Fixed Frames and Body Frames:

In standard Newtonian mechanics, there is a fixed frame of reference that motion of objects is defined with respect to. In rigid body dynamics, it is sometimes helpful to define a body reference frame, which is a coordinate system that is fixed with respect to the orientation of the rigid body itself. When the rigid body is rotating with respect to the fixed frame, the body frame is rotating in precisely the same way, with its origin at the center of mass of the rigid body. The rigid body thus never translates or rotates in the body frame. The body frame and the fixed frame can be related through various coordinate transformations.

One common way of transforming between 3d reference frames is by use of Euler angles. Three successive transformations are made, each involving rotation of the coordinate system about a differently axis. The three Euler angles (a,b,c) are defined by the operations of the transformation:

1. Rotate counterclockwise about the z axis an angle a. This yields an x* axis and y* axis to replace the x and y axes, and leaves the z axis unaffected.

2. Rotate counterclockwise about the newly formed x* axis by an angle b.This creates a z* axis from the z axis, as well as a y** axis from the y* axis, leaving x* unaffected.

3. Last, rotate the system counterclockwise an angle c about the z* axis, creating finally a y*** axis from y**, and an x** axis from x*, with no effect on z*.

For convenience of notation, let x**=x',y***=y', and z*=z'.

From those three rotations one can obtain a point (x',y',z') from an initial point (x,y,z)  

The combined transformations yield the following trigonometric relationships between(x,y,z)and(x',y',z'):

x'=(cos(c)cos(a)-cos(b)sin(a)sin(c))x+(cos(c)sin(a)+cos(b)cos(a)sin(c))y+(sin(c)sin(b))z

y'=(-sin(c)cos(a)-cos(b)sin(a)cos(c))x+(-sin(c)sin(a)+cos(b)cos(a)cos(c))y+(cos(c)sin(b))z

z'=(sin(b)sin(a))x+(-sin(b)cos(a))y+(cos())z

This transformation can be implemented in the form of a 3x3 matrix, often denoted A, with the three parenthesized coefficients of equation as the three elements in each of three rows. Premultiplying that matrix with a column vector r=xi +yj+zk would yield a transformed column vector, r'=x'i+y'j+z'k as follows: