Suppose a rotation by is performed, followed by a translation
by . This can be used to place the robot in any desired
position and orientation. Note that translations and rotations do not
commute! If the operations are applied successively, each
is transformed to

(3.33) 
The following matrix multiplication yields the same result for the
first two vector components:

(3.34) 
This implies that the
matrix,

(3.35) 
represents a rotation followed by a translation. The matrix will
be referred to as a homogeneous transformation matrix. It is
important to remember that represents a rotation followed by
a translation (not the other way around). Each primitive can be
transformed using the inverse of , resulting in a transformed solid
model of the robot. The transformed robot is denoted by
, and in this case there are three degrees of
freedom. The homogeneous transformation matrix is a convenient
representation of the combined transformations; therefore, it is
frequently used in robotics, mechanics, computer graphics, and
elsewhere. It is called homogeneous because over
it is just a
linear transformation without any translation. The trick of
increasing the dimension by one to absorb the translational part is
common in projective geometry [804].
Steven M LaValle
20200814