In Chapter 3, we learn representations of configurations, velocities, and forces that we'll use throughout the rest of the book. As discussed in the last chapter, we'll use implicit representations of configurations, considering the C-space as a surface embedded in a higher-dimensional space. In other words, our representation of a configuration will not use a minimum set of coordinates, and velocities will not be the time derivative of coordinates. This approach may be new to you if you haven't taken a course in three-dimensional kinematics before.

Rigid-body configurations are represented using frames. A frame consists of an origin and orthogonal x, y, and z coordinate axes. All frames are right-handed, which means that the cross product of the x and y axes creates the z-axis. You can create a right-handed frame using your right hand: your index finger is the x-axis, your middle finger is the y-axis, and your thumb is the z-axis.

If I want to represent the position and orientation of a body in space, I fix a frame to the body and fix a frame in space. The configuration of the body is given by the position of the origin of the body frame and the directions of the coordinate axes of the body frame, expressed in the space-frame coordinates.

In this book, all frames are considered to be stationary. Even if the body is moving, when we talk about the body frame, we mean the stationary frame coincident with the frame attached to the body at a particular instant in time.

Positive rotation about an axis is defined by the right-hand rule. If you align the thumb of your right hand with the axis of rotation, positive rotation is the direction that your fingers curl.

With those preliminaries out of the way, in the next video we move on to representing the orientation of a rigid body.