
One unique characteristic of the angular kinematics is that the angular position is not unique. This is due to the nature of the angular motion: rotation about an axis of rotation. For example, the object shown in Figure 1 can have several possible angular positions:
where n = arbitrary integer. This is because the object returns to the same position after each revolution of rotation. If one uses angle ranges of p to p rad, or 0 to 2p rad to describe the angular position, one will likely to observe discontinuity in the angletime curve if multiple revolutions of the object is involved. This discontinuity in the angletime curve is not acceptable because it causes problems in the computation of the angular velocity. As long as the object rotates in the same direction, the angular position of the object must either increase (counterclockwise rotation) or decrease (clockwise rotation). Note here that the angular distance data are free from this problem since the angular distance between two lines stays within the range of 0 to p. The discontinuity problem can be solved by a twostep approach:
Let the true angular positions of two adjacent frames be a_{i1} and a_{i}. and the computed angular positions be a'_{i1} and a'_{i}. Let's assume both a'_{i1} and a'_{i} are described in the range of p to p. It is always possible for the object to cross the discontinuity point (p or p) between these two frames. The change in angular position suffices
since
where n = an arbitrary integer. [2] is in fact equivalent to
because the difference in the angular position is equivalent to the relative angular position. Da_{i} shown in [2] and [4] can be computed using the inverse tangent function:
where
As a result, Da_{i} is given within the range of
In reality, Da_{i} shown in [5] definitely stays within the range shown in [7] if the sampling rate (frame rate) is sufficiently high. If not, one must have the socalled aliasing problem due to insufficient frame rate. A positive Da_{i} means an increase in the angular position while a negative value means a decrease. From [2] and [5]:
[2], [5] & [7] guarantees a continuous angletime curve. 
© YoungHoo Kwon, 1998 