University of Notre Dame
Aerospace and Mechanical Engineering

ME 698: Geometric Nonlinear Control
Homework 7

B. Goodwine
Fall, 1999
Issued: October 28, 1999
Due: November 4, 1999

1.
As presented in class, the stereographic projection is a map, $\psi$ from the punctured sphere $S^2-\{N\}$ onto $\mathbb R^2$, where N is the north pole, (0,0,1). Prove that $\psi : S^2-\{N\}
\rightarrow \mathbb R^2$ is a diffeomorphism by writing $\psi$explicitly in coordinates and solving for $\psi^{-1}$.

2.
(Sastry, 3.27) A Lie group is a manifold which is also a group. This problem explores several Lie groups.
(a)
Consider the set of all $n \times n$ nonsingular matrices. Denote it GL(n), which stands for general linear group. Prove that GL(n) is a Lie group. What is its dimension?
(b)
Consider the set of all $n \times n$ unitary matrices, i.e., matrices such that AT = A-1, with determinant +1. Denote it by SO(n) for special orthogonal group. Prove that SO(n) is a Lie group. What is its dimension? What can you say about unitary matrices with determinant -1?
(c)
Consider matrices of the form

\begin{displaymath}\left[ \begin{array}{cc} A & b \\ 0 & 1 \end{array} \right]
\end{displaymath}

with $A \in SO(n)$ and $b \in \mathbb R^n$. Denote it by SE(n) for special Euclidean group. Prove that SE(n) is a Lie group; in particular, be sure to write a formula for the inverse of elements in SE(n) (which is necessary to prove that it is a group). What is the dimension of SE(n)?
3.
Recall from class that a coordinates on a manifold were defined as a diffeomorphism from any point $x \in M$ to $\mathbb R^m$, i.e.,

\begin{displaymath}\psi: U \subset M \subset \mathbb R^k \mapsto \tilde U \subset \mathbb R^m,
\end{displaymath}

where If M is embedded in $\mathbb R^k$, then this definition may be confusing at first glance, because, for instance, it would seem that $\psi$ should map an open set of $\mathbb R^k$ onto an open set of $\mathbb R^m$, but this is impossible if $\psi$ is a diffeomorphism. The proper interpretation is that $\psi$ smoothly extends to a map $\Psi: V \mapsto \mathbb R^m$ (where $U = M \cap V$, $V \subset
\mathbb R^k$ is open), such that $\left. \Psi \right\vert _U$ is a diffeomorphism onto $\tilde U$. We will explore these concepts using M = S1.

Let $S^1 \subset \mathbb R^2$ be the set given by

\begin{displaymath}S^1 = \{ x \in \mathbb R^2 \vert x^T x = 1 \}.
\end{displaymath}

(a)
Construct the stereographic projection $\Psi: \mathbb R^2
\mapsto \mathbb R$ which maps $S^1-\{N\}$ to the real line as shown in the figure below. $\{N\}$ denotes the ``north pole,'', i.e., x = (0,1).
(b)
Let $U = S^1-\{N\}$ and define $\psi = \left. \Psi \right\vert _U$. Construct the map $\psi^{-1}: \mathbb R \mapsto U \subset \mathbb R^2$and show that it is a smooth mapping.
(c)
Show that $\psi$ is a bijection by verifying that
i.
the map $\Psi \circ \psi^{-1}: \mathbb R \mapsto \mathbb R$ is the identity; and,
ii.
$\psi^{-1} \circ \Psi$ is the identity when restricted to U.
(d)
The tangent space at a point $x \in U$ is defined as

\begin{displaymath}T_xS^1 = \operator{Im} \left(D \psi^{-1}_{\psi(x)} \right).
\end{displaymath}

Show that this subspace of $\mathbb R^2$ can be identified with vectors tangent to S1 at x. Hence, when talking about a tangent vector of $S^1 \subset \mathbb R^2$ it makes sense to only consider vectors in $\operator{Im} \left( D \psi^{-1} \right)$.
(e)
Show that $D \psi_x : T_xS^1 \mapsto \mathbb R$ is an isomorphism by verifying that
i.
$D \Psi_x \cdot D \psi^{-1}_{\psi(x)} = I \in \mathbb R$; and,
ii.
$\left. \left( D \psi^{-1}_{\psi(x)} \cdot D \Psi_x \right)
\right\vert _{T_xS^1} = \operator{id}$.



An engineer? I had grown up among engineers, and I could remember the engineers of the twenties very well indeed: their open, shining intellects, their free and gentle humor, their agility and breadth of thought, the ease with which they shifted from one engineering field to another, and, for that matter, from technology to social concerns and art. Then, too, they personified good manners and delicacy of taste; well-bred speech that flowed evenly and was free of uncultured words; one of them might play a musical instrument, another dabble in painting; and their faces always bore a spiritual imprint.

--- Aleksandr Solzhenitsyn, The Gulag Archipelago.


Last updated: October 28, 1999.
B. Goodwine (jgoodwin@nd.edu)