Positioning is a fundamental issue for mobile robots. Therefore, a performance analysis is suitable to determine the behavior of a system, and to optimize its working. Unfortunately, some systems are only evaluated experimentally, which makes the performance analysis and design decisions very unclear.
In , we have proposed a new angle measurement system, named BeAMS, that is the key element of an algorithm for mobile robot positioning. BeAMS introduces a new mechanism to measure angles: it detects a beacon when it enters and leaves an angular window. A theoretical framework for a thorough performance analysis of BeAMS has been provided to establish the upper bound of the variance, and to validate this bound through experiments and simulations. It has been shown that the estimator derived from the center of this angular window provides an unbiased estimate of the beacon angle.
This document complements our paper by going into further details related to the code statistics of modulated signals in general, with an emphasis on BeAMS. In particular, the probability density function of the measured angle has been previously established with the assumption that there is no correlation between the times a beacon enters the angular window or leaves it. This assumption is questionable and, in this document, we reconsider this assumption and establish the exact probability density function of the angle estimated by BeAMS (without this assumption).
The conclusion of this study is that the real variance of the estimator provided by BeAMS was slightly underestimated in our previous work. In addition to this specific result, we also provide a new and extensive theoretical approach that can be used to analyze the statistics of any angle measurement method with beacons whose signal has been modulated. To summarize, this technical document has four purposes:
(1) to establish the exact probability density function of the angle estimator of BeAMS,
(2) to calculate a practical upper bound of the variance of this estimator, which is of practical interest for calibration and tracking (see Table 1, on page 13, for a summary),
(3) to present a new theoretical approach to evaluate the performance of systems that use modulated (coded) signals, and
(4) to show how the variance evolves exactly as a function of the angular window (while remaining below the upper bound).