PID with Pacejka’s Magic Tire Formula for Snow.  
DESCRIPTION: The plot shows the slip ratio being controlled to achieve 
the desired value for snow. 

 



  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 
 
 
 
 

 
 
 
  

FIGURE 13: Step Response Using PID.  
DESCRIPTION: The slip ratio was successfully controlled  
using the PID controller. However, it overshot the desired 
slip ratio, and it created many oscillations. 

FIGURE 14: Tracking Error Using PID.  
DESCRIPTION: The error plot shows the delay in the PID  
controller’s response and the oscillations present. 

FIGURE 15: Step Response Using Sliding Mode.  
DESCRIPTION: The plot shows the step response of the  
sliding mode controller. The controller had a critically 
damped response with a faster rise and settling time  
compared to the PID. 

FIGURE 16: Tracking Error Using Sliding Mode.  
DESCRIPTION: The plot shows the tracking error of the  
sliding mode controller. The error decreases rapidly and 
does not oscillate like the PID’s tracking error. 



  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

5 CONCLUSION  

Slip-based traction control is a very useful 
tool for autonomous rovers. These techniques can 
be applied for a wide-range of purposes: planetary 
exploration, rescue missions, and commercial and 
military endeavors. The simulations were the begin-
nings of creating an autonomous rover with a trac-
tion control system. The results showed the validity 
of the mathematical modeling and the controller  
design, which opened a doorway for furthering the 
complexity of this research. To summarize, the neg-
ative feedback-traction control system was broken 
down into three components: the plant, the actua-
tor, and the controller. The plant was the mathemat-
ical model of the vehicle and terrain interaction 
(mostly the deformable soil-rigid tire model). The  
actuator was the hypothetical DC motor that pow-
ered the wheels (modeled according to a simple  
Armature Circuit). The (final) controller was the slid-
ing mode, which worked more efficiently than the 
PID. All of these combined with the correct initial 
conditions simulated a slip-based traction control 
system for a rover on Mars. 

 In conclusion, the method utilized led to a 
better understanding of the physics of deformable 
soil and the limitations of a model-based traction 
control system. In addition, the results confirmed the 
hypothesis that a sliding mode controller would be 
more effective than a PID controller. This project  
further confirms the use of a sliding mode controller 
for a traction control system, which has been tested 
in recent research[6]. This project can be a trailblazer 
for sliding mode controller research specifically for a 
Martian rover. The computational results will be used 
on a physical rover in order to make Mars explora-
tion easier and more efficient.  

For future works, the traction control system 
will be tested on a physical rover (prototype being 
assembled in the picture above) and on a  
deformable test bed. In addition, theoretical work 
about using a sliding mode controller integrated 
with data-driven modeling and control will be re-
searched in order to advance this technique. These 
results will be compared to that of the Taylor series 
approximation∎ 
 

 

6 ACKNOWLEDGEMENTS  

I would like to thank my research advisor  
Dr. Annalisa Scacchioli for her phenomenal  
guidance. She has always been so supportive of me, 
and without her this project would never have come 
to fruition. Also, I would like to thank the Aresty  
Research Program for giving me this amazing oppor-
tunity. Finally, I would also like to thank Arjun Singh, 
Vishal Rachapudi, and Galileo Wang for their  
impeccable assistance, which led to the creation of 
the prototype rover. 

 

7 REFERENCES  
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FIGURE 17: Rover Prototype.  
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  ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

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https://www.mathworks.com/matlabcentral/fileexchange/52429-example-on-sliding-mode-control
https://www.mathworks.com/matlabcentral/fileexchange/52429-example-on-sliding-mode-control
https://www.mathworks.com/help/simulink/slref/modeling-an-anti-lock-braking-system.html
https://www.mathworks.com/help/simulink/slref/modeling-an-anti-lock-braking-system.html
https://mars.nasa.gov/msl/spacecraft/rover/summary/

