System and method for relative positioning of an object with respect to a target surface based on vision and navigation system of said object

Abstract

Vision-based relative positioning system and method with respect to a target surface, characterized in that it comprises: - at least three reference markers (1) located on the target surface at the vertices of an equilateral triangle (2) as seen from a bird's-eye view of said target surface, - at least one imaging device located on the object, - a processing means configured to receive and process the captured images (4) configured to: detect the reference markers (1) in the received image (4), identify the centroid (5) of a triangle (7) formed by the reference markers (1) in the received image (4), circumscribe the reference markers (1) by means of a circumscribed ellipse (6), Determine the relative position with respect to the target surface by the relationship between the circumscribed ellipse (6) and the position of the reference markers (1).

Description

System and method for relative positioning of an object with respect to a target surface based on vision and navigation system of said object Field of invention The invention relates to a system or method for estimating the position of an object relative to a target surface. The method could be used as part of vehicle navigation systems, such as those for aircraft, to achieve precise positioning and / or provide redundancy to traditional positioning systems in complex environments, for example, with a denied Global Navigation Satellite System (GNSS). Background of the invention Vision-based navigation (VBN) refers to the use of optically acquired data, via cameras, for navigation purposes. The three main steps of vision-based navigation (VBN) are typically the following: a) Capture an image of a portion of the environment surrounding an object. This image will contain the information that will allow a position measurement to be generated. b) Extract and process information from the captured image to generate the position measurement. This step typically uses computer vision techniques such as pattern recognition, object detection and identification, or motion estimation with visual odometry. c) Filter the measurements to calculate an optimal estimate of the position. It is known that landing on unpaved runways makes the operation more demanding than usual. In these cases, instrument approaches with very high frequency omnidirectional range (VOR), distance measuring equipment (DME), and, even less so, instrument landing systems (ILS) are not typically available. At airports, Ground Level Augmentation Systems (ILS / GBAS) that enable assisted landings are currently the primary way to facilitate piloting during this phase. Thanks to these systems, aircraft deviations from the ideal approach can be calculated and used in guidance rules. However, these systems are expensive, and their availability is limited to airports that have deployed the necessary infrastructure. Furthermore, in the case of unpaved runways, relying solely on the Global Positioning System (GPS) presents some problems. Known vision-based systems typically have a limited range, and the aerial vehicle must be close to the target to achieve accurate localization. Furthermore, a vision-based system can be affected by lighting conditions such as glare and shadows, which make it difficult to see some features on unpaved surfaces. In certain environments, such as flat terrain, snow, and deserts, vision-based methods cannot be used because there are no distinctive visual features in the environment. Summary of the invention The objective of the invention is to estimate the position of an object, for example, an aircraft, relative to a target surface. To this end, the system of the invention allows the relative position of said object to be estimated in spherical coordinates, providing angles and a distance, thus positioning the object three-dimensionally. More specifically, the system and method of the invention allows the distance, elevation, and azimuth of the object to be provided relative to a pattern located on the target surface. According to the foregoing, the present invention relates to a vision-based system for estimating the relative position of an object with respect to a target surface, comprising: - At least three reference markers configured to be located on the target surface such that they are situated at the vertices of an equilateral triangle as viewed from a top view of that target surface, where the three vertices share a centroid. The centroid of this triangle corresponds to the origin of the reference system to be used for relative positioning. Viewed from a top view, the three reference markers could be circumscribed by a circle called the circumscribed circle. - At least one imaging device, configured to be located on the object, the object being capable of spatially localizing the reference markers, and capturing images of the target surface including the reference markers, wherein the imaging device has a focal length. - A processing means configured to receive and process images captured by at least one imaging device and configured to: • detect reference markers in the received image, • Identify the centroid of a triangle formed by the reference markers in the received image, • circumscribe the reference markers in the received image using a circumscribed ellipse centered on the centroid of the triangle and passing through the vertices of the triangle, • Determine the relative position of the object with respect to the target surface according to its spherical geometric coordinates distance, elevation and azimuth by means of the geometric relationship in the received image between the semi-axes and the angle of the semi-major axis of the circumscribed ellipse and the position of the reference markers and the radius of its circumscribed circle. According to the above, the processing medium has the input data provided by the three reference markers that constitute two-dimensional coordinates in pixels in the image taken by the image formation device. According to the claimed invention, the equilateral triangle formed by the three reference markers can be circumscribed by a circle. When this triangle is viewed in perspective, i.e., from an object, for example, an aircraft approaching a runway, the triangle ceases to be equilateral. The circumscribed circle is then perceived as an ellipse, whose parameters can be analytically determined from the image. This ellipse is called the Steiner circumscribed ellipse, which is the only ellipse that passes through the three vertices and is centered at the centroid of the triangle. According to the above, the objective of the claimed invention is to obtain an estimate of the relative position of an object. The invention therefore solves the estimation of the relative 3D position of an object by means of the image capture device and the objects observed in the captured image. Furthermore, the installation of reference markers on the target surface, for example, on the target terrain, could enable the creation of a low-cost and easily transportable landing system. Various types of objects could benefit from the invention's system: airplanes, helicopters, drones, reusable first-stage rockets, unmanned vehicles, etc. A specific application of the system and method of the invention could be an aid to landing on unpaved runways, thereby reducing the pilot's workload. Other applications could include relative positioning during in-flight refueling maneuvers, landing on small or mobile helipads, landing guidance for the retropropulsive vertical landing of a reusable rocket stage, as an aid to the landing of multicopters, etc. Another application where vision-based landing (VBL) is extremely helpful is when the landing surface is in motion, for example, on the decks of ships or aircraft carriers, since even a few centimeters of difference can jeopardize and compromise the mission. The invention is capable of determining the relative position in 3D in these scenarios where landing aids are often unavailable. Furthermore, the ability to land accurately on unpaved terrain and in environments without GPS would facilitate the logistics of humanitarian missions. The claimed system could reduce crew fatigue and training costs, thus providing more efficient operations in, for example, in-flight refueling. To determine the number of reference markers, the following two aspects are taken into account: • Use the fewest possible number of markers. • Enable the detection of the object's 3D position. Placing three reference markers as if they were the vertices of an equilateral triangle is sufficient to solve the 3D positioning analytically, provided the assumption that the object is located within a 120° azimuth sector is met, since there are three possible azimuth solutions and it is necessary to resolve the ambiguity. This assumption is generally applicable to many of the scenarios described. The system of the invention provides accurate results even with a basic camera as the image capture device. The present invention includes a method for converting the 2D pixel coordinates of three reference points in the captured image into a 3D position relative to the centroid of the triangle formed by the reference points. Therefore, the method determines the 3D relative position with respect to the centroid of the equilateral triangle formed by the reference points, based on the 2D pixel coordinates of said reference points. In this way, the reference markers facilitate the image processing task. Therefore, the claimed system and method comprise a solution that provides both accurate and fast-to-calculate results. The claimed system has the following advantages: • It is capable of using images from one or more imaging devices, for example, cameras, installed on the object to estimate its 3D position, infrared sensors, etc. • It is a system that is potentially certifiable under aeronautical regulations. This is due to the deterministic nature of the method. • Initially, it is intended for use as an external navigation aid system, in addition to certified sensor architectures such as ADIRUS (Air Data Inertial Reference Unit) and GPS on civil platforms. • It is compatible with current Flight Management System protocols, allowing for integration with various avionics architectures. • It is lightweight. • It has low energy consumption. • It is low cost. • It allows real-time processing, meaning the algorithm has a low processing time. • He is reliable and honest. • Easy to install and maintain. • The markers are capable of being detected from a great distance. As previously mentioned, this positioning system has been applied to develop a solution that allows for precise aircraft landings without the need for complex infrastructure and at low cost. This innovative landing system enables specific applications such as landing on unpaved runways, as already indicated. Additionally, the claimed relative positioning system could also be used as part of an object's navigation systems to achieve precise positioning and / or provide estimation redundancy in addition to traditional positioning sensors. The invention also includes a navigation system comprising a vision-based relative positioning system for an object with respect to a target surface, as explained above. The present invention also relates to a method for estimating the position of an object with respect to a target surface, comprising the following steps: - provide at least three reference markers located on the target surface such that they are situated at the vertices of an equilateral triangle viewed from a top view, where the equilateral triangle has a centroid and a circumscribed circle, - provide at least one imaging device located on the object to capture images of the target surface including reference markers, wherein the imaging device has a focal length, - provide a processing means that receives and processes captured images received from at least one imaging device that: • Detect reference markers in the received image, • Identify the centroid of the triangle formed by the reference markers in the received image, • circumscribe the reference markers on the received image using a circumscribed ellipse centered on the centroid of the triangle and passing through the vertices of the triangle, • Determine the position of the object with respect to the target surface according to its spherical geometric coordinates: distance, elevation and azimuth, by means of the geometric relationship in the received image between the semi-axes, the angle of the semi-major axis of the circumscribed ellipse and the position of the reference markers and the radius of its circumscribed circle. Description of the figures To complete the description and provide a better understanding of the invention, a set of drawings is provided. These drawings form an integral part of the description and illustrate preferred embodiments of the invention. The drawings comprise the following figures. Figure 1 shows a schematic representation of three reference markers located at the vertices of an equilateral triangle having a centroid, the three vertices being circumscribed by a circle. Figure 2 shows an image captured from an aircraft of a runway and its surroundings, showing three reference markers located on the runway as seen from a bird's-eye view. Figure 3 shows an image captured from an aircraft of the runway in Figure 2 and its surroundings, in which three reference markers located on the runway can be seen from an aircraft approaching the runway. Figure 4 shows a schematic representation of three reference markers located at the vertices of an isosceles triangle; the three vertices are circumscribed by an ellipse. Figure 5 shows the three-dimensional polar coordinates that position an object: distance, elevation, and azimuth. Figure 6 is a schematic representation of a camera, the camera's image plane, and the object observed in 3D space. Detailed description of the invention Figure 1 shows a schematic representation of three reference markers (1) located at the vertices of an equilateral triangle (2) having a centroid (5). The three vertices are circumscribed by a circle (3). The three reference markers (1) are located on the target surface. The target surface can be an airstrip, as shown in Figures 2 and 3. Alternatively, the decks of ships or aircraft carriers can also be a target surface. Figure 2 shows an image of an airstrip (8) and its surroundings captured from an aircraft (10). The three reference markers (1) are located on the airstrip (8) and are viewed from a bird's-eye view. The three reference markers (1) form an equilateral triangle (2) as seen from this bird's-eye view. The images (4) captured from Figures 2 and 3 are taken from at least one image capture device located on an aircraft (10). From the images (4) captured from figures 2 and 3, it is clear that, viewed from above, i.e., from a zenithal perspective, it is an equilateral triangle (2). There would be a circumscribed circle (3) passing through the vertices of the equilateral triangle (2). However, viewed in perspective, as in figure 3, the equilateral triangle (2) would no longer be equilateral. In that case, the circumscribed circle (3) would become a circumscribed ellipse (6), as seen in figure 4. Accordingly, the processing means are configured to receive and process the captured images (4) received from at least one imaging device and configured to: • detect the reference markers (1) in the received image (4), • Identify the centroid (5) of the triangle (7) formed by the reference markers (1) in the received image (4), • circumscribe the reference markers (1) on the received image (4) by means of a circumscribed ellipse (6) centered on the centroid (5) of the triangle (7) and passing through the vertices of the triangle (7), • Determine the relative position of the object with respect to the target surface according to its spherical geometric coordinates: distance, elevation and azimuth, by means of the geometric relationship in the received image (4) between the semi-axes and the angle of the semi-major axis of the circumscribed ellipse (6) and the position of the reference markers (1) and the radius of its circumscribed circle (3). These spherical geometric coordinates are represented in Figure 5 with respect to the aircraft object (10). The spherical geometric coordinates are distance, elevation, and azimuth. The parameters that define the circumscribed ellipse (6), Steiner's ellipse, have an analytical solution. It is the only ellipse that passes through the vertices of a triangle and has as its center the centroid of the triangle (7). Steiner's circumscribed ellipse (6) is also the ellipse of smallest area that passes through the vertices. The parameters of the circumscribed ellipse (6) are known. With them, as explained above, an analytical estimation of the relative position can be performed. Visual position estimation is the process of estimating the object's position based on data provided, for example, by a camera (9). The semi-major axis of the circumscribed ellipse (6) has the same length for any azimuth, considering a constant distance to the object. The semi-minor axis of the circumscribed ellipse (6) varies in length depending on the object's elevation above the target for the same distance. The distance is the distance from the camera to the center of the ellipse. Knowing the camera's focal length and the actual measurements of the pattern and those perceived in the image (4), it is possible to calculate the distance at which I am viewing that point. In the implementation example in Figure 6, the x-y-z frame of the imaging device is a three-dimensional frame with the origin as its projection center and its z-axis pointing in the target direction. The u-v image frame (4) is a two-dimensional frame with the u and v axes parallel to the x and y axes of the imaging device frame, respectively. The three-dimensional coordinates (x, y, z) can be related to the image coordinates (u, v) as follows: either Where f is the focal length of the imaging device, uov is the larger size of the semi-axes of the circumscribed ellipse (6) in the received image (4), xoy are the actual radii of the circumscribed circle size (3), and yz is the distance from the imaging device to the reference markers (1). f is a constant for a specific camera model and, for the equations above, its dimensions are in pixels. A parameter from the reference marker (1) will be selected to introduce into the equation both its actual dimension (x) and its length measured on the image (4) (u). This parameter is the semi-major axis of the circumscribed ellipse (6). Therefore, in an example implementation for estimating the distance of the object from the reference markers (1), the processing medium is configured to calculate the geometric relationship using the following equation: where f is the focal length of the imaging device, v is the size of the semi-major axis of the circumscribed ellipse (6) in the received image (4), y is the actual radius of the circumscribed circle size (3) and z is the distance from the object to the reference markers (1). In another example of implementation for estimating the elevation of the object with respect to the reference markers (1) the processing means is configured to calculate the geometric relationship by means of the arcsine of the ratio between the semi-minor axis and the semi-major axis of the circumscribed ellipse (6). In another example of implementation for estimating the azimuth of the object with respect to the reference markers (1) the processing means is configured to calculate the angle between the semi-major axis of the circumscribed ellipse (6) and one of the sides of the triangle (7) formed by the reference markers (1) in the image (4) received from a top-down view of the target surface. The semi-major axis of the circumscribed ellipse provides information about the object's azimuth. This semi-major axis is perpendicular to the vector connecting the object to the origin projected onto the plane, but only when viewed from a top-down perspective. This is no longer the case when the image is distorted by a different viewpoint. In one implementation example, the processing system is configured to undo this distortion and attempt to replicate the top-down view. The calculation of this angle can be performed in various ways. In one embodiment, the processing medium is configured to calculate the Steiner inellipse of the triangle (7) from the reference markers (1) and apply Marden's theorem to calculate the angle between the semi-major axis of said Steiner inellipse and one of the sides of the triangle (7), the base for example, which coincides with the angle of the semi-major axis of the circumscribed ellipse (6) and the base of the triangle (7). More specifically, based on Marden's Theorem: Suppose that the three complex zeros of a third-degree polynomial p(z) are z1, z2, and z3, and suppose that these three points in the complex plane are not collinear. Then there exists a unique ellipse inscribed in the triangle with vertices z1, z2, z3 and tangent to the sides at their midpoints: the Steiner inellipse. The foci of this ellipse are precisely the zeros of the derived polynomial p'(z). The Steiner inellipse is an inellipse tangent to the sides of the triangle at their midpoints and has the maximum area of ​​any inellipse. It is also the image of the Steiner circumellipse under the homothety with homothetic center G and similarity ratio 1 / 2. This means that the axis of symmetry of the Steiner inellipse will have the same direction as that of the Steiner circumellipse. The foci of any ellipse lie on the major axis, and, using Marden's theorem, obtaining its coordinates is straightforward. Finally, to obtain the angle between the major axis of the ellipse and the axis of a side of the triangle, for example, the base of the triangle, the foci of the Steiner inellipse are subtracted, and the angle of the resulting complex number is calculated. This is the angle that defines the azimuth of the object. In one embodiment, the system includes a fourth reference marker (1) located on one side of the equilateral triangle (2) to uniquely determine the azimuth. This resolves the ambiguity in determining the azimuth, since, according to calculation methods, three azimuth solutions can be obtained if three reference points are used. One of the first aspects to resolve is determining what the reference markers should be like: what type, what size, how many, etc. In one implementation example, the three reference markers (1) are passive cooperative targets. Passive cooperative targets, as opposed to active cooperative targets, are simpler, that is, mechanically less complex. Active cooperative targets actively illuminate the scene, have flashing targets, or some other active mechanism. An example of such an active reference marker (1) could be infrared lights. More specifically, the three reference markers (1) can be circles, as shown in Figures 2 and 3. And more specifically, the circles can be white circles. The advantages of using white circles as passive cooperative targets are as follows: • Cooperative objectives are an optical navigation aid. The objective to be found is known in advance. Therefore, the algorithms can be adapted to them, resulting in greater efficiency and accuracy. • Circles are detected reliably and are, for the most part, insensitive to translations and rotations. • White circles are not a common geometric shape that can appear in aerial images. • It is desired that the reference marker (1) be visible from the greatest possible distance. A smooth circle, with no elements inside, fulfills this requirement. In one example of implementation, the centers of the circles form the vertices of the triangles (3, 7). In one implementation example, the pattern is recursive; for instance, within the three circles there would be three more circles, and so on. This is because, as the object approaches the target, the three outer circles may already be outside its field of view. In another example of implementation, the passive cooperative targets are painted with a paint containing titanium dioxide. Thus, the white circles are painted with titanium dioxide. In another implementation example, the passive cooperative targets are part of a sleeve that is placed over the target surface. Thus, the sleeve, comprising, for example, the white circles, is laid on the floor. The circles can be sewn to the sleeve, painted, glued, etc. According to the above, in one implementation example, to determine the 3D position of an aircraft (10), the white circles of the reference marker (1) must be detected. In particular, the centers of the circles are the parameters sought. To obtain them, computer vision techniques are used. Therefore, it is expected that, if the threshold that discriminates whether a pixel is white or black is correctly set, the white circles will appear in a binarized image. Once the centers have been obtained at this point, it is necessary to calculate the circumscribed ellipse (6) that passes through the vertices of the triangle (7). In one embodiment, the imaging device is located on an aircraft (10), preferably installed on a longitudinal axis of the aircraft (10). In one embodiment, the target surface is a runway (8) and the reference markers (1) are located at the start of the runway (8) on its longitudinal axis as shown in Figures 2 and 3. The imaging device can be a camera. There are several types of cameras (9) that utilize structured light technologies. However, for vision-based landing (VBL), the preferred choice is monocular and stereoscopic cameras (9). The selection of the camera (9) will influence the design of the reference markers (1). Preferably, a monocular camera (9) is used. In one implementation example, camera (9) is a grayscale camera. If the installed camera (9) captured color images, an intermediate step in the computer vision process would be to convert the RGB image into a grayscale intensity image. Therefore, the best option is to install a grayscale camera (9) on the aircraft (10). This is because: • When loaded into memory, a grayscale image occupies one-third of the space required for an RGB image. • Because a grayscale image has one-third of the data, it requires less computing power to process and can reduce processing time. • A grayscale image is conceptually simpler than an RGB image, so developing an image processing algorithm can be easier when working with grayscale. To fulfill the exact definition of estimating the position of an object, for example, an aircraft (10), the orientation of the aircraft (10) must also be obtained. The system of the invention allows obtaining the orientation of the aircraft, for example, in the form of roll, pitch, and yaw angles. - Warping: the angle could be obtained using trigonometry. - Pitch: the center of the circumscribed ellipse (6) will appear higher in the captured image (4) as the pitch of the aircraft (10) increases. The angle is obtained using trigonometry. - Yaw: As the aircraft (10) yaws, the center of the circumscribed ellipse (6) moves to the left or right in the captured image (4). The angle is obtained using trigonometry. The invention also includes a navigation system comprising a vision-based relative positioning system for an object with respect to a target surface, as explained above.

Claims

1. A vision-based relative positioning system for an object with respect to a target surface, characterized in that it comprises: - at least three reference markers (1) configured to be located on the target surface such that they are situated at the vertices of an equilateral triangle (2) as viewed from a top view of said target surface, wherein the equilateral triangle (2) has a centroid (5) and a circumscribed circle (3), - at least one imaging device configured to be located on the object and to capture images (4) of the target surface that include the reference markers (1), wherein the imaging device has a focal length, - a processing means configured to receive and process the images (4) captured by the at least one imaging device and configured to: • detect the reference markers (1) in the received image (4),• Identify the centroid (5) of a triangle (7) formed by the reference markers (1) in the received image (4), • Circumscribe the reference markers (1) in the received image (4) using a circumscribed ellipse (6) centered on the centroid (5) of the triangle (7) and passing through the vertices of the triangle (7), • Determine the relative position of the object with respect to the target surface according to its spherical geometric coordinates: distance, elevation, and azimuth, using the geometric relationship in the received image (4) between the semi-axes and the angle of the major semi-axis of the circumscribed ellipse (6) and the position of the reference markers (1) and the radius of its circumscribed circle (3). 2.- Vision-based relative positioning system of an object with respect to a target surface, according to claim 1,characterized in that, to estimate the object's distance from the reference markers (1), the processing means is configured to calculate the geometric relationship using the equation: where f is the focal length of the imaging device, v is the size of the semi-major axis of the circumscribed ellipse (6) in the received image (4), and is the radius of the circumscribed circle (3) on the target surface and z is the object's distance to the reference markers (1). 3.- A vision-based relative positioning system for an object with respect to a target surface, according to any one of the preceding claims,characterized in that, for calculating the elevation of the object with respect to the target surface, the processing means is configured to calculate the arcsine of the ratio between the semi-minor axis and the semi-major axis of the circumscribed ellipse (6).

4. A vision-based relative positioning system for an object with respect to a target surface, according to any one of the preceding claims, characterized in that, for calculating the azimuth of the object with respect to the target surface, the processing means is configured to calculate the angle between the semi-major axis of the circumscribed ellipse (6) and one of the sides of the triangle (7) formed by the reference markers (1) in the image (4) received in a top-down view of the target surface.

5. A vision-based relative positioning system for an object with respect to a target surface, according to claim 4.characterized in that, to calculate the angle between the semi-major axis of the circumscribed ellipse (6) and the side of the triangle (7) formed by the reference markers (1), the processing means is configured to calculate the Steiner nelipse of the triangle (7) of the reference markers (1) and apply Marden's theorem to calculate the angle between the semi-major axis of the Steiner nelipse and the side of the triangle (7) that coincides with the angle of the semi-major axis of the circumscribed ellipse (6) and the side of the triangle (7).

6. A vision-based relative positioning system for an object with respect to a target surface, according to any one of the preceding claims, characterized in that it comprises a fourth reference marker (1) located on one of the sides of the equilateral triangle (2) for the unique determination of the azimuth.

7. A vision-based relative positioning system for an object with respect to a target surface,according to any one of the preceding claims, characterized in that the three reference markers (1) are passive cooperative targets.

8. A vision-based relative positioning system for an object with respect to a target surface, according to claim 7, characterized in that the three reference markers (1) are circles.

9. A vision-based relative positioning system for an object with respect to a target surface, according to claim 8, characterized in that the centers of the circles form the vertices of the triangles (3, 7).

10. A vision-based relative positioning system for an object with respect to a target surface, according to claim 8 or 9, characterized in that the circles are white circles.

11. A vision-based relative positioning system for an object with respect to a target surface, according to any one of claims 7 to 10 above.characterized in that the passive cooperative targets are painted with a paint comprising titanium dioxide.

12. A vision-based relative positioning system for an object with respect to a target surface, according to any one of claims 7 to 11 above, characterized in that the passive cooperative targets form part of a sleeve configured to be positioned on the target surface.

13. A vision-based relative positioning system for an object with respect to a target surface, according to any one of the preceding claims, characterized in that the object is an aircraft (10) and the imaging device is configured to be installed on a longitudinal axis of the aircraft (10).

14. A vision-based relative positioning system for an object with respect to a target surface, according to claim 13,characterized in that the target surface is a runway (8) and the reference markers (1) are configured to be located at the beginning of the runway (8) on its longitudinal axis.

15. Navigation system, characterized in that it comprises a vision-based relative positioning system for an object with respect to a target surface according to any one of the preceding claims.

16. Method of vision-based relative positioning of an object with respect to a target surface, characterized in that it comprises the following steps: - providing at least three reference markers (1) located on the target surface such that they are situated at the vertices of an equilateral triangle (2) viewed from a top view, wherein the equilateral triangle (2) has a centroid (5) and a circumscribed circle (3),- provide at least one imaging device located on the object to capture images (4) of the target surface including the reference markers (1), wherein the imaging device has a focal length, - provide a processing means that receives and processes the captured images (4) received from the at least one imaging device that: • detects the reference markers (1) in the received image (4), • identifies the centroid (5) of the triangle (7) formed by the reference markers (1) in the received image (4), • circumscribes the reference markers (1) in the received image (4) by means of a circumscribed ellipse (6) centered on the centroid (5) of the triangle (7) and passing through the vertices of the triangle (7), • determines the position of the object with respect to the target surface according to its spherical geometric coordinates: distance, elevation, and azimuth,through the geometric relationship in the received image (4) between the semi-axes and the angle of the major semi-axis of the circumscribed ellipse (6) and the position of the reference markers (1) and the radius of its circumscribed circle (3).

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