<title>Comprehensive study of three-object triangulation</title>

Abstract

Given three landmarks whose location in Cartesian space is known, and a robot with the ability to detect each landmark's heading with respect to the robot's internal orientation, the task is to determine the robot's location and orientation in Cartesian space. Although methods are mentioned in literature, none are completely detailed to the point where useful algorithms to compute the solution are given. This paper presents four methods: (1) Iterative search, (2) Geometric circle intersection, (3) Geometric triangulation, and (4) Newton-Raphson iterative method. All four methods are presented in detail and compared in terms of robustness and computation time. For example, circle intersection fails when all three landmarks lie on a circle, while the Newton-Raphson method fails when the initial guess of the robot's position and orientation is beyond a certain bound. The shortcomings and strengths of each method are discussed. A sensitivity analysis is performed on each method, with noise added to the landmark locations. The authors were involved in a mobile robotics project that demanded a robust three landmark triangulation algorithm. None of the above methods alone was adequate, but an intelligent combination of each method served to overcome their individual weaknesses. The implementation of this overall absolute positioning system is detailed.