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.
