Abstract
Localization, finding the coordinates of an object with respect to other objects with known coordinates—hereinafter, referred to as anchors, is a nonlinear problem, as it involves solving circle equations when relating distances to Cartesian coordinates, or, computing Cartesian coordinates from angles using the law of sines. This nonlinear problem has been a focus of significant attention over the past two centuries and the progress follows closely with the advances in instrumentation as well as applied mathematics, geometry, statistics, and signal processing. The Internet-of-Things (IoT), with massive deployment of wireless tagged things, has renewed the interest and activity in finding novel, expert, and accurate indoor self-localization methods, where a particular emphasis is on distributed approaches. This paper is dedicated to reviewing a notable alternative to the nonlinear localization problem, i.e., a linear-convex method, based on Khan et al. ’s work. This linear solution utilizes relatively unknown geometric concepts in the context of localization problems, i.e., the barycentric coordinates and the Cayley–Menger determinants. Specifically, in an $m$ -dimensional Euclidean space, a set of $m+1$ anchors, objects with known locations, is sufficient (and necessary) to localize an arbitrary collection of objects with unknown locations—hereinafter, referred to as sensors, with a linear-iterative algorithm. To ease the presentation, we discuss the solution under a structural convexity condition, namely, the sensors lie inside the convex hull of at least $m+1$ anchors. Although rigorous results are included, several remarks and discussion throughout this paper provide the intuition behind the solution and are primarily aimed toward researchers and practitioners interested in learning about this challenging field of research. Additional figures and demos have been added as auxiliary material to support this aim.
Highlights
The Internet-of-Things (IoT) can be thought of as a massive network of objects including but not limited to sensors, machines, robots, humans, and/or animals, interconnected via software, communication, and electronics
The communication radius is increased until each sensor finds a triangulation set. These sets lead to the barycentric coordinates using the Cayley-Menger determinant; DIstributed sensor LOCalization (DILOC) is implemented with zero initial conditions and Fig. 6 (Middle) shows the intermediate locations estimates over 50 DILOC iterations
We provide a linear theory for localization, a linear solution to this nonlinear problem, that is applicable to networks of arbitrary number of sensors
Summary
The Internet-of-Things (IoT) can be thought of as a massive network of objects including but not limited to sensors, machines, robots, humans, and/or animals, interconnected via software, communication, and electronics. Reference [44] discusses the concepts on congruency, equivalence, rigidity, and global rigidity Another direction is the completion of partially specified distance matrices, considered in [45] and [46]; the algorithms calculate the unspecified distances under the geometrical constraints of the underlying network. Reference [54] uses trilateration to solve the localization/tracking problem that requires a large number of close by anchors to have a reasonable location estimate. The emphasis on this paper is to describe the fundamental properties of the aforementioned linear approach and provide the underlying intuition behind this construct that could be of significant interest to theoreticians and practitioners alike Towards this effort, we provide detailed illustrations, figures, and demos that further encompass several generalizations to dynamic topologies, uncertain environments, and mobile networks [1], [65]–[69]. Illustrations, demos, and examples are included throughout the text and several historical remarks are added during the exposition
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