Elsevier

European Journal of Control

Volume 49, September 2019, Pages 116-130
European Journal of Control

Long baseline navigation with explicit pseudo-range clock offset and propagation speed estimation

https://doi.org/10.1016/j.ejcon.2018.12.010Get rights and content

Abstract

This paper proposes a novel one-way-travel-time (OWTT) long baseline (LBL) navigation concept that departs from previous approaches in that: (i) the clock of the receiver in the vehicle needs not to be synchronized with the clocks of the emitters; and (ii) the speed of propagation of the signals is assumed unknown. The nonlinear system dynamics are considered in a continuous-discrete time framework, taking advantage of the pseudo-range measurements obtained at low update rates and the data from other sensors obtained at high rates. An augmented system is proposed whose observability is analyzed and that is shown to be equivalent to the original nonlinear system under appropriate conditions. A Kalman filter provides the estimation solution, with globally exponentially stable error dynamics, in spite of the original nonlinear nature of the system dynamics. The performance of the proposed solution is evaluated with numerical simulations resorting to Monte Carlo runs. The comparison with the extended Kalman filter and the Bayesian Cramér-Rao bound are also included.

Introduction

The global positioning system (GPS) is one of the most popular options when it comes to the choice of aiding devices for navigation purposes. However, in some scenarios, the GPS is unavailable, such as in underwater environments, where the attenuation that the electromagnetic waves suffer prevents its use, which fosters the development of alternative solutions. In underwater, long baseline (LBL) acoustic positioning systems have been widely explored, usually in one of two alternative modes: (i) in one-way-travel-time (OWTT) LBL configurations, the clocks are assumed synchronized, the speed of propagation of the acoustic waves is assumed available, and the distances between several known emitters and the receiver installed on-board the vehicle are computed based on time tags that are sent along the acoustic signals; or (ii) in two-way-travel-time (TWTT) LBL configurations, a transponder installed on-board the vehicle interrogates a set of known transponders that are fixed in the mission scenario and computes the distances between the vehicle and each of these transponders based on the round-trip travel time assuming that the speed of propagation of the acoustic signals is available.

Some of the early work on underwater acoustic positioning systems can be found in [13], [25]. An extended Kalman filter (EKF) with a Rauch-Tung-Striebel smoother is implemented in [34] to solve the 2D navigation problem with three acoustic transponders. A conventional LBL positioning system is merged with a Doppler sonar, with bottom-lock, in [39]. It includes a magnetometer and roll/pitch sensors, and a complementary low-pass/high-pass filter approach is implemented to show that the Doppler precision is effectively improved. In [35] two different LBL alternatives are presented: (i) fix computation approach; and (ii) filtering approach. In the first, dead-reckoning is performed between acoustic fixes, which reset the vehicle position whenever available. In the second, dead-reckoning is performed but valid travel times are used, whenever available, to correct for dead-reckoning drift. In [18] some preliminary field trials are shown of a navigation system that consists of a LBL acoustic positioning system, a Doppler sonar, a fiber-optic North seeking gyro, pressure sensors and magnetic compasses, whereas the main navigation algorithm resorts to the least-squares method. In [15] the problem of navigation with pseudo-range measurements, considering additive bias, is addressed. The approach that is proposed therein is new in that it uses a three-stage filter. In particular, a linearized Kalman filter is proposed that avoids divergence as the linearization is made about estimates of another sub-optimal Kalman filter that is based on a valid quasi-linear measurement model. A similar approach is followed in [17], [32] but considering a multiplicative unknown parameter, which accounts for the unknown wave speed. A different concept was proposed in [16], where one aims to estimate a segment of the trajectory instead of the current position. To that purpose, diffusion-based trajectory observers are considered. Another interesting setup was presented in [21], where distance measurements to a single acoustic source are considered, combined with dead-reckoning between distance measurements, to obtain a so-called synthetic long baseline. A related approach is proposed in [20], where simulations of a so-called virtual long baseline (VLBL) navigation algorithm for autonomous underwater vehicles are presented. In these two approaches some kind of trilateration is performed, even though not instantaneous. The direct use of distance measurements to a single source, without explicit trilateration, was evaluated in [11], where the distance measurements are fused with the data obtained from an inertial navigation system (INS) in an EKF. The topic of single range navigation has seen many contributions, see e.g. [1], [8], [22], [27], [37]. Finally, in what concerns acoustic-based positioning and navigation systems, it is also important to refer the ultra-short baseline (USBL) acoustic positioning systems, which are rapidly increasing in popularity and number, see e.g. [26], [28], and references therein. For interesting discussions and detailed surveys on underwater vehicle navigation techniques and challenges, see [19], [24], [33].

In OWTT navigation, the beacons and receiver clocks are assumed synchronized and the time of signal emission is either predefined or encoded and sent through communication modems, see [7], [38], and references therein. However, high-grade clocks must be employed, to mitigate clock drift, and clock synchronization at the beginning of each mission must be performed, which poses an additional and very heavy burden. Finally, even with high-grade clocks, clock drift is inevitable in long missions unless synchronization is performed periodically. On the other hand, a common assumption throughout the literature is that the speed of propagation of the waves in the medium is known or measured. This quantity depends on several characteristics such as the salinity, pressure, and temperature and it is either measured or profiled, often prior to the experiments. If that is not the case, or even for small errors of the sound velocity profile, the range measurements can carry large errors, particularly when the distances become large, thus putting into question the entire navigation data. While that is out of the scope of the paper, there exist many algorithms in the literature to handle the presence of outliers, see e.g. [27], [40].

In previous work by the authors a novel filtering solution was proposed for LBL navigation [2] based on an extension of the framework for single range measurements detailed in [1]. In these approaches, the range measurements were assumed to be available, either resorting to two-way-travel-time or to OWTT coupled with synchronous clocks and acoustic communications. In short, the system dynamics are augmented, including as system states the range measurements and identifying nonlinear terms with new system states, until the system can be regarded as linear. A careful observability analysis follows and, due to its constructive nature, the Kalman filter provides the estimation solution with globally exponentially stable (GES) error dynamics. Previous works applying state augmentation to applications with LBL positioning can be found in [23], [29], [31], where past measurements were included in the state. In the present work the differences between pseudo-range measurements are considered as additional states. More recently, in [4], [5], the author has addressed the issues of clock offset and propagation speed estimation separately. The main contribution of this paper is the design and analysis of a novel navigation filter that unifies both concepts. In short, a OWTT navigation setting is assumed but the clock of the receiver installed on-board does not need to be synchronized with those of the emitters. Additionally, the speed of propagation of the acoustic waves in the medium is also unknown. A filtering solution is proposed, effectively bringing together the concepts introduced in [4], [5], that yields estimates of the inertial position, inertial ocean current, clock offset, and speed of propagation of the acoustic waves. This setting potentially reduces the required hardware and simplifies deployment, as the clock of the vehicle does not need to be synchronized and the speed of sound does not need to be profiled. The discrete-time nature of the pseudo-range measurements is taken into account, the observability of the system is carefully analyzed, and the errors of the estimates, provided by a Kalman filter, converge exponentially fast to zero for all initial conditions. Higher rate attitude and velocity measurements drive the system dynamics between pseudo-range measurement updates. Initial work was presented in [3], where the novel problem framework was first introduced and the overall system dynamics proposed. This paper presents a unified and thorough presentation and analysis of the solution and it includes all the theorems and proofs that had been omitted in the conference paper. It also includes additional theoretical results, the discussion of several issues, and an additional solution without state augmentation. Finally, extensive and realistic simulations are also presented in detail, including cases with slowly time-varying quantities. The performance is evaluated also with Monte Carlo runs and the comparison with the EKF and the Bayesian Cramér-Rao bound is also included.

The paper is organized as follows. The problem statement and the nominal system dynamics are introduced in Section 2. The proposed filter design is detailed in Section 3, including a thorough observability analysis and the discussion of multi-rate implementations. Extensive simulation results are presented in Section 4, including the comparison with the EKF and the Bayesian Cramér-Rao bound (BRCB), and Section 5 summarizes the main results of the paper.

Throughout the paper, the symbol 0n × m denotes a n × m matrix of zeros and In an identity matrix of dimensions n × n. A square matrix of zeros of dimension n × n is simply represented by 0n. A block diagonal matrix is represented by diag(A1,,An). For xR3 and yR3, x · y and x × y represent the inner and cross products, respectively. The Special Orthogonal Group is denoted by SO(3):={XR3×3:XXT=XTX=I3det(X)=1}. For convenience, define also the transpose operator (.)T, and notice that x·y=xTy, x,yR3.

Section snippets

Problem statement

Consider a scenario where a set of acoustic emitters are installed in an LBL configuration and an underwater vehicle, equipped with an acoustic receiver, operates. The emitters are fixed, their clocks are synchronized and their inertial positions are available to the vehicle. Periodically, the emitters send acoustic signals, which are received by the vehicle. Additionally, the vehicle is also equipped with a Doppler velocity log (DVL) and an attitude and heading reference system (AHRS). In a

Filter design

The two main issues regarding LBL navigation that are addressed simultaneously in this paper, namely (i) dealing with unknown speeds of propagation of the acoustic waves in the medium; and (ii) dealing with unknown clock offsets between the emitters and the receivers have been addressed separately in the past. In particular, the latter was considered in [4], where a novel long baseline navigation solution was proposed considering that there is an unknown clock offset between the acoustic

Simulation results

In order to evaluate the performance of the novel OWTT LBL solution, simulation results resorting to the Monte Carlo method are presented herein. In Section 4.1 the simulation setup is introduced, whereas a theoretical performance limit is described in Section 4.2. The proposed solutions are detailed in Section 4.3, whereas the EKF is briefly described in Section 4.4. Finally, extensive Monte Carlo runs are detailed and discussed in Section 4.5, where the comparison with the theoretical

Conclusions

A novel concept for long baseline navigation in a one-way-travel-time operational mode was proposed in this paper. It departs from previous approaches in that: (i) the clock of the receiver mounted on-board the vehicle does not need to be synchronized with the clocks of the emitters; and (ii) the speed of propagation of the signals is assumed unknown. This significantly lowers the operational burden and hardware requirements in LBL navigation. In the proposed framework, the system dynamics were

Acknowledgments

The author would like to acknowledge the valuable discussions and suggestions of Profs. Carlos Silvestre and Paulo Oliveira mainly during the first phases of development of this work.

This work was supported by the Fundação para a Ciência e a Tecnologia (FCT) through ISR under FCT UID/EEA/50009/2013.

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