Axis Transformation
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Most of the mechanical quantities one has to deal with in motion analysis, such as linear & angular position, velocity and acceleration of the markers & segments, are vectors. Because a vector has both magnitude and direction, one can describe the same vector in several different perspectives, depending on the intention or objective of the analysis. Describing a vector in a particular perspective is in essence equivalent to computing its components based the coordinate system of the particular perspective.

In other words, one can not only describe the same vector in several different perspectives, but also change the perspective from one to another depending on the situation and needs. This changing perspective of describing a vector is called vector transformation or axis transformation.

The issues covered in this section are:

Reference Frames
Transformation Matrix
Rotation Matrix

One thing to note is the difference between describing a vector and observing a vector. Observation means one moves with the perspective, and as a result, the outcome is the motion of the body relative to the observer's perspective. The motion (vector) of an object observed in one perspective is different from the same motion observed in another perspective.

For example, to the spectators, a gymnast's airborne maneuver is perceived as a complex motion with both translation of the body CM and rotation (somersault & twist) of the body about its CM. But the same maneuver can be perceived slightly differently by people watching it on TV since the motion of the camera compensates translation of the body CM. Thus, one can only see the rotation of the body on the screen. Moreover, the gymnast's own perception of the maneuver is very different from those of the spectators and TV watchers. The gymnast only sees the motions of the body parts relative to his/her perspective.

Observation of a vector in a perspective is simply a process of computing the vector relative to the perspective employed. In this process, one can still describe the relative vector in different perspectives. See Motion in a Rotating Reference Frame for the details of the quantification of the relative motion.

 

Young-Hoo Kwon, 1998-