ME227-Assignment 5: Getting Sideways Solved

Problem 1 – Phase Portraits
In this problem, we will look at phase portraits of Marty generated using PPLANE, a MATLAB plotting tool for phase plane analysis. There are code and instructions zipped up on Canvas. You will run the script called pplane2017.m. You can either run PPLANE locally or in MATLAB Online.

Parameters you will need for Marty are given in the table below:

Parameter
Value
Units
Meaning
m
1450
kg
Mass
Cαf
60,000
N/rad
Front cornering stiffness (lumped)
Cαr
160,000
N/rad
Rear cornering stiffness (lumped)
µ
1.1
 
Front/rear tire-road friction coefficient
L
2.4
m
Wheelbase
Iz
2300
kg m2
Yaw inertia
Wf
0.33
 
Static front axle weight fraction
Table 5.1: Marty Parameters

Question 1.A – Predicting the Phase Portrait (Gradescope)
If we use a linear tire model and the linear bicycle model equations of motion for Marty, how many equilibria do you expect to see? Compute the understeer gradient of Marty – based on this, what stability characteristics do you expect for the linear model?

Give a number of equilibria, the understeer gradient, and expected stability characteristics.

Question 1.B – Linear Tire Phase Portrait (Gradescope)
Generate a phase portrait for Uy and r and include the result. Is this a stable equilibrium? Explain why.

Include your phase portrait.               Describe whether this is a stable equilibrium.                Explain your reasoning.


Question 1.C – Nonlinear Tire Phase Portrait (Gradescope)
Generate a phase portrait for Uy and r and include your plot. How many equilibria are there? How would you classify these equilibria? Are they stable? Explain why.

Include your phase potrait. How many equilibria are there? Describe what types of equilibria these are and if they are stable. Explain your reasoning.


Question 1.D – Drifting Location (Gradescope)
Go to the YouTube link here and watch Marty on the course. First, enjoy how super cool that is. Second, see if you can understand how this works based on what we saw in class.

Which of the equilibria in the phase portrait you generated in Problem 1C corresponds to the drifting condition of turning right to go left? You can see Marty doing this around 40 seconds into the video. Explain why it corresponds to that equilibrium.

Explain which equlibrium corresponds to the described condition and how you know.


Problem 2 – Simulated Drifting
Now that we have explored the equilibria of a two-state model with nonlinear tires, let’s see if we can make Marty drift in simulation.

Question 2.A – Equations of Motion (Gradescope)
We are going to simulate Marty with a nonlinear bicycle model with lateral velocity Uy and yaw rate r as the two lateral states, and a third state for longitudinal velocity Ux. Since Marty is rear-wheel drive, assume that a longitudinal drive force Fxr is applied at the lumped rear tire (there is no braking force at the front axle), and neglect drag and rolling resistance. What are the (nonlinear) equations of motion? (Do not make any small angle or linear tire approximations)

U˙x =? U˙y =? r˙ =?

Give the nonlinear equations of motion for Marty.

Question 2.B – Constant Speed From Rear Longitudinal Force (Gradescope)
The model in PPLANE assumed Ux was constant and we used that to generate the phase portraits. However, our model includes longitudinal dynamics. What value of Fxr allows us to obtain a constant speed of 8 m/s in steady-state? Use the equilibrium lateral velocity and yaw rate you found in Problem 1D with a steer angle of δ = −10◦

Find the value of Fxr that gives a constant forward velocity of 8 m/s.

Question 2.C – Simulating from Equilibrium (Gradescope)
For this problem we will use the simple coupled tire model where Fx is assumed to be known. Incorporate this into your simulation (We will supply verification code). Run your simulation of the three-state bicycle model for 4 seconds using δ = -10° and Fxr equal to the value you computed in Problem 2B. Use the drift equilibrium found in Problem 1D as the initial condition. Plot Uy, r, and Ux on the same plot. Does Marty hold the drift? What happens? Did you expect this? If you’d like, visualize using animateDrift.m.

Include your plot and an explanation of what you see.

Question 2.D – Drifting Intuition (Gradescope)
In problem 2.E, you will implement a longitudinal and lateral controller to stabilize Marty in a drift. Let’s think briefly about the forces at play while drifting in this problem. In the case of a left-handed drift (e.g. yawrate is positive), you, the driver, have stabilized the vehicle with a negative steering wheel angle. In each ”Observed Behavior” listed in the table, select the correct input to stabilize the vehicle. When δ is held constant, select the correct Fx trend to stabilize the vehicle. When Fx is held constant, select the correct steering input trend to stabilize the vehicle. Provide an explanation for the input you selected.

Circle the correct δ and Fx inputs where applicable. Provide an explanation for your choice.

Observed Behavior
δ
Fx
Explanation
r too large
more negative / less negative
constant
 
 
 
r too large
constant
increase decrease
Fx

Fx
/
 
|Uy| too large
more negative / less negative
constant
 
 
 
|Uy| too large
constant
increase decrease
Fx

Fx
/
 
Question 2.E – Controlled Drifting (Gradescope)
To sustain the drift let’s add feedback terms to the values of δ and Fxr. Use a simple longitudinal controlled to track the desired longitudinal speed:

Fxr = Kx(UX,des − Ux) + Fx,ff

Where Kx = 2,000 N/(m/s), Ux,des = 8 m/s, and Fx,ff is the value you found in Problem 2B. For the feedback steering, using proportional control on Uy and r:

δ = kr(rdes − r) + ky(Uy,des − Uy) + δff

The absolute value kr = 1 s. The absolute value of ky = 0.5 rad/(m/s), and δff = -10°. Based on your observations in Problem 2.D, select the sign for kr and ky. Simulate for 9 seconds using the drift equilibrium as the initial condition. Plot Uy, r, and Ux and visualize the animation. Are we drifting now? What is the steady state sideslip angle?

Plot of the states with an explanation of whether we are drifting and why. A calculation for the steady state sideslip angle.