Finite Rotations
In kinematics, there are two commonly used sets of transformations for describing the change in orientation of a body in space or the rotation of a set of reference axes with respect to another. These transformations are categorized as either space fixed and body fixed rotations. In the first case, each rotation is about an axis in which remains fixed in space with respect to the body or axes being rotated. Regardless of the change in orientation of a body or axes, space fixed rotations always occur about the same set of axes. Body fixed rotations, however, are rotations that occur about axes fixed in the reference of the body. Thus, the absolute orientation of each axis depends on the previous rotation. In both space fixed and body fixed rotations, the final orientation of a body or set of reference axes depends on both the sequence of the rotations and their magnitude.
Although space fixed and body fixed rotations generally provide different results, the matrix transformations used to describe the rotations are the same. What differs is the order or sequence in which they are applied. The following matrices are used to describe rotations about each axis in a Cartesian coordinate system
An obvious characteristic of the above matrices is that they
are orthogonal or better yet orthonormal. This means that
the determinate of the matrix is unity and that the magnitude of any vector
transformed by these matrices remains the same
(i.e. no scaling). Also, orthogonal/orthonormal matrices have the
characteristic that their transpose is equal to their inverse. In short
There are two common uses of the above transformations. The first is to be able to describe a set of basis vector in one frame of reference in terms of another. In any problem, just like working with a consistent set of units, one must make sure there is consistent set of basis vectors used in the formulation of the problem.
The second common use for the above transformations is transforming a point or body in a reference system to a new orientation. These two uses are very closely related in that the former can be viewed as transformation of axes relative to a fixed point or frame and then latter viewed a transformation of a point or frame relative to a fixed set of axes. The transformations are nearly identical except for a change in sign for the rotation angles.
The following is an example of using the transformations for
a change in coordinate system. Given two coincident frames A and B,
let’s rotate a reference frame B, with basis vector ,
about the
axis in frame A, which has basis vectors
.
The transformation of
to obtain
is accomplished with the following matrix
operation.
Expanding this gives us the relationship between of the
basis vectors in frame B to the basis vectors
in frame A. The opposite
transformation(applying a negative value for the rotation angle) gives us the
transformation of the
basis vectors in frame B in terms of
the basis
in frame A. Together we have
and
This process is repeated as many times as necessary for a given problem depending on the number of frames of reference used and the final reference frame chosen for completing the solution.
Addressing the second most common use of these
transformations we want to look at the transformation of axes and points and
see also see how they are related. Again, let’s assume we have a fixed
reference system A and reference system B which we will rotate 45°
about the axis. Let’s also have a point
.
First, we will rotate the B axes to yield the point relative to
reference frame B. Similarly to previous example, the transformation of
basis vector between frame A and B is
the reverse transformation yields
To obtain the point in the B reference frame all we need to do is
substitute the
basis vector using the relationship in (1.3).
From this result it is obvious that the transformation is a
transformation of the axes of frame B relative to the stationary point in frame A
In many applications, it is desired to actually transform or
rotate the point instead of the axes. In this instance, let’s perform the same
transformation above but this time we’ll rotate the point instead of the axes.
To perform this operation, we will fix point in frame B prior to the rotation about
such that
now becomes
and remains stationary relative to frame B.
Again, substituting the transformations of the basis vectors in (1.2)
yields
.
In general, it’s all a matter of perspective. In this case
we wanted to rotate point but what we actually did was rotate frame A
45° in
the opposite direction and obtain the new position of the point relative to
frame A. In short, to rotate the coordinate system, use the matrices in (1.1)
with positive values for the angles. If it is desired to obtain the
transformation of a point, then use equations (1.1)rotating
the axes in the opposite direction by substituting negative values for the
angles.
Body Fixed Rotations
As mentioned, body fixed rotations are rotations which occur
about the axes fixed a body frame of reference. In other word, regardless of a
change in position or orientation, the reference axis system moves with the
body and all rotations are about the axes in body. For example, a body
undergoes a sequence of rotations ,
,
about the x, y, and z body axes,
respectively. The first rotation,
,
is about the x-axis in the body frame’s coordinate system. Now with the body in its new orientation,
the second rotation,
,
occurs about the y-axis fixed in the body. Finally, the third rotation
occurs about the body’s z-axis. Although the
absolute orientation of the axes have changed relative to the fixed reference
system for each rotation, the axes about which the body rotates remain fixed
when viewed from the body’s frame of reference.
For a sequence of body fixed rotations, the matrix transformations are combined in the order in which the rotations occur with each successive rotation pre-multiplying the previous. A final transformation matrix of n body fixed rotations has the form
For the above example, a final transformation for the
sequence of rotations ,
,
and
,
would be
It should be noted that, in general for a Cartesian coordinate systems, the sequence of rotations most commonly used occur about the z-axis first, then the y-axis, then the x-axis. As you’ll see in the next section, reversing the order correspond to a completely different set of rotations with a completely different intent.
Space Fix Rotations
Space fixed rotations are very similar to body fixed rotation with the exceptions noted above. Each rotation occurs about a fixed set of axes regardless of the orientation of the body. Like body fixed rotations, a sequence of space fixed rotations can be combined into a single transformation; however, the order in which the transformations are applied is in reverse. For a sequence of n space fixed rotations, a final matrix [T] has the following form.
The similarities between equations (1.4)and (1.5) can be both a source of convenience and a significant source for errors and care should be taken to make sure the proper sequence is used for the desired set of transformations.