University of Notre Dame
Aerospace and Mechanical Engineering

ME 698: Geometric Nonlinear Control
Final Exam

B. Goodwine Fall, 1999

Issued: December 2, 1999 Due: December 17, 1999

This problem will consider feedback linearization of two versions of the shimmying wheel problem.

1.

If we assume that the tire is rigid, i.e., q(x,t) = 0 and that it rolls without slipping, the equations of motion are

  $$\begin{eqnarray*}\ddot{\theta} &=& \frac{ - \frac{v}{l} \left( \mbox{sec}^2 \the... ...ec } \theta \left( v + l \dot \theta \sin \theta \right) }{r}. \end{eqnarray*}$$

(a)

Follow the procedure outlined in class to find an output function which makes this system feedback linearizable. Since the wheel rotation angle is cyclic, we can ignore the third equation, and full state linearization requires a relative degree of three.

(b)

Simulate the system (both controlled and uncontrolled) using the following parameter values: m_c = 1.5kg, m_w = 2.75kg, l = 0.152 m, r = 0.1m, k = 75N/m and v = 1m/s.

2.

Now, if we consider the spring to be rigid, but the tire to be elastic, the equations of motion are:

$$\begin{eqnarray*}I \ddot \psi &=& -l F(q_o) - M(q_0) \\ \dot q_0 &=& v \sin \psi + (l - a) \dot \psi - \frac{v}{\sigma} q_o \cos \psi \end{eqnarray*}$$

where I is the mass moment of inertia of the assembly about the attachment point and the other variables are the same as discussed previously. The force, F, and moment, M, due to the contact line deflection are given by

 

$$\left\{ \begin{array}{ll} 22.2 \frac{c a^2}{\sigma} q_0, \qquad &... ... \\ \mu N \operatorname{sgn} q_0,& \vert q_0\vert \geq q^* \end{array} \right.$$ F(q₀) =

$$\left\{ \begin{array}{ll} 0.16 \mu N a \sin \left( 69.8 \frac{c a... ...uad & \vert q_0\vert < q^* \\ 0, & \vert q_0\vert \geq q^* \end{array} \right.$$ M(q₀) =

(a)

Follow the procedure again to verify that the system is full state feedback linearizable. Determine the output function. Will this approach work? Why or why not?

(b)

Use the output function $y = h(x) = \psi$ to partially linearize the system. Are the zero dynamics stable?

(c)

(optional) Simulate the system with the following parameter values: $I=0.3576\,[kg\,m^2],\ c=10^5\,[N/m^2],\ a=0.04\,[m],\ \sigma=0.12\,[m], \ l=0.1\,[m],\ v=10\,[m/s],\ \mu F_z=800\,[N]$.

3.

Pick two of the subjects covered by the student lectures (excluding your own) and submit, in complete detail, an example problem, including simulation results, if applicable.

I suppose it is possible that the stereotype of the wicked scientist dissuades some youngsters from entering the profession, but the world today is so topsy-turvy that perhaps as many are attracted as are repelled by the prospect of a career of malefaction.

--- Sir Peter Medawar, Advice to a Young Scientist.

Last updated: December 2, 1999.
B. Goodwine (jgoodwin@nd.edu)