Any orientation can be achieved from an initial
orientation aligned with the space frame by
rotating about some unit axis by a particular
angle. We call the unit axis omega-hat and
the rotation distance theta. If we multiply
these two together, we get the 3-vector omega-hat
theta. This is a 3-parameter representation
of orientation. We call these 3 parameters
the exponential coordinates representing the
orientation of one frame relative to another.
This is an alternative representation to a
rotation matrix.
We call these exponential coordinates because
of the connection to linear differential equations.
In particular, we should view omega-hat as
an angular velocity that is followed for theta
seconds, and we have to integrate the angular
velocity from the initial orientation to find
the final orientation.
Before solving that problem, let's look at
a familiar problem in linear ordinary differential
equations in a single variable: x-dot = a
times x, where a is a constant. The solution,
as you learn in any course on differential
equations, is e to the a t times x at time
zero, where the exponential function e to
the a t is defined by the series expansion
shown here.
This scalar linear differential equation has
an analogous vector linear differential equation,
where x is now an n-vector and A is a constant
n by n matrix. The solution to this differential
equation has the same form as the single-variable
case. The term e to the A t is called a matrix
exponential. As we'll see in the next video,
this equation can be used to integrate an
angular velocity, where the matrix A is the
3 by 3 skew-symmetric representation of the
angular velocity.
