Description
ME227: Vehicle Dynamics and Control
Purpose
There is a lot we can learn about the design and control of vehicles using simple models. To start off, we have
a few exercises to familiarize yourself with the notation of coordinate transformations and rotation matrices
introduced on the first day of class and to gain some more insight into the kinematic model. You will also
build a simple simulation and test controllers capable of tracking a path with the kinematic model.
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. We will provide three different homework
formats on Canvas (a latex template, a pdf template, and a condensed pdf template). 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 assessments 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.
1
Assignment 1: The Kinematic Model 2
Problem 1 – Rotation Matrices
Rotation matrices are an important component of any control of dynamical systems. They provide easy
transformations of locations, velocities, and commands that occur in different frames. This problem will
help you gain some intuition into rotation matrices as you practice creating them.
Question 1.A – Combining Rotation Matrices (Gradescope)
Show that rotating from the vehicle frame to the inertial frame and then into the path frame is equivalent to
rotating directly from the vehicle frame to the path frame. In other words, demonstrate the result we used
in class that
AP OAOV = AP V
Write the expression showing AP OAOV = AP V . Please include your work.
Question 1.B – Creating Rotation Matrices (MATLAB Grader)
Implement a function that returns the rotation matrix Avo for some heading angle ΨV E. You can use this
function to translate a point from the inertial frame, O, to the vehicle frame, V .
Assignment 1: The Kinematic Model 3
Problem 2 – Lateral Velocity and Sideslip Angle
In the kinematic bicycle model, the lateral velocity at the rear axle is zero under the assumption that the
tire velocity is aligned with the tire’s longitudinal axis. The lateral velocity is nonzero at other points along
the centerline of the kinematic vehicle, however.
Question 2.A – Velocity (Gradescope)
Derive an expression for the velocity at any point along the centerline of the kinematic model starting at the
rear axle (x = 0) and ending at the front axle (x = L). You should express this only in terms related to the
vehicle (the steer angle, the velocity and the wheelbase) and not the radius of the path. In our notation, we
want an expression for the vector
v
e
ox,V
at any point from 0 to L along the centerline.
Write the expression for v
e
ox,V . Please include your work.
Question 2.B – Sideslip Angle (Gradescope)
Another way to think about the lateral velocity is in terms of the sideslip angle, β, of the vehicle which is
the angle between the velocity vector and the vehicle’s centerline:
β = arctan
Uy
Ux
Using the kinematic bicycle model and assuming small angles, calculate an expression for the sideslip angle
at any point on the vehicle centerline from x = 0 to x = L.
Write the expression for β. Please include your work.
Question 2.C – Model Intuition (Gradescope)
Based on this velocity scenario and the geometry of the model, if the vehicle is tracing out a constant radius
with point v on the centerline of the rear axle, is the front axle tracing the same path? If not, what is the
shape of the path traced by the front axle and does it lie inside or outside the path traced by the rear axle?
Select the correct answers below and give your explanation for these answers.
The front axle will track a path which is .
( Circular / Ellipsoidal / Same As The Rear Axle / Other)
The front axle will trace a path which is the path traced by the rear axle.
( Inside / The Same As / Outside )
Assignment 1: The Kinematic Model 4
Problem 3 – Tracking a Path
Let’s start off on a straight road and assume that the vehicle will be close enough to the desired path
that small angle assumptions are valid. Assume that we begin pointing along the centerline of the road
(∆Ψ(0) = 0) but with an offset from our desired lateral position, (e(0) 6= 0).
Question 3.A – Lookahead Controller (Gradescope)
A common method for path tracking with vehicles is to use a lookahead controller. This can either look at
the projected error some distance ahead on the path or, more commonly, approximate this error as a linear
combination of the lateral error and heading error:
δ = Klaela
ela = e + dla∆ψ
Here dla functions as the “lookahead distance” ahead of the reference point at which the error is calculated.
This is exactly the projected error on a straight road and approximately the projected error on a curved
road if the curvature is not too large relative to the lookahead distance.
During lecture, we also derived these equations assuming a small heading error and a small curvature:
s˙ = V
e˙ = V ∆ψ
∆˙ψ =
V δ
L
− κV
Perform several Laplace transforms assuming small angles, a straight road, and constant velocity. Arrange
this into a form where we can look at the response to the lateral error from the initial condition. In other
words, calculate the relationship:
E(s)
e(0)
The denominator of this relationship is the characteristic equation of the system that we can use to design
the performance of our tracking controller.
Note: Where does the initial condition e(0) come from? Remember that initial conditions are a part of the
Laplace transformation, we just often set them to zero when forming a transfer function. Leave them in your
expression here.
Write the resulting relationship E(s)
e(0) . Please include your work.
Assignment 1: The Kinematic Model 5
Question 3.B – Controller Stability (Gradescope)
If the lookahead distance (dla) is a positive number, what can you immediately say about the sign of the
lookahead gain (Kla) in order for the tracking system to be stable?
Select the correct answer below.
Solution:
The lookahead gain (Kla) must be .
(Positive / Negative)
Question 3.C – Lookahead vs PD Control (Gradescope)
Show that for a constant vehicle speed, the lookahead controller is equivalent to a PD controller of the form:
δ = Kpe + Kd
de
dt
Calculate the parameters Kp and Kd in terms of Kla and dla. You should see that the lookahead control law
provides a simple way of scaling, or gain scheduling, proportional and derivative gains as the vehicle speed
changes.
Write the parameters Kp and Kd in terms of Kla and dla.
Question 3.D – Controller Design (Gradescope)
Calculate values of Kla and dla that will give the system a rise time of 2 seconds and a peak overshoot of
10% if the vehicle has a wheelbase, L, of 2.5m and travels at a speed, V , of 10m/s.
Write the calculated values of Kla and dla. Please include your work.
Assignment 1: The Kinematic Model 6
Problem 4 – Building a Simulator
Now we want to see how well this controller developed using simple analysis works in simulation. You will
build up a simulation environment in MATLAB by following the MATLAB Grader prompts available in
Canvas. The prompts are also shown for reference in the questions below.
Question 4.A – Equations of motion (MATLAB Grader)
Follow the prompts in MATLAB Grader to create a function which calculates expressions for the vehicle
state derivatives ˙s, ˙e, and ∆Ψ. Do ˙ not assume small angles when writing expressions for the state derivative.
You are to modify the state derivatives s_dot, e_dot, and dpsi_dot in the function template on MATLAB
Grader.
Question 4.B – Euler Integration (MATLAB Grader)
Follow the prompts in MATLAB Grader to create a function which implements a simple Euler integration
scheme with a given time step. For this function, don’t use built in MATLAB functions such as trapz. You
are to modify the expression for the state at the next time step x1.
Question 4.C – Lookahead Controller (MATLAB Grader)
Follow the prompts in MATLAB Grader to create a function which implements the lookahead controller
you designed in Problem 3. Use the lookahead gain and distance calculated in Question 3.D. You are
to modify the lookahead controller parameters K_la and d_la, the lookahead error e_la, and the steering
angle command delta.
In this function you should assume small angles, and that the road is straight.
Assignment 1: The Kinematic Model 7
Problem 5 – Simulate Various Road Geometries
Now we’ll use the simulator built in Problem 4 to analyze the controller’s performance. Follow the MATLAB
Grader prompts available in Canvas for each question below. The prompts are copied here for reference.
Once the simulations are working, answer the qualitative questions about each simulation case.
Question 5.A – Simulation with a Straight Road (MATLAB Grader)
Now we’ll use the simulator built in Problem 4 to analyze the controller’s performance. We’ll begin on a
straight road with no intial heading error (∆Ψ) and a small path-tracking error (e). The simulation will use
parameters from our research vehicle Niki which is a Volkswagen GTI. The simulation is set to run for 10
seconds. After the simulation is complete, plot the lateral error (e), the heading error (∆Ψ), and lookahead
error (ela) as a function of time. Follow the prompts in MATLAB Grader to create and run this simulation.
Question 5.B – Simulation with a Straight Road (Gradescope)
Examine the plots you created in Question 5.A. Is the vehicle able to track the path with zero error?
Describe the controller’s performance in terms of tracking error.
Question 5.C – Simulation with a Curved Road (MATLAB Grader)
Now we are going to let the road curve. Simulate your controller here for the same gains and initial conditions
as Question 5.A on a curved road with a constant radius of 20 meters. After the simulation is complete,
plot the lateral error (e), the heading error (∆Ψ), and lookahead error (ela) as a function of time. Follow
the prompts in MATLAB Grader to create and run this simulation.
Question 5.D – Simulation with a Curved Road (Gradescope)
Examine the plots you created in Question 5.C. Is the vehicle able to track the path with zero error?
Describe the controller’s performance in terms of tracking error compared to Question 5.A.
Assignment 1: The Kinematic Model 8
Question 5.E – Simulation with an Undulating Road (MATLAB Grader)
Now we’ll run the simulation on an undulating road which curves left and right. Simulate your controller
here for the same gains and initial conditions as Question 5.A. After the simulation is complete, plot the
lateral error (e), the heading error (∆Ψ), and lookahead error (ela) as a function of time. Follow the prompts
in MATLAB Grader to create and run this simulation.
Question 5.F – Simulation with an Undulating Road (Gradescope)
Examine the plots you created Question 5.E. Is the vehicle able to track the path with zero error?
Describe the controller’s performance in terms of tracking performance compared to Question
5.A and Question 5.C.
Question 5.G – Compensating for Non-Zero Curvature (MATLAB Grader)
One way to handle tracking in the event of non-zero (and possibly changing) curvature is to use nonlinear
feedback to compensate. The yaw dynamics take the form of:
d∆Ψ
dt =
V
L
δ −
κV
1 − eκ
Compensate for the curvature term in this equation by changing the commanded steer angle to:
δ = Klaela +
Lκ
1 − eκ
,
Create a function called lookahead_lateral_control_curvature that implements this modified lookahead
controller. Follow the MATLAB Grader prompts to create this function.
Question 5.H – Simulation with Curvature Compensation (MATLAB Grader)
Now use your updated version of the lookahead controller with curvature compensation to control the vehicle
on the undulating road. Simulate your controller here for the same gains and initial conditions as Question
5.A. After the simulation is complete, plot the lateral error (e), the heading error (∆Ψ), and lookahead error
(ela) as a function of time. Follow the prompts in MATLAB Grader to create and run this simulation.
Question 5.I – Simulation with Curvature Compensation (Gradescope)
Examine the plots you created in Question 5.H. Is the vehicle able to track the path with zero error?
Describe the controller’s performance in terms of tracking error compared to Question 5.E.
Assignment 1: The Kinematic Model 9
Question 5.J – Understanding Curvature Compensation (Gradescope)
Explain how the additional term in our control law, from Question 5.G, is compensating for curvature.
Please provide both a mathematical and qualitative explanation of our curvature compensation.
Assignment 1: The Kinematic Model 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
Table 1.1: Vehicle Parameters and Values