Ground Vehicles and Ranging Sensors: Structural Properties for Estimation and Control

In this thesis we address the constructibility problem for a ground vehicle moving across an environment instrumented with ranging sensors. When the measurements collected by the vehicle along the trajectory are sufficiently informative, the global constructibility property is achieved and the vehicle is able to localise itself in the environment without relying on prior information on its state. When this condition is not met, the system can still achieve local (or weak) constructibility, where localising the robot requires some initial information on the state, such as a sufficiently small set containing the initial position of the robot, or some inaccessible areas of the Cartesian plane. First, we address the global problem: we show that extending the well--known solutions for the positioning problem, e.g. trilateration, is not trivial and leads to unintuitive results where constructibility is not attained. By building an abstract trajectory, which contains all the relevant information to reconstruct the actual trajectory followed by the vehicle, we analyse how global constructibility properties are affected by the shape of the abstract trajectory, the number of sensors, their deployment in the environment, and the distribution of measurements among the beacons. To describe local constructibility, we build the Constructibility Gramian for a robot described by the unicycle kinematic model. We rely on this tool for a twofold aim: (a) we build the same abstract trajectory presented for the global analysis and define necessary and sufficient conditions to attain local constructibility, and (b) in an environment instrumented with two beacons and for straight trajectories followed by the vehicle, we measure local constructibility by means of the smallest eigenvalue of the Constructibility Gramian, and we analyse how this metric is affected by the geometry of the scenario, e.g. the distance between anchors, and the distance between the trajectory and the line joining the anchors. Lastly, we extend the devised results to multiagent systems, both for constructibility analysis and for trajectory planning algorithms. We build the Constructibility Gramian for the multiagent system with relative ranging measurements and assess local constructibility property. Then, we propose a trajectory planning algorithm where a pair of vehicles without a priori information achieve global constructibility with both absolute and relative measurements. Moreover, we propose a variation of the Constructibility Gramian, limited to the position variable and hence called Position Gramian, and use this tool in a Model Predictive Control framework to plan the trajectory of a tracker vehicle aiming at simultaneously localising itself and a collaborative target through ranging measurements.