Editing Screw theory (section)

== Twists as elements of a Lie algebra ==
Consider the movement of a rigid body defined by the parameterized 4x4 homogeneous transform,
: <math> \textbf{P}(t)=[T(t)]\textbf{p} = 
\begin{Bmatrix} \textbf{P} \\ 1\end{Bmatrix}=\begin{bmatrix} A(t) & \textbf{d}(t) \\ 0 & 1\end{bmatrix}
\begin{Bmatrix} \textbf{p} \\ 1\end{Bmatrix}.</math>
This notation does not distinguish between '''P''' = (''X'', ''Y'', ''Z'', 1), and '''P''' = (''X'', ''Y'', ''Z''), which is hopefully clear in context.

The velocity of this movement is defined by computing the velocity of the trajectories of the points in the body,
: <math> \textbf{V}_P = [\dot{T}(t)]\textbf{p} =
\begin{Bmatrix} \textbf{V}_P \\ 0\end{Bmatrix} = \begin{bmatrix} \dot{A}(t) & \dot{\textbf{d}}(t) \\ 0 & 0 \end{bmatrix}
\begin{Bmatrix} \textbf{p} \\ 1\end{Bmatrix}.</math>
The dot denotes the derivative with respect to time, and because '''p''' is constant its derivative is zero.

Substitute the inverse transform for '''p''' into the velocity equation to obtain the velocity of ''P'' by operating on its trajectory '''P'''(''t''), that is
: <math>\textbf{V}_P=[\dot{T}(t)][T(t)]^{-1}\textbf{P}(t) = [S]\textbf{P},</math>
where
: <math>[S] =  \begin{bmatrix} \Omega & -\Omega\textbf{d} + \dot{\textbf{d}} \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} \Omega & \mathbf{d}\times\omega+ \mathbf{v} \\ 0 & 0 \end{bmatrix}.</math>
Recall that [Ω] is the angular velocity matrix.  The matrix [''S''] is an element of the Lie algebra [[Lie algebra|se(3)]] of the Lie group [[Euclidean group|SE(3)]] of homogeneous transforms.  The components of [''S''] are the components of the twist screw, and for this reason [''S''] is also often called a twist.

From the definition of the matrix [''S''], we can formulate the ordinary differential equation,
: <math>[\dot{T}(t)] = [S][T(t)],</math>
and ask for the movement [''T''(''t'')] that has a constant twist matrix [''S''].  The solution is the matrix exponential
: <math>[T(t)] = e^{[S]t}.</math>

This formulation can be generalized such that given an initial configuration ''g''(0) in SE(''n''), and a twist ''ξ'' in se(''n''), the homogeneous transformation to a new location and orientation can be computed with the formula,
: <math> g(\theta) = \exp(\xi\theta) g(0),</math>
where ''θ'' represents the parameters of the transformation.