Some of my colleagues noticed oddities when tracking with an optical tracking system so I decided to try some things and see if I could explain them. The pose is defined as position and orientation in the form
[q0 qx qy qz tx ty tz] where q0, qx, qy, qz correspond to the quaternion and tx ty to the position in millimetres.
Say you have two targets and Target 1 is tracked relative to Target 2. In this example, the two targets are far apart (~ 1 meter). If you look at the pose of Target 1 (relative to Target 2) over a few time steps, there is a large variability in the reported position (note the last three columns):
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Target 1 (relative to Target 2)
At first glance, this looks like bad news. But is it really? The target pose is consistent so there must be something with the transformation that we are applying to Target 1 to get it into the coordinate system of Target 2.
Recall that transformation is a rotation followed by a translation so small changes in the rotation will produce large changes in position; a lever arm effect of sorts. It’s an artefact of the order of transformations.
If this is all true, then we would expect to see poses in which this is not visible. If we orient our targets in the plane of the camera (facing the camera) such that we reduce the rotation in the relative pose, then we find that the phenomenon goes away.
Target 1 (relative to Target 2)
Target 2
In both of these experiments, the targets were placed far apart. Do we observe the same effect when the targets are close together? Let’s test.
Target 1 (relative to Target 2)
0.509549975 -0.046162441 0.131538942 0.849073231 16.716178894 -392.597290039 146.070953369
0.509573936 -0.046189453 0.131330892 0.849089622 16.749166489 -392.610046387 146.034591675
0.509699881 -0.046255570 0.130519107 0.849135578 16.694667816 -392.674804688 145.853637695
0.510617197 -0.045960952 0.127721161 0.849025905 15.691468239 -393.516571045 143.797225952
0.510697603 -0.046687875 0.127098173 0.849031329 15.635018349 -393.507110596 143.815322876
Target 2
0.580050111 0.115894221 -0.028374750 0.805794775 131.511703491 380.052520752 -3522.971679688
0.580013394 0.116067760 -0.028516831 0.805791199 131.514755249 380.058593750 -3522.985839844
0.580016553 0.115881808 -0.028601188 0.805812776 131.509735107 380.056640625 -3522.963378906
0.579708576 0.113154754 -0.030199394 0.806363404 131.506240845 380.047027588 -3523.080078125
0.579732120 0.113021135 -0.030011872 0.806372225 131.498870850 380.044982910 -3523.052490234
Note that the last column of Target 1 shows some variability, not the size of variability in the first experiment but variability nonetheless. Is there a relationship between the distance between the targets?
I should point out that to properly do this experiment, I would need a coordinate measurement machine or cartesian robot for precise manipulation of the targets with respect to the camera. All measurements were taken with a Certus (manufactured by Northern Digital Inc.).
There isn’t really a practical implication of this given that the tracking system is accurately tracking the positions of both targets.
UPDATE 10 PM
It makes sense to me now that the variability in the pose would be greater the further the targets are apart. The effect is likely exaggerated the further from the camera system that you get.
That doesn’t change the fact that the tracking system is still accurately tracking the targets. (This is why this phenomenon isn’t noticeable when we use a navigation system.)
Let’s say that you want your transformation parameters to look better. Then you could simply define the rotation to be a rotation about the centre of the target or the origin of the target (instead of the tracking system centre), followed by the translation. This will make the translation more consistent since the translation is just the centre (or origin) of the two targets. Mathematically, it’s still the same overall transformation.
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