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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 [4], 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).

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 [4], 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).

Positioning is a fundamental issue in mobile robot applications, and it can be achieved in multiple ways. Among these methods, triangulation based on angle measurements is widely used, robust, accurate, and flexible. This paper presents BeAMS, which is a new active beacon-based angle measurement system used for mobile robot positioning. BeAMS introduces several major innovations. One innovation is the use of a unique unsynchronized channel with on–off keying modulated infrared signals to measure angles and to identify the beacons. We also introduce a new mechanism to measure angles: Our system detects a beacon when it enters and leaves an angular window. We show that the estimator resulting from the center of this angular window provides an unbiased estimate of the beacon angle. A theoretical framework for a thorough performance analysis of BeAMS is provided. We establish the upper bound of the variance and validate this bound through experiments and simulations; the overall error measure of BeAMS is lower than $\hbox{0.24}^\circ$ for an acquisition rate of $10\,\hbox{Hz}$ . In conclusion, BeAMS is a low-power, flexible, and robust solution for angle measurement and a reliable component for robot positioning.

Positioning is a fundamental issue in mobile robot applications. It can be achieved in many ways. Among them, triangulation based on angles measured with the help of beacons is a proven technique. Most of the many triangulation algorithms proposed so far have major limitations. For example, some of them need a particular beacon ordering, have blind spots, or only work within the triangle defined by the three beacons. More reliable methods exist; however, they have an increasing complexity, or they require to handle certain spatial arrangements separately. In this paper, we present a simple and new three object triangulation algorithm, known as ToTal, that natively works in the whole plane and for any beacon ordering. We also provide a comprehensive comparison between many algorithms and show that our algorithm is faster and simpler than comparable algorithms. In addition to its inherent efficiency, our algorithm provides a very useful and unique reliability measure that is assessable anywhere in the plane, which can be used to identify pathological cases, or as a validation gate in Kalman filters.