## 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.

#### Introduction to Rigid-Body Motions

This video introduces rotation about an axis by the right-hand rule and right-handed frames, including the body frame and the space frame.

#### 3.2.1. Rotation Matrices (Part 1 of 2)

This video introduces the space of rotation matrices SO(3), a Lie group, and properties of rotation matrices.

#### 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 1 of 2)

This video introduces the 3-vector exponential coordinates of rotation and the matrix exponential using the so(3) representation of exponential coordinates.

#### 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.