The robot, , can be rotated counterclockwise by some angle
by mapping every
as
|
(3.30) |
Using a
rotation matrix,
|
(3.31) |
the transformation can be written as
|
(3.32) |
Using the notation of Section 3.2.1,
becomes , for which
. For linear transformations,
such as the one defined by (3.32), recall that the column
vectors represent the basis vectors of the new coordinate frame. The
column vectors of are unit vectors, and their inner
product (or dot product) is zero, indicating that they are orthogonal.
Suppose that the and coordinate axes, which represent the body
frame, are ``painted'' on . The columns of can be
derived by considering the resulting directions of the - and
-axes, respectively, after performing a counterclockwise rotation
by the angle . This interpretation generalizes nicely for
higher dimensional rotation matrices.
Note that the rotation is performed about the origin. Thus, when
defining the model of , the origin should be placed at the
intended axis of rotation. Using the semi-algebraic model, the entire
robot model can be rotated by transforming each primitive, yielding
. The inverse rotation,
, must be applied to
each primitive.
Steven M LaValle
2020-08-14