Phase Based Localization for Underwater Vehicles Using Interval Analysis

International audienceThis paper addresses the common issue of locating an Underwater Vehicle (UV). Usually, the positioning of a vehicle is based on the propagation of electromagnetic waves, using systems such as the Global Positioning System (GPS). However, water, and particularly salty water, makes the use of electromagnetic waves impractical due to the attenuation caused by the high conductivity of the medium. Hence, the most reliable way to transmit information underwater is by using sound waves and many technologies have emerged to solve the positioning problem in this way. Technologies such as Long BaseLine (LBL), Short BaseLine (SBL) and Ultra-Short BaseLine (USBL) are the most frequently used underwater. These technologies are based on the use of multiple emitters in the case of LBL and SBL, or multiple receivers in the case of USBL. This paper describes a way of finding a vehicle location, on-board, based on the measurement at the vehicle by a single receiver of the phase of an acoustic sine wave transmitted from a single emitter that is at a fixed and known location. The method also uses other proprioceptive measurements: vehicle's velocity and heading. An algorithm based on contractors and bisections scatters the solution space searching for all possible solutions (positions in this case) to the set of equations. Moreover, this paper introduces the Time Constraint Satisfaction Problem (TCSP). Indeed, the proposed algorithm does not compute the solutions from measurements at a single point in time, rather it uses a set of measurements taken over a time window and stored in a buffer. As a result, the location is not only known at the latest instant but the past locations can be tracked back over the length of the chosen time window

Computing capture tubes

International audienceA dynamic system can often be described by a state equation ˙x = h(x, u, t)where x ∈ Rn is the state vector, u ∈ Rm is the control vector and h :Rn × Rp × R → Rn is the evolution function. Assume that the control lowu = g (x, t) is known (this can be obtained using control theory), the systembecomes autonomous. If we define f (x, t) = h(x, g (x, t) , t), we get the followingequation.˙x = f (x, t) .The validation of some stability properties of this system is an important anddifficult problem [2] which can be transformed into proving the inconsistency of aconstraint satisfaction problem. For some particular properties and for invariantsystem (i.e., f does not depend on t), it has been shown [1] that the V-stabilityapproach combined interval analysis [3] can solve the problem efficiently. Here,we extend this work to systems where f depends on time