3.2.2. Angular Velocities

This video introduces 3-vector angular velocities and the space of 3×3 skew-symmetric matrices called so(3), the Lie algebra of the Lie group SO(3). Any 3-vector angular velocity has a corresponding so(3) representation.

Let's say that R is the rotation matrix of a frame {b} relative to a frame {s}. A simple idea is to define the rate of rotation of {b}, which is also called the angular velocity, to be R-dot, the time rate of change of R. But this has 9 variables, and we should be able to find a good representation of angular velocity using only 3 variables. Unlike the curved space of orientations, SO(3), represented here as a sphere, at any given orientation, the space of angular velocities is a flat 3-dimensional vector space tangent to SO(3) at that orientation. A 3-dimensional vector space can be represented globally, without any singularities, by three coordinates.

This tinkertoy coordinate frame represents the body frame {b}. Now imagine a rotation axis passing through the origin, and the motion of the frame as it rotates about that axis according to the right-hand rule. Any angular velocity can be represented by a rotation axis and the speed of rotation about it. We can express the axis as a unit vector in the {s} frame, writing it as omega-hat_s. The hat means that the vector has unit length. We call the rate of rotation theta-dot, and we can multiply the unit axis omega-hat_s by the rate of rotation theta-dot to get the angular velocity vector omega s, expressed in the {s} frame.

As the frame rotates about the axis, the b-frame x-axis traces out a circle. The linear velocity of the x-axis is in a direction tangent to this circle, and is calculated as omega_s cross x-hat_b. A similar relationship holds for the other two coordinate axes. Since we will often take the cross product of a vector with another vector, we define a bracket notation that allows us to write x crossed with y as bracket-x times y, where bracket-x is a 3 by 3 matrix representation of the 3-vector x. The matrix bracket-x is called a skew-symmetric matrix because bracket-x is equal to the negative of its transpose. The set of all 3 by 3 skew-symmetric matrices is called little so(3), due to its relationship to big SO(3), the space of rotation matrices.

With the bracket notation, we can write the relationship between R-dot and the angular velocity omega_s as R-dot = bracket omega_s times R.

The angular velocity vector can be expressed in other frames, not just the {s} frame. For example, we could write it in the {b} frame coordinates. Using our change of reference frame subscript cancellation rule from the previous video, we get omega-b equals R_bs times omega_s, or R_sb inverse times omega-s. R usually indicates the body frame relative to the space frame, so we can drop the subscripts and write the relationship between the body angular velocity and spatial angular velocity as omega_b equals R inverse times omega_s and omega_s equals R times omega_b. We will also find the little so(3) matrix representation of the angular velocity to be useful.

In the next video we will begin to learn how to integrate a constant angular velocity for a given time to find a rotational displacement.