This repository contains tools for exploring the position uncertainty that exists when using Multidimensional Scaling (MDS) for localization.
The report on the effort to quantify MDS uncertainty can be found here
in the docs directory.
To learn more about Multidimensional Scaling, please refer to [1].
example plot from the mds gui
Install python package dependencies from the requirements.txt file.
Installation instructions for pytorch can be found here.
The MDS Uncertainty GUI can be run with: python3 simulate_mds.py.
In the plotted scenario, ranging measurements are being exchanged between all four robots marked with stars. The range measurements are then input into the multidimensional scaling algorithm and then Procrustes anlaysis (also mentioned in [1]) is used to align the MDS result with the true point positions.
The Meas. Std. slider increase the standard deviation of all ranging measurements.
The Meas. Bias slider increases the measurement bias for all
ranging measurements.
The X Position slider moves the X position of the fourth purple robot.
The Y Position slider moves the Y position of the fourth purple robot.
The bottom right graph shows the distribution of bias/noise added to the
ranging measurements.
The top right plots show the distributions of the final estimated X and
Y positions for each robot.
The left plot shows a top-down view of the estimated X-Y positions
of each robot with the 95% confidence interval of the Cramer Rao Lower
Bound ellipse plotted as the black ellipse and centered at the true
robot's position.
If plotting is too slow when moving sliders, you can decrease the k
variable near the top of simulate_mds.py (here) to decrease the number of samples that are plotted.
[1] Dokmanic, R. Parhizkar, J. Ranieri, and M. Vetterli, “Euclidean Distance Matrices: Essential theory, algorithms, and applications,”IEEE Signal Processing Magazine, vol. 32, no. 6, pp. 12–30, nov 2015.