A Directed Project Final Report
Submitted to the Faculty
Purdue School of Engineering and Technology
Tyler Stover
In partial fulfillment of the requirements for the
Degree of Master of Science in Technology
Committee Member
Approval Signature
Dr Peter Hylton,
Motorsports Engineering
Andrew Borme
Motorsports Engineering
Elaine Cooney
Electrical and Computer
Engineering Technology
Due to resource restrictions and regulations limiting on-track time, driver-in-loop
simulation is used increasingly in the motorsports industry. Race Like A Pro Simulators (RLAPS)
is developing a simulator package to meet the needs of developing drivers looking to practice
fundamental driving skills in virtual simulation. An essential element to delivering this simulation
experience is a robust vehicle physics model.
As part of the development of the RLAPS Simulation Software Program (SSP) a vehicle
physics model was developed around four subsystems – chassis and suspension, aerodynamics,
powertrain, and tires. Tires are the most complex model, and have the most direct impact on the
performance and feel of the vehicle model. A complex algorithm governing vehicle physics was
presented in a generalized form to guide the programming of the RLAPS SSP. From the
generalized algorithm, a practical model was implemented using Unity 3D game creation software
(Unity, 2017). The simulation was tested and evaluated against data from numeric lap-time
simulation software. Various parameters were opened for tuning to refine the performance and
behavior of the vehicle in simulation. The tuned vehicle model performed in such a manner as to
exercise the steering, braking, and throttle application skills of drivers using the simulator.
Simulation in a virtual environment is an increasingly important training tool in many
industries and pursuits. Improving technology allows simulations to improve their complexity and
fidelity. This has been particularly true for motorsports applications. Several race sanctioning
bodies have regulations limiting on-track testing. Beyond regulatory limitations, track testing is
time-consuming, expensive, and carries the risk of damage and injury from crashes.
Manufacturers, racing teams, and drivers alike are turning to simulators to address this limitation.
There are multiple potential uses for simulators in motorsports applications, and simulators
can fall into several categories. Applications can be categorized into driver-in-loop (DIL), also
known as operator-in-loop, or numeric algorithm simulations. The first type, driver-in-loop
simulators, utilize a human driver to provide control input into the simulation program, provide an
environment that mimics a vehicle driving position , and generate visual, audio, and tactile
feedback of the simulated environment to the driver. The second type, numeric algorithms,
numerically solve mathematic representations of vehicle dynamics over time without input from a
human operator. DIL simulators can be further categorized as entertainment, human-factors, or
engineering grade (Ansible, 2016). Entertainment grade simulators are frequently used in personal
applications, utilizing a single personal computer or gaming console, basic controller hardware,
and a cockpit environment ranging from a simple chair in front of a desk to a compact mockup of
a vehicle cockpit on a motion platform. As the name implies this type of simulator is frequently
used for entertainment purposes by providing average consumers the chance to experience motor
racing in a safe, controlled, and affordable environment, but are also used by racing drivers, even
professional racing drivers, as a means to learn cars and tracks and to practice their craft outside
of on-track activity. Human factors simulators are more hardware intensive, typically requiring
advanced computing resources and employing a full-scale mockup of an actual vehicle interior.
The focus of simulators in this category is to provide an environment where driver behavior is
studied, such as distracted driving experiments and evaluating the ergonomics and intuitiveness of
controls. Finally engineering-grade simulators are the most complex category, requiring the
highest level of computing resources to operate extremely detailed vehicle models. The fidelity of
these simulators make them useful in vehicle development where the nuances of design details can
be evaluated for their impact to the driving dynamics and performance of the whole vehicle. They
are also used at the highest level of motorsports to provide the most accurate and immersive
environment for training drivers and a testbed for engineers to rapidly evaluate new component
designs and vehicle setups.
Race Like A Pro Simulators (RLAPS) is a motorsport simulator company with a plan to
bring a different kind of simulation experience to the growing marketplace. The company is
developing a simulator that will fall into the entertainment grade segment for the purpose of
allowing drivers to develop and practice fundamental driving and racing skills in a virtual
environment. The RLAPS simulator hardware consists of a single PC, multiple viewing screens
arranged in a semi-circle, and a cockpit mockup on a motion platform offering three degrees of
freedom: pitch, roll, and yaw. This hardware setup allows drivers to run several simulation
software programs currently available on the market. It is also appropriately scaled to fit into a
home office or small team workshop environment. In addition to hardware, the RLAPS simulator
comes equipped with RLAPS’ own proprietary Simulation Software Program (SSP). Other
simulation programs in this category attempt to replicate real vehicles and tracks and are used by
drivers to practice the nuances of particular cars and tracks ahead of real-world races with the same
combination. The focus of the RLAPS SSP, on the other hand, is to train and evaluate drivers on
the fundamental skills associated with driving a race vehicle. Generic cars and tracks are modeled
for the simulation environment. The driver can choose from a number of scenarios related to
evaluating a particular skill or set of skills. The SSP simulates the driving environment and vehicle
dynamics, while recording metrics related to the driver’s performance. These metrics are
catalogued and evaluated against a set of standards that indicate levels of proficiency in the related
skill area. Drivers and their coaches can review the metrics generated by the SSP to inform their
training program, assess progress, and determine next steps. The RLAPS simulator is aimed
initially at amateur or young racers, particularly those competing in stock cars on paved ovals in
local and regional series. Consequently, the initial RLAPS vehicle modeled for the simulation is
similar to a late model stock car-type race car, and the simulation environments are ¼ mile high
banked and 7/8ths mile flat ovals, layouts typical of tracks raced on by drivers in the target market.
A significant part of the development work for the RLAPS SSP has been related to the
vehicle physics modeling. Although the purpose is to model a generic car rather than a specific
real world car, developing a model that would accurately reproduce the performance and feel of a
typical late model stock car was important in crafting a simulation that would serve as an adequate
tool for driver development. The model would have to behave in a manner that would be
demanding on the driver’s steering, brake, accelerator, and gear shift inputs to highlight the skills
the simulator is attempting to exercise. In order to craft this model, research was undertaken to
develop a complex and comprehensive vehicle modeling outline that would guide the design and
implementation of the vehicle model in the RLAPS SSP.
Problem Statement
Real time, driver-in-loop vehicle simulation is computationally complex. The
computational resource requirements for calculating several equations governing the dynamics and
state of a multi-body vehicle, simulating computer-controlled vehicles in the environment, and
tracking performance metrics are high. The RLAPS SSP must handle those three tasks using a
single personal computer.
The market for driver-in-loop simulators is growing at a high rate. The range of the market
is broad, from racing games on desktop computers and game consoles to professional engineering
grade full scale mockups on complex motion platforms. Sanctioning bodies are further limiting
allowed on-track testing time so simulators are taking on increasing importance in developing
driver skills and helping drivers learn the nuances of particular cars and tracks. As this industry
segment matures, the expectation for increasing realism and fidelity grows.
Improvement in vehicle modeling for simulation benefits the wider automotive and
motorsports industries. Simulations, both numerical and driver-in-loop, can be used to reduce
development time by allowing rapid testing and validation of design concepts. Increasing the
complexity and scope of the vehicle model can allow for greater fidelity and detail in the simulated
outputs, while simplifying the model through assumptions and parameterization can reduce the
computing requirements. The ability to run a high-fidelity simulation with modest hardware,
including a single personal computer, is crucial to expanding simulation to racers below the toplevel professional categories of racing. Lower-level racers are limited in resources but need
extensive practice and coaching to improve their skills, making a small-scale driver-in-loop
simulator an important tool to have at their disposal.
Simulation is multidisciplinary, incorporating learnings from several areas in the
motorsports engineering curriculum, including vehicle dynamics, aerodynamics, data acquisition
and analysis, computer modeling and coding, vehicle control systems, and project management.
This project represents a comprehensive use of the broad range of subjects studied in the course of
earning a Master’s of Technology degree concentrating in Motorsports. It also highlights practical
experience relevant to careers in the motorsports industry. Simulation Engineers are increasingly
in demand by sanctioning bodies, racing teams, constructors, and manufacturers.
Literature Review
Several technical papers, studies, research, and white papers have been published on the
subject of vehicle models, or portions of vehicle models, for simulations. The scope of this
literature is broad, ranging from discussions of full vehicle and track modeling techniques to
detailed analysis of individual subsystems within the vehicle model. Literature also includes
studies with methods of evaluating results from driver-in-loop and numeric algorithm-based
The literature review process began with a search through the Society of Automotive
Engineers (SAE) technical papers archives, using search keywords such as simulation, vehicle
model, and more specific search terms such as tire model. Literature found in this search included
citation of related literature that was reviewed as well. The researcher signed up for distribution of
literature including white papers from other companies involved in the simulation industry. Faculty
in the Motorsports Engineering program in the Purdue School of Engineering and Technology at
Indiana University Purdue University Indianapolis recommended relevant studies and materials.
Finally, additional literature was provided by the partners within RLAPS from their own research
and affiliation with simulation-oriented organizations and trade groups.
A good place to start the literature review was a high level overview of simulator options
and categories, the benefits of utilizing simulation, and the equipment necessary to develop a
simulator. The previously noted categories of entertainment grade, human factors, and
engineering-grade simulators was reinforced by Norfleet, Wagner, Alexander, & Pidgeon (2009)
as well as the range of hardware complexity from a simple desktop computer station to a full range
motion platform with full scale cockpit mockup. The breakdown of the vehicle model into
subsystems including chassis, engine, and transmission, as well as an environment module was
proposed. The review noted higher complexity vehicle models typically require greater computing
resources, with detailed multi-body models found in engineering grade and human-factors grade
simulators typically requiring multiple computers to operate.
One of the primary judgments of the vehicle model created in this project would be the
perception of fidelity held by the drivers operating the simulator. Literature suggests experienced
operators tend to rate the fidelity of simulators poorly. In a test of two flight simulators, one simple
with relatively low fidelity and one complex with high fidelity, test pilots gave higher ratings to
the high fidelity simulator, but both simulators were given low marks across the board (Schreiber,
Bennett, & Gehr, 2006).
Andrzej Nalecz (1989) outlined and demonstrated methods for sensitivity analysis of
vehicle dynamics systems in simulation. These methods allowed the effect of changes to vehicle
parameters on vehicle performance to be analyzed in time and frequency domains. Two
dimensionless functions that enable the comparison and ranking of the influence of parameters on
whole system performance are identified. The methods outlined were most useful for using
numeric simulation algorithms to assist in the design and tuning of a real physical vehicle, but
could provide useful insights for using similar algorithms to assist in the design of a virtual vehicle.
Dols and Pardo offered a comprehensive look at the development and evaluation of a
driver-in-loop simulator for the purpose of driver training (Dols, Pardo, Verwey, & De Waard,
2001). This study touched on software development (in C++), hardware specification, and the
deconstruction of driving into a series of tasks. The project largely mirrored the RLAPS project
but within the context of general on-the-road driving rather than race driving and motorsports
applications. The results of this study indicate the methods and philosophy underlining the RLAPS
concept, and the software and hardware used to implement it, can be successful.
Since motorsport simulation is an established and growing industry, examination of other
simulation software titles can be instructive. iRacing is one of the premier entertainment-grade
driver-in-loop simulation programs commercially available, and is used by many professional
racing drivers. The developers of iRacing documented the process of creating their car and track
models (Kaemmer, Benwick, & Porter, 2008). iRacing has in one sense the opposite objective as
RLAPS. iRacing attempts to recreate real cars and tracks to create a virtual simulation that is useful
for drivers to learn the nuances of particular cars and tracks. RLAPS, on the other hand, is less
concerned about exactly recreating real world cars and tracks as it is in crafting a simulation
environment that emphasizes driving techniques. Still, many of the considerations in crafting a
vehicle model as detailed in this paper are relevant to the development of the RLAPS vehicle
model. The structure of modeling chassis, aerodynamic, powertrain, and tire subsystems and
applying forces to a rigid body model is used in the RLAPS project. The testing and validation
procedure, evaluating not only objective performance metrics but also subjective feel and
behavior, mirrors the validation procedure of this project.
A paper outlined a comprehensive mathematical and computer model for vehicle dynamics
situation (Gim & Nikravesh, 1991). The tire model component was the central and most developed
part of the model. The other components such as steering, suspension, powertrain, and brakes,
were far simpler, not as thoroughly developed, and were used primarily to determine the inputs for
the tire model. The tire model takes into account terrain condition such as physical profile and
friction coefficient, tire geometry and construction especially as relating to tire contact patch, and
empirical data for forces developed related to slips (lateral and longitudinal) and camber.
In simulators intended for driver development, the ability to capture data and metrics for
driver behavior is important, even in non-motorsports applications. One such study looked at
methods for designing simulation software and hardware to accommodate data acquisition at
sufficiently high sampling rates (Campelo, Martí, Pardo, & Serrano, 2006). As with the RLAPS
project considerations for integrated hardware and software compatibility with the data acquisition
method is a primary concern in the study and communication protocols throughout the system are
The study of human-factors and engineering or vehicle dynamics-grade simulators can be
useful in the development of an entertainment-grade simulator by defining an upper-limit of
expectations for the fidelity of the simulator. Although it was intended for human-factors and
hardware-in-loop studies, the simulator of the Korea Automotive Technology institute has many
similar characteristics to those desired for the RLAPS simulator. The vehicle model is a complex
multi-body model with subsystems for powertrain, transmission, brakes, and a Pacejka-based tire
model with parameters tuned by comparing to real-world testing results (Yu, Lee, & Kim, 2007).
This simulator also utilized a motion based and associated control software alongside the
simulation software.
One of the requirements for the vehicle model was to be simple enough to allow the RLAPS
simulation software program to have acceptable computational and rendering performance.
Defining acceptable performance would be necessary for designing to this objective. A study of
graphically-intensive simulation performance in a cloud environment identified 18 frames per
second (fps) as the minimum objective performance standard for such simulations (Benslay,
Osborne, Clapis, Matthews, & Parrish, 2016). Subjective qualities of look and feel were also
discussed, and the observation that more experienced operators are more sensitive to latency was
Factors other than the accuracy and complexity of the vehicle model can impact the
perceived fidelity of a simulator. In a motion-based simulator designed for replicating drift
maneuvers, it was observed that both the type and number of degrees of freedom used in the motion
of the simulator cockpit heavily influenced test driver performance and perception of accuracy
(Nozaki, Shimizu, & Sakuno, 2009). The vehicle model used in this study was created in the preexisting CarMaker simulation software. Tracking of the test-driver’s gaze and eye movements was
used in the study which was an interesting suggestion for further metric tracking and performance
In most of the literature reviewed vehicle models consisted of various sub-systems,
typically some combination of tires, chassis, suspension, powertrain, brakes, and occasionally
aerodynamics. Universally, tires were presented as the primary system through which the vehicle
model is defined. The tires are the primary form of interaction between the vehicle and track
models in the simulation environment. Further study of tire models for simulation was warranted.
Tire models used in driver-in-loop simulations vary from using empirically-based models such as
the Pacejka model to physical models that attempt to derive tire performance from construction
and materials (Heusinkveld, 2012). Brach goes further and identified complex physical models,
simplified physical models, similarity methods, and empirical models. This study notes empirical
models are only truly accurate in steady state conditions and do not model response lag and the
popular friction ellipse concept can underestimate tire forces in combined slip conditions (Brach
& Brach, 2009). Some tire models have been designed that combine physical and empirical models
so that longitudinal and lateral forces and aligning torque are not only functions of slip and camber
as with a typical empirical model, but tire spring rate, pressure, and physical dimensions as with a
physical model (Salaani, 2007).
The purpose of this project was to develop a vehicle model for the RLAPS simulation
program that would meet the technical and marketing objectives for the simulator while working
with the resources allocated to running the software program. This work would assist the
programmers of the simulation software in the design of the program, then evaluate and adjust the
program to achieve the desired characteristics of the simulation.
The assumptions underlying this project were driven by objectives for the RLAPS
simulator. It was assumed simulation software would be developed in Unity 3D game development
software. The simulator would be used for driver training purposes but would fall under the
definition of an entertainment-grade driver-in-loop simulator. A single high-end personal
computer would be used to run the simulation program, record driver metrics, and operate the
hardware including motion platform for the simulator.
The specific application of this simulator is for personal driver training. It is a driver-inloop, motion-based simulator. Although the purpose is for serious driver training, by the
definitions discussed earlier it can best be categorized as an entertainment-grade simulator in terms
of complexity and realism. The simulation must be run using a single personal computer that is
simultaneously controlling the motion platform and providing metrics output, limiting the
computational resources available for handling the vehicle dynamics model. The specifications for
this computer are listed in Table 1 below. The RLAPS SSP was developed in Unity 3D using
resources from an existing driving simulator for non-high-performance-driving as a basis. The
specific scope of this project was to first create a generalized vehicle dynamics algorithm.
Table 1 Computer Hardware Specifications for RLAPS simulator
Power Supply
Intel® Core™ i5-4690K Processor (4x 3.50GHz/6MB L3 Cache)
16 GB [8 GB x2] DDR3-2133 Memory Module - G.SKILL Ripjaws X
Dual AMD R9 380 4GB
24x Dual Layer DVD±R/±RW + CD-R/RW Drive
850 Watt - EVGA SuperNOVA 850 - 80 PLUS Bronze
Microsoft Windows 10 Professional
NZXT Source 210 Mid Tower Case - Blue
Gigabyte GA-Z97X-SLI
Asetek 550LC 120mm Liquid CPU Cooler - Standard 120mm Fan
This project developed an initial generalized vehicle dynamics framework which was then
applied to a simulation program developed in Unity 3D. The limitations of the Unity 3D
development software, the hardware specifications that would run the software, and the additional
tasks, such as controlling motion hardware and tracking metrics, determined the complexity of the
vehicle model that would be implemented. This model reduces vehicle characteristics to scalar
parameters, and neglects some finer details of vehicle modeling such as camber and toe gain for
wheels experiencing vertical articulation in suspension compression, or ride height, yaw, steer,
and roll sensitivity for aerodynamic characteristics. The vehicle model was also designed around
emphasizing skills and tasks for driver practice as part of the RLAPS training methodology, so
direct reproduction of specific real-world vehicle behavior was not the ultimate objective.
The development, tuning, and evaluation of the vehicle model in the RLAPS SSP for this
project was constrained by time, the schedule for RLAPS business deliverables, and resources
available for testing. Versions of the simulation software with parameters open to the researcher
for adjustment and displays indicating lap time and vehicle speed were made available beginning
January 2017. From that time to the scheduled completion of this report it was possible for the
researcher to personally test the software and evaluate it against performance estimates from
numeric lap time simulation software. Real world data and professional test drivers were not
available within the time frame of this project, but are anticipated for later phases of work.
The scope of work and responsibilities for the writer of this report was to advise on the
development of the vehicle physics model for the RLAPS SSP. Tasks included researching vehicle
modeling methods, providing governing equations, feature requirements, and parameter values to
the programming team, adjust parameters in software builds, and evaluate the simulated
performance of the vehicle against objective and subjective criteria.
The timeframe and scope of this project took the vehicle model from a general conceptual
framework, into implementation in a simulation program created in Unity 3D, and concluded with
evaluation and fine tuning of the simulation program against estimates produced from lap time
numeric simulation software. The first stage of the project consisted of providing to the
programmers guidelines describing how a race vehicle behaves and how it can be mathematically
modeled. The literature reviewed earlier and previous numeric vehicle dynamics simulation
algorithms were used to compile and organize a list of governing equations organized as an
algorithm. Accompanying this algorithm, descriptions of the interactions and relationships
between components comprising a whole vehicle system were written. The algorithm and
descriptions were passed on to the programming team to inform how the vehicle in the RLAPS
SSP should be modeled to meet the requirements for the simulation program.
The programming team applied the provided vehicle dynamics outline to the modeling
tools and physics engine in the Unity 3D game creation software. The performance objectives for
the software, the hardware it would be run on, as well as the requirement to track and report various
driver and vehicle performance metrics and operate a motion platform were taken into
consideration. The programming team delivered to the researcher a simulation software program
that featured a vehicle model based on the principles dictated to them crafted using the tools
provided in Unity. The simulation program also included two oval track environments to drive the
vehicle model in, graphics displaying lap time, engine speed, vehicle speed, engaged gear number,
and lap number, rudimentary computer controlled opponent vehicles, and support for the Logitech
G27 steering wheel and pedals peripherals for controlling the vehicle. A file containing several
parameters governing vehicle behavior was included with the simulation program installation. The
researcher could edit the parameters in this document to alter the characteristics of the vehicle
model in the simulation.
While the programming team was developing the simulation software, the researcher
developed simulation models in the Optimum Lap numeric lap time simulation software. This
software uses two-dimensional track models and simple, parameter-driven vehicle models to
estimate an ideal lap time in a given vehicle around a given course without using input from a
human driver. The performance estimates generated by this software were used to generate a
baseline figure to evaluate reasonable, realistic performance against. The researcher acted as an
initial test driver and operated the simulation software. Objective comparisons of the performance
of the simulated vehicle as well as subjective assessments of the feel and character of the vehicle
were compared to the Optimum Lap estimates of vehicle behavior. Vehicle parameters were
adjusted and the results compared again. This process continued until the vehicle performance in
the RLAPS SSP was similar to the performance predicted by Optimum Lap while also behaving
in a manner that would highlight and exercise the skills the RLAPS simulator was intended to test.
Model Development
The vehicle physics model for a real-time driver-in-loop simulation can be developed by
integrating four sub-models: Chassis and Suspension, Powertrain, Aerodynamics, and the Tire
model. The vehicle’s motion and interaction with the simulated environment is primarily acted
through the tire model. All forces that govern the acceleration of a vehicle through space, except
for aerodynamic drag, are ultimately resolved through contact patches formed between the tires
and the track surface. As such, in the vehicle model the chassis and suspension, powertrain, and
aerodynamic models determine the force and geometry inputs into the tire model, which produces
the force outputs that determine vehicle state. The aerodynamic model also produces aerodynamic
drag, which influences vehicle state.
The limitation of using a single personal computer to run the simulation program, including
vehicle modeling, environment modeling, and metrics tacking, dictates the scope of complexity of
the vehicle model. For computational efficiency it is necessary to parameterize the models to a
rather high level. For example, rather than model suspension kinematics in real time to determine
the orientation of the wheel for a given amount of suspension compression or extension, linear
gain values can be used to compute or look up camber and toe values for a given wheel vertical
translation value.
Generalized Vehicle Model
A generalized outline of vehicle dynamics governing equations and relationships was
prepared to guide the programmers as they developed the RLAPS SSP. This outline can then be
used as a reference when applied to specific game development software programs and physics
engines. This model arranged and outlined as an algorithm can be found in Appendix A. Figure 1
illustrates the proposed vehicle model in block diagram form and specifies inputs and outputs for
each sub-system. Interaction with the environment and input from the operator through controls is
also modeled in the diagram.
Figure 1 Block Diagram of proposed generalized vehicle model
Chassis/suspension. The chassis and suspension subsystem contains most of the
parameters relating to the vehicle’s physical dimensions. Mass and mass distribution (laterally,
longitudinally, and vertically), front and rear track and wheelbase and rotational inertia of the
vehicle body (particularly about the vertical axis) are defined here. When forces are generated by
the tires and aerodynamic drag they are resolved with the parameters in the chassis model to
determine accelerations of the vehicle that in turn determine its velocities, rotation, and ultimately
translation in the simulation environment.
Suspension is modeled in terms of spring and damping rates, as well as functions that relate
the kinematics of the suspension geometry to changes in the wheel orientation. Spring and damping
rates include what is traditionally regarded as the suspension spring, whether that is in the form of
a coil, flexure (cantilever) or torsion spring, anti-roll bars, third springs, the spring rate of the tire
at its current temperature and pressure, and even the stiffness of the chassis itself. All of these
springs are to be resolved with motion ratios so as to produce spring rates at the wheel, where
wheel rate is given as 𝑊𝑅 = 𝑀𝑅2 where SR is the nominal spring rate and MR is the motion ratio,
or ratio of spring compression to wheel vertical translation. As the wheel experiences vertical
translation in bump (compression) and rebound (extension) there could be corresponding changes
in the wheel’s camber or inclination angle and toe induced by the suspension kinematics. These
changes can be quantified in terms of gain values, such as a certain degree of camber gained for
every unit of suspension compression, for the sake of computational efficiency. These spring and
damping forces, reacting to weight transfer in the chassis model, and the wheel orientation
parameters, are sent as inputs to the tire model.
Powertrain. Powertrain is a relatively simple system to model. Maximum engine torque
available is a function of engine speed and this function can be found by curve fitting results from
real world dynomometer tests. The actual torque delivered can be determined by mapping throttle
position to a percentage of maximum power available. A gear stack is defined by specifying a list
of gear ratios. The selected gear ratio is determined from a simple control input, and this ratio is
multiplied by the final gear ratio found in the differential or rear end of the powertrain. The
effective combined gear ratio acts as a multiplier for the torque and rotational speed translated to
the driven wheels. Efficiency ratios can be applied at the engine, transmission, driveshaft, and
differential, and all losses are regarded as torque losses. Brakes can also be placed in the powertrain
module, mapped as a percentage of maximum brake torque (a fixed parameter) to brake pedal
position. Brake torque acts as a torque on all tires in the opposite direction as drive torque.
Aerodynamics. The aerodynamics model produces as outputs drag and lift or downforce.
Drag is a force applied directly to the body of the vehicle and acts in opposition to the longitudinal
forces generated by the tires in acceleration, or in conjunction with tire longitudinal forces in
braking. Lift or downforce acts as an input into the tire model, combining with vehicle weight to
determine the vertical or normal force on each tire. This force can be distributed on the front and
rear axles as defined by a balance percentage. Aerodynamic downforce and drag are calculated,
respectively, by:
𝐿 = 1/2𝜌𝐶𝐿 𝐴𝐹 𝑉 2
𝐷 = 1/2𝜌𝐶𝐷 𝐴𝐹 𝑉 2
Where L is downforce, D is drag, ρ is the air density, Af is the frontal area of the vehicle, and V is
the vehicle’s velocity. Air density is determined by the simulation environment/track model,
frontal area is a fixed parameter of the car, and velocity is determined from the vehicle state.
Coefficients of downforce and drag can be fixed parameters but are more accurately functions of
front and rear ride height, yaw, pitch, and roll, as given by the chassis and suspension model. The
coefficients can be computed from lookup tables by those parameters or by a multidimensional
fitting equation. The coefficients can also depend on adjustments to the aerodynamic settings of
the vehicle such as wing angle and gurney flap heights, if such adjustment factors are made part
of the simulation setup.
Tire. The tire model is the most complex system in the vehicle model and has the most
direct impact on the vehicle’s interaction with the simulation environment. The tires generate the
lateral and longitudinal forces that create accelerations in the vehicle state. They also generate an
aligning torque that provides feedback into the steering controls for the driver. As discussed in the
literature review there are several possible methods for modeling tires. Proposed for this project is
a method based off of a typical empirical model such as the Pacejka model (Bakker, Nyborg, &
Pacejka, 1988). Tire forces are primarily functions of slip and vertical (normal) tire loads.
Longitudinal forces are determined by slip ratio, or the difference between the velocity of the tire
carcass, or the rotational velocity of the tire translated into linear velocity, and the velocity of that
tire through space. Lateral forces are determined by slip angle, or the angle between the tire’s
heading and direction of travel. Empirical curve fits can be used to find tire friction coefficients as
a function of slip angle, slip ratio, or combined slip. The product of the resulting coefficient and
the normal load on the tire is the force produced. However, this coefficient can also be saturated
by vertical load, so every unit increase in vertical load also results in a decrease to the friction
coefficient. Toe or bump steer from the suspension model influences slip angle, while camber can
be considered an additive force. Camber force can be found as a simple function of a tire’s camber
angle, again through an empirical curve fit, and added to lateral force produced by slip.
RLAPS Vehicle Model
The programming team developing the RLAPS simulation software program took the
proposed generalized vehicle dynamics model and attempted to implement it within the specific
framework of the Unity 3D game creation software. Early on computing and rendering
performance were found to be the major limiting factors in what could be accomplished with
implementing the vehicle model. A complex model adhering closely to the proposed model was
found by the programmers to require too much memory and slowed the simulation down to an
unacceptable level. A simplified model that utilized the Rigidbody and Wheel Collider
components within the unity framework and relied heavily upon Unity’s built-in physics engine
was found to perform acceptably. This model was heavily parameterized, and a list of the
parameters made available to tuning by the researcher and their initial as-delivered values is found
in Appendix B
Chassis/suspension. The chassis is modeled using the Unity Rigidbody component. This
component defines the vehicle’s mass, and can be offset laterally, longitudinally, and vertically. It
represents a solid single mass that can be located within the dimensions of the wheelbase and track.
A massless mesh collider creates the actual physical dimensions of the vehicle for collision
detection purposes. The suspension allows for vertical translation of the wheels. Spring and
damping rates determined at the wheel are parameterized. Each wheel is given its own mass so
unsprung mass and a multibody approach can be executed. The tire also has its own damping rate,
and suspension travel range is defined. The wheels articulate vertically in response to the terrain
and impart forces through the springs and dampers onto the sprung body mass. Steering geometry
including maximum steer angle and relation to control input are defined.
Powertrain. The powertrain model incorporated into the RLAPS SSP differs slightly from the
proposed model. A maximum torque value is defined, and then ranges of engine speed are assigned
a percentage of maximum torque output. Maximum and minimum engine speeds are set. A vector
containing gear ratios including reverse and neutral is defined and used as a multiplier. At any
given point in the simulation the torque to the driven wheels is a product of the maximum torque
value, the engine speed, and the gear ratio. A top speed limiter can be set so torque to the drive
wheels is interrupted at the maximum allowed speed. Traction control which limits drive torque
based on wheel slip is also optional, and the degree of interference can be adjusted.
Aerodynamics. The aerodynamics model in the RLAPS SSP remains an underdeveloped area. As
the intent was the model a late model stock car type vehicle, aerodynamics are not a primary driver
of performance but this is an area that will need further development if more aerodynamic21
dependent vehicle types are modeled. The programming team was not able to explain how the
Unity physics engine models aerodynamic forces. A downforce value and drag coefficient are
present in the vehicle parameters but it is not understood how these are applied to the vehicle, their
dependency on any other factors. Since downforce is given as a singular value and not a coefficient,
it is not known if this value is speed dependent or a constant.
Tire. Unity uses a feature called “Wheel Colliders” to model vehicle tires and it is a robust
slip-based tire model combined with collision detection and wheel physics ("Unity Manual Wheel Collider," 2017). Friction is computed separately from the rest of the physics engine, using
a slip-based method similar to a Pacejka model. A tire friction coefficient is determined from a
spline fit relating friction coefficient to slip, both laterally and longitudinally. This spline is defined
by two developer-set points, one for what is referred to extremum, the other referred to as the
asymptote point. Extremum is the point of maximum friction coefficient and asymptote is the point
where the decline in friction coefficient at higher slip values begins to level off. Both points are
defined by their corresponding slip and friction values. The plot begins at point (0,0) and the slope
of the tangent at both extremum and asymptote points is zero. The spline curve associated with the
Wheel Collider model is illustrated in Figure 2. Accompanying both lateral and longitudinal
friction/slip curves is a stiffness value that acts as a global multiplier for the curve at all slip values.
This value can be modified with code at run-time to have a continuously varying stiffness
multiplier. This could be used to apply varying levels of track surface friction depending on the
wheel’s physical location on track. Attempts were made to use this feature to vary stiffness based
on normal load so as to model load saturation characteristics of tires, but this resulted in unstable
code so the attempt was put on hold for later development.
Figure 2 Illustration of the spline fit used in the Unity Wheel Collider model
Optimum Lap Models
After the RLAPS SSP vehicle model was developed, the testing plan called for it to be
evaluated against data produced by the Optimum Lap simulation software. In order to form this
data, vehicle and track models were created in the Optimum Lap software by specifying vehicle
parameters and track physical dimensions. The Optimum Lap software has pre-made track and
vehicle models available for use so these were tested to gauge the accuracy of the software.
Optimum Lap is a mathematical vehicle dynamics simulation (OptimumG, 2017). Rather
than a driver-in-loop simulation, it removes the driver as a variable, and produces an estimate of
vehicle state and dynamic performance over a lap of a specified race course. This estimate can be
regarded as a mathematical ideal, what would be achievable from a perfect driver operating the
vehicle to its fullest dynamic potential at all times over the course of a lap.
Optimum Lap utilizes a number of key simplifications in its modeling which represent
some limitations to the estimates it produces. It utilizes a point model for the vehicle, so wheelbase,
track, suspension, and weight transfer are neglected. The track models are two dimensional, so
elevation and banking are unable to be represented. It assumes braking is traction limited. In this
assumption the brakes are strong enough to lock up the tires at any time, so the limitation to braking
performance is the grip of the tires, not the capacity of the brakes. In light of these limitations, the
estimates produced by Optimum Lap are regarded as being accurate to within 10% of real-world
After verifying the accuracy of the Optimum Lap software using existing assets, a vehicle
model was made from scratch to represent the late model stock car set up for a short oval as will
be used in the RLAPS Simulation Software Program (SSP). A “mass” (weight) of 2,200 pounds
was specified because that is the minimum weight required by regulations in several sanctioning
bodies. A drag coefficient of 0.45 is typical of a stock car, and an initial downforce coefficient of
0.55 was selected as a reasonable estimate that would minimize the influence of aerodynamic
downforce. Frontal area for the vehicle is estimated at 17.8 square feet based on a straight-on front
view photograph of a representative stock car used in RLAPS promotional material, imported into
a CAD program, outlined, and measured. Air density of 0.078 pounds per cubic foot is
representative of standard temperature and pressure.
Tire radius is about 1 foot in this category of car. A coefficient of rolling resistance can
depend on inflation pressure, vehicle velocity, driving surface, and slip angle, but this software
requires a single coefficient so an average value of 0.015 was selected based on data provided by
Hucho (Hucho, 1998). Longitudinal and lateral friction coefficients are also dependent on several
factors but since a single value is required for each and this simulation assumes maximum
performance at all times, peak friction coefficients from representative tire data are selected. From
sample data found in Milliken peak coefficients for short oval stock car tires can be determined to
be around 1.250 for longitudinal friction and 1.350 for lateral friction (Milliken & Milliken, 1995).
Engine data is provided by specifying torque outputs at engine speed intervals. Engine
dynomometer results of a typical Chevrolet crate engine were used to fill this table. Optional
modifiers for thermal efficiency and fuel energy density were ignored because this level of detail
is unavailable and perhaps too nuanced for this early stage of testing. A two speed sequential
gearbox is initially set with a 1.26 ratio for first gear, 1.00 ratio for second gear, and 3.80 for final
drive, but these ratios are typically changed as a common tuning parameter for specific tracks and
conditions, so these can be revisited based on testing data. Drivetrain efficiency was set to 100%
Since banking and elevation are not considered in Optimum Lap, the RLAPS ¼ mile high
banked oval track is not suitable for testing. Metrics tracking has also not yet been implemented
for this track in the RLAPS SSP. The 7/8 mile flat oval is however perfectly suited to the
requirements of Optimum Lap and includes some early implementation of metrics tracking at the
current release build in the RLAPS SSP. This track was recreated in Optimum Lap based on
specifications provided in Appendix C of the RLAPS System Design Document. The track consists
of two 1,154.86 foot long straights, with one straight divided into two equal-length sections to
create a start/finish line midway down the straight, and two 1,154.86 foot long corners with radii
of 367.454 feet. The track is divided into four sectors, with sector one beginning at the start/finish
line and ending at the midpoint of the first turn. Sector two begins at the midpoint of the first turn
and ends at the midpoint of the back straight. Sector three begins at the midpoint of the back
straight and ends at the midpoint of the second turn. Sector four begins at the midpoint of the
second turn and ends at the start/finish line. A diagram of the track including locations of the
start/finish line and sector dividers is included as Figure 3.
Figure 3 Diagram of Optimum Lap model representation of RLAPS Generic 7/8 mile flat oval track layout showing start/finish line
and timing segments
Results and Discussion
The first step in the validation process was to produce results in Optimum Lap to serve as
the target for RLAPS SSP vehicle model performance. To verify the accuracy of Optimum Lap, a
test simulation was conducted with pre-fabricated vehicle and track models from Optimum Lap’s
library and compared to real world results. The vehicle selected was a generic LMP2 style car and
the track was the Autodromo Internazionale Enzo e Dino Ferrari, also known as Imola. With this
combination Optimum Lap estimated a lap time of 81.82 seconds. This was compared to
qualification results from the 2016 4 Hours of Imola European Le Mans Series event. In these
results the fastest qualifying time for an LMP2 car was 93.78 seconds. The difference between the
two is 11.96 seconds or 12.75%. The average speed estimated by Optimum Lap is 226.89 km/hr
while the real world average speed was 188.2 km/hr. This represents a 17% difference, indicating
a source of error in the Optimum Lap model may be the length of the track. If the track length was
accurate, the average speed would have the same margin of error as lap time.
A second test was performed to more closely match the conditions RLAPS will attempt to
simulate. A generic stock car was chosen from Optimum Lap’s vehicle library. The lone oval in
the Optimum Lap track library, Indianapolis Motor Speedway, was selected for the location. Since
Indianapolis is a relatively flat, low-banked track, the limitations of the two-dimensional constraint
are minimized with this track. An initial run indicated the stock configuration of the car resulted
in the car operating at its maximum engine RPM for the entire lap, an unrealistic behavior.
Lowering the final gear ratio resulted in a simulation that changed engine speed over the course of
the lap. With no other changes, an estimated lap time of 50.78 seconds was produced. The pole
time for the most recent NASCAR Brickyard 400 race at the same venue was 48.745 seconds. The
difference in lap times for this test is 4.1%. Without knowledge of the exact aerodynamic
configuration, engine performance, and gearing of the pole-winning car the parameters of the car
selected from the library cannot be tuned to produce a more accurate result, but this test indicates
the estimate of a 10% margin of error is reasonable.
After setting up the RLAPS Stock Car and 7/8 Mile Flat Oval models, a simulation was
run using these vehicle and track parameters. According to Optimum Lap, a perfect driver can
achieve a lap time in this car on this track from a flying (as opposed to standing) start of 33.00
seconds. The car is in second gear for the entire lap. It reaches a top speed of 103.34 miles per
hour before braking before the entry into the first turn down to a steady cornering speed of 89.66
miles per hour. It accelerates back up to maximum speed on the exit of the turn. Average lap speed
is 95.89 miles per hour. A track map in Figure 4 with a color overlay indicating speed illustrates
the vehicle’s speed over the lap as described. These are the metrics the RLAPS SSP can currently
report. Optimum Lap braking and acceleration points, as well as cornering speed, appear
reasonable. The results from the Optimum Lap simulations are presented in Table 2 along with
comparisons to their real-world counterpart results.
Table 2 Results from testing of Optimum Lap software and initial test of RLAPS simulation. Note: "Real World" values for RLAPS
car represent performance from RLAPS SSP
Real World
Lap Time (s)
Lap Time (s)
% difference
Real World
Speed (km/hr)
% Difference
OL [email protected]
Stock Car
Flat Oval
Figure 4 Track Map output from Optimum Lap with color range overlay indicating speed at any point on track.
After adjusting the vehicle model in the RLAPS SSP to match the Optimum Lap vehicle
model, the researcher acted as test driver to gauge the initial match of the RLAPS SSP to the results
predicted by Optimum Lap. The test driver is not a professional driver, but has competitive driving
experience in karting and autocross, computer simulated driver-in-loop simulator experience, and
motorsports engineering background.
In the initial test using vehicle parameters that should closely mimic the Optimum Lap
model, the test driver produced lap times of 33.00, 32.66, and 32.62 seconds. Although these times
compared favorably to the times predicted by Optimum Lap, the nature in which they were
achieved was not similar. The RLAPS SSP did not require any braking on corner entry, nor any
steering correction to manage slip. The accelerator could be fully depressed the entire time, and
slight constant steering input for each turn. The initial assumption of the test driver is that the
RLAPS SSP exhibits too much lateral grip, not enough longitudinal grip, and no perceptible body
motion cues. During this test the on-screen simulated speedometer did not work, so no estimates
of velocity at various points on the track are available. A subsequent software build included a fix
for the frozen speedometer. Testing again with the new software build and repaired speedometer,
similar lap times (32.52 to 32.72 seconds) were produced with the same driving character of full
throttle throughout the lap and neutral handling requiring no correction to steering. The
speedometer indicated a relatively constant speed between 95 and 100 miles per hour throughout
the lap, with minor amounts of speed scrubbed off in the turns and regained down the straights.
From the initial impressions that the RLAPS SSP vehicle demonstrated too much lateral
grip relative to longitudinal grip and forward thrust, attention turned to adjusting the parameters
that would have the most direct impact on those factors. First, adjustments were made to the lateral
and longitudinal stiffness values that were multipliers for the whole slip/friction spline curve.
Longitudinal gain was increased from 2 to 3 and lateral gain was decreased from 10 to 8. The result
was slower lap times, ranging from 33.1 to 33.3 seconds, but the same speed, handling, and
controls characteristics as before. The car could still be driven at full throttle for the entire lap. The
next adjustment disabled completely the steering help and traction control driver aids. Again this
reduced lap times to the range of 34.1 to 34.7 seconds without changing the subjective character
of the vehicle model’s behavior. Further, more severe adjustment was made to the stiffness values,
increasing longitudinal gain value to 6 and reducing lateral gain value to 4. This change was made
to see if dramatic changes in the tire model would produce results in the desired direction. With
this change lap times fell to a range of 30.9 to 31.7 seconds. The behavior changed slightly as well.
The throttle was still at 100% for the entire lap, but higher steering inputs could induce slight
oversteer behavior, the car in general felt edgier and more difficult to control, and the speed
differential was greater, reaching a maximum of 120 miles per hour before the turns, and scrubbing
speed down to 80 miles per hour in the turns without braking.
The results from the previous changes strongly indicated tire parameters had a significant
influence on both objective performance measures and subjective evaluation of feel. The
parameters for the Wheel Collider model were comprehensively altered from the initial values set
by the programmers, setting stiffness to 1 and extremum values to 1.2 for both lateral and
longitudinal. Longitudinal asymptote value was set to 0.8 occurring at slip of 0.8, lateral value to
0.5 occurring at slip of 0.5. Longitudinal extremum was set to occur at 0.2 slip, lateral extremum
at 0.4 slip. These values more closely aligned with the inputs for the Optimum Lap model and with
the data found in Milliken. With these tire settings lap times rose to a range of 37.3 to 38.1 seconds.
Maximum speed reached 120 miles per hour. Now the throttle could not be maintained for the
entire lap, and indeed significant braking down to 70 miles per hour entering the turns was
required. Throttle could be applied mid-turn, raising speed to 80 miles per hour. The handling
behavior of the car comprehensively changed. On power the car exhibited pronounced oversteer
tendencies, while tending to understeer off power. Significant steering adjustment was required to
navigate a turn, and near spins could be recovered from with generous amounts of counter-steer.
These character changes brought the car much closer to the goals the RLAPS SSP is
intended to meet for driver development, but performance lagged on objective standards and the
car was in a way too difficult to drive, with many spins and crashes encountered on the way to
setting representative lap times. To address both of these issues, stiffness was increased globally
to 1.5 laterally and longitudinally. This lowered lap times to a range of 34.8 to 35.1 and made the
car easier to control. Maximum torque was increased from 200 to 300, which actually increased
lap time to 35.2 to35.7 second range. The car became difficult to drive and produce clean
sustainable laps. Maximum speed rose to 130 miles per hour. The lateral asymptote slip value was
then moved to 0.7, broadening the usable grip curve and making the transition from grip to sliding
less severe. This made the car much more forgiving, controllable, and easier to recover from nearspins. Lap time fell to a range of 31.4 to 33.1 seconds. Finally, maximum torque was reduced to
275, which lowered lap times to 32.4 to 34.9 second range. This setup produced a car that required
significant braking before corner entry, modulation of throttle and steering through the corner, and
smooth transition back to full power exiting the corner.
The results of each evaluation following a parameter change are summarized in the table
below. The quality of the change, whether positive (better) or negative (worse), toward the
objective and subjective criteria are specified. For example, a change that resulted in lap times and
speeds that were closer to the values predicted by Optimum Lap than the previous test, but behaved
with a less realistic feel and character would be classified as positive for objective criteria and
negative for subjective criteria.
Objective Results
Subjective Results
Initial parameters as
Mixed – lap time
positive, speeds
Decrease tire lateral
No change
No change
No change
and increase
longitudinal stiffness
Turn traction control
and steering help off
Decrease tire lateral
Mixed – lap time Positive
and increase
longitudinal stiffness
Change all tire
Mixed – lap time Positive
parameters to match
real world data
Increase lateral and
longitudinal stiffness
Increase maximum
Increase lateral
asymptote value
Decrease power to
level still greater than
initial value
The RLAPS Simulation Software Program was intended to serve as a driver training and
development tool by allowing drivers to practice and receive feedback on fundamental driving
skills such as braking upon corner entry, managing a line through a turn, consistent braking and
throttle points, managing wheel spin and vehicle yaw, and combining these skills to complete a
fast lap without spinning out or crashing. In order to allow drivers to practice these skills while
performance metrics were recorded and motion and control feedback was provided to inform the
driver, the simulation program required a high fidelity but low complexity vehicle physics
model. The model produced as outlined here using tools provided in Unity 3D was designed to
meet objective performance requirements. After some initial tuning the vehicle performed in
such a way that highlighted the exact skills that a driver using this simulator would want to
exercise. The vehicle model meets the purpose for which the simulator was intended.
Evaluation and development does not end with this phase of the project. A major
limitation of this study was a lack of objective real world data to measure the simulator’s true
fidelity against. Only one non-professional test driver was used. Further testing will involve
multiple professional test drivers and real data from actual late model stock cars on
representative tracks. From the existing model, beyond fine tuning based on the feedback from
test drivers and data, further development can focus on renewing attempts to add details such as
load saturation and camber force to increase the accuracy of the already refined tire model.
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Appendix A Proposed Algorithm to Programmers
Model Parameters
Coordinate System for this algorithm
o X: oriented parallel to car’s longitudinal centerline, from front of car to back, positive
towards front
o Y: oriented parallel to car’s lateral centerline, from left to right, positive towards right
o Z: oriented vertically from car, positive upward
o Origin at vehicle’s center of gravity
o cD: 0.45
o cL: 0.55
o A: 17.85 ft2
o Aerodynamic forces act at vehicle Center of Gravity
o Torque Curve
 𝑇𝑒 = 5 ∗ 10−9 𝑛𝑒3 + 4 ∗ 10−5 𝑛𝑒2 − 0.0775𝑛𝑒 + 383.13
o Linear throttle ie Effective Torque = Throttle %*Te.
o Max RPM 5500
o Ie=.7 Ia=20 (includes both rear wheels) in lb/ft2
o Gears:
 Neutral: 0
 1st: 1.26
 2nd: 1.00
 Final: 5.88
o On/off clutch – clutch either 100% engaged, or 0% engaged if clutch depressed past
some cutoff point
o With no load (clutch disengages) engine deceleration due to friction and air pumping is
1,000 RPM/sec
o Max force: 4000 lbs
o Tires have 1 ft radius, so max torque = 4000 lb/ft
o Linear braking ie Effective force = brake%*max braking force
o Initial Bias 60% front, 50-50% left-right
o Weight distribution
 55% front
 50-50 left-right distribution
 20 inch height of COG
 Iyaw=790 lb/ft2
o Dimensions
 103 inch wheelbase
 66 inch track front and rear
 2850 pounds
 Assume 0 degree toe and camber
 Rigid suspension
24 inch effective diameter
No growth model
cRR – 0.055
Iw=9 lb/ft2
Note: Any subscript “xx” denotes a value that would be calculated for each individual wheel. For example
if there was some value X that would have a value individually calculated for each wheel, it would
appear as Xxx. If fully written out, this value would be calculated as XFL for the front left wheel, XFR for the
front right wheel, etc.
Determine Drive and Braking Torque to wheels
o Engine Torque
 Terms
 Ne=engine speed
 TP=throttle position percentage. Throttle pedal at least depressed
position TP=0.1 (may adjust to create acceptable idle speed), Throttle at
fully depressed position TP=1
 Te=engine torque produced
 If ne<nemax
 𝑇𝑒 = 𝑇𝑃 ∗ (5 ∗ 10−9 𝑛𝑒3 + 4 ∗ 10−5 𝑛𝑒2 − 0.0775𝑛𝑒 + 383.13)
 If ne>nemax
 Te=0 for 0.05 sec (may adjust timeframe, see what feels like a typical
real-life rev limiter)
o Equivalent Drivetrain Inertia
 Terms
 Ie=engine inertia
 Ia=drivetrain inertia including rear wheels
 Z=overall gear ratio = selected gear (1st, 2nd, neutral) ratio * final drive
𝐼𝑒𝑞 = 𝐼𝑒 + 𝑍𝑎2
Braking Torque
 Terms
 BP=brake position percentage. Brake pedal at least depressed position
BP=0, Brake at fully depressed position BP=1
BD= brake distribution as a percent of braking force apportioned to the
front brakes. For example, a front brake bias of 60% would result in a
 Tb=braking torque produced
 Front Brake Torque
 TbFX=0.5*BD*BP*Max Brake Torque
 Rear Brake Torque
 TbRX=0.5*(1-BD)*BP*Max Brake Torque
Net Torque to rear wheels
 Term
 Tw is the net torque to the rear wheel
 𝑇𝑤 = 𝑇𝑒 − (𝑇𝑏𝑅𝐿 + 𝑇𝑏𝑅𝑅 )
Engine Speed and Acceleration
 Terms
 𝑛̇ 𝑒 is engine speed acceleration
 FXR is the net longitudinal force produced by the rear tires
 rt is the effective radius of the rear tire
 δt is the time frame between calculations.
 fe is the engine deceleration due to friction and air pumping when clutch
 Engine Speed Acceleration
𝑛̇ 𝑒 = (𝑇𝑤 − ((𝐹𝑋𝑅𝐿 + 𝐹𝑋𝑅𝑅 ) 𝑡 )) ∗ (
If clutch depressed past cutoff point, 𝑛̇ 𝑒 =
− 𝑓𝑒
New Engine Speed
 𝑛𝑒 = 𝑛𝑒 + 𝑛̇ 𝑒 ∗ 𝛿𝑡
Determine Wheel Speeds
o Terms
 VX is the velocity of the vehicle in the X axis direction
 ωw is the wheel’s rotational velocity
 δs is the steering angle, the angle between the front wheel longitudinal
centerline and the vehicle X axis, positive counterclockwise
 Iw is the inertia of the individual front wheel
 Vc = circumferential velocity of tire tread base, function of rotational velocity,
defined as 𝑉𝑐 = 𝑟𝑡 𝜔𝑤
 ac = circumferential acceleration of velocity of tire tread base
o Front wheels
 Free Rolling
 If no braking force applied (TbFX=0), assume front wheels are free rolling
with no longitudinal slip
𝑉𝑐𝐹𝑥 = cos𝑋𝛿
If braking force applied, brake torque creates acceleration (negative) for
acFx =-(TbFX/Iw)
Rear Wheels
 If clutch engaged
𝑛𝑒∗𝑟 ∗𝜋
 𝑉𝑐𝑅𝑥 = 30𝑍𝑡
If clutch disengaged and brakes applied
 acRx =-(TbRX/0.5Ia)
 VcRx=VCRx+acRxδt
 If clutch disengaged and no brakes, rear wheels free rolling
 𝑉𝑐𝑅𝑥 = 𝑉𝑋
 VcRx=VXRx
Determine Wheel Slip
o Traction
𝑉𝑐𝑥𝑥 −𝑉𝑋
𝑠𝑥𝑥 =
𝑉𝑋 −𝑉𝑐𝑥𝑥
𝑠𝑥𝑥 =
Determine Slip Angle
o Terms
 α is the slip angle, the angle between the wheel’s longitudinal centerline and
the wheel’s direction of travel. Positive counterclockwise
 VY is the velocity of the vehicle in the Y axis direction
 ωyaw is the angular velocity of yaw
 b is the distance from the front wheel centerline to the center of gravity of the
 a is the distance from the rear wheel centerline to the center of gravity of the
𝑉𝑌 +𝜔𝑦𝑎𝑤 𝑏
Front 𝛼𝐹𝑥 = tan−1 (
Rear 𝛼𝑅𝑥 = tan−1
) − 𝛿𝑠
𝑌𝑌 −𝜔𝑦𝑎𝑤 𝑐
Determine tire friction coefficients
o Terms
 Wxx is the normal force load on the individual wheel
 µXxx is the lateral friction force coefficient developed by the wheel
 µXxx is the lateral friction force coefficient developed by the wheel
o 𝜇𝑋𝑥𝑥 = (48.685𝑠𝑥𝑥
− 139.31𝑠𝑥𝑥
+ 148.85𝑠𝑥𝑥
− 73.17𝑠𝑥𝑥
+ 16.17𝑠𝑥𝑥 ) − (0.25𝛼𝑠) −
(0.00105𝑊𝑥𝑥 𝑠)
o 𝜇𝑌𝑥𝑥 = −(0.0007218𝛼𝑥𝑥
− 0.02728𝛼𝑥𝑥
+ 0.3232𝛼𝑥𝑥 ) + (0.3𝛼𝑠) + (0.000035𝑊𝑥𝑥 𝛼)
Determine Forces
o Individual wheels
 Longitudinal
 Front: 𝐹𝑋𝐹𝑥 = cos 𝛿𝑠 𝜇𝑋𝐹𝑥 𝑊𝐹𝑥
 Rear: 𝐹𝑋𝑅𝑥 = 𝜇𝑋𝑅𝑥 𝑊𝑅𝑥
 Lateral
 Front: 𝐹𝑌𝐹𝑥 = sin 𝛿𝑠 𝜇𝑌𝐹𝑥 𝑊𝐹𝑥
 Rear: 𝐹𝑌𝑅𝑥 = 𝜇𝑌𝑅𝑥 𝑊𝑅𝑥
Total Resultant
 Longitudinal
 𝐹𝑋 = ∑ 𝐹𝑋𝑥𝑥
 Lateral
 𝐹𝑌 = ∑ 𝐹𝑌𝑥𝑥
Aligning torque front wheels
 𝑀𝑧 = (−17.798𝛼𝐹2 + 147.68𝛼𝐹 )(𝑊𝐹𝐿 + 𝑊𝐹𝑅 ) + (0.00001(𝑊𝐹𝐿 + 𝑊𝐹𝑅 )𝛼)
Determine Aerodynamic forces
o Terms
 ρ is the density of air
 CD is the drag coefficient
 CL is the downforce coefficient
 Af is the frontal area of the vehicle
o 𝐷 = 0.5𝜌𝐶𝐷 𝐴𝑓 𝑉 2
o 𝐿 = 0.5𝜌𝐶𝐿 𝐴𝑓 𝑉 2
Determine Acceleration
o Terms
 aX is longitudinal acceleration of the vehicle
 aY is lateral acceleration of the vehicle
 crr is the coefficient of rolling resistance
 W is the weight of the vehicle
 g is acceleration due to gravity (ie 32.2 ft/s2)
o 𝑎𝑋 = (𝑊 (𝐹𝑥 − (𝑐𝑟𝑟 ∗ (𝑊 + 𝐿)) − 𝐷))
𝑎𝑌 = (𝑊 𝐹𝑦 )
Determine Velocity
o 𝑉𝑋 = 𝑉𝑋 + 𝑎𝑋 𝛿𝑡
o 𝑉𝑌 = 𝑉𝑌 + 𝑎𝑌 𝛿𝑡
Determine Sideslip
𝛼𝐹 = 2 (𝛼𝐹𝐿 + 𝛼𝐹𝑅 )
Β = tan−1 𝑉𝑦
Turn Radius and yaw
o Terms
 l is the wheelbase
 R is the radius of the turn
 ωyaw is the yaw angular velocity
 𝜔𝑦𝑎𝑤
̇ is the yaw angular acceleration
 Iyaw is the yaw moment of inertia of the vehicle
o Low Speed 15 MPH or less
sin 𝛿𝑠
Turn Angular Velocity 𝜔𝑦𝑎𝑤 =
No slip angle at low speed so no lateral velocity is produced. Turn is simply a
function of geometry and the changing direction of the vehicle’s longitudinal
velocity vector relative to the track
High Speed
 Yaw
 Torque
o Front: 𝑇𝑦𝑎𝑤𝐹 = cos 𝛿𝑠 (𝐹𝑌𝐹𝐿 + 𝐹𝑌𝐹𝑅 )𝑏
o Rear: 𝑇𝑦𝑎𝑤𝑅 = −(𝐹𝑌𝑅𝐿 + 𝐹𝑌𝑅𝑅 )𝑐
o Net: 𝑇𝑦𝑎𝑤 = 𝑇𝑦𝑎𝑤𝐹 + 𝑇𝑦𝑎𝑤𝑅
 Acceleration
𝜔𝑦𝑎𝑤̇ =
o 𝜔𝑦𝑎𝑤 = 𝜔𝑦𝑎𝑤 + 𝜔𝑦𝑎𝑤
̇ 𝛿𝑡
Vehicle position and orientation
o Terms
 X is the vehicle’s X coordinate on the track’s coordinate system
 Y is the vehicle Y coordinate on the track’s coordinate system
 RZ is the vehicle’s rotation about the Z axis on the track’s coordinate system
o 𝑋 = 𝑋 + 𝑉𝑋 𝛿𝑡
o 𝑌 = 𝑌 + 𝑉𝑌 𝛿𝑡
o 𝑅𝑍 = 𝑅𝑍 + 𝜔𝑦𝑎𝑤 𝛿𝑡
Weight Transfer
o Terms
 t is the track
 h is the height of the center of gravity
o Static
Front distribution 𝑊𝑆𝐹𝑥 =
Rear distribution 𝑊𝑆𝑅𝑥 =
Load transfer
Longitudinal: Δ𝑊𝑙𝑜𝑛𝑔 =
Lateral: Δ𝑊𝑙𝑎𝑡 =
𝑊𝑎𝑋 ℎ
𝑊𝑎𝑌 ℎ
Downforce application
Front 𝐿𝐹𝑥 =
Rear 𝐿𝑅𝑥 =
Final Distribution
 𝑊𝐹𝐿 = 𝑊𝑆𝐹𝐿 − Δ𝑊𝑙𝑜𝑛𝑔 − Δ𝑊𝑙𝑎𝑡 + 𝐿𝐹𝐿
 𝑊𝐹𝑅 = 𝑊𝑆𝐹𝑅 − Δ𝑊𝑙𝑜𝑛𝑔 + Δ𝑊𝑙𝑎𝑡 + 𝐿𝐹𝑅
 𝑊𝑅𝐿 = 𝑊𝑆𝑅𝐿 + Δ𝑊𝑙𝑜𝑛𝑔 − Δ𝑊𝑙𝑎𝑡 + 𝐿𝑅𝐿
 𝑊𝑅𝑅 = 𝑊𝑆𝑅𝑅 + Δ𝑊𝑙𝑜𝑛𝑔 + Δ𝑊𝑙𝑎𝑡 + 𝐿𝑅𝑅
Appendix B Parameters of RLAPS Vehicle Model From Unity
0 Mass = 1800
1 Drag = .05
2 Angular Drag = 1
3 Center of Mass Offset = 0, 0, 0
4 Max Steer Angle = 40
// 0 is raw physics , 1 the car will grip in the direction it is facing
5 Steer Helper = .2
// 0 is no traction control, 1 is full interference
6 Traction Control = 1
7 Full Torque Over All Wheels = 200
8 Reverse Torque = 500
9 Downforce = 100
// Fastest the car can go
10 Topspeed = 150
// Forward and Sideways slip limit until car starts to slip
11 Slip Limit = .3
12 Brake Torque = 20000
13 Mass = 20
14 Wheel Damping Rate = .35
15 Suspension Distance = .001
16 Spring = 70000
17 Damper = 45
18 Extremum Slip = .4
19 Extremum Value = 1
20 Asymptote Slip = .8
21 Asymptote Value = .5
22 Stiffness = 2
23 Extremum Slip = .4
24 Extremum Value = 1
25 Asymptote Slip = .5
26 Asymptote Value = .75
27 Stiffness = 10
28 Gear Ratios = -10, 0, 9, 6, 4.5, 3, 2.5
// The Curve
// Table of efficiency at certain RPM, in tableStep RPM increases, 1.0f is 100% efficient
29 Efficiency Table = 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.85, 0.9, 0.95, 1, 1, 1, 0.95, 0.95,
0.95, 0.95, 0.8, 0.75, 0.7, 0.65
//The scale of the indices in table, so with 250f, 750RPM translates to efficiencyTable[3].
30 Efficiency Table Step = 250
31 Max RPM = 5500
32 Min RPM = 500
33 RPM to Shift Up = 5000
34 RPM to Shift Downn = 1500
Appendix C Optimum Lap Track Report
RLAPS Generic 78 Flat
Track Information
Type of Track
Oval Circuit
Track Direction
Forward Direction
Total Track Length
4619.42 ft
Percent Left Corners
50 %
Percent Right Corners
Percent Straights
50 %
Average Corner Radius
367.45 ft
Minimum Corner Radius
367.45 ft
Longest Straight
1154.86 ft
Sector 1
From 0 To 1154.86 ft
Sector 2
From 1154.86 To 2309.71 ft
Sector 3
From 2309.71 To 3464.57 ft
Sector 4
From 3464.57 To 4619.42 ft
Track Map
Friday, March 24, 2017
Appendix D Optimum Lap Vehicle Report
RLAPS Generic Stock Car
March 24,
Vehicle Configuration
Total Mass
2200 lb
Max Torque
400 ft.lbf @ 4000 rpm
Type of Fuel
Type of Transmission
Sequential Gearbox
Max Power
371.76 hp @ 5500 rpm
Power Mass Ratio
0.17 hp/lb
Downforce @ 62 mph
98.56 lbf
Drag @ 62 mph
115.13 lbf
Performance Metrics
Top Speed
103.34 mph
Time for 0 to 62 mph
1.32 s
Time for 62 to 0 mph
2.2 s
Lateral Acceleration - Skidpad 164 ft
45.19 ft/s^2
Charts Summary
Tractive Force at Wheel
Appendix E Numeric Results from Optimum Lap Simulation
[Result Details]
Result Name
Date Created
Track Configuration
[KPI Values]
Lap time [s]
Percent in Corners [%]
Percent Accelerating [%]
Percent Braking [%]
Percent Coasting [%]
Percent 100% Throttle [%]
Percent TCS Enabled [%]
Lowest Speed [mph]
Highest Speed [mph]
Average Speed [mph]
Energy Spent [kJ]
Fuel Consumption [lb]
Gear Shifts [-]
Maximum Lateral Acceleration [ft/s^2]
Maximum Longitudinal Acceleration [ft/s^2]
Maximum Longitudinal Deceleration [ft/s^2]
Time in Sector 1 [s]
Time in Sector 2 [s]
Time in Sector 3 [s]
Time in Sector 4 [s]
Maximum Speed in Sector 1 [mph]
Maximum Speed in Sector 2 [mph]
Maximum Speed in Sector 3 [mph]
Maximum Speed in Sector 4 [mph]
Minimum Speed in Sector 1 [mph]
Minimum Speed in Sector 2 [mph]
Minimum Speed in Sector 3 [mph]
Minimum Speed in Sector 4 [mph]
Percent in Gear 2 [%]
[Vehicle Parameters]
Mass Longitudinal Friction [lb]
Vehicle Mass [lb]
Drag Coefficient [-]
[33.00] RLAPS Generic Stock
RLAPS Generic 78
3/24/2017 13:23
RLAPS Generic 78 Flat
Closed Circuit
RLAPS Generic Stock Car
Downforce Coefficient [-]
Aero Efficiency [-]
Frontal Area [ft^2]
Drivetrain Efficiency [%]
Tire Rolling Radius [ft]
Air Density [lb/ft^3]
Rev Limit [rpm]
Longitudinal Friction [-]
Lateral Friction [-]
Final Drive Ratio [-]
Fuel Energy Density [J/kg]
Engine Thermal Efficiency [%]
Tire Rolling Drag [-]
Power Scaling Factor [%]
Aero Scaling Factor [%]
Grip Scaling Factor [%]
Mass Lateral Friction [lb]
Load Sensitivity Lateral Friction [-]
Load Sensitivity Longitudinal Friction [-]
Gear Ratio Data [-]
Gear Shift Points [rpm]
Torque Data [ft.lbf]
RPM Data [rpm]
[Vehicle KPIs]
Shift Point [1 to 2] [rpm]
Deceleration Time For Speed [27.778 mph] [s]
Deceleration Distance For Speed [27.778 km/h]
Top Speed [mph]
Acceleration Time To Speed [27.778 mph] [s]
Acceleration Time For Distance [100 ft] [s]
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