# FBM3 - “Control”

## Clarifications

### Parameters

The parameters for the FBM3 are as follow:

For the line:

x = 0.15

m = 0.3

For the sine:

x = 0.15

a = 0.075

c = 12.5 Pi rad\m

### Accuracy Evaluation

To reduce confusion on the accuracy of the robot here is a detailed explanation on the robot path reconstruction from the ground truth system:

Because of the great amount of positions provided by the ground truth system, a subset of these is enough to precisely approximate the robot's path.

Since the ground truth system introduces a measurement error on every position that it provides, if the path is simply composed by all the measured positions (or even a subset of positions sampled at a certain frequency), the measured path results in the real path plus a random distribution.

This means that the measured path would consist in a tangle when the robot stays still and in a sort of "telephone cord" when the robot is moving. This effect tends to be less relevant the faster the robot is moving, but it also depends by the sampling frequency (a higher fs cause a tangler measured path).

Assuming the distribution is constant in time and has a mean error of epsilon, then even when the robot stays still, the path grows by epsilon for each sample. This means that the path's length suffers a constant growth summed to the real one.

It gets worse: if the mean error varies in time, the length error does too.

This constitutes a problem since the path is "indexed" by its (normalised) length. In fact, lengthwise, the weigh of any small segment on the path would depend by the measurement error distribution on that segment.

Solution: Since the benchmark focuses on the path (sequence of positions) rather than the trajectory (position, speed, acceleration), it suffices to only include in robot_path sampled positions that are at least d_min distant. More precisely the samples' minimum distance d_min, have to be greater than c·epsilon, with c > 1 and big enough to make sure that the measurement error never exceeds d_min.