Continuous Angle-Time Curve
Up ] Distance vs. Position ] [ Continuous Angle-Time Curve ]

 

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:

    Figure 1

,    [1]

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 angle-time curve if multiple revolutions of the object is involved. This discontinuity in the angle-time 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 two-step approach:

First, compute the angular positions and describe them within the angle range of choice: -p to p rad (bipolar), or 0 to 2p (unipolar).
Then, compute the change in angle between two adjacent frames and update the angle based on the difference.

Let the true angular positions of two adjacent frames be ai-1 and ai. and the computed angular positions be a'i-1 and a'i. Let's assume both a'i-1 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

,    [2]

since

    [3]

where n = an arbitrary integer. [2] is in fact equivalent to

,    [4]

because the difference in the angular position is equivalent to the relative angular position.

Dai shown in [2] and [4] can be computed using the inverse tangent function:

,    [5]

where

.    [6]

As a result, Dai is given within the range of

    [7]

In reality, Dai 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 so-called aliasing problem due to insufficient frame rate. A positive Dai means an increase in the angular position while a negative value means a decrease. From [2] and [5]:

    [8]

[2], [5] & [7] guarantees a continuous angle-time curve.

 

Young-Hoo Kwon, 1998-