Basic Characteristics RAPID

What is a Quaternion?
The orientation of a coordinate system (such as that of a tool) can be described by a
rotational matrix that describes the direction of the axes of the coordinate system in
relation to a reference system (see Figure 1).
x
z
Reference
coordinate
system
Figure 1 The rotation of a coordinate system is described by a quaternion.
The rotated coordinate systems axes (x, y, z) are vectors which can be expressed in the
reference coordinate system as follows:
x = (x1 , x2 , x3 )
y = (y1, y2, y3) z = (z1, z2, z3) y
x
Rotated
coordinate
system
This means that the x-component of the x-vector in the reference coordinate system
will be x 1 , the y-component will be x 2, etc.
These three vectors can be put together in a matrix, a rotational matrix, where each of
the vectors form one of the columns:
x 1 y 1 z 1
x 2 y 2 z 2
x 3 y 3 z 3
A quaternion is just a more concise way to describe this rotational matrix; the
quaternions are calculated based on the elements of the rotational matrix:
x1 + y2 + z3 + 1
q1 = -----------------------------------------
2
q2
q3
x1 – y2 – z3 + 1
= ----------------------------------------
2
y2 – x1 – z3 + 1
= ----------------------------------------
2
z3 – x1 – y2 + 1
q4 =
----------------------------------------
2
sign q2 = sign (y 3-z 2)
sign q3 = sign (z 1-x 3)
sign q4 = sign (x 2-y 1)
7-orient-2 RAPID Reference Manual
y
z