Data-Driven Strategies for Coping with Incomplete DVL Measurements

An autonomous underwater vehicle (AUV) is a self-propelled, untethered robot designed to operate underwater without direct human control. These vehicles are equipped with sensors, navigation systems, and sometimes manipulators, enabling them to perform various tasks such as oceanographic research, underwater mapping, pipeline inspection, environmental monitoring, and search and rescue operations. AUVs are valuable tools for exploring and studying the ocean’s depths, offering efficiency, flexibility, and the ability to access areas that may be difficult or dangerous for manned vehicles [1].
The primary navigation sensors equipped in AUVs are the inertial navigation system (INS), which provides a high-rate navigation solution characterized by accumulating error over time, and the Doppler velocity log (DVL), a low-rate sensor that offers accurate velocity updates to mitigate error accumulation [2, 3]. Navigating through the underwater domain poses significant challenges, particularly for AUVs, whose sole purpose is to accomplish this task. Addressing these challenges is paramount for the successful operation of such vehicles. One of the challenges in underwater navigation is the inability of global navigation satellite system (GNSS) signals to penetrate water, rendering them unreliable for underwater navigation. As a result, the DVL becomes the sole sensor relied upon for correcting the inertial navigation errors. In real-world scenarios, such as passing over trenches in the seabed, encountering obstacles that obstruct the sensor’s view, or executing extreme maneuvers, the DVL may fail to operate correctly. This is because its mechanism relies on four beams being transmitted and reflected back to the sensor to estimate speed. In situations where fewer than three beams are reflected back, accurate speed estimation becomes impossible [4].
Since incomplete DVL measurements pose a critical problem for AUV navigation, several solutions have been proposed in the literature to address this challenge. In [5, 6], an extended loosely coupled (ELC) strategy is introduced as a solution for this circumstance. By utilizing the restricted DVL beam measurements along with external data, the ELC approach computes the three-dimensional velocity of the AUV and integrates it into the navigation filter. Liu et al. [7] proposed a method grounded in the tightly coupled approach for INS/DVL integrated navigation. This method directly incorporates DVL beam measurements, bypassing the need to transform them into 3-D velocity. It incorporates limited DVL beam measurements alongside depth updates obtained from a pressure sensor. In [8] Initially, the vehicle’s translational model is deduced, capturing the dynamics of translational velocity across three directions in the body frame. The integration of this model produces the vehicle’s translational velocity in the body frame. This velocity, referred to as pseudo-DVL, functions to offset the lack of DVL in INS/DVL navigation systems. Additional solutions for scenarios involving complete DVL outages were introduced in [9, 10].
As hardware capabilities advanced to handle heavy computational loads, the utilization of machine learning (ML) and deep learning (DL) techniques emerged to enhance AUV navigation [11]. In [12], a DL approach was applied to enhance DVL velocity estimation, while in [13, 14], data-driven methods were employed to estimate the process noise covariance in INS/DVL fusion. In situations of partial or complete DVL outage, various ML and DL approaches have been explored to address this challenge and maintain INS/DVL solutions [15, 16, 17, 18, 19]. Specifically, Yona et al. proposed a deep learning approach based on long short-term memory (LSTM) networks in [20, 21]. This method, known as MissBeamNet, aims to regress the missing beams using past DVL measurements. Furthermore, in [12, 22, 23], a series of architectures called BeamsNet were introduced to tackle scenarios involving complete, partial, and absent DVL measurements. Notably, LiBeamsNet utilized a one-dimensional convolutional neural network (1DCNN) to regress two missing beams.
In this study, we undertake a comparative analysis between LiBeamsNet and MissBeamNet, alongside a model-based average estimator, to ascertain whether DL approaches surpass model-based methods and whether long-term dependencies offer superior solutions in LSTM networks compared to the feature extraction from nearby windows used in a 1DCNN. We evaluate these strategies using real-world recorded data from an AUV experiment conducted in the Mediterranean Sea, which is available online at https://github.com/ansfl/A-KIT.
The remainder of the paper is structured as follows: Section II formulates the problem, while Section III introduces and discusses the competitive strategies. Section IV provides an overview of the dataset used in the study, and Section V presents the results of the comparison. Finally, Section VI discusses the conclusions drawn from the results.

The DVL operates by transmitting four acoustic beams to the sea floor, and once they are reflected back to the sensor, it calculates the frequency shift, thereby determining the velocity of the AUV in the beam directions. Given that the configuration of the transducers is known and provided by the manufacturer, solving a least squares problem is necessary to obtain the velocity in the DVL frame. The DVL velocity vector estimation based on the beam measurements are [12]:

In (1), $s$ and $c$ represent $\sin$ and $\cos$, respectively, $\boldsymbol{v}_{\text{beam}}\in\mathbb{R}^{4\times 1}$ denotes the velocity in the beam directions, and $\boldsymbol{v}^{d}\in\mathbb{R}^{3\times 1}$ is the velocity in the DVL frame. Each beam transducer undergoes a rotational transformation with a yaw angle $\psi_{\text{beam}}$ and a pitch angle $\alpha$, where $\alpha$ is a constant predefined by the manufacturer and $i$ is the beam index. The measurements, prone to inherent errors, are modeled in (3), with $\boldsymbol{b}_{\text{DVL}}\in\mathbb{R}^{4\times 1}$ as the bias vector, $\boldsymbol{s}_{\text{DVL}}\in\mathbb{R}^{3\times 1}$ as the scale factor vector, and $\boldsymbol{n}\in\mathbb{R}^{4\times 1}$ representing zero-mean white Gaussian noise. Upon obtaining raw measurements, the subsequent step involves extracting $\boldsymbol{v}^{d}$ by filtering the data according to the cost function in (4). The solution of 4 is the velocity in the DVL frame as detailed in (5).
In real-world scenarios, the DVL may face challenges leading to the reflection of acoustic beams in directions other than toward the sensor. When two or more beams are unavailable, the DVL cannot perform the task of estimating velocity. Examples of such scenarios may include passing over trenches on the sea floor, sea creatures obstructing the view of the sensor, and extreme roll and pitch maneuvers, including the AUV’s initial operation, diving. Illustrations of these scenarios are visualized in Fig. 1.

To address incomplete DVL measurements, we focused on scenarios where two beams are missing, and as a consequence, the velocity vector cannot be estimated. We examine two state-of-the-art deep learning architectures that employ different techniques yet utilize only DVL information. The first one is LiBeamsNet, which employs 1D CNN on past complete DVL measurements combined with the current observation of the partial beam measurements to regress the missing beams. The second one is MissBeamNet, which uses LSTM architecture to regress the missing beams given past measurements and the current observation. Both approaches demonstrated good performance in estimating the missing beams. However, a question arises regarding which of them performs better: the 1DCNN architectures that work by extracting features from the window of past measurements or the LSTM network that considers larger time dependencies.

Both networks underwent training for 100 epochs with a learning rate of 0.001, a learning rate decay of factor 0.1 after 50 epochs, a mini-batch size of 4, and the ADAM optimizer. In addition, for both networks, $N=3$ past DVL measurements were employed.
In LiBeamsNet, the measurements undergo processing through a 1D convolutional layer comprising six filters of size $2\times 1$ aimed at extracting features from the data. Following this, the features extracted are flattened and then subjected to a dropout layer with a dropout rate of 0.2. Subsequently, a sequence of fully connected layers is employed, and the current partial beam measurement is integrated before passing through the last fully connected layer, ultimately producing a $4\times 1$ vector representing the estimated DVL four-beam velocity. The architecture and activation functions after each layer are illustrated in Fig. 2. In MissBeamNet, after the LSTM with a hidden size of 500 performs feature extraction, these features are concatenated with the available beam measurements into a fully connected layer. This layer executes the final processing, resulting in the output of the regressed missing beams, as can be seen in Fig. 3. For comparison, we examined a conventional average estimator, which takes the average of the $N=3$ past DVL beam measurements to regress the missing ones.

In the initial phase of the analysis, we examined the loss values as a function of the number of epochs. In Fig. 4, both the training and testing loss values for LiBeamsNet are depicted, while Fig. 5 illustrates the loss values for MissBeamNet. Both graphs show a convergence of the training loss, with LiBeamsNet achieving a lower value. Examining the test loss values reveals that MissBeamNet converges more rapidly and exhibits a monotonically decreasing trend, whereas LiBeamsNet takes a longer time to converge. This suggests that MissBeamNet is better able to generalize the problem.

N.C. is supported by the Maurice Hatter Foundation and University of Haifa presidential scholarship for outstanding students on a direct Ph.D. track.