Additional experiments for the paper: Automatic self-contained calibration of an industrial dual-arm robot with cameras using self-contact, planar constraints, and self-observation http://karlastepanova.cz/publications/self-calibration-industry/)
In our approach, we assume that the noise in our measurements is relatively small and is normally distributed, which is in this type of calibration a standard approach. In general, noise might only increase the rank of the Jacobian. Therefore, for the setups where the observability and rank of Jacobian is low, it would be even lower without noise. On the other hand, for the setups where the rank is high, we expect that the noise is small enough and is not the only cause of the high rank of the Jacobian.
To test that our calibration is not working only because of the presence noise, we prepared a simulated experiment, where noise of various levels can be added to several parameters. We perturbed either the self-touch distance, distance to a plane, or robot kinematic parameters. From these experiments, we verified that for perturbations up to S.D. 5mm self-touch distance (noise has a normal distribution), the calibration results and corresponding observability is similar to the case where no noise is added. Both for calibration of offsets and all DH parameters.
In the plots below, we show the simulated results for self-touch with and without measurement noise. Top: Jacobian for the case without noise; Bottom: Jacobian with noise (S.D. 1mm).
We also tested the other case – when the Jacobian rank is low even with measurement noise, without the noise it would be even lower. To show this, we run the experiment for the vertical plane – as shown and analyzed in our paper (and also other former works), using a single plane is not sufficient for estimating all DH robot parameters. See the figure below, where it can be seen that the rank of Jacobian is in this case increased only thanks to the noise even when the parameters are actually not observable.
Here we show Jacobian for all DH parameters without noise.
This figure shows the Jacobian when noise to the data is added (normal distribution with S.D. 1 mm).
