A variational Bayesian approximation based adaptive single beacon navigation method with unknown ESV
Introduction
Autonomous Underwater Vehicles (AUVs) are widely used in underwater operations, such as exploration, oceanographic survey, or rescue mission, and accurate underwater positioning is essential for these operations (Qin et al., 2019a, Qin et al., 2019b, Emami and Taban, 2018, Guerrero et al., 2019, Paull et al., 2014, Xiang et al., 2017). Unlike land and air operations, lack of electromagnetic signals is a special issue in underwater environment. Range-based acoustic navigation systems are mainstream methods in underwater operations. The long baseline (LBL) underwater localization systems, which depend on the slant range between vehicle and multi-beacon, are the most widely used underwater navigation methods. LBL systems are expensive and time-consuming due to the need of calibrating multi-beacon positions before underwater missions (Paull et al., 2014, Chen et al., 2016, Miller et al., 2010). In addition to LBL systems, the single beacon navigation systems, which fuse the slant range information of single beacon and the dead-reckoning data, have also attracted the attention of researchers. Existing studies on single beacon navigation system mainly focus on system design (Fallon et al., 2010, Claus et al., 2018, Kepper et al., 2018, Jin et al., 2019, Vaulin et al., 2017), information fusion algorithm design (Wang et al., 2013, Walls and Eustice, 2014, Webster et al., 2013, Webster et al., 2012) and model observability analysis (Gadre and Stilwell, 2004, Gadre and Stilwell, 2005, De Palma et al., 2017, Indiveri et al., 2016). Comparing with LBL systems, the decrease in number of beacons reduces the cost and time consumption of single beacon navigation system, and improves the utilization potential of the system in underwater environment.
The slant range is obtained from the effective sound velocity (ESV) and the acoustic transit time measured by a hydrophone. Most existing single beacon navigation methods treat the ESV as a known constant value (Fallon et al., 2010, Claus et al., 2018, Kepper et al., 2018, Jin et al., 2019, Wang et al., 2013, Walls and Eustice, 2014, Webster et al., 2013, Webster et al., 2012, Gadre and Stilwell, 2004, Gadre and Stilwell, 2005, De Palma et al., 2017, Indiveri et al., 2016). However, in the practical application of underwater navigation, ESV is time varying and hard to be accurately determined because of changing underwater environment. The setting error of ESV will lead to the ranging error, and consequently deteriorate the performance of single beacon navigation system. There are two common methods to deal with the unknown ESV in single beacon navigation systems: the state augmented method and the expectation–maximization (EM) method. The state augmented based navigation method was originally proposed in Zhu and Hu (2018) and Zhu et al., 2016a, Zhu et al., 2016b, in which Zhu et al. treated the unknown ESV as a part of state vector, and estimated it along with the vehicle position by extended Kalman filter (EKF). This method introduces an additional noise parameter related to the ESV uncertainty, which cannot be precisely obtained in practical application. Performance of navigation method proposed in Zhu and Hu, 2018, Zhu et al., 2016a, Zhu et al., 2016b is sensitive to the noise statistic parameter (Deng et al., 2018), which limits its application. The EM-based single beacon navigation method was recently proposed by Qin et al. (2019c), which treated the ESV as an unknown deterministic system parameter, and used EM method to simultaneously estimate the parameter and system state. This method avoids the additional tuning parameter, and achieves an ideal performance in the absence of initial position offset. However, a large initial position offset will induce large positioning error and ESV estimated error.
To overcome the deficiencies of single beacon navigation methods proposed in Zhu and Hu, 2018, Zhu et al., 2016a, Zhu et al., 2016b and Qin et al. (2019c), this paper proposes a variational Bayesian (VB) approximation based adaptive single beacon navigation method. VB approximation, which has been widely concerned by the state estimation community, is mainly used to estimate the unknown noise statistic parameters of state space model, and design the corresponding adaptive Kalman filter (Sarkka and Nummenmaa, 2009, Huang et al., 2018, Xu et al., 2018, Xu et al., 2019). This paper treats the ESV as a random variable with unknown statistic parameters, and estimates the state vector, ESV and ESV uncertainty parameters by VB approximation. Comparing with the state augmented method (Zhu and Hu, 2018, Zhu et al., 2016a, Zhu et al., 2016b), the ESV uncertainty is modeled as a white Gaussian process with unknown mean and variance, which avoids the difficult Kalman tuning process and enhances the adaptiveness of the navigation system. Comparing with the EM-based method (Qin et al., 2019c), the robustness of the proposed method to the initial position offset is improved with the help of Bayesian treatment of ESV. Field data will be used to study the possible improvement of the proposed VB-based adaptive navigation method.
Section snippets
Single beacon underwater navigation model
We roughly review the basic single beacon navigation model based on known ESV in this section. Several studies describe the details of navigation model (Gadre and Stilwell, 2004, Gadre and Stilwell, 2005, Casey et al., 2007).
In this paper, the subscript “” represents the th epoch. Denote as the discrete state vector, in which and are the horizontal coordinates of the hydrophone in the and directions, respectively; and and are the corresponding ocean
VB-Based adaptive single beacon navigation method
In this section, VB method is used to estimate unknown ESV and ESV uncertainty parameters. The measurement model is reformulated in Section 3.1, and consequently the optimal filtering of the problem under consideration is described subsection in 3.2. The prediction and correction steps of VB-based adaptive single beacon underwater navigation method are described in Sections 3.3 Prediction step, 3.4 Correction step, respectively. The proposed method is summarized in Section 3.5. Finally, the
Numerical studies
The field data collected from surface boat is utilized to evaluate the performance of proposed VB-based navigation method (referred to as the “PM”) against that of the existing state augmented method (Zhu and Hu, 2018, Zhu et al., 2016a, Zhu et al., 2016b) and EM-based navigation method (Qin et al., 2019c) (referred to as the “SAM” and “EMM”, respectively).
Several studies describe the methods of collecting field data (Zhu and Hu, 2018, Zhu et al., 2016a, Zhu et al., 2016b). The boat is equipped
Concluding remarks
The ESV setting error will deteriorate the performance of single beacon navigation system. The existing state-of-the-art single beacon navigation methods include the state augmented method and the EM based method. The former is sensitive to the noise statistic parameter related to the ESV uncertainty that is difficult to tune, whereas the latter is sensitive to vehicle initial position offset. To overcome the deficiencies of existing state-of-the-art single beacon navigation methods, this paper
CRediT authorship contribution statement
Hong-De Qin: Conceptualization, Funding acquisition, Project administration. Xiang Yu: Conceptualization, Methodology, Validation, Writing original draft. Zhong-Ben Zhu: Resources, Supervision, Writing - review & editing. Zhong-Chao Deng: Investigation, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work was funded by the National Natural Science Foundation of China under Grant 51939003 and 51879062.
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