Problem statement: Given the simultaneous information of sonobuoys positions and the TOAs at different sonobuoys, and assuming the measurement performed each time affected by a bounded error accounting for all the uncertainties, then the simultaneous Least Squared (LS) navigation algorithm could be proposed for cooperative navigation for heterogenous AUVs, and the observation equations could be made. The Taylor-series will give a Least Squared error solution to a set of simultaneous nonlinear algebraic position equations, and the position of AUV will be achieved via the iterative refinement scheme. When Time of Arrivals from the pinger on-board the vehicle to the sonobuoys combined with the DGPS positions of the sonobuoys themselves are used to identify the position of an AUV, it is possible for the sonobuoys to get the positions of multiple underwater vehicles within different time slots, namely the time-multiplexed navigation. Then, the central control ASV can collect all the positions of underwater vehicles via radio link to sonobuoys. It gives a clear understanding that the navigation algorithm for acoustic navigation in coordinated underwater vehicles regresses to the navigation algorithm of a single target. The detailed algorithm will be shown as follows.
Consider an absolute earth fixed reference system (O}:=(X, Y, Z} with the Z-axis pointing upward from the sea surface, and n sonobuoys(usually n is equal to four) at the sea surface with hydrophones at DGPS positions given by vector Pi=[xi, yi, Zi]T; i=1,2,...n. Let Po=[x y z]T denote the position of one of the interested underwater vehicles with respect to the reference frame. The navigation problem considered in this chapter can then be concisely stated as follows: obtain estimates of the AUVs position based on information provided by the sonobuoys, which compute the flight time of the acoustic signals emitted periodically by a pinger installed on-board the underwater vehicle. It belongs to a passive acoustic navigation system but not an active navigation system according to (Freitag et al., 2001). Further let f=[f1...f] ; 1=1,2,...n, denote the ranges between the dedicated underwater vehicle and the sonobuoys.
Before formulating the measurement, there are three assumptions based on practical but also reasonable principle.
1. Each sonobuoy position will be known at differential GPS accuracy, and the uncertainty of the DGPS position can be treated as an additional uncertainty in the totally measurement.
2. The sonobuoys freely drift at the sea surface due to waves and current and are assumed with a much slower drifting speed compared with the instantaneous flight time of the acoustic signals, which means the sonobouy position is relatively static and the movement of the sonobuoy can be treated again as an additional uncertainty in the totally measurement.
3. The sound speed c(z) in the area of interest is assumed known and the sound speed is assumed to vary only with depth in the area of interest is considered in the most common situation. Although the different sea temperature do affect the sound speed in different water column, the slightly varied sound speed due to heterogeneous temperature in different current layers can be treated again as an additional uncertainty in the totally measurement.
With the simultaneous information of sonobuoys positions and the TOAs at different sonobuoys, the simultaneous Least Squared (LS) navigation algorithm can be followed as: The n equations describing the distance between the unknown position of the AUV and the sonobuoy location are:
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