 Original Article
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Underwater inertial error rectification with limited acoustic observations
Satellite Navigation volume 5, Article number: 3 (2024)
Abstract
Underwater inertial navigation is particularly difficult for the longdurance operations as many navigation systems such global satellite navigation systems are unavailable. The acoustic signal is a marvelous choice for underwater inertial error rectification due to its underwater penetration capability. However, the traditional Acoustic Positioning Systems (APS) are expensive and incapable of positioning with limited acoustic observations. Two novel underwater inertial error rectification algorithms with limited acoustic observations are proposed. The first one is the single acousticbeacon Rangeonly Matching Aided Navigation (RMAN) method, which is inspired by matching navigation without reference maps and presented for the first time. The second is the improved single acousticbeacon Virtual Long Baseline (VLBL) method, which considers the impact of indicated relative position increments on virtual beacon reconstruction. Both RMAN and improved VLBL are further developed when multi acousticbeacons are available, named mABRMAN and mABVLBL. The comprehensive simulations and field investigations were conducted. The results demonstrated that the proposed methods achieved excellent accuracy and stability compared to the baseline, specifically, the mABRMAN and mABVLBL can reduce the inertial error by more than 90% and 98% when using single and double acousticbeacons, respectively. These proposed techniques will provide new perspectives for underwater positioning, navigation, and timing.
Introduction
The increasing marine exploration and activities have placed high demands on underwater Positioning, Navigation and Time (PNT). Unlike aerial and land navigation, the Global Satellite Navigation Systems (GNSS) is unavailable underwater due to its poor penetration capability (Yang, 2018). Typical underwater positioning and navigation techniques include Inertial/Dead Reckoning Navigation System (INS/DRNS) (ElSheimy & Youssef, 2020; Paull et al., 2014), Acoustic Positioning Systems (APS) (Qin et al., 2022; Tang et al., 2023; Zou et al., 2023), and Geophysical Matching Aided Navigation (GMAN) using gravity, terrain, and magnetic (Wang et al., 2023; Zhang et al., 2022), and etc.. However, an individual positioning and navigation method is often insufficient to meet the demands of underwater PNT for the activities such as ocean exploration, monitoring, and military operations (Yang & Qin, 2021), particularly for the longduration and longdistance underwater missions (Xu, 2017).
Underwater positioning commonly utilizes inertialacoustic integrated navigation, which combines Inertial Navigation System (INS) and APS (Claus et al., 2018; Liu et al., 2021; Masmitja et al., 2019; Wang et al., 2022a, 2022b; Zou et al., 2023). Inertial navigation provides selfsufficiency, concealment, and allaround output with a high rate, but it is prone to accumulating errors over time, which needs external position correction periodically. The Doppler Velocity Log (DVL) has limitations in deepsea areas and can only measure the velocity relative to water (Wang et al., 2020). INS/DVL integrated navigation only partially restrains INS errors and cannot guarantee bounded positioning errors (Tang et al., 2023). Acoustic positioning includes Long Baseline (LBL), Short Baseline (SBL), UltraShort Baseline (USBL), and single acousticbeacon rangeonly positioning. LBL measures the RoundTrip Time (RTT) to perform triangulation on vehicle using asynchronous queryresponse. A typical LBL system includes multi acousticbeacons that need calibration and supervision (LaPointe, 2006). It is an expensive and timeconsuming task, particularly during largescale underwater operations in multiple areas that require separate deployment, calibration, and recovery for the acousticbeacons (Masmitja et al., 2019). Additionally, LBL's queryresponse approach restricts the acousticbeacons service capacity and cannot work for multiple underwater vehicles in parallel. Moreover, complicated underwater environments may limit acoustic observations that needs improving to meet the requirement for the triangulation.
Due to its simple structure and considerable scalability, the single acousticbeacon rangeonly navigation has attained increasing attention (Jakuba et al., 2021; Rypkema et al., 2018; Zhao et al., 2022). The OneWay Travel Time (OWTT)based passive inverted USBL (piUSBL) system estimates the OWTT slant range and azimuth by acquiring broadcast signals from a timesynchronized acousticbeacon (Wang, et al., 2022a). However, the piUSBL system has limited potential to support highaccuracy positioning in extensive operating spaces as the restricted size of the receiver. The filterbased rangeonly navigation techniques correct the position errors of the vehicle by employing extended Kalman filters or particle filters as estimators while taking the slant range as observation (Jankovic et al., 2023; Masmitja et al., 2019). Most filterbased methods rely on linearized error models, prior system knowledge, and measurement noise (Claus et al., 2018). Unreasonable linearized error models or filter parameter settings may cause the filter diverge. The Virtual/Synthetic Long Baseline navigation (VLBL/SLBL) technique is commonly used in single acousticbeacon rangeonly navigation (LaPointe, 2006; Scherbatyuk, 1995). The SLBL approach can be developed for mobile LBL systems (Vaganay et al., 2004; Webster et al., 2012) for cooperative navigation with the help of reliable acoustic communication (Huang et al., 2018). In these systems, the acoustic beacons are fixed on the leader equipped with a highprecision navigation system, and their positions are broadcast to other vehicles through acoustic communication. This method offers excellent flexibility, but it does result in a higher system cost and may impact the system concealment.
The existing literature has primarily assumed that the indicated Relative Position Increment (RPI) is reliable for constructing virtual beacons and achieving VLBL positioning. However, this assumption is not always accurate, especially when the vehicle enters the functional area of the acousticbeacon after a longduration snorkeling. Both heading and scale errors will directly affect the position accuracy of virtual beacons. Traditional VLBL algorithms cannot eliminate these errors, and increasing the length of observations will also not improve algorithm accuracy. To our best knowledge, no publicly available literature reported the VLBL, which considers the indicatedRPI error.
The GMAN utilizes the intrinsic (slowly varying) physical features of the Earth for navigation purposes (Wang et al., 2023; Zhang et al., 2022). This kind of navigation exhibits the features such as allweather capability, strong concealment, and nonaccumulative error. The GMAN maximizes the correlation between the retrieved and measured sequences by matching the indicated track with the most suitable track in a precollected background reference map. Traditional matching aided algorithms include Terrain Contour Matching (TERCOM), Iterative Closest Contour Point (ICCP), and Sandia Inertial TerrainAided Navigation (SITAN). Among them, the ICCPbased algorithm maintains the advantages of TERCOM and SITAN algorithms. It addresses their shortcomings, such as the linearization of terrain required by SITAN and the accurate yaw information required by TERCOM. However, establishing highresolution background maps for the GMAN requires expensive and laborintensive effort and the deployment of costly measurement instruments like gravity meters on underwater vehicles. Furthermore, its matching aided navigation accuracy is limited, restricting its extensive application.
Motivated by the limitations mentioned above, this work aims at inertial error rectification with limited acoustic observations. For clarity, an abbreviation list with full explanations is given in abbreviation. The main contributions of this paper are as follows:

1.
A single AcousticBeacon (sAB) Rangeonly Matching Aided Navigation (RMAN) algorithm, inspired by the GMAN without a reference map, is presented. It is the firstever incorporation of matching aided navigation into rangeonly navigation, potentially opening up new perspectives in underwater navigation.

2.
An improved sAB VLBL algorithm considering the systemindicated RPI error (both rotational and scaling) is investigated.

3.
Both sABRMAN and improved sABVLBL are further developed to accommodate the wider conditions with multi acousticbeacons, denoted as mABRMAN and mABVLBL, and achieve higher reliability.

4.
Comprehensive simulation and field tests are conducted to evaluate the proposed algorithms and the results show excellent accuracy and stability are obtained compared to the baselines.
The remainder of this paper is organized as follows. Preliminaries are presented first. Section “Acousticbeacon rangeonly matching aided navigation” derived the sABRMAN and mABRMAN. The details of sABVLBL and mABVLBL are provided in the “Improved VLBL by considering indicatedRPI error” section. Comprehensive simulation and field tests are conducted in the “Performance verification” section. Conclusion and future work are given in the last section.
Preliminaries
We use upper (lower) boldface letters to denote matrices (vectors). An \(n \times n\) identity matrix will be marked as \({\varvec{I}}_{n}\). The special group \({\mathcal{SO}}(n): = \{ {\varvec{A}} \in {\mathbb{R}}^{n \times n}  {\varvec{AA}}^{{\text{T}}} = {\varvec{I}}_{n} ,{\text{det}}{\varvec{A}} = {\text{1}}\}\) is rotation matrices in \({\mathbb{R}}^{n}\), where det is for the determinant. \({\varvec{A}}_{i}\) is the ith row of matrix \({\varvec{A}}\). The operators \(\cdot\) and \({}\cdot{{}}\) are the L_{1} and L_{2} norms. Given \(\vartheta \in [0,2{\uppi })\), we define the unit orthonormal vectors \({\varvec{w}}_{\vartheta } : = [\cos (\vartheta )\;\;\sin (\vartheta )]^{{\text{T}}}\),\({\varvec{w}}_{\vartheta }^{ \bot } : = [  \sin (\vartheta )\;\;\cos (\vartheta )]^{{\text{T}}}\), and define the map \(f:\vartheta \mapsto {\varvec{R}}_{\vartheta }^{{}} : = [{\varvec{w}}_{\vartheta } \;\;{\varvec{w}}_{\vartheta }^{ \bot } ] \in {\mathcal{S}\mathcal{O}}(2)\). The body, navigation, earth, and inertial frame are denoted by lower letters b, n, e, and i, respectively. The “East (E)North (N)Up (U)” frame plays as the nframe in this paper. Denote \({\varvec{R}}_{b}^{n} \in {\mathcal{S}\mathcal{O}}(3)\) as the attitude matrix from bframe to nframe, the \({\varvec{v}}_{eb}^{n} = {[}v_{{eb,{\text{E}}}}^{n} \;\;v_{{eb,{\text{N}}}}^{n} \;\;v_{{eb,{\text{U}}}}^{n} {]}^{{\text{T}}}\) as velocity, and the \({\varvec{p}}^{n} = [L\;\;\lambda \;\;h]^{{\text{T}}}\) as position in nframe, where \(L\), \(\lambda\) and h is the latitude, longitude, and height, respectively.
Inertial/dead reckoning navigation system
The Strapdown Inertial Navigation System (SINS) kinematics can be described in terms of attitude, velocity, and position (Chang et al., 2023).
where \({\varvec{\omega}}_{ib}^{b}\) is the body angular rate expressed in the bframe w.r.t the iframe, measured by gyroscopes with inevitable drift, bias, and random noise. \({\varvec{\omega}}_{in}^{n}\) is nframe angular rate w.r.t iframe, \({\varvec{\omega}}_{ie}^{n} = [0\;\;\omega_{ie} \cos L\;\;\omega_{ie} \sin L]^{{\text{T}}}\) is the earth angular rate expressed in nframe w.r.t the iframe and \(\omega_{ie}\) is the earth rotation rate. \({\varvec{\omega}}_{en}^{n}\) is nframe angular rate expressed in nframe w.r.t eframe. \({\varvec{f}}_{ib}^{b}\) is the specific force measured by accelerometers. \({\varvec{g}}_{ib}^{n}\) is the gravity vector in nframe. The matrix R_{c} is the local curvature matrix.
The \({\varvec{p}}^{n}\) is not strictly belongs to Euclidean space and can be converted to the Cartesian coordinate system by
where \(R_{N}\) is the transverse radius of curvature of the WGS84 reference ellipsoid.
The underwater vehicle position can also be obtained by a Dead Reckoning Navigation System (DRNS)
where \({\varvec{v}}_{bd}^{d}\) is the ground velocity measured by DVL and expressed in DVL frame (dframe) w.r.t bframe. \({\varvec{R}}_{d}^{b} \in {\mathcal{S}\mathcal{O}}(3)\) is the constant misalignment attitude transformation matrix from dframe to bframe. \(\kappa\) is the scale factor of DVL. They can be calibrated in advance (Li et al., 2022). \({\varvec{R}}_{b}^{n}\) is the attitude transformation matrix and can be determined by a gyrocompass or Inertial Measurement Unit (IMU).
In the following derivations, an underwater Two Dimensional (2D) scenario will be employed as the Three Dimensional (3D) slantrange measured by acoustic modems can be transformed into a 2D range by incorporating the vehicle's depth provided with pressure sensor and the prior knowledge of the depth of the acousticbeacon.
Virtual long baseline
The VLBL navigation algorithm enables an underwater vehicle to determine its globally referenced position using an external single acousticbeacon with a known global position (LaPointe, 2006). As shown in Fig. 1 with colored version, by manipulating multiple asynchronous ranges from the same acousticbeacon, a long baseline of virtual beacons situated in various locations at a single point in time is created. Consequently, the underwater vehicle computes its global location using these virtual beacons like the method employed in a traditional LBL system.
Based on the geometric relationship between the virtual beacons and the vehicle, the acoustic observation equations can be derived as
where \(R_{i} = c\Delta t_{i} \in {\mathbb{R}}\) is the ith 3D slantrange between the physical beacon and the vehicle position, \(r_{i}\) is the 2D range, \(c\) is the Effective Sound Velocity (ESV), \(\Delta t_{i}\) is the OWTT, \(\Delta h_{i}\) is the depth difference between the vehicle and the single acousticbeacon, and \({\varvec{x}}_{N}\) is the current position to be solved. The location of the ith virtual beacon p_{v,i} can be denoted as
where \(\Delta {\varvec{x}}_{i}^{N} = {\varvec{x}}_{N}  {\varvec{x}}_{i}\) is the Relative Position Increment (RPI) from time \(t_{i}\) to \(t_{N}\). The \({\varvec{p}}_{r}\) is the position of the physical beacon. Note that \({\varvec{p}}_{v,N} = {\varvec{p}}_{r}\) as \(\Delta {\varvec{x}}_{N}^{N} = 0\).
According to the quadratic nonlinear equations shown in (4), the current position \({\varvec{x}}_{N}\) can be determined by the Least Squares (LS) when the matrix AA^{T} is nonsingular
where \({\varvec{A}}_{i} = [{\varvec{p}}_{{v,i{ + }1}}  {\varvec{p}}_{v,i} ]^{{\text{T}}}\), \({\varvec{B}}_{i} = {{(r_{i}^{2}  r_{i + 1}^{2} + d_{i + 1}^{2}  d_{i}^{2} )} \mathord{\left/ {\vphantom {{(r_{i}^{2}  r_{i + 1}^{2} + d_{i + 1}^{2}  d_{i}^{2} )} 2}} \right. \kern0pt} 2}\), and \(d_{i} = { }{\varvec{p}}_{v,i} {}\). Note that \({\varvec{p}}_{{v,N{ + }1}} = {\varvec{p}}_{v,1}\), \(r_{N + 1} = r_{1}\) and \(d_{N + 1} = d_{1}\).
Iterative closest contour point
Iterative closest contour point, a typical correlation matching method, was developed from the Iterative Corresponding Point (ICP) algorithm for image registration (Zhang et al., 2022). The feasibility of using this algorithm for gravity and terrain matching aided navigation was demonstrated by many researchers (Wang et al., 2018; Zhang et al., 2017). The fundamental concept of the ICCP matching algorithm is to find an optimal Rigid Transformation (RT) parameter \({\varvec{\psi}}_{r}^{ + }\) under the specified objective function \({\mathcal{L} }( \cdot , \cdot )\). The mathematical formula of ICCP is given by
where \({\varvec{\psi}}_{r} \triangleq [\vartheta ,{\updelta }{\varvec{x}}^{{\text{T}}} ] \in {\mathbb{R}}^{3}\) the RT parameter, \(\vartheta\) and \({\updelta }{\varvec{x}}\) are the rotational and translation parameter, respectively. The RT is given by
where \(\overline{\user2{X}} = {[}\overline{\user2{x}}_{1} {,}...{, }\overline{\user2{x}}_{N} {]} \in {\mathbb{R}}^{2 \times N}\) is the indicated track. \({\varvec{R}}_{\vartheta }\) is the rotation matrix w.r.t \(\vartheta\). \(\tilde{\user2{X}} = {[}\tilde{\user2{x}}_{1} {,}...{, }\tilde{\user2{x}}_{N} {]}\) is the rigid track of indicated track \(\overline{\user2{X}}\) determined by \({\varvec{\psi}}_{r}\). The retrieval operator \({\mathcal{M} }{(} \cdot {)}\) retrieves the Closest Neighboring Grid (CNG) sequence of the input track according to the prestored reference map and the input track. The correction between the measured sequence \({\varvec{Y}}\) and the retrieved sequence \(\tilde{\user2{Y}} = {\mathcal{M} }(\tilde{\user2{X}})\) will be evaluated by the correction analysis function \({\mathcal{L} }( \cdot , \cdot )\), such as Mean Absolute Difference (MAD), Mean Squared Difference (MSD), Cross Correction (COR), etc.
After the optimal parameter \({\varvec{\psi}}_{r}^{ + }\) is obtained, the optimal matched track \(\hat{\user2{X}}\) can be acquired by Eq. (8). In GMAN algorithms, the builtin reference map is critical for retrieval operation \({\mathcal{M} }{(} \cdot {)}\) to find the CNG sequence and the optimal matched track. However, developing a highresolution and highprecision physical reference map is an essential and indispensable task, albeit timeconsuming and laborintensive. To some extent, these limitations are the underlying factors that GMAN is yet to be fully widespread.
Acousticbeacon rangeonly matching aided navigation
Marine complexity, acousticbeacon integrity, and vessel maneuverability can limit acoustic observations, and further hinder the efficacy and accuracy of the APS. The sABRMAN is developed for single acousticbeacon, and its workflow is presented. The advantages of the RMAN over the GMAN are discussed in detail. Then, extensions of RAMN with multiavailable acousticbeacons (mABRMAN) can accommodate more application scenarios. Finally, an intuitive analysis on mABRMAN performance as the available acoustic beacons increase is presented.
Single acousticbeacon RMAN (sABRMAN)
The SINS(DRNS)indicated track suffers from the rigid and scaling transformation due to the irreversible accumulated errors caused by the IMU or Odometer. Thus, it is more reasonable to model the relationship between the indicated track and the reference track using an Affine Transformation (AT), which fully covers the RT when no scale error exists. Therefore, the affine rather rigid one will be adopted in RMAN.
The indicated track \(\overline{\user2{X}}\) will suffer an AT determined by \({\varvec{\psi}}_{a}\) before feeding it into the retrieval operation \({\mathcal{M} }(\cdot)\). The AT is given by
where \(\tilde{\user2{X}}\) is the affined track w.r.t the variable \({\varvec{\psi}}_{a} \triangleq [{\updelta }\kappa ,{\varvec{\psi}}_{r} ] \in {\mathbb{R}}^{4}\), where \({\updelta }\kappa\) the scale error while \({\varvec{\psi}}_{r}\) is in line with the definition of Eq. (7).
Using weightedMAD (wMAD) as the correction analysis function, the optimal \({\varvec{\psi}}_{a}^{ + }\) can be determined by solving the following problem.
where \(\odot\) is elementwise multiply, \({\varvec{r}} = {[}r_{1} ,...,r_{N} {]}^{{\text{T}}}\) is the measured range sequence, and \(\tilde{\user2{r}} \triangleq {\mathcal{M} }(\tilde{\user2{X}}) = { [}\tilde{r}_{1} ,...,\tilde{r}_{N} {]}^{{\text{T}}}\) the retrieved range sequence w.r.t the affined track \(\tilde{\user2{X}}\). \({\varvec{w}} \in {\mathbb{R}}^{N \times 1}\) is the weight that can be assigned by \(w_{i} = \tfrac{1}{N}\) for \(i = 1,...,N\). The retrieval operator \({\mathcal{M} }( \cdot )\) is given by
The framework of the proposed sABRMAN is shown in Fig. 2. The specific rangeonly matching aided procedure is as follows.
Step 1 Record the slant ranges and the corresponding indicated track (including the depth difference measured by depthmeter).
Step 2 Conduct the affine transformation determined by \({\varvec{\psi}}_{a}\) on the indicated track to generate candidate tracks and evaluate the correction between the measured range sequence and retrieved range sequence.
Step 3 Employ an optimization algorithm to search iteratively the global minimal value of the correlation function and find the optimal affine parameter \({\varvec{\psi}}_{a}^{ + }\). A proper function tolerance or max iteration number can be set as the stop criterion of this search process.
Step 4 Determine the optimal matching track \(\hat{\user2{X}}\) with respect to \({\varvec{\psi}}_{a}^{ + }\) and correct the indicated track.
The proposed RMAN does not require expensive sensors and advanced technologies, and only a hydrophone is needed if the vehicle and acoustic beacon are timesynchronized. Moreover, the RMAN does not require the reference map, while the GMAN needs a substantial investment and effort. The retrieval operation \({\mathcal{M} }( \cdot )\) of the RMAN is substituted by an efficient mathematical manner rather than a database lookup. Consequently, the map resolution will no longer limit the CNG of affined tracks obtained by retrieval operation. The prior knowledge on acousticbeacon required by the retrieval operation can be obtained through pattern recognition or acoustic communication.
The sABRMAN constructs a rangefield akin to an electrostatic field using the beacon location for matching aided navigation. Although there are infinite matched results on the rangefield due to the infinite CNG of each indicated point if no constraint is employed, the constraints among elements of the indicated track will rapidly reduce the solution space, enabling the incorporation of the ranges for matching aided purpose.
Extension of RMAN with multi acousticbeacons
When the vehicle is moving within the effective zone of the acousticbeacon array, multi acousticbeacons may be available. Assuming that the acousticbeacon location corresponding to the slantrange \(R_{i}\) obtained at \(t_{i}\) is \({\varvec{p}}_{i}\), the retrieval operation of the sABRMAN can be substituted simply by a dynamic one.
The dynamic retrieval operation enables the RMAN for the scenarios with multi acousticbeacons without additional modifications.
The matching accuracy will improve with the increased number of the acousticbeacons. An intuitive proof, take double acousticbeacon RMAN (dABRMAN) as example, is presented. As shown in Fig. 3, the initial error between the indicated position \(I_{i} ,i \in [1,N]\) and it’s the reference position T_{i} is given by
When the matching aided algorithm is employed for the indicated track on the rangefield #01 generated by the acousticbeacon #01, the matched track \(\hat{\user2{X}}_{\# 01}\) is obtained as
The remaining matching error of \(I_{i} ,i \in [1,N]\) on the rangefield #01 is decreased as
where \(\theta_{1,i} = \angle C_{1,i} I_{i} T_{i}\). Obviously, \({\varvec{e}}_{1,i} \le {\varvec{e}}_{0,i}\).
Applying the matching algorithm for the matched track \(\hat{\user2{X}}_{\# 01} \,\) on the rangefield #02 that generated by the acousticbeacon #02, and the rematched track \(\hat{\user2{X}}_{\# 02}\) can be obtained:
Based on the twostep matching process, the matching error for indicated track can be written as
where \(\theta_{2,i} = \angle C_{2,i} C_{1,i} T_{i}\). No doubt that the \({}{\varvec{e}}_{2,i} {} \le {}{\varvec{e}}_{1,i} {} \le {}{\varvec{e}}_{0,i} {}\) is always hold, which means that additional acousticbeacon can further improve the matching accuracy.
It is not difficult to generalize that the theoretical matching error of indicated point \(I_{i} ,i \in [1,N]\) with n available beacons can be expressed as
Therefore, more rangefield generated by acousticbeacon can be introduced into the mABRMAN to further minimize the matching error.
Improved VLBL by considering indicatedRPI error
The construction process of the virtual beacon in Eq. (5) relies entirely on the indicated RPI and physical beacon's location. The localization of underwater fixed beacons have been extensively studied and can achieve centimeterlevel accuracy (Yang & Qin, 2021). The accumulated errors of SINS (or DRNS) over time will impact the accuracy of the RPI, and the indicated RPI errors will propagate directly to the virtual beacon's location, as shown in Fig. 1 with the indicated track and virtual acoustic beacon in grey. The desired performance of VLBL will only be guaranteed if the indicated RPI error is compensated. This section presents an improved single acousticbeacon VLBL (sABVLBL) considering the indicated RPI error compensation. Then, the sABVLBL is developed to mABVLBL to adapted to the scenario with multi acousticbeacons.
Improved single acousticbeacon VLBL (sABVLBL)
To enhance the positional accuracy of virtual acousticbeacons, it is crucial to initiate the error compensation of the indicated RPI. One natural and intuitive approach is to take the affine transformation with zero translation to model the relation of the true RPI \(\Delta {\varvec{x}}\) and the indicated RPI \(\Delta \overline{\user2{x}}\). The ith virtual acousticbeacon in Eq. (5) can be rewritten as follows while considering RPI error compensation.
where \(\Delta \overline{\user2{x}}_{i}^{N} = \overline{\user2{x}}_{N}  \overline{\user2{x}}_{i}\) is the indicated RPI from time \(t_{i}\) to \(t_{N}\), while \({\updelta }\kappa\) and \(\vartheta\) are the rotational and scaling error compensation factors for the indicated RPI.
Let the tobesolved current position and the error compensation factors be \({\varvec{x}}^{  } \triangleq [{\varvec{x}}_{N}^{{\text{T}}} ,{\updelta }\kappa ,\vartheta ]^{{\text{T}}}\), which can be obtained by solving the following problem:
where
The abovementioned problem can be viewed as a Nonlinear Least Squares (NLS) problem. The Levenberg–Marquardt (LM) algorithm is adopted to find the optimal solution in this paper. The LM, also called the damped least squares method or GaussianNewtown method using a trust region approach, is a powerful means of resolving NLS problems.
Extension of the VLBL with multi acousticbeacons
Like the RMAN extension, it is easy to extend the sABVLBL for multi acousticbeacon scenario by reformulating the Eq. (21) as
where \({\varvec{p}}_{r,i}\) is the acousticbeacon position w.r.t the ith measured range. Takes the double acousticbeacon VLBL (dABVLBL) as a particular example of mABVLBL for explanation, noting \({\varvec{p}}_{{r,\{ 1,2,...,N  1,N\} }} = {\varvec{p}}_{{r,\{ 1,2,...,2,1\} }}\). As shown in Fig. 4, the position of virtual acousticbeacon directly generated by the indicated track (green) deviates significantly from the theoretical position of the virtual acousticbeacon generated by the reference track (blue). Compared with the traditional VLBL without error compensation on the indicated RPI, the dABVLB will construct the virtual acousticbeacon with higher position accuracy. It evidently leads to a more accurate outcome as shown in the shadow area.
Traditional VLBL requires the vehicles not traveling in a straight line, because it can create ambiguity (Koshaev, 2020). The accuracy of traditional VLBL will improve with the observation length N increased if the indicatedRPI is errorfree. Unfortunately, the accuracy of traditional VLBL does not improve with N increased but somewhat decreases. The proposed improved VLBL with considering indicatedRPI error, especially when multi acousticbeacons are available, reduces the requirement for vehicle maneuverability and enables higher positioning accuracy and robustness.
Performance verification
The performance of our proposed algorithms will be evaluated via simulation experiments and field trials, compared with traditional algorithms. The simulations were conducted on a computer with processor AMD Ryzen 7 5800H CPU @3.2 GHz with 32 GB of memory, all of the algorithms are implemented in Matlab 2020b. Abbreviations for different methods refer to Abbreviations section. In this paper, mABICCP and mABRMAN are classified as matchbased algorithms, with the former being the baseline for matchbased comparison. Similarly, the LBLbased algorithm includes the traditional VLBL, the mABVLBLrot (only the rotational error is compensated), and the proposed mABVLBL (both the rotational and scaling error are compensated). Among them, the VLBL will play as the baseline for the LBLbased algorithm. The lowercase letter m in the notations will be substituted by the lowercase letters s and d when single and double acousticbeacons are available. In the following discussion, unless otherwise specified, the statements with respect to the mABRMAN (or mABVLBL) are applicable to both single and double acousticbeacon scenarios.
Simulation and field experiment setup
We adopt DRNS as the primary navigation system in the following simulation experiments. The simulation parameters for DRNS are detailed in Table 1. DRNS updates the indicated position iteratively based on Eq. (3). The gyroscope drift is \({{0.03\;(^\circ )} \mathord{\left/ {\vphantom {{0.03\;(^\circ )} {\text{h}}}} \right. \kern0pt} {\text{h}}}\), and the scale error of odometer is 5%. The installation angle errors between the gyroscope and the odometer are (0.25°, 0°, 0.17°). To simulate the accumulated heading error of the DRNS over time more realistically, the initial errors (1°, 0°, 1°) are set. To evaluate the effectiveness of the proposed algorithms when single and double acousticbeacons are available, two acousticbeacons, numbered with AB#01 and AB#02, were deployed at coordinates (32.034°, 118.020°, 30 m) and (32.050°, 18.035°, 40 m), respectively, as shown in Fig. 5. These acousticbeacons emit fixedformat acoustic signals with frequency 0.25 Hz. The onboard hydrophone can measure the OWTTbased slantrange with ranging accuracy 2 m. The reference and DRNSindicated track are depicted in Fig. 5, with a duration of about one hour.
The whole track is divided into five segments to cover the relationship between the track and acousticbeacon as much as possible. They are marked as Track#0105, abbreviated as T1T5, with the duration [800,1300], [1400,1900], [2000,2800], [2900,3600] and [800,3600], respectively. Note that Track#05 covers Track#01–04. This division scheme is reasonable for the performance verification on various test conditions. It is not always possible for the hydrophone to periodically acquire acoustic observations due to the complex underwater environment and vessel mobility. Hence, two scenarios, abbreviated as S1 and S2, were simulated, where S1 indicates the acoustic range received periodically while S2 randomly. For the brevity, the TnS1 and TnS2 represent that the hydrophone obtains acoustic observations periodically and randomly, respectively, under the track Tn (n = 1, …, 5).
The field experiment was conducted at Lake Qiandao, and the experimental instruments and their deployment are shown in Fig. 6. The main experimental instruments and their characteristics are detailed in Table 2. It should be noted that the USBL system is utilized to acquire the slantrange between the vehicle and the fixed acousticbeacon rather than for USBL positioning. The relative displacement between SINS and USBL was compensated carefully. The underwater single acousticbeacon location was determined using the method described in (He et al., 2023). Due to the experimental conditions and limitations of the reference baseline, a shipbased experiment was conducted rather than underwater AUV. In this field experiment, only single acousticbeacon was deployed underwater. The SINS equipped with Fiber Optics Gyroscopes (FOGSINS) was installed as the primary navigation system, operating in pure inertial mode. The drift and bias of the laserIMU are better than \({{0.02\;(^\circ )} \mathord{\left/ {\vphantom {{0.02\;(^\circ )} {\text{h}}}} \right. \kern0pt} {\text{h}}}\) and \(50 \times 10^{  6} \;g{ (}g{ = }9.7803{\text{ m/s}}^{{2}} )\), respectively. In addition, an ultraprecision SINS/ Global Positioning System (GPS) integrated navigation system was deployed to offer accurate reference, where the SINS is equipped with Laser gyroscopes (LaserSINS) and GPS operating in RealTime Kinematic (RTK) mode. Figure 7 shows the reference track and the indicated track. The whole track duration is approximately 1 700 s. Like the simulation experiments, we picked three subtracks with duration [344, 908], [948, 1260] and [1356, 1684], respectively, marked as Track#01–03, to evaluate the performance of algorithms under different tracks. The ESV of the simulation and lake trial is 1 473 m/s.
In addition to the lake trial, a set of sea trial data was also collected to assess the performance of the proposed algorithms. Approximately 100 acoustic observations were obtained from two acousticbeacons during onehour voyage. Due to the confidentiality of this sea trial, the specific implementation procedures and equipment details are not disclosed to the public, and the positions of these acousticbeacons, indicated track, and reference track used in the evaluation have undergone meticulous declassification procedures. Therefore, only the relative positions were provided rather than the absolute positions.
Analysis of simulation results
The statistical results of positioning accuracy and efficiency in various test conditions based on 100 times Monte Carlo tests with different algorithms are reported in Table 3. The DRNSindicated error is determined based on the positioning output when acoustic observation is acquired. Consequently, the indicated errors under S1 and S2 for the same track in Table 3 are slightly different. This error computation manner is suitable to evaluate the performance as these algorithms offer positioning output only when the acoustic observation is received. This approach does not introduce an explicit error as the indicated errors under S1 and S2 are identical as shown in Table 3 but facilitates more reasonable quantitative analysis of the algorithm's performance. For different test tracks under S1 and S2, the minimum and maximum mean Positioning Error (PE) of DRNS are 205.58 m and 305.02 m, respectively, and the Average PE (APE) of all the tracks is 235.60 m.
The baseline algorithm mABICCP is effective for the rangeonly matching aided navigation. The sABICCP demonstrated a substantial improvement in positioning accuracy under all test conditions. The minimum and maximum mean PE of sABICCP is 42.90 m (in T1S2) and 84.51 m (in T5S2), respectively. The APE of all test conditions decreased from 235.60 m to 56.11 m, resulting in a 75.7% improvement compared with the indicated accuracy. The dABICCP achieved a minimum and maximum mean PE of 33.94 m (in T2S2) and 84.52 m (in T5S1), respectively, with an APE of 50.50 m, improvement by 77.8% over indicated accuracy. Table 3 shows that dABICCP reduced the APE by an additional 5.61 m compared to the sABICCP, which means that more acousticbeacons can improve the matching accuracy (improvement by 11.8% over sABICCP) even though inaccurate matching model is adopted. In terms of computational efficiency, the average time consumed by the oncematching operation of the sABICCP and dABICCP is 0.97 s and 1.65 s, respectively. The efficiency tradeoff of the dABICCP brings worthwhile improvement in positioning accuracy.
The proposed mABRMAN achieved much better accuracy than the mABICCP while maintaining an acceptable computational efficiency. Under all the test conditions, the maximum mean PE of sABRMAN and dABRMAN is only 10.19 m and 3.59 m, respectively, with minimum mean PE of only 5.37 m and 1.53 m. Undoubtedly, these are satisfactory dynamic positioning accuracy. Additionally, the APE of sABRMAN and dABRMAN is 8.40 m and 2.15 m, respectively, an improvement by 96.38% and 99.04% compared to indicated accuracy. Notably, dABRMAN improved the positioning accuracy by 73.6% compared to sABRMAN. It can be concluded that the matching accuracy will be increased with more acousticbeacons available. This experimental conclusion coincides with our theoretical analysis. Most importantly, the accuracy and stability of the proposed mABRMAN are superior over the mABICCP, which demonstrates that the matching model of mABRMAN is more reasonable and effective. In terms of computational efficiency, the average time consumed by the sABRMAN and dABRMAN is 1.61 s and 2.53 s, respectively. Considering the significant improvement in positioning accuracy, mABRMAN is completely acceptable for navigation computers though the computational efficiency is lower.
In addition, since the Cumulative Distribution Function (CDF) can describe the probability distribution of random variables and is the integral of the probability density function, it also used as one of the positioning performance metrics to evaluate the PE of different methods. The average CDF curves of 100 Monte Carlo tests for different methods under different conditions are shown in Fig. 8. From these figures, it can be verified that the proposed mABRMAN can reduce significantly the PE over DRNS and mABICCP under all conditions. Take the Track#05 as example, Fig. 9 plots the positioning error of 100 Monte Carlo tests. The max PE of sABRMAN and dABRMAN is less than 30 m and 10 m, respectively, while the mABICCP over 150 m. The above validation and analysis demonstrates that the mABRAMN is effective and accurate.
The statistical results of APE for LBLbased algorithms are plotted in Fig. 10(a)–(h). To investigate the inherent correlation between the positioning accuracy and the observation length N, we tested the parameter N in Eq. (4), which ranges from 15 to 30. The VLBL indicates lower APEs on complex maneuvering tracks (such T4) than smoother ones (such T1) when single acousticbeacon is available. This finding is consistent with the fundamental requirement of the VLBL, which relies on vehicle maneuverability. Therefore, applying the LBLbased algorithm in a vehicle moving along a straight line may deliver a lower convincing outcome.
Additionally, under different test conditions (with the same N and acousticbeacons), the positioning accuracy of VLBL and mABVLBLrot is significantly affected by the effectiveness of the acousticbeacon, i.e., whether the hydrophone can periodically receive acoustic observations has a direct impact on these two methods. Approximately, these two algorithms will exhibit better positioning accuracy if acoustic observation can be acquired periodically, indicating that the application requirements for VLBL and mABVLBLrot are more stringent. There are two reasons for this phenomenon. Firstly, under the same track, periodic working will obtain more acoustic observations for positioning, which will inevitably impact the positioning accuracy. On the other hand, using the same N to solve the current position requires walking a longer distance in S2, which undoubtedly leads to more uncertainty in the location of virtual beacons, resulting in a larger solution space, as shown in Fig. 4. However, the positioning accuracy of the proposed mABVLBL algorithm does not exhibit the aforementioned phenomenon. This is entirely attributed to the reasonable error compensation for the indicated RPI. Moreover, this improvement brings considerable benefits to our positioning accuracy. It can be concluded that under any condition, mABVLBL not only has better positioning accuracy but also better adaptability.
Table 4 summarized the statistical results of the APE and efficiency of the LBLbased algorithms under different N. The APEs of the baseline VLBL are 28.82 m and 10.08 m, which means the decline of 87.96% and 96.00% compared to DRNSindicated error. Besides this, when single and double acousticbeacons are available, the time consumed by the singlepoint positioning of VLBL is 21 ms and 23 ms, respectively. It is worth noting that the singlepoint positioning time of the sABVLBLrot is longer by over 65.85% than that of the VLBL, but the expected improvement in accuracy caused by the rotational compensation of indicatedRPI is only observed under certain testing conditions, as shown in Fig. 10. In other words, only performing rotational compensation for the virtual beacons does not guarantee a complete improvement in positioning accuracy and may even lead to a slight degradation. This phenomenon can be attributed to the pronounced scale error (5%) that we have set, and this unexpected phenomenon will significantly diminish in practical application, just as shown in the field result in following.
The proposed mABVLBL offers significant improvement in accuracy compared to both VLBL and mABVLBLrot, with APE only 7.45 m and 3.50 m, respectively, which improves the DRNSindicated accuracy by 96.88% and 98.75%. In addition, dABVLBL can further enhance the positioning accuracy by 53% compared to sABVLBL. Compared to mABVLBLrot, the additional compensation on scale factor in the mABVLBL brings about substantial gains in accuracy, more than 77.1% and 62.9% when single and double acousticbeacons are available, respectively. Although the efficiency of mABVLBL is only half of that of the baseline, this reduction in efficiency will not cause an intolerable burden on the navigation computer as its singlepoint positioning time is only 43 ms, which is completely tolerable when compared to the acoustic observation rate (typically 0.1–1 Hz). This compromise in efficiency ensures a higher accuracy.
Another significant finding revealed from Table 4 is that the positioning accuracy of VLBL and mABVLBLrot is not improved with N increased, even though the latter incorporates the rotational compensation on the indicatedRPI. Only the proposed mABVLBL shows consistence with expectation. The explanation for this phenomenon lies in the inherent rotational and scaling errors in indicated RPI, which will propagate into virtual acousticbeacons. Although the redundant observations can increase reliability, the VLBL and mABVLBLrot still fail to eliminate (ultimately) the position errors of the virtual beacons from indicated RPI. In contrast, the mABVLBL simultaneously tackles rotational and scaling error corrections on indicatedRPI, theoretically improving the accuracy of virtual beacon positions. Consequently, increasing the observations will improve the positioning accuracy, highlighting the advantage of the proposed method. This result is theoretically explainable and consistently supported as well by the following field trial data.
Analysis of the field test results
Figure 11 depicts the positioning results and error curves for the lake trail, and the corresponding statistical results are presented in Table 5 (left part). For the lake trail, all the LBLbased algorithms were compared with observation length N = 15.
The matchbased baseline sABICCP successfully reduced the mean PE for Track#01–03 from 54.93 m, 98.18 m, and 163.93 m to 17.57 m, 4.40 m and 22.66 m, respectively, with an average reduction of 83.23%. The proposed sABRMAN outperforms sABICCP by achieving lower mean PE of 6.41 m, 6.81 m, and 5.69 m, respectively. The sABRMAN achieves an impressive mean PE reduction of 92.64% compared to the indicated accuracy, with a minimum decrease of 88.33% (Track#01) and a maximum of 96.53% (Track#03). Furthermore, the sABRMAN improves the positioning accuracy by 27.87% compared to sABICCP. It is important to note that the sABICCP achieved a higher positioning accuracy than sABRMAN in the Track#02, and the characteristics of the track may potentially explain this result. However, it exhibits significantly lower stability across different test tracks than the sABRMAN. The results of the matchbased algorithms in both simulation and field tests demonstrate that incorporating the affine transformation into RMAN will significantly improve positioning accuracy compared to rigid one. This finding highlights the necessity and effectiveness of affine correction for the indicated track.
In comparison of LBLbased methods, the proposed sABVLBL demonstrates superior performance. The mean PE is only 8.69 m, 5.48 m, and 8.94 m, with an average reduction of 91.04% compared to indicated error. Furthermore, VLBL and sABVLBLrot show average reductions of 73.71% and 84.73% compared to indicated error, respectively. Due to the compensation on the indicated RPI, both sABVLBLrot and sABVLBL achieve higher positioning accuracy than the baseline. Compared with the baseline and sABVLBLrot, the sABVLBL can further enhances average positioning accuracy by 63.94% and 43.15%, respectively. The additional compensation on scale factors compared to sABVLBLrot enhances accuracy by minimum 38.09% (on Track#01) and maximum 47.16% (for the Track#02), respectively. Note that VLBL performed worst in Track#02, demonstrating that applying the traditional VLBL for a vehicle moving along a straight line will deliver a less convincing outcome. Given the inability of underwater vehicles to navigate in perfectly straight paths, the benefits of compensating for indicatedRPI are immediately apparent, and this assertion can be easily derived from Table 5. These evidences and analysis demonstrate the effectiveness and superiority of the proposed mABVLBL.
The results of the sea trial presented in Table 5 (right part) show that the proposed algorithms can correct the indicated RPI effectively, significantly improving the performance over the baseline. The matchbased baseline dABICCP can reduce the indicated error by 93.41%, while the proposed dABRMAN can reduce up to 99.92%, which means a positioning accuracy close to the acoustic measuring accuracy. The baseline dABICCP achieved a positioning error of 46.53 m, even though it is already a good improvement on the indicated error (705.74 m). This residual error in dABICCP can be attributed to the 2.59% scale error (refer to dABRMAN), which has not been compensated yet. In the LBLbased comparison, the dABVLBL holds a clear advantage over the VLBL and dABVLBLrot, with an impressive mean PE reduction of 99.30% compared to the indicated accuracy. The results of the positioning, positioning error, and correction parameters for the matchbased and LBLbased algorithms are reported in Figs. 12 and 13, where the LBLbased algorithm is employed with N = 15. It can be observed that the estimated error compensation coefficients of LBLbased algorithms closely align with the matchbased ones, though they appear to be somewhat nonsmooth. It should be acknowledged that we do not report the pointbypoint matching algorithm in this paper, but a sequencebased matching one. However, this is already included in our agenda.
A more accurate positioning solution cannot be accomplished solely by increasing the observation length N without compensating the indicated RPI. This assertion is also supported by the positioning results of LBLbased algorithms presented in Table 5 (right part) for different N. Surprisingly, VLBL and dABVLBLrot perform worse despite a bigger N is adopted to address the current position. These sea trial evaluations are consistent with the statistical results in Table 4 exactly. This outcome can be explained as that the position of the virtual beacons is erroneous as it is derived directly from the inaccuracy indicated RPI. The dABVLBLrot only compensates for the rotational error of the RPI, yet it improves accuracy by 42.8% compared to VLBL. The proposed dABVLBL attains an average gain of 83.4% and 70.9% in accuracy compared to the baseline and dABVLBLrot, respectively. Furthermore, with an increase in observations, the desired positioning accuracy can be obtained. This indicates that compensating for the indicated RPI will improve the accuracy of the virtual beacons, thereby naturally getting accurate positioning.
Conclusion
This paper investigated two classes, matchbased and LBLbased, of underwater inertial error rectification algorithms with limited acoustic observations. Firstly, a novel matchbased mABRMAN algorithm is proposed for acoustic beacon rangeonly navigation by introducing the matchingaided concept. Furthermore, an improved LBLbased mABVLBL algorithm that considers the indicated RPI error is proposed. Comprehensive simulation and field tests were conducted to verify the effectiveness and accuracy of the proposed methods, including sABRMAN, dABRMAN, sABVLBL, and dABVLBL. We arrive at the following conclusions.

1.
The mABRMAN gives a novel perspective for acoustic beacon rangeonly navigation and is effective for inertial error rectification. The mABRMAN achieved excellent performance compared to the baselines. In simulation, the sABRMAN and dABRMAN improved accuracy by 96.38% and 99.04%, respectively. In field tests, they performed well with accuracy improvements by 92.64% and 99.92%, respectively.

2.
The mABLBL demonstrated excellent ability in compensating for the indicated RPI, which can significantly improve positioning accuracy by using more acoustic observations, but not for VLBL. The proposed mABLBL method exhibited superior in terms of accuracy and stability compared to the baseline. In simulation, the sABVLBL and dABVLBL achieved an improvement of 96.88% and 98.75%, respectively. In field tests, their improvement is 91.04% and 99.30%, respectively.
We are convinced that the proposed algorithms have a great potential in acoustic beacon rangeonly navigation. Future work is as follows.

1.
Collecting more extensive field datasets to facilitate a more comprehensive assessment of algorithm performance.

2.
A more appropriate matching algorithm for rangeonly matchingaided navigation will be explored while considering realtime performance.
Availability of data and materials
The data analyzed during the current study are not publicly available due to the management regulations of relevant organizations but are partly available from the corresponding author on reasonable request.
Abbreviations
 GNSS:

Global navigation satellite system
 INS:

Inertial navigation system
 SINS:

Strapdown inertial navigation system
 DRNS:

Dead reckoning navigation system
 DVL:

Doppler velocity log
 RTT:

Roundtrip time
 OWTT:

Oneway travel time
 APS:

Acoustic positioning system
 LBL:

Long baseline
 USBL:

Ultrashort baseline
 GMAN:

Geophysical matching aided navigation
 RMAN:

Rangeonly matching aided navigation
 ICCP:

Iterative closest contour point
 CNG:

Closest neighboring grid
 SITAN:

Sandia inertial terrain aided navigation
 SLBL:

Synthetic long baseline
 VLBL:

Virtual long baseline (without only error compensation for the virtual beacons)
 PNT:

Positioning, navigation, and timing
 mAB(sAB, dAB)ICCP:

Multi acousticbeacon ICCP (the prefix will substitute with sAB and dAB when using single and double acousticbeacon)
 mAB(sAB, dAB)RMAN:

Multi acousticbeacon RMAN (the prefix will substitute with sAB and dAB when using single and double acousticbeacon)
 mAB(sAB, dAB)VLBL:

Multi acousticbeacon VLBL (the prefix will substitute with sAB and dAB when using single and double acousticbeacon, if the abbreviation is followed by a suffix 'rot', it indicates that only rotational error has been considered).
 RPI:

Relative position increment
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Acknowledgements
The authors would like to thank Dr Enfan Lin from Peking University, and the Institute of Acoustics, Chinese Academy of Sciences for the experimental collaboration.
Funding
The funding was provided by Natural Science Foundation of China (Grant numbers 42004067, 62373367, 42176195)
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Tang, H., He, H., Li, F. et al. Underwater inertial error rectification with limited acoustic observations. Satell Navig 5, 3 (2024). https://doi.org/10.1186/s43020023001234
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DOI: https://doi.org/10.1186/s43020023001234