Observers for position estimation using bearing and biased velocity information

This chapter considers the questions of observability and design of observers for position estimates of an object moving in Rⁿ. The scenario considered assumes a possibly biased measurement of velocity along with bearing or direction measurements to one or multiple fixed points in the environment. The motivating example is a robot moving in either two- or three-dimensional space with a sensor, such as a camera, capable of providing bearing but not distance to observed fixed points in the environment. We provide a comprehensive observability analysis, and discuss stability and convergence of the observer design and observer error dynamics, under persistence of excitation conditions on the vehicle motion. Interestingly, we can show uniform observability even for a single direction measure and with unknown velocity bias as long as the vehicle trajectory is persistently exciting. Some extensions to the case of multiple directions with known- or unknown-observed point location are addressed and observers endowed with exponential stability of error dynamics are derived. Simulation results are presented that demonstrate the performance of the proposed approach.