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Dead Reckoning for Trajectory Estimation of Underwater Drifters under Water Currents ^{†}

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^{‡}

## Abstract

**:**

## 1. Introduction

- Inertial Navigation Systems (INSs): An INS uses accelerometers and gyroscopes and requires initial conditions to calculate the device state through dead reckoning (DR). Although the full state can be determined by the INS, it suffers from an inherent drift. This is because the INS-measured quantities contain noises and biases that are integrated to obtain the device state [4]. Therefore, INSs are usually fused with external sensors [5] or information about the environment [6,7] to compensate for this drift.
- Acoustic Localization: Acoustic localization provides the navigation system with position fixes by measuring the device’s range to nodes of known positions, referred to as anchors. Acoustic ranging is based on measuring the time-of-flight (TOF), the time-difference-of-flight (TDOF), or the signal strength of an acoustic signal from the anchor to the submerged device. Ranging can be carried out passively or actively, but in either case requires the existence of at least one anchor in the acoustic range [8,9].

#### 1.1. Scope of Work

#### 1.2. Contribution

- a compensation for the directional angles when DR navigation is required;
- the estimation of directional angles using acceleration measurements only for short time periods of a few seconds between two successive position updates;
- a simplified DR approach for submerged floaters under the effect of directional angles for online/offline trajectory estimation.

## 2. Preliminaries

#### 2.1. Approaches for Underwater Dead Reckoning

#### 2.2. Common Approaches for Sideslip Angle Estimation

#### 2.3. Coordinate Frames and Transformations

#### 2.4. Principle Component Analysis (PCA)

## 3. Applying the PCA Approach for Underwater Navigation

#### 3.1. System Model

#### 3.2. Estimating the Directional Angles

## 4. PCA-DR Navigation

#### 4.1. The DR Solution

#### 4.2. Summary of PCA-DR Approach

#### 4.3. Impact of Errors in Estimating the Directional Angles

## 5. Analysis and Results

#### 5.1. Numerical Investigation

^{2}and $\delta {\gamma}_{h}=15$ deg and a scenario of 10s between two position updates. In the DR implementation, we assume perfect knowledge of the initial position and velocity of the drifter. The comparison between the navigation performance over time of the DR and the PCA-DR approaches for this scenario is shown in Figure 10. On the left panel, we show the estimated position of the drifter using the two approaches relative to the true position. Notice that, using the DR approach, the position remains in a straight line. This is excepted due to the assumption of no water current. Differently, the performance of the PCA-DR approach depends on the values of $\delta {a}_{p}$ and $\delta {\gamma}_{h}$. On the right panel of Figure 10, we show the position error of the two approaches. We observe that PCA-DR offers a significant improvement of the position error by more than $70\%$ compared to that of the traditional DR approach.

#### 5.2. Experimental Investigation

#### 5.3. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Illustration of the acceleration triangle experienced by a subsea device. The force induced by the water current over the drifter is translated into a shift in the heading direction, thus creating an horizontal and/or vertical directional angles between the drifter’s moving direction and its body frame. (

**a**) Horizontal plane; (

**b**) vertical plane.

**Figure 2.**Illustration of the projected velocity triangle. (

**a**) Horizontal plane; (

**b**) vertical plane.

**Figure 3.**Dead reckoning (DR) position error when the horizontal directional angle is not compensated.

**Figure 6.**Average of the RMS of $\delta {a}_{p}$ as a function of ${\gamma}_{h}$. Curves represent different accelerometers’ noise variance $\sigma $ in units of $\mathsf{\mu}\mathrm{g}/\sqrt{(\mathrm{Hr})}$. The inner plot zooms in an area of interest in the figure.

**Figure 7.**RMS of $\delta {\gamma}_{h}$ as a function of ${\gamma}_{h}$. Curves represent different accelerometer variance values $\sigma $ in units of $\mathsf{\mu}\mathrm{g}/\sqrt{(\mathrm{Hr}).}$

**Figure 8.**RMS of $\delta {\gamma}_{h}$ as a function of ${\gamma}_{h}$. Use of 150 acceleration measurements. Curves represent different accelerometer variance values $\sigma $ in units of $\mathsf{\mu}\mathrm{g}/\sqrt{(Hr)}$.

**Figure 10.**Comparison between traditional DR and PCA-DR. Left panel: actual position of the two approaches relative to the true position. Right panel: position error of the two approaches.

**Figure 11.**Pictures of the deployed drifter with the self-made INS in the test tank (

**a**)) and at sea (

**b**)).

**Figure 12.**Results from the sea experiment comparing PCA-DR with the traditional DR approach. (

**a**) Trajectory A. Data collected for 3 s. (

**b**) Trajectory B. Data collected for 10 s.

**Figure 13.**Position error results from the sea experiment comparing PCA-DR with the traditional DR approach.

Time [s] | 3 | 4 | 5 | 6 | 7 | 8 | 10 |
---|---|---|---|---|---|---|---|

True angle [deg] | 0 | 2.03 | 2.04 | 2.35 | 2.34 | 2.81 | 2.89 |

PCA angle [deg] | 0.02 | 1.57 | 1.59 | 1.60 | 1.57 | 1.64 | 0.34 |

Error [deg] | 0.02 | 0.46 | 0.45 | 0.75 | 0.77 | 1.16 | 2.55 |

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**MDPI and ACS Style**

Klein, I.; Diamant, R. Dead Reckoning for Trajectory Estimation of Underwater Drifters under Water Currents ^{†}. *J. Mar. Sci. Eng.* **2020**, *8*, 205.
https://doi.org/10.3390/jmse8030205

**AMA Style**

Klein I, Diamant R. Dead Reckoning for Trajectory Estimation of Underwater Drifters under Water Currents ^{†}. *Journal of Marine Science and Engineering*. 2020; 8(3):205.
https://doi.org/10.3390/jmse8030205

**Chicago/Turabian Style**

Klein, Itzik, and Roee Diamant. 2020. "Dead Reckoning for Trajectory Estimation of Underwater Drifters under Water Currents ^{†}" *Journal of Marine Science and Engineering* 8, no. 3: 205.
https://doi.org/10.3390/jmse8030205