# Method for the Coordination of Referencing of Autonomous Underwater Vehicles to Man-Made Objects Using Stereo Images

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### Problem Statement

## 2. Method for Coordinate Referencing

#### 2.1. SPS Model

#### 2.2. Identification of the SPS Feature Points

_{1}…C

_{M}} (3D cloud), seen by the camera, is formed in the current position of the AUV trajectory. The absolute coordinates of the points from the 3D cloud are calculated using the visual navigation method according to the known procedure:

- FPs matching on images of two stereo pairs;
- Calculation of 3D coordinates of the corresponding points in $C{S}^{AUV\_i}$;
- Computation of the local matrix to transform the coordinates at the current stage;
- Computation of matrix ${H}_{C{S}^{AUV\_0},C{S}^{AUV\_i}}$ via combining local matrices from the previous stages;
- Calculation of absolute coordinates, sequentially applying two transformations—${({H}_{C{S}^{AUV\_0},C{S}^{AUV\_i}})}^{-1}\xb7{H}_{C{S}^{AUV\_0},WCS}$.

#### 2.2.1. Stage 1

#### 2.2.2. Stage 2

- On set ${S}^{ob\_id}$ of the points of the object model (see stage 1), a set of samples ${s}_{m}^{ob\_id}\left\{{P}_{{i}_{1}},\dots {P}_{{i}_{q}}\right\}$ is generated, where m—the sample number, q—the sample length. The number of possible samples is defined by the number of permutations of n − q at a given time—${A}_{n}^{q}=\frac{n!}{(n-q)!}$;
- The set of distances ${D}_{m}^{ob\_id}\left\{{d}_{{i}_{1},{i}_{2}}^{ob\_id},\dots {d}_{{i}_{q-1},{i}_{q}}^{ob\_id}\right\}$ is constructed, where ${d}_{{i}_{k},{i}_{s}}^{ob\_id}$ is the distance between points ${P}_{{i}_{k}}$ and ${P}_{{i}_{s}}$. Here, ${i}_{k}$ and ${i}_{s}$ are the numbers of points in the object model ${M}^{ob\_id}$, and indices k and s are related to the numbering of points in sample ${s}_{m}^{ob\_id}$, which is linked to each sample ${s}_{m}^{ob\_id}\left\{{P}_{{i}_{1}},\dots {P}_{{i}_{q}}\right\}$. There are $\frac{q(q-1)}{2}$ elements in set ${D}_{m}^{ob\_id}$;
- The set of samples ${c}_{n}^{cloud}\left\{{c}_{1},\dots {c}_{p},\dots {c}_{q}\right\}$, comprised of the 3D cloud points, is generated for each sample ${s}_{m}^{ob\_id}\left\{{P}_{{i}_{1}},\dots {P}_{{i}_{p}},\dots {P}_{{i}_{q}}\right\}$ of the object model. Here, n is the number of samples. The point from list ${l}_{p}\left\{{C}_{{j}_{1}},\dots {C}_{{j}_{m}}\right\}$, connected to point ${P}_{{i}_{p}}$ (see stage 1), is taken as the ${c}_{p}$ element of sample ${c}_{n}^{cloud}$. The number of the generated samples ${c}_{n}^{cloud}$ is defined by the number of lists q and the lengths of these lists. For example, if q = 3, and the lengths of the corresponding lists are length1, length2, length3, the number of samples will be length1⋅ length2 ⋅ length3;
- The set of distances ${D}_{n}^{cloud}\left\{{d}_{1,2}^{cloud},\dots {d}_{k,s}^{cloud},\dots {d}_{q-1,q}^{cloud}\right\}$ is constructed, where ${d}_{k,s}^{cloud}$ is the distance between points ${C}_{k}$ and ${C}_{s}$—here, indices k and s are related to the numbering of points in the sample, and ${c}_{n}^{cloud}$ is linked to each sample ${c}_{n}^{cloud}$. There are $\frac{q(q-1)}{2}$ elements in set ${D}_{n}^{cloud}$;
- For a sample ${s}_{m}^{ob\_id}\left\{{P}_{{i}_{1}},\dots {P}_{{i}_{q}}\right\}$ (step 1) from the object model, the sample ${c}_{n}^{cloud}\left\{{c}_{1},\dots {c}_{q}\right\}$ (par.3) from the 3D cloud is sought, such that ${D}_{m}^{ob\_id}={D}_{n}^{cloud}$. Here, the equivalence means the equivalence between all the corresponding pairs of elements: $\left|{d}_{{i}_{k},{i}_{s}}^{ob\_id}-{d}_{k,s}^{cloud}\right|\le \Delta $. The error ∆ is determined by the accuracy of measuring the coordinates of the 3D cloud points (depending on the resolution of pictures and the distance between the camera and the points). In that case, with consideration for the above-described rules of forming samples, the determined correspondence between sample ${c}_{n}^{cloud}$ and sample ${s}_{m}^{ob\_id}$ enables the unambiguous identification of the points of the 3D cloud that belong to the SPS object, and for them to be matched with the object model points;
- If there are no corresponding points found in the 3D cloud for the specified length q of sample ${s}_{m}^{ob\_id}$, the correspondence for a smaller sample shall be searched for, i.e., for $q=q-1$. It should be noted that the implementation of searching, aimed at detecting the maximum number of points matched to the SPS object model’s points, in the 3D cloud increases the degree of certainty of object identification. Subsequent to the identification of several points (three as a minimum) belonging to the SPS object, in the 3D cloud, the coordinate referencing of the AUV to the SPS can be performed. Using more FPs would improve the accuracy of the method.

#### 2.3. Calculation of the Matrix of the Geometric Transformation of the Points from the AUV CS to the SPS Object CS

_{1}, C

_{2}and C

_{3}be the object points identified (applying the algorithm as described above) in the 3D cloud. Let the auxiliary CS ($C{S}^{ad}$) be constructed on the identified object points, according to the rule shown in Figure 1; i.e., let unit vectors e1_AUV, e2_AUV and e3_AUV of the $C{S}^{ad}$ coordinate system be constructed in the $C{S}^{AUV\_i}$ coordinate system.

_{1}) origin, specified in $C{S}^{AUV\_i}$.

_{1}, C

_{2}, C

_{3}in $C{S}^{ob\_id}$ are known, which means that the constructed unit vectors of the $C{S}^{ad}$ coordinate system can be defined in $C{S}^{ob\_id}$ as well. Let e1_ob_id, e2_ob_id, e3_ob_id denote these vectors. Accordingly, the matrix of transformation from $C{S}^{ad}$ to $C{S}^{ob\_id}$ can be formed from the unit vectors specified in $C{S}^{ob\_id}$:

#### Other Methods for Calculating the Transformation from the AUV CS to the CS of the SPS Object

#### 2.4. Calculation of the AUV Coordinates in the SPS CS

## 3. Experiments

#### 3.1. Experiments with a Virtual Scene

#### 3.2. Experiments with the Karmin2 Camera

#### 3.3. The Discussion of the Results and Comparison with Other Approaches

## 4. Conclusions

- The object recognition algorithm uses a predetermined 3D point model of the object, in which there are a limited number of characteristic points with known absolute coordinates;
- The method uses a structural coherence criterion when comparing the 3D points of an object with a model;
- The method references the AUV coordinate matrix to the object using the matched points;
- High accuracy during the continuous movement of an AUV in SPS space is ensured by regular referencing to the SPS object coordinate system.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Construction of an object coordinate system based on points P

_{1}, P

_{2}, and P

_{3}specified in the WCS: the X axis is determined by points P

_{1}P

_{3}, the Z axis is normal to the plane of the P

_{1}P

_{2}and P

_{1}P

_{3}vectors, and the Y axis is normal to the ZX plane.

**Figure 3.**Coordinate systems (WCS, CS SPS, object CS, CS associated with an AUV in the initial and current positions) used and geometric transformations between them.

**Figure 4.**Points in min–max-shells in the 3D cloud, marked in black, are identified by the search algorithm as belonging to the SPS object.

**Figure 5.**Calculation of the geometric transformation matrix from the AUV CS (CS

^{AUV}) to the CS of the SPS object (CS

^{ob_k}) using the auxiliary CS (CS

^{ad}).

**Figure 7.**Identification of points belonging to the SPS object: (

**a**) extraction of characteristic points by the Harris angle detector in the image taken by the camera; (

**b**) a set of 3D points constructed from 2D images. All selected points in the scene are marked in white. Points belonging to SPS objects are marked in black. Six points belonging to SPS have been identified.

**Figure 8.**In the photo taken by the Karmin2 camera, the desired objects are marked: A–F. Points belonging to the desired objects (model) are marked in black. Their number is 48, as indicated by the operator showing eight on each box. Of these, 33 points fell into the camera’s field of view: on object A—7, on B—7, on C—6, on D—2, on E—4, on F—7. The coordinate system (CS), in which all points of objects (model) were set, was built on three corner points of object A. The points built by the Harris detector are marked with white circles. There are 89 of them in the scene.

**Figure 9.**The figure shows the points identified in the 3D cloud (marked in black) as belonging to the sought objects. Their number was 13: on object A—2, on B—3, on C—3, on D—1, on E—1, on F—3. The matrix connecting the CS of objects with the CS of the camera was calculated by 3 points (they are marked with numbers 1, 2, 3), which were selected by the algorithm from the found points.

$\mathit{W}\mathit{C}\mathit{S}$ | – | World Coordinate System |
---|---|---|

$C{S}^{AUV\_i}$ | – | Coordinate system associated with AUV in position i. |

$C{S}^{AUV\_0}$ | – | Coordinate system associated with AUV in the initial position. |

$C{S}^{SPS}$ | – | SPS coordinate system. |

$C{S}^{ob\_id}$ | – | Coordinate system of object id, belonging to SPS. |

${H}_{C{S}^{AUV\_0},C{S}^{AUV\_i}}$ | – | Transformation matrix from the coordinate system in the initial AUV position to the coordinate system in position i. This matrix is formed by multiplying out local matrices of relative displacement, each of which connects the css of the two adjacent positions. |

${H}_{C{S}^{AUV\_i},C{S}^{ob\_id}}$ | – | Transformation matrix from the coordinate system of AUV in position i to the coordinate system of object No.id. |

${H}_{C{S}^{ob\_id},WCS}$ | – | Transformation matrix from the coordinate system of object id to the world coordinate system. |

${H}_{C{S}^{AUV\_0},WCS}$ | – | Transformation matrix from the coordinate system of AUV in position 0 to the world coordinate system. |

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**MDPI and ACS Style**

Bobkov, V.; Kudryashov, A.; Inzartsev, A. Method for the Coordination of Referencing of Autonomous Underwater Vehicles to Man-Made Objects Using Stereo Images. *J. Mar. Sci. Eng.* **2021**, *9*, 1038.
https://doi.org/10.3390/jmse9091038

**AMA Style**

Bobkov V, Kudryashov A, Inzartsev A. Method for the Coordination of Referencing of Autonomous Underwater Vehicles to Man-Made Objects Using Stereo Images. *Journal of Marine Science and Engineering*. 2021; 9(9):1038.
https://doi.org/10.3390/jmse9091038

**Chicago/Turabian Style**

Bobkov, Valery, Alexey Kudryashov, and Alexander Inzartsev. 2021. "Method for the Coordination of Referencing of Autonomous Underwater Vehicles to Man-Made Objects Using Stereo Images" *Journal of Marine Science and Engineering* 9, no. 9: 1038.
https://doi.org/10.3390/jmse9091038