Book Chapter: Chapter 3

Rigid-Body Motions

Chapter 3 Autoplay

Autoplay of the YouTube playlist for all videos in this chapter.  This description box will not be updated with information about each video as the videos advance.

3.2.1. Rotation Matrices (Part 2 of 2)

This video introduces three common uses of rotation matrices: representing an orientation, changing the frame of reference of a vector or a frame, and rotating a vector or a frame.

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.

3.2.3. Exponential Coordinates of Rotation (Part 2 of 2)

This video describes how the solution of a vector linear differential equation calculates the rotation achieved after rotating a given time at a constant angular velocity. The matrix exponential maps the so(3) matrix representation of the 3-vector of exponential coordinates of rotation to a rotation matrix in SO(3), and the matrix logarithm maps a rotation

3.3.1. Homogeneous Transformation Matrices

This video introduces the 4×4 homogeneous transformation matrix representation of a rigid-body configuration and the special Euclidean group SE(3), the space of all transformation matrices. It also introduces three common uses of transformation matrices: representing a rigid-body configuration, changing the frame of reference of a frame or a vector, and displacing a frame or a

3.3.2. Twists (Part 1 of 2)

This video introduces the 6-vector twist, a representation of the linear and angular velocity of a rigid body. A twist can be represented as a normalized screw axis, a representation of the direction of the motion, multiplied by a scalar speed along the screw axis. A screw axis or twist can be represented in any

3.3.2. Twists (Part 2 of 2)

This video introduces the 6×6 adjoint representation of a 4×4 SE(3) transformation matrix and shows how it can be used to change the frame of reference of a twist or a screw. The 4×4 se(3) matrix representation of a 6-vector twist is also introduced. This se(3) representation is used in the matrix exponential in the

3.3.3. Exponential Coordinates of Rigid-Body Motion

Any rigid-body transformation can be achieved from any other by following some 6-vector twist for unit time. The six coordinates of this twist are called the exponential coordinates. This video shows how the rigid-body transformation can be calculated using a matrix exponential with the se(3) matrix representation of the exponential coordinates. The matrix exponential maps

3.4. Wrenches

This video introduces the wrench 6-vector representation of forces and moments in three-dimensions, and it shows how to change the frame of representation of a wrench.