Description
Purpose
In this assignment, we will add longitudinal and lateral weight transfer to our bicycle model and see how
handling at the limits of tire friction can be influenced by load transfer. We’ll consider load transfer caused
by braking, acceleration, and suspension roll stiffness.
We’ll explore these concepts while using a new tire model, the Pacejka Magic Formula (MF). The MF tire
model is a semi-empirical model that has stood as a reference point in tire modelling since its introduction
in the late 1980s.
Download the source code zip folder hw7.zip from Canvas. In it you’ll find the code you’ll need for Problems
1, 3 and 4.
Instructions
This and following homework assignments will be submitted using two different tools, Gradescope and
MATLAB Grader. Throughout the assignment, each prompt will be marked with either Gradescope or
MATLAB Grader to make it clear where you should be submitting the answer to that problem. MATLAB
Grader questions will also be shown in Blue to make them easy to see.
All written portions must be turned in through Gradescope. See the Piazza post on homework guidelines
for more instructions on the different homework resources available to you. Whatever format you decide to
use, please BOX all of your final answers.
Some problems will make use of the MATLAB Grader suite discussed in class. These problems are available
directly from Canvas by clicking on each individual question. You are allowed to submit code to MATLAB
Grader as many times as you want before the due date without penalty. In this way you can be sure each
function or script passes all of the assesments that go along with it before moving on to the next problem.
We encourage you to write your own test cases as well to ensure your code is working as expected.
Additionally, some problems will make use of MATLAB Online or a normal MATLAB desktop installation.
For these problems we will give you most of the source code needed for the problem. See our Piazza post
for more information about how to setup MATLAB Online.
When writing functions and running simulations, use the set of parameters given to you for Niki. These will
be available in each MATLAB Grader problem where they are appropriate. The values of these paramters
for Niki are also given in Appendix A and B at the end of this document.
1
Assignment 7: Navigating the Hairpin Corner 2
Exploring the Magic Formula
In this assignment, we will move away from the Modified Fiala Tire Model that we’ve been using and explore
the Pacejka Magic Formula tire model. The Modified Fiala Model is a good, simple way to represent tire
force generation, and we can derive it nicely from physical observations. However, the Magic Formula is able
to represent some more complex tire characteristics that often interest manufacturers and racing teams so
it has evolved into the industry standard since its inception in the late 1980s. In its more complex form,
we can use it to calculate Fx and Fy using longitudinal slip, lateral slip, tire load, friction coefficient and
wheel camber. We’ve decided to introduce it here to give you exposure to what you may see in industry, not
because the Fiala Model is flawed. Quite the opposite, The Fiala Model does a very good job of representing
our tire dynamics given its simplicity.
If you’d like to read more about the Magic Formula, we recommend checking out Pacejka’s Tire and Vehicle
Dynamics (https://www.sciencedirect.com/science/article/pii/B9780080970165000048). For the
context of this homework, we will use a simplified Magic Formula of the form Fy = f(µ, Fz, Fx, α) to
calculate lateral force analogously to the simplified Fiala Model.
Let’s compare Fyf and Fyr from the two models:
If you’re looking at the plots and saying, ”these two models look pretty much the same,” that is the point.
We’ve simply substituted one mathematical representation for our tires for another, both tuned for the
same set of tires. We can notice some difference between the curves in the slip range between the linear
and saturated regions, and that is where these models tend to differ slightly with the Magic Formula more
closely matching recorded data.
The Teaching Team has taken care of implementation of the Magic Formula in MF_tire.m. Nonetheless, we
do recommend checking out the magic behind the Magic Formula in some of Pacejka’s papers.
Problem 1 – How to Race Through the Hairpin Corner
In this problem we’ll gain some intuition about how we’ll manuever through the hairpin corner described
below:
hairpin_corner.mat contains path information about a 180 degree hairpin turn in the same format we
used in the project. The path consists of a straight of 40m, a clothoid segment 100m long, a 75m radius arc
segment with a length of 135m, another 100m clothoid, and finally another 40m straight.
The file also contains the desired acceleration and velocity profiles for the corner. These were calculated
using a simple friction circle model, so they aren’t necessarily what the vehicle will be able to achieve.
Assignment 7: Navigating the Hairpin Corner 3
Download and unzip the hw7.zip file from Canvas. Open up the viewPathAndSpeedProfile.m script in the
Problem1/ directory. Run this script in MATLAB Online or MATLAB Desktop to plot the path and speed
profile described in the problem info. You don’t need to add or modify any code here, you just need the
plots to answer the following question.
Question 1.A – Understanding the Hairpin Corner (Gradescope)
Refer to the plots generated in viewPathAndSpeedProfile.m and answer the following questions:
1. At what speed does the car travel on the constant radius arc?
2. What lateral acceleration (in g’s) does this correspond to?
3. At what level of acceleration (in g’s) is the car braking on the initial straight?
4. At what level of acceleration (in g’s) is the car accelerating on the final straight?
Answer the questions based on the plots of the path and speed profile for the hairpin corner.
Assignment 7: Navigating the Hairpin Corner 4
Problem 2 – Calculating Weight Transfer and Effective Friction
Coefficient
When modeling vehicle behavior at the limits, there are three new phenomena we need to account for which
we have not simulated previously. These are:
• Longitudinal load transfer
• Lateral load transfer
• Friction as a function of tire normal load
In class we derived equations for lateral and longitudinal weight transfer. To combine them, simply calculate
the longitudinal weight transfer from the longitudinal acceleration and the lateral weight transfer from the
lateral accleration and sum the effects on each wheel.
Be careful when calculating the lateral load transfer term proportional to the roll center height. Think about
what the proper weight or normal force is to use for this term. If you understand the static roll model and
its underlying assumptions, there is a clear answer here.
Question 2.A – Calculating Weight Transfer (MATLAB Grader)
Follow the prompts on MATLAB Grader to create a function to calculate the normal load on each tire and
the body roll angle given longitudinal and lateral accelerations.
Question 2.B – Calculating Effective Friction (MATLAB Grader)
Follow the prompts on MATLAB Grader to create a function to calculate the effective friction available at
each axle.
You should use the following relationship for friction coefficient as a function of normal load to get the
friction at each tire:
µ = 1.25 − (4.0 × 10−5
)Fz
You can then find the effective friction coefficient for each axle by taking a weighted sum of the left and right
tires:
µef f =
µlFzl + µrFzr
Fzl + Fzr
Assignment 7: Navigating the Hairpin Corner 5
Question 2.C – How Does Weight Transfer Affect Friction (Gradescope)
In the ”Run Function” section of the MATLAB Grader assignment Question 2.B, there are three combinations of roll stiffness as listed in Table 7.1 below.
Combination Parameter Value
Combination A Front Roll Stiffness Kφf 40,000 Nm/rad
Rear Roll Stiffness Kφr 80,000 Nm/rad
Combination B Front Roll Stiffness Kφf 90,000 Nm/rad
Rear Roll Stiffness Kφr 30,000 Nm/rad
Combination C Front Roll Stiffness Kφf 70,000 Nm/rad
Rear Roll Stiffness Kφr 50,000 Nm/rad
Table 7.1: Roll Stiffness Combinations
A quick way to gain intuition about the effects of roll stiffness is to see how the car behaves at a constant
lateral acceleration. Here we’ve set up a scenario where the vehicle is cornering at 0.9g. Run your function
and answer the following questions for each combination of roll stiffnesses:
1. What is the roll angle at 0.9g lateral acceleration?
2. How much load is on each tire when the vehicle is cornering at 0.9g?
3. What are the front and rear friction coefficients when the vehicle is cornering at 0.9g?
Answer the questions about the vehicle’s behavior for each combination of roll stiffnesses.
Assignment 7: Navigating the Hairpin Corner 6
Problem 3 – Simulating and Tuning Vehicle Setup
Through previous homeworks, you’ve built up a nonlinear vehicle simulator to analyze vehicle behavior and
design autonomous controllers. As you saw in the project, this simulator captures a good amount of detail
of the vehicle’s real behavior, but not all of it. In this question we will add one more level of complexity by
accounting for weight transfer and roll effects. We have supplied all of the code you need for this Problem
in the Problem3/ directory included in the hw7.zip file on Canvas.
In this problem we will simulate Niki racing through the hairpin corner you saw in Problem 1. It is
important to note that Niki is front wheel drive; this will frame your analysis of Niki’s behavior through
the manuever. The simulation will run for 14 seconds. By keeping the simulation time the same, we can
compare the value of s reached when the simulation ends. A higher value of s means we made greater
progress through the corner, and consequently did a better job navigating the corner.
The simulator included here is essentially your nonlinear vehicle simulator with the weight transfer and
effective friction calculation functions you wrote in Problem 2. This is all conducted using the MF tire
model, not the modified Fiala Model we’ve used in previous assignments.
When asked you will just need to comment/uncomment different roll stiffness and brake distribution combinations in the simulator.m script and run the simulation to see how a different setup affects you performance
through the corner. You will then need to implement aerodynamic effects in aero_effects.m from a splitter
and wing that we’ll mount on Niki.
Question 3.A – Benchmarking Our Setup (MATLAB Online / Gradescope)
Begin with a brake distribution of 64%/36% between the front and rear axle, and roll stiffness Combination
A (from Table 7.1). If we want to be able to follow the path within 2m of the desired path, how are we
doing at this point? Which tire saturates at the start of the turn? Is this limit understeer or limit oversteer
behavior? What evidence do we have that this is the case?
Run the simulation and answer the questions about the vehicle’s behavior. Include the tire
force plots.
Question 3.B – Tuning Roll Stiffness (MATLAB Online / Gradescope)
Let’s try to address our handling problems by adjusting roll stiffness. Combination A has the majority of
the stiffness on the rear axle which is not common for passenger vehicles. Let’s try the simulation again with
Combination B to see if we do any better.
Did this correct the problem we saw with Combination A? Did it create another issue, and if so, what was
it? Did we make better progress through the corner (in other words, is our final distance along the path
greater at the end of our 14 second simulation time)?
Run the simulation and answer the questions about the vehicle’s behavior. Include the tire
force plots.
Assignment 7: Navigating the Hairpin Corner 7
Question 3.C – Tuning Roll Stiffness – Continued (MATLAB Online / Gradescope)
Finally, try roll stiffness Combination C in the simulation. This combination is closest to what you would
see on the production vehicle. Describe what is happening to the car through the corner and why that is
different from Combination A or Combination B. Which combination allows us to cover the greatest distance
in our 14 second simulation period? What is the maximum absolute lateral error?
Run the simulation and answer the questions about the vehicle’s behavior. Include the tire
force plots.
Question 3.D – Tuning Brake Proportioning (MATLAB Online / Gradescope)
Now that we have an idea of how to best setup our roll stiffness from Question 3.C, let’s take a look at
changing the brake setup. Set up the simulation with roll stiffness Combination C.
Change the brake proportioning to 36%/64% between the front and rear axle. Describe what is happening
to the vehicle with this setup, and what problems we’ve created by reversing the brake proportioning.
NOTE: You will get warnings about tire forces when running this setup. Why do these warnings crop up?
This should give you a hint as to what is happening.
Run the simulation and answer the questions about the vehicle’s behavior. Include the tire
force plots.
Assignment 7: Navigating the Hairpin Corner 8
Problem 4 – Tricking Out Niki
We’ve decided that Niki needs an upgrade. You are to mount a full aerodynamic-enhancing body kit on
Niki. Let’s envision Niki looking something like this:
https://www.rtheorymotorsports.com/shop/mk7-golf-tsigtir-rear-wing-kit
We added a front splitter and a rear wing. The additional downforce for each of these can be modelled as:
Fdown =
1
2
∗ CLA ∗ ρ ∗ U
2
x
CLA is the coefficient of lift for that wing. ρ is the air density.
This may look familiar to one of the feedforward components of the longitudinal controller you implemented
in the project. And in fact, we can use the same equation with different coefficients (where CDA is the
coefficient of drag) to model the additional drag caused by the new wings:
Fdrag =
1
2
∗ CDA ∗ ρ ∗ U
2
x
Question 4.A – Adding Aerodynamics (MATLAB Online / Gradescope)
Modify aero_effects.m to implement the added downforce and drag from the wing. We can assume that
the front wing aero forces act at the center of the front wheel and that the rear wing aero forces act at a
distance veh.hRwing directly above the rear wheel. We can also assume that the front wing acts equally on
the left and right front wheels and that the rear wing acts equally on the left and right rear wheels. Here is
a free body diagram of the GTI with the forces we added with the two wings.
Assignment 7: Navigating the Hairpin Corner 9
https://www.vw.com/en/models/golf-gti.html
Note: The drag on the rear wing will have an impact on the downforce on the front axle. Be sure to account
for this.
Restore Niki’s brake distribution to the original 64/36. Now, simulate the same curve as in the previous
problem. How does the distance covered compare to Question 3C? How does the maximum lateral error
compare to 3C? Plot the front and rear tire forces. Provide the plots and describe what is happening. Does
it make sense to add an aero kit to Niki?
Run the simulation and answer the questions about the vehicle’s behavior. Was it a good idea
to upgrade Niki? Include the tire force plots.
Assignment 7: Navigating the Hairpin Corner 10
Appendix A – Vehicle Parameters
Variable Name Value Units Description
veh.m 1926.2 kg Mass (Includes 4 passengers)
veh.Iz 2763.49 kg m2 Yaw Moment of Inertia
veh.a 1.264 m Distance from Center of Mass to Front Axle
veh.b 1.367 m Distance from Center of Mass to Rear Axle
veh.L 2.631 m Wheelbase
veh.Wf 9817.9 N Static front axle weight
veh.Wr 9078.1 N Static rear axle weight
Table 7.2: Vehicle Parameters and Values
Assignment 7: Navigating the Hairpin Corner 11
Appendix B – Tire Parameters
Linear Tire Model
Variable Name Value Units Description
f_tire.ca_lin 80,000 N/rad Front Cornering Stiffness
r_tire.ca_lin 120,000 N/rad Rear Cornering Stiffness
Table 7.3: Linear Tire Model Parameters
Assignment 7: Navigating the Hairpin Corner 12
Appendix C – Aero Parameters
Variable Name Value Units Description
veh.hRwing 1.5480 m Distance from rear wheel center to rear wing
veh.cdAf 0.0180 m2 Drag coefficient of the front wing
veh.cdAr 0.0315 m2 Drag coefficient of rear wing
veh.clAf 0.1080 m2 Lift coefficient of front wing
veh.clAr 0.2331 m2 Lift coefficient of rear wing
Table 7.4: Aero Parameters and Values
