CN113253285A - Method for upgrading fixed-point three-dimensional panoramic imaging sonar system into navigation system - Google Patents

Method for upgrading fixed-point three-dimensional panoramic imaging sonar system into navigation system Download PDF

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CN113253285A
CN113253285A CN202110510947.7A CN202110510947A CN113253285A CN 113253285 A CN113253285 A CN 113253285A CN 202110510947 A CN202110510947 A CN 202110510947A CN 113253285 A CN113253285 A CN 113253285A
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sonar
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CN113253285B (en
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陈槐
朱立俊
王建中
范红霞
王乃茹
魏裕翀
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Nanjing Institute Of Water Conservancy Sciences State Energy Bureau Ministry Of Transportation Ministry Of Water Conservancy
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Nanjing Institute Of Water Conservancy Sciences State Energy Bureau Ministry Of Transportation Ministry Of Water Conservancy
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01SRADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S15/00Systems using the reflection or reradiation of acoustic waves, e.g. sonar systems
    • G01S15/88Sonar systems specially adapted for specific applications
    • G01S15/89Sonar systems specially adapted for specific applications for mapping or imaging
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01SRADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S19/00Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
    • G01S19/38Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system
    • G01S19/39Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system the satellite radio beacon positioning system transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/42Determining position
    • G01S19/421Determining position by combining or switching between position solutions or signals derived from different satellite radio beacon positioning systems; by combining or switching between position solutions or signals derived from different modes of operation in a single system
    • YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02ATECHNOLOGIES FOR ADAPTATION TO CLIMATE CHANGE
    • Y02A90/00Technologies having an indirect contribution to adaptation to climate change
    • Y02A90/30Assessment of water resources

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Abstract

The invention discloses a method for upgrading a fixed-point three-dimensional panoramic imaging sonar system into an aerial system, and belongs to the technical field of water conservancy measurement. The underwater structure real-time measurement system comprises a fixed-point sonar measurement system (such as BV5000-1350), a working ship, a satellite positioning module, an inertial sensing module and a computer, wherein the fixed-point sonar measurement system is installed and calibrated, the satellite positioning module and the inertial sensing module are additionally arranged, the coordinate, the azimuth angle, the pitch angle and the roll angle of the sonar system are obtained, the underwater structure detail form generated is displayed on the computer in real time by combining a space coordinate conversion formula, and therefore the underwater structure sailing type real-time measurement is economically, safely and high-qualitatively.

Description

Method for upgrading fixed-point three-dimensional panoramic imaging sonar system into navigation system
Technical Field
The invention relates to the technical field of water conservancy measurement, in particular to a method for upgrading a fixed-point three-dimensional panoramic imaging sonar system into an aerial system.
Background
China has wide range of members, numerous rivers and wide sea area. Under the influence of natural disasters such as typhoons, floods, earthquakes and the like, hydraulic buildings are seriously damaged, dikes break-off and reservoir emergency frequently occur, and in order to ensure national economy and the safety of lives and properties of people, underwater detailed structures at hidden dangers need to be rapidly detected in time so as to facilitate the establishment of emergency rescue and treatment schemes.
At present, the detection of a hydraulic building or an underwater detail structure mainly comprises technical means such as underwater visual detection, underwater laser imaging, underwater sonar imaging and the like. The first method is a basic method for detecting the quality of underwater buildings, and comprises visual inspection (also called visual inspection), underwater photography, underwater video recording and the like, wherein although the method is simple in operation, the influence of subjective factors is large, and the underwater imaging effect of a camera is poor; secondly, the underwater laser scanning imaging is to use a laser to emit a laser beam, a detector to receive central reflected light of the laser beam, and the laser beam and the central reflected light are scanned and imaged synchronously, but the detection distance is short and the imaging quality is poor due to the scattering and energy consumption of the laser in water; and thirdly, underwater sonar imaging is divided into single-beam sonar imaging and multi-beam sonar imaging, and the multi-beam sonar imaging can be divided into a multi-beam depth sounding system, a side scan sonar and a three-dimensional imaging sonar.
The three-dimensional imaging sonar system has the characteristics of high efficiency, high precision and high resolution, and is an advanced means for detecting underwater detail structures at present. BV5000-1350 is a three-dimensional multi-beam sonar system developed by Teledyne Marine, capable of directly acquiring data of the outline of an object in the horizontal (X), vertical (Y) and height (Z) directions, and simultaneously acquiring detailed description of the material (attribute) of the object, and generating a real-time three-dimensional image of the object, and the system can work in a water area environment with large sand content, low visibility and complex underwater topography.
The BV5000 is a fixed-point measurement system, which is sold at a price of about 20 million dollars, but can only measure objects within a range of about 30m around the BV at a fixed point, and if a large area is to be measured, the position needs to be frequently changed, and the time and the labor are consumed; although Teledyne Marine also sells airborne measuring equipment, the price is very high, about 40 million dollars. The BV5000-1350 system has small size and light weight, and can be easily loaded on various working ships, underwater robots and other equipment, but no relevant report is found at present on how to upgrade the BV5000-1350 system into an aerial measurement mode.
Disclosure of Invention
The invention provides a method for upgrading a fixed-point three-dimensional panoramic imaging sonar system into a navigation system, which achieves the acquisition of real-time coordinate position and azimuth information and improves the measurement precision.
In order to achieve the purpose, the invention provides the following technical scheme: the method for upgrading the fixed-point three-dimensional panoramic imaging sonar system into the air-navigation system comprises the following steps:
s1, firstly, fixing a work ship provided with a fixed point sonar measurement system, wherein a satellite positioning module and an inertia sensing module are added in the fixed point sonar measurement system, the satellite positioning module comprises a fixed base station and a mobile base station, and the inertia sensor comprises a gyroscope and an accelerator;
s2, calibrating parameters of the fixed point sonar measurement system in a satellite inertial navigation module coordinate system O-XYZ, and selecting three calibration points;
s3, removing the fixed constraint of the workboat;
s4, carrying out navigation measurement, recording the GPS position coordinate of the installation point of the mobile base station in the satellite positioning module by using a computer at any moment, recording three Tait-Bryan rotation angles of the working ship at the moment, which are obtained by the inertial sensing module, opening the fixed point sonar measurement system to carry out riverbed terrain measurement while recording the GPS coordinate, and recording the bed surface coordinate and the echo intensity measured by the fixed point sonar measurement system by using the computer;
s5, after the target measurement is finished, closing the satellite positioning module, the inertia sensing module and the fixed point sonar measurement system;
s6, performing space coordinate conversion, and obtaining real bed surface topography or building form at any time by calculating real bed surface coordinates at any time and performing space coordinate conversion by a mathematical method;
and S7, drawing all the topography of the bed surface or the shape of the building calculated at any time, and visually displaying the measurement result of the sailing type fixed point sonar measurement system.
Preferably, in S1, the fixed point sonar measurement system is specifically a fixed point measurement mode of BV5000-1350, the satellite positioning module is used to obtain coordinates and an azimuth angle in the fixed point sonar measurement system, and the inertial sensor is used to obtain a pitch angle and a roll angle in the sonar system.
Preferably, in S2, the parameter calibration includes:
(1) calibrating coordinates (X) of a built-in coordinate system O' -abc origin of the quantitative sonar measuring system in a coordinate system O-XYZ0,Y0,Z0):
Figure BDA0003060355610000031
In the formula (x)AZ,yAZ,zAZ) Is the GPS position coordinates of the installation point of the mobile base station in the satellite positioning module, and (x)BV,yBV,zBV) A GPS coordinate corresponding to the origin of the fixed-point sonar measurement system;
(2) calibrating three Tait-Bryan rotation angles alpha, beta and gamma of a built-in coordinate system O' -abc of the fixed-point sonar measurement system and a coordinate system O-XYZ:
Figure BDA0003060355610000032
in the formula, the Tait-Bryan rotation is defined as O-XYZ translation (X) of the coordinate system0,Y0,Z0) Then, rotating along X, Y, Z coordinate axes according to a fixed sequence, rotating by alpha, beta and gamma angles respectively, and then coinciding with a coordinate system o' -abc; (a)1,b1,c1)、(a2,b2,c2) And (a)3,b3,c3) Coordinates of the three calibration points in a built-in coordinate system of the fixed point sonar measurement system are respectively set; (X)1,Y1,Z1)、(X2,Y2,Z2) And (X)3,Y3,Z3) Coordinates of the three calibration points in a satellite positioning module coordinate system are respectively; the matrix M is a 3X 3 matrix, Mi1=Xi-X0,Mi2=Yi-Y0,Mi3=Zi-Z0Wherein i is 1, 2, 3; matrix M-1Is the inverse of matrix M;
Figure BDA0003060355610000041
Figure BDA0003060355610000042
preferably, in S4, the time is t, and the GPS position coordinates are (x)0t,y0t,z0t) And the three Tait-Bryan rotation angles are respectively
Figure BDA0003060355610000043
θt、ψtThe echo coordinate is (a)t,bt,ct,It)。
Preferably, in S6, the real bed surface coordinate at the arbitrary time t is (x)t,yt,zt) The space coordinate conversion and solving steps comprise:
(1) the relationship between the coordinate system O-XYZ and the coordinate system O-XYZ is as follows:
Figure BDA0003060355610000044
in the formula
Figure BDA0003060355610000045
(2) The relationship between the coordinate system O' -abc and the coordinate system O-XYZ is as follows:
Figure BDA0003060355610000046
in the formula
Figure BDA0003060355610000047
The real bed surface topography or building form at any time can be obtained by solving the two formulas:
Figure BDA0003060355610000048
in the formula R-1Representing the inverse of the respective matrix.
Compared with the prior art, the invention has the beneficial effects that:
1. aiming at the defects of the fixed-point three-dimensional panoramic imaging sonar system, a satellite positioning module and an inertial sensing module are added in the fixed-point three-dimensional panoramic imaging sonar system to obtain the real-time coordinate position and orientation information of the sonar system;
2. the accuracy of the navigation type measuring result is ensured through mathematically strict space coordinate conversion;
3. displaying the riverbed topography or hydraulic building form measured in an air-navigation mode in real time through a computer;
4. the method is economical, practical, safe, reliable and high in measurement precision.
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The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention.
In the drawings:
FIG. 1 is a schematic view of a measurement system arrangement of the present invention;
reference numbers in the figures: 1. a fixed base station; 1-1, a first satellite antenna; 1-2, a first portable battery; 1-3, a first wireless data transmission antenna; 2. a mobile base station; 2-1, a second satellite antenna; 2-2, a third satellite antenna; 2-3, a second portable battery; 2-4, a second wireless data receiving antenna; 3. an inertial sensing module; 4. a sonar system mounting bracket; 5. a fixed-point sonar measurement system; 6. a work boat; 7. and (4) a computer.
Detailed Description
The preferred embodiments of the present invention will be described in conjunction with the accompanying drawings, and it will be understood that they are described herein for the purpose of illustration and explanation and not limitation.
Example (b): as shown in FIG. 1, a method layout for upgrading a fixed-point three-dimensional panoramic imaging sonar system into an aerial measurement system comprises a fixed base station 1 and a moving base station 2 in a satellite positioning module, an inertial sensing module 3, a sonar system mounting bracket 4, a fixed-point sonar measurement system 5, a work ship 6 and a computer 7, wherein the satellite positioning module can realize a positioning accuracy of 2.5 cm by using a difference principle, the fixed base station 1 comprises a first satellite antenna 1-1, a first portable battery 1-2 and a first wireless data transmission antenna 1-3, the first portable battery 1-2 is used for providing electric energy for the first satellite antenna 1-1 and the first wireless data transmission antenna 1-3, the first satellite antenna 1-1 can receive Beidou, GPS and GLONASS satellite signals and rate and correct the satellite signals in a difference mode, the corrected result is transmitted to the mobile base station 2 through the first wireless data transmission antenna 1-3, so that the positioning accuracy of the mobile base station 2 is improved; the mobile base station 2 comprises a second satellite antenna 2-1, a third satellite antenna 2-2, a second portable battery 2-3 and a second wireless data receiving antenna 2-4, wherein the second portable battery 1-2 is used for supplying electric energy to the second satellite antenna 2-1, the third satellite antenna 2-2 and the second wireless data receiving antenna 2-4; the second wireless data receiving antenna 2-4 is used for receiving the correction signal sent by the first wireless data transmission antenna 1-3; the second satellite antenna 2-1 and the third satellite antenna 2-2 can receive Beidou satellite signals, GPS satellite signals and GLONASS satellite signals, coordinates are corrected according to the correction signals, and coordinates of the two points are measured at the same time, so that a connecting line of the second satellite antenna 2-1 and the third satellite antenna 2-2 can indicate an azimuth angle (namely a course and an angle rotating around an x axis) of the working ship 6; the inertial conduction module 3 is composed of a micro-inertial measurement unit (gyroscope and accelerometer) and is used for obtaining a pitch angle (an angle rotating around a y axis) and a roll angle (an angle rotating around a z axis) of the working ship 6, and because the fixed point sonar measurement system 5 and the working ship 6 are rigid bodies, 2 measured angles are also the pitch angle and the roll angle of the sonar system; wherein, the sonar system mounting bracket 4 is used for fixedly mounting a sonar fixed-point measurement system 5 (namely a fixed-point three-dimensional panoramic imaging sonar system) on a working ship 6; the fixed-point sonar measurement system 5 is used for measuring the topography of a riverbed or the shape of a hydraulic building; wherein the working vessel 6 is used for providing navigation service; wherein the computer 7 is used for recording measurement data.
The working principle of the invention is that the fixed point sonar measurement system 5 is arranged on a working ship 6, and a satellite positioning module and an inertia sensing module 3 are added to obtain the real-time coordinate position and the azimuth information of the installation point of the fixed point sonar measurement system 5, and the fixed point measurement result is converted into the sailing type measurement result through strict space coordinate conversion.
The specific layout of this embodiment is shown in fig. 1, and the method of the present invention specifically includes the following steps:
s1, firstly fixing a working ship 6 provided with a fixed point sonar measurement system 5, installing a fixed base station 1 on a river bank, and fixedly installing a mobile base station 2, an inertia conduction module 3 and the fixed point sonar measurement system 5 on the working ship 6;
s2, calibrating parameters of the fixed-point sonar measurement system 5 in a satellite inertial navigation module coordinate system O-XYZ, and selecting three calibration points, wherein the parameter calibration comprises:
(1) calibrating coordinates (X) of a built-in coordinate system O' -abc origin of the quantitative sonar measuring system in a coordinate system O-XYZ0,Y0,Z0):
Figure BDA0003060355610000071
In the formula (x)AZ,yAZ,zAZ) Is the GPS position coordinates of the installation point of the mobile base station in the satellite positioning module, and (x)BV,yBV,zBV) A GPS coordinate corresponding to the origin of the fixed-point sonar measurement system;
(2) calibrating three Tait-Bryan rotation angles alpha, beta and gamma of a built-in coordinate system O' -abc of the fixed-point sonar measurement system and a coordinate system O-XYZ:
Figure BDA0003060355610000072
in the formula, the Tait-Bryan rotation is defined as O-XYZ translation (X) of the coordinate system0,Y0,Z0) Then, rotating along X, Y, Z coordinate axes according to a fixed sequence, rotating by alpha, beta and gamma angles respectively, and then coinciding with a coordinate system o' -abc; (a)1,b1,c1)、(a2,b2,c2) And (a)3,b3,c3) Coordinates of the three calibration points in a built-in coordinate system of the fixed point sonar measurement system are respectively set; (X)1,Y1,Z1)、(X2,Y2,Z2) And (X)3,Y3,Z3) Coordinates of the three calibration points in a satellite positioning module coordinate system are respectively; the matrix M is a 3X 3 matrix, Mi1=Xi-X0,Mi2=Yi-Y0,Mi3=Zi-Z0Wherein i is 1, 2, 3; matrix M-1Is the inverse of matrix M;
Figure BDA0003060355610000073
Figure BDA0003060355610000074
s3, removing the fixed constraint of the workboat 6;
s4, making an aerial survey, and recording the GPS position coordinates (x) of the installation point of the mobile base station 2 in the satellite positioning module at any time t by using a computer0t,y0t,z0t) And recording three Tait-Bryan rotation angles of the working ship 6 at the moment obtained by the inertial sensing module 3
Figure BDA0003060355610000085
θt、ψt(from coordinate system O-XYZ to coordinate system O-XYZ), while recording GPS coordinates, opening the fixed point sonar measurement system 5 to perform riverbed terrain measurement, and recording the coordinates of the bed surface (or the hydraulic structure) and the echo intensity (a) measured by the fixed point sonar measurement system 5 using the computer 7t,bt,ct,It);
S5, after the target measurement is finished, closing the satellite positioning module, the inertia sensing module 3 and the fixed point sonar measurement system 5;
s6, performing space coordinate conversion, calculating the real bed surface coordinate at any time t, and performing space coordinate conversion by a mathematical method to obtain the real bed surface terrain or building form at any time t, wherein the real bed surface coordinate at any time t is (x)t,yt,zt) The space coordinate conversion step comprises the following steps:
(1) the relationship between the coordinate system O-XYZ and the coordinate system O-XYZ is as follows:
Figure BDA0003060355610000081
in the formula
Figure BDA0003060355610000082
(2) The relationship between the coordinate system O' -abc and the coordinate system O-XYZ is as follows:
Figure BDA0003060355610000083
in the formula
Figure BDA0003060355610000084
The real bed surface topography or building form at any time t can be obtained by solving the two formulas:
Figure BDA0003060355610000091
in the formula R-1An inverse matrix representing the respective matrix;
and S7, drawing all the topography of the bed surface or the morphology of the building calculated at any time t, and visually displaying the measurement result of the sailing type fixed point sonar measurement system 5.
In addition, in practical application, if the relative installation positions of the satellite positioning module and the fixed-point sonar measurement system 5 are not changed, calibration is only needed once, but once the relative positions of the satellite positioning module and the fixed-point sonar measurement system are changed, calibration is needed again.
Finally, it should be noted that: although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that changes may be made in the embodiments and/or equivalents thereof without departing from the spirit and scope of the invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (5)

1. The method for upgrading the fixed-point three-dimensional panoramic imaging sonar system into the air navigation system is characterized by comprising the following steps of: the method comprises the following steps:
s1, firstly, fixing a work ship provided with a fixed point sonar measurement system, wherein a satellite positioning module and an inertia sensing module are added in the fixed point sonar measurement system, the satellite positioning module comprises a fixed base station and a mobile base station, and the inertia sensor comprises a gyroscope and an accelerator;
s2, calibrating parameters of the fixed point sonar measurement system in a satellite inertial navigation module coordinate system O-XYZ, and selecting three calibration points;
s3, removing the fixed constraint of the workboat;
s4, carrying out navigation measurement, recording the GPS position coordinate of the installation point of the mobile base station in the satellite positioning module by using a computer at any moment, recording three Tait-Bryan rotation angles of the working ship at the moment, which are obtained by the inertial sensing module, opening the fixed point sonar measurement system to carry out riverbed terrain measurement while recording the GPS coordinate, and recording the bed surface coordinate and the echo intensity measured by the fixed point sonar measurement system by using the computer;
s5, after the target measurement is finished, closing the satellite positioning module, the inertia sensing module and the fixed point sonar measurement system;
s6, performing space coordinate conversion, calculating the real bed surface coordinate at any moment, performing space coordinate conversion by a mathematical method, and solving the real bed surface topography or building form at any moment;
and S7, drawing all the topography of the bed surface or the shape of the building calculated at any time, and visually displaying the measurement result of the sailing type fixed point sonar measurement system.
2. The method for upgrading the fixed-point three-dimensional panoramic imaging sonar system to the air-navigation system according to claim 1, is characterized in that: in S1, the fixed point sonar measurement system is specifically a fixed point measurement mode of BV5000-1350, the satellite positioning module is used to obtain coordinates and azimuth angles in the fixed point sonar measurement system, and the inertial sensor is used to obtain pitch angles and roll angles in the sonar system.
3. The method for upgrading the fixed-point three-dimensional panoramic imaging sonar system to the air-navigation system according to claim 1, is characterized in that: in S2, the parameter calibration includes:
(1) calibrating coordinates (X) of a built-in coordinate system O' -abc origin of the quantitative sonar measuring system in a coordinate system O-XYZ0,Y0,Z0):
Figure FDA0003060355600000021
In the formula (x)AZ,yAZ,zAZ) Is the GPS position coordinates of the installation point of the mobile base station in the satellite positioning module, and (x)BV,yBV,zBV) A GPS coordinate corresponding to the origin of the fixed-point sonar measurement system;
(2) calibrating three Tait-Bryan rotation angles alpha, beta and gamma of a built-in coordinate system O' -abc of the fixed-point sonar measurement system and a coordinate system O-XYZ:
Figure FDA0003060355600000022
in the formula, the Tait-Bryan rotation is defined as O-XYZ translation (X) of the coordinate system0,Y0,Z0) Then X, Y are arranged in a fixed order,The three coordinate axes Z rotate respectively by the angles of alpha, beta and gamma and then coincide with a coordinate system o' -abc; (a)1,b1,c1)、(a2,b2,c2) And (a)3,b3,c3) Coordinates of the three calibration points in a built-in coordinate system of the fixed point sonar measurement system are respectively set; (X)1,Y1,Z1)、(X2,Y2,Z2) And (X)3,Y3,Z3) The coordinates of the three calibration points in the coordinate system of the satellite positioning module, wherein the matrix M is a 3 x 3 matrix, Mi1=Xi-X0,Mi2=Yi-Y0,Mi3=Zi-Z0Wherein i is 1, 2, 3; matrix M-1Is the inverse of matrix M;
Figure FDA0003060355600000023
Figure FDA0003060355600000024
4. the method for upgrading the fixed-point three-dimensional panoramic imaging sonar system to the air-navigation system according to claim 1, is characterized in that: in S4, the time is set to t, and the GPS position coordinates are (x)0t,y0t,z0t) And the three Tait-Bryan rotation angles are respectively
Figure FDA0003060355600000025
θt、ψtThe echo coordinate is (a)t,bt,ct,It)。
5. The method for upgrading the fixed-point three-dimensional panoramic imaging sonar system to the air-navigation system according to claim 1, is characterized in that: in S6, the real bed surface coordinate at the arbitrary time t is (x)t,yt,zt) The space coordinate conversion and solving steps comprise:
(1) the relationship between the coordinate system O-XYZ and the coordinate system O-XYZ is as follows:
Figure FDA0003060355600000031
in the formula
Figure FDA0003060355600000032
(2) The relationship between the coordinate system O' -abc and the coordinate system O-XYZ is as follows:
Figure FDA0003060355600000033
in the formula
Figure FDA0003060355600000034
The real bed surface topography or building form at any time can be obtained by solving the two formulas:
Figure FDA0003060355600000035
in the formula R-1Representing the inverse of the respective matrix.
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