In this video we'll see how the matrix exponential can be applied to integrate the angular velocity

of a rotating rigid body.

Let's start with this frame of coordinate axes, which will rotate about a unit angular

velocity axis.

To understand the motion of the coordinate axes, it suffices to consider just one of

the coordinate axes, since the same reasoning applies to any axis.

Let's call this remaining vector p.

As the vector p rotates about the rotation axis, it traces out a circle.

The purpose of this video is to determine the final location of the vector if it rotates

an angle theta about the rotation axis.

We will do this by integrating the differential equation of motion describing the motion of

p.

Here is a picture of our initial vector, p at time 0, and the unit rotation axis omega-hat.

As p begins to rotate, it traces out a circle around the rotation axis.

The 3-vector linear velocity is tangent to the circle at any time, and is given by omega-hat

cross p.

After rotating an angle theta, the vector ends up at p at time theta.

At any instant of time, the time derivative of p is given by p-dot = omega-hat cross p.

We can write this as a differential equation p-dot of t equals omega-hat cross p of t.

The angular velocity is constant.

Using our 3 by 3 skew-symmetric matrix notation, this becomes p-dot of t equals bracket omega-hat

times p of t.

This is a vector differential equation, whose solution, as we saw in the last video, is

calculated using the matrix exponential.

In general, a matrix exponential can be calculated using a series expansion, but when the matrix

is 3 by 3 and skew symmetric, the series expansion has a simple closed form: the 3 by 3 identity

matrix plus sin of theta times bracket omega-hat plus 1 minus cosine of theta times bracket

omega-hat squared.

In other words, the matrix exponential takes the skew-symmetric representation of the exponential

coordinates omega-hat theta and calculates the corresponding rotation matrix.

This equation is often called Rodrigues' formula.

Essentially, exponentiation integrates the angular velocity omega-hat for time theta

seconds, going from the identity matrix to the final rotation matrix R. We can also define

the inverse of the matrix exponential, the matrix logarithm, which takes a rotation matrix

R and returns the skew-symmetric matrix representation of the exponential coordinates that achieve

it, starting from the identity orientation.

Just as the matrix exponential is like integration, the matrix log is like differentiation: it

returns the unit angular velocity and the integration time that achieves the rotation

matrix R. The matrix log is an algorithm that inverts Rodrigues' formula.

Later, when we're studying the kinematics of robots, the matrix exponential and log

will become very useful.

Basically, for a revolute joint, the unit angular velocity omega-hat represents the

axis of rotation of the joint, and theta represents how far that joint has been rotated.

Before we get to robot kinematics, though, we have to generalize the matrix exponential

and log to cases where frames both rotate AND translate.

In other words, general rigid-body motion.

We'll start the process of generalizing from rotations to general rigid-body motions in

the next video.