Large circle route guidance and control method based on Mercator chart

Abstract

The invention relates to a major circle route guidance and control method based on a Mercator chart, which belongs to the field of ship track tracking and comprises the following steps of: establishing a ship motion nonlinear mathematical model; designing a guidance strategy of a great circle route, and enabling the ship to sail along the great circle route according to an expected course; based on the ship motion nonlinear mathematical model, a final controller is designed by adopting a third-order closed-loop gain shaping algorithm and is used for controlling the ship to track an expected course; and converting the plane graph of the great circle route in the ship driving process into a Mercator projection chart to obtain the navigation track of the ship along the great circle route under Mercator projection. Through simulation verification, the ship can be effectively controlled to sail in a large arc line, the track error is reduced, and the sailing efficiency is improved.

Description

Technical Field

[0001] The invention belongs to the field of ship track tracking and relates to a great circle route guidance and control method based on a Mercator chart. Background Art

[0002] Currently, ship track tracking involves tracking a straight route. However, when sailing at high latitudes, ships often follow great circle routes, which limits the effectiveness of existing track tracking methods. Furthermore, track tracking results are currently mostly displayed on a flat surface, which is not intuitive enough for practical ship applications. Guidance based on great circle routes on Mercator charts can better meet the needs of ships sailing in high latitudes. Applying track information to Mercator projections has significant practical implications for ship navigation and track control.

[0003] Existing methods include treating the route between two locations as a straight line, using rhumb lines. Another approach involves dividing a great circle route into several straight segments, connecting them together, a method known as a great circle approximation. While these methods can meet navigation needs, they still rely on the same rhumb line method and are complex and cumbersome. Therefore, enabling ships to navigate along great circle routes in high latitudes and applying this to Mercator projection charts is crucial for advancing navigation practice and research. Summary of the Invention In order to solve the above problems, the technical solution adopted by the present invention is: a great circle route guidance and control method based on Mercator chart, comprising the following steps: Establish a nonlinear mathematical model of ship motion; Design guidance strategies for great circle routes to enable ships to navigate along the great circle routes according to the desired course; Based on the nonlinear mathematical model of ship motion, a third-order closed-loop gain shaping algorithm is used to design the final controller to control the ship to track the desired heading. The plan view of the great circle route during the ship's navigation process is converted into a Mercator projection nautical chart to obtain the ship's navigation trajectory along the great circle route under the Mercator projection.

[0004] Furthermore, the process of establishing the nonlinear mathematical model of ship motion is as follows: When analyzing ship maneuvering, if only the drift speed is concerned, and bow angular velocity The linear equations corresponding to these two degrees of freedom for the ship are: (1) In this formula, Represents the rudder angle input, represent Taking the first derivative with respect to time, Respectively represent First derivative with respect to time; coefficient It is determined by the basic parameters of the ship. Formula (1) is converted into a simple equation describing the steering response to the bow roll, namely: (2) In this formula, Represents the rudder angle input, Indicates rudder angle input Taking the derivative with respect to time, Indicates the bow angular velocity Taking the second derivative with respect to time, All are manipulative indicators; Perform Laplace transform on Equation (2) and assume that the initial value Established, the following transfer function is obtained : (3) In this formula is the Laplace operator, represents the Laplace transform of the rudder angle input, represents the Laplace transform of the bow angular velocity, All are manipulative indicators; According to the relationship , and get the corresponding equation: (4) In formula (4), is the heading angle, is the Laplace operator, represents the Laplace transform of the rudder angle input, represents the Laplace transform of the heading angle, All are manipulative indicators; Simplify formula (4) and derive the following formula: (5) In the formula , is the Laplace operator, is the ship maneuverability index, and Equation (5) is written as the following differential equation (6) in, is the rudder angle, Represents the heading angle Taking the first derivative with respect to time, Represents the heading angle Taking the second derivative with respect to time, is the ship maneuverability index; Divide both sides of the equation (6) by have to: (7) in, is the rudder angle, Represents the heading angle Taking the first derivative with respect to time, Represents the heading angle Taking the second derivative with respect to time, is the ship maneuverability index; For a statically unstable ship, the second term on the left side of Equation (7) Need to be replaced by nonlinear terms Replace, and: (8) Thus, the nonlinear mathematical model of ship motion is established: (9) In formula (9): is the heading angle, is the rudder angle, Represents the heading angle Taking the first derivative with respect to time, Represents the heading angle Taking the second derivative with respect to time, is the ship maneuverability index, is a nonlinear parameter.

[0005] Furthermore, the process of designing the guidance strategy for the great circle route so that the ship can sail along the great circle route according to the desired course is as follows: S31: Calculate the course and distance of a great circle route: Assume that the starting coordinates of the ship are , the end point coordinates are , use the spherical triangle formula to find the starting point coordinates The initial great circle heading at , the formula for calculating the great circle range and heading is: (10) (11) in, For great circle voyages, is the starting latitude, is the latitude of the arrival point, is the longitude difference between the starting point and the arrival point, is the initial great circle heading; S32: Calculate the coordinates of the point closest to the current ship position. The specific process is as follows: Assume the current ship position is , then the current ship position is The great circle distance between ,but: (12) (13) Where: is the longitude difference from the starting point to the current ship position; is the latitude of the current ship position; is the longitude of the current ship's position; is the great circle distance from the starting point to the current ship position; for Point relative to The great circle azimuth of the point, with the current position as Point, to the great circle route Draw a perpendicular line with the foot of , forming a spherical right triangle ,but: (14) (15) (16) Where: for arrive The vertical distance; is the great circle distance from the starting point to the current ship position; for arrive The great circle distance of a point; Two large arcs and The angle between for Point relative to The great circle azimuth of the point; is the initial great circle heading; The coordinates of a point and the great circle heading are calculated using the following formula: (17) (18) (19) (20) Where: is the initial great circle heading; for arrive The great circle distance of a point; Starting point to The longitude difference of a point; for The latitude of the point; yes The longitude of the point; is the latitude of the starting point; is the longitude of the starting point; is the latitude of the arrival point; is the longitude of the arrival point; for The great circle heading of the point; From the starting point to Great circle range of points; S33: Calculate the target point coordinates and the desired heading; Let the tracking target point be located at the perpendicular point Great circle distance ahead at point; (twenty one) (twenty two) (twenty three) (twenty four) Where: for point to The longitude difference of a point; for and The great circle distance between for The latitude of the point; yes The longitude of the point; for The latitude of the point; yes The longitude of the point; for The great circle heading of the point; for The great circle heading of the point, that is, the desired heading; is the latitude of the arrival point; is the longitude of the arrival point.

[0006] Furthermore, the process of designing the final controller using a third-order closed-loop gain shaping algorithm based on the nonlinear mathematical model of ship motion is as follows: Assume that the bandwidth frequency of the closed-loop system of the ship is , then the compensation sensitivity function of the ship heading keeping control system is That is, the closed-loop transfer function of the control system is shown in formula (25): (25) The controller for: (26) From Equation (26), the controller is in the form of a typical PD controller connected in series with an oscillating link. Under the PD controller, a constant , the ship heading keeping control system will greatly reduce the adjustment time without changing the stability. Adding a positive constant to the proportional part of the PD link in Equation (26), the final controller is in the form of Equation (27): (27) Furthermore, the process of converting the plan view of the great circle route during the ship's navigation process into a Mercator projection chart to obtain the ship's navigation trajectory along the great circle route under the Mercator projection is as follows: Plane coordinates of the planned route , based on the following formula, the plane coordinates of the planned great circle route are Convert to Mercator projection latitude and longitude using the following formula: (28) (29) in: is the radius of the Earth, is the horizontal coordinate of the rectangular coordinate on the plane, is the ordinate of the rectangular coordinate on the plane; is the Mercator projection coordinate longitude, is the Mercator projection coordinate latitude; The longitude and latitude obtained above are in radians and are converted according to the following formula: (30) (31) in, Represents Mercator projection coordinate longitude in radians, Represents the Mercator projection coordinate latitude in radians, represents the Mercator projection coordinate longitude in degrees, Represents Mercator projection coordinate latitude in degrees.

[0007] The present application discloses a great circle route guidance and control method based on a Mercator chart. Compared with other methods, the method of the present application can shorten the navigation time of a ship during long-distance navigation. Moreover, the method of the present application is no longer limited to applying the navigation trajectory to an ordinary plane, but applies the operation trajectory to a Mercator projection map, making it more suitable for practical applications of ship navigation. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative labor.

[0009] Figure 1 A flow chart of the method of this application; Figure 2 This is the principle diagram of great circle route guidance; Figure 3 is a diagram of a spherical right triangle; Figure 4 This is a track comparison chart of this method and other methods; Figure 5 The following is a comparison chart of the heading of this method and other methods; Figure 6 This is a comparison chart of the rudder angle between this method and other methods; Figure 7 Comparison chart of track errors between this method and other methods. DETAILED DESCRIPTION

[0010] It should be noted that, unless there is any conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0011] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. The following description of at least one exemplary embodiment is actually only illustrative and is in no way intended to limit the present invention and its application or use. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0012] Figure 1 A flow chart of the method of this application; A great circle route guidance and control method based on Mercator chart, S1: Establish a nonlinear mathematical model of ship motion, which is a nonlinear Nomoto mathematical model; S2: Design a guidance strategy for the great circle route to enable the ship to navigate along the great circle route according to the desired course; S3: Based on the nonlinear mathematical model of ship motion, a third-order closed-loop gain shaping algorithm is used to design the final controller to control the ship to track the desired heading; S4: Convert the plan view of the great circle route during the ship's navigation process into a Mercator projection nautical chart to obtain the ship's navigation trajectory along the great circle route under the Mercator projection.

[0013] Steps S1 / S2 / S3 / S4 are performed sequentially; This application is implemented using MATLAB software; Furthermore, the process of establishing the nonlinear Nomoto mathematical model of ship motion is as follows: When analyzing ship maneuvering, if only the drift speed is concerned, and bow angular velocity The linear equations corresponding to these two degrees of freedom for the ship are: (1) In this formula, Represents the rudder angle input, represent Taking the first derivative with respect to time, Respectively represent Take the first derivative with respect to time, the coefficient It is determined by the basic parameters of the ship. Formula (1) is converted into a simple equation describing the steering response to the bow roll, namely: (2) In this formula, Represents the rudder angle input, Indicates rudder angle input Taking the derivative with respect to time, Indicates the bow angular velocity Taking the second derivative with respect to time, All are manipulative indicators; Perform Laplace transform on Equation (2) and assume that the initial value Established, the following transfer function is obtained : (3) In this formula is the Laplace operator, represents the Laplace transform of the rudder angle input, represents the Laplace transform of the bow angular velocity, All are manipulative indicators; According to the relationship , and get the corresponding equation: (4) In formula (4), is the heading angle, is the Laplace operator, represents the Laplace transform of the rudder angle input, represents the Laplace transform of the heading angle, All are manipulative indicators; Simplify formula (4) and derive the following formula: (5) In the formula , is the Laplace operator, is the ship maneuverability index, and Equation (5) is written as the following differential equation (6) in, is the rudder angle, and Represents the heading angle Find the first and second derivatives with respect to time, is the ship maneuverability index.

[0014] Divide both sides of the equation (6) by have to: (7) in, is the rudder angle, and Represents the heading angle Find the first and second derivatives with respect to time, is the ship maneuverability index.

[0015] For some statically unstable ships, the second term on the left side of Equation (7) Need to be replaced by nonlinear terms Replace, and: (8) Thus, the nonlinear mathematical model of ship motion is established: (9) In formula (9): is the heading angle, is the rudder angle, and Represents the heading angle Find the first and second derivatives with respect to time, is the ship maneuverability index, is a nonlinear parameter.

[0016] Figure 2 Great circle route guidance principle diagram; Figure 3 is a diagram of a spherical right triangle; Furthermore, the process of designing a great circle route guidance strategy to make the ship sail along the great circle route according to the desired course is as follows: S31: Calculate the great circle route heading and distance: Assume that the starting coordinates of the ship are , the end point coordinates are , use the spherical triangle formula to find the starting point coordinates The initial great circle heading at , the formula for calculating the great circle range and heading is: (10) (11) in, For great circle voyages, is the starting latitude, is the latitude of the arrival point, is the longitude difference between the starting point and the arrival point, is the initial great circle heading; S32: Calculate the coordinates of the point closest to the current ship position: Assume the current ship position is , then the current ship position is The great circle distance between ,but: (12) (13) Where: is the longitude difference from the starting point to the current ship position; is the latitude of the current ship position; is the longitude of the current ship's position; is the great circle distance from the starting point to the current ship position; for Point relative to The great circle azimuth of the point, with the current position as Point, to the great circle route Draw a perpendicular line with the foot of , forming a spherical right triangle ,but: (14) (15) (16) Where: for arrive The vertical distance; is the great circle distance from the starting point to the current ship position; for arrive The great circle distance of a point; Two large arcs and The angle between for Point relative to The great circle azimuth of the point; is the initial great circle heading; The coordinates of a point and the great circle heading are calculated using the following formula: (17) (18) (19) (20) Where: is the initial great circle heading; for arrive The great circle distance of a point; Starting point to The longitude difference of a point; for The latitude of the point; yes The longitude of the point; is the latitude of the starting point; is the longitude of the starting point; is the latitude of the arrival point; is the longitude of the arrival point; for The great circle heading of the point; From the starting point to Great circle range of points; S33: The process of calculating the target point coordinates and the desired heading is as follows: Let the tracking target point be located at the perpendicular point Great circle distance ahead at point; (twenty one) (twenty two) (twenty three) (twenty four) Where: for point to The longitude difference of a point; for and The great circle distance between for The latitude of the point; yes The longitude of the point; for The latitude of the point; yes The longitude of the point; for The great circle heading of the point; for The great circle heading of the point, that is, the desired heading; is the latitude of the arrival point; is the longitude of the arrival point; Furthermore, the process of designing the final controller based on the nonlinear mathematical model of ship motion and using the third-order closed-loop gain shaping algorithm is as follows: Controller design using a third-order closed-loop gain shaping algorithm , assuming the bandwidth frequency of the ship's closed-loop system is , then the compensation sensitivity function of the ship heading keeping control system is That is, the closed-loop transfer function of the system is shown in formula (56): (25) The controller is: (26) From Equation (57), we can see that the controller is a typical PD controller connected in series with an oscillating link. Under the PD controller, a constant , the ship heading keeping control system will significantly reduce the adjustment time without changing the stability. Analogously to this method, a positive constant can be added to the proportional part of the PD link in Equation (57). This application takes , the final controller is in the form of formula (58) (27) Furthermore, the process of converting the plan view of the great circle route during the ship's navigation process into a Mercator projection chart to obtain the ship's navigation trajectory along the great circle route under the Mercator projection is as follows: Plane coordinates of the planned route , based on the following formula, the plane coordinates of the planned route are Convert to Mercator projection latitude and longitude using the following formula: (28) (29) in: is the radius of the Earth, is the horizontal coordinate of the rectangular coordinate on the plane, is the ordinate of the rectangular coordinate on the plane; is the Mercator projection coordinate longitude, is the Mercator projection coordinate latitude; The longitude and latitude obtained above are in radians and are converted according to the following formula: (30) (31) in, Represents Mercator projection coordinate longitude in radians, Represents the Mercator projection coordinate latitude in radians, represents the Mercator projection coordinate longitude in degrees, Represents Mercator projection coordinate latitude in degrees.

[0017] In order to verify the effectiveness of the method of the present invention, the following simulation experiment was conducted to simulate the navigation path of the Yukun ship from the Bahamas Islands (approximate position: latitude 24°48.0′N, longitude 75°48.0′W) to the Strait of Gibraltar (approximate position: latitude 36°00.0′N, longitude 5°12.6′W). The main parameters of the Yukun ship are as follows.

[0018] Table 1 Main design parameters of "Yukun"

[0019] The initial ship heading was set to 070° during the simulation, and a Beaufort Scale 6 wind condition was set. To reduce the simulation time, the radius of the Earth was reduced by a factor of 1000. The simulation tested the performance of three navigation methods, including the great circle route guidance method proposed in this study, the LOS-based constant direction line guidance method, and the LOS-based approximate great circle segmented guidance method. The simulation results are shown in Figure 2. Figure 4 — Figure 7 shown.

[0020] Figure 4 The following is a comparison chart of the tracks of this method and other methods. Whether it is the great circle route guidance method designed by the present invention, the guidance method based on the LOS constant direction line, or the approximate great circle segmented guidance method based on LOS, they can all navigate stably along the planned route under the influence of wind speed of level 6 on the Beaufort scale.

[0021] Figure 5 The following figure compares the heading of this method with other methods. The great circle method begins to stably track the reference heading after approximately 100 seconds, and then the ship's heading gradually and evenly increases as the ship's position changes. The rhumb line method stably tracks the reference heading after approximately 100 seconds, and then the ship's heading remains stable. In contrast, the approximate great circle segmented method requires a heading adjustment for each segment, so the heading is continuously adjusted throughout the simulation.

[0022] Figure 6 The following figure compares the rudder angles of this method with those of other methods. The first 100 seconds of the great circle and rhumb line methods are primarily focused on the controller adjustment phase, resulting in significant rudder angle variations. After approximately 100 seconds, the controller begins to steadily track the reference signal, and the rudder angle stabilizes, with only minor adjustments made when subjected to high-frequency wave interference. However, the segmented method, which approximates a great circle, requires frequent course changes during each segment, requiring the vessel to make frequent adjustments using larger rudder angles throughout the simulation.

[0023] Figure 7 The following chart compares the track errors of this method and other methods. Throughout the entire voyage, the maximum track error of the LOS-based rhumb line guidance method is approximately 25 meters, while the maximum track error of the LOS-based approximate great circle segmented guidance method is approximately 40 meters. In contrast, the great circle route guidance and control strategy designed by the present invention achieves a maximum track error of only 6 meters throughout the entire voyage, demonstrating a significant advantage.

[0024] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A great circle route guidance and control method based on a Mercator chart, characterized by: The following steps are involved: Establish a nonlinear mathematical model of ship motion; Design guidance strategies for great circle routes to enable ships to navigate along the great circle routes according to the desired course; Based on the nonlinear mathematical model of ship motion, a third-order closed-loop gain shaping algorithm is used to design the final controller to control the ship to track the desired heading. The plan view of the great circle route during the ship's navigation process is converted into a Mercator projection nautical chart to obtain the ship's navigation trajectory along the great circle route under the Mercator projection.

2. The great circle route guidance and control method based on the Mercator chart according to claim 1, characterized in that: The process of establishing the nonlinear mathematical model of ship motion is as follows: When analyzing ship maneuvering, if only the drift speed is concerned, and bow angular velocity The linear equations corresponding to these two degrees of freedom for the ship are: (1) In this formula, Represents the rudder angle input, represent Taking the first derivative with respect to time, Respectively represent First derivative with respect to time; coefficient It is determined by the basic parameters of the ship. Formula (1) is converted into a simple equation describing the steering response to the bow roll, namely: (2) In this formula, Represents the rudder angle input, Indicates rudder angle input Taking the derivative with respect to time, Indicates the bow angular velocity Taking the second derivative with respect to time, All are manipulative indicators; Perform Laplace transform on Equation (2) and assume that the initial value Established, the following transfer function is obtained : (3) In this formula is the Laplace operator, represents the Laplace transform of the rudder angle input, represents the Laplace transform of the bow angular velocity, All are manipulative indicators; According to the relationship , and get the corresponding equation: (4) In formula (4), is the heading angle, is the Laplace operator, represents the Laplace transform of the rudder angle input, represents the Laplace transform of the heading angle, All are manipulative indicators; Simplify formula (4) and derive the following formula: (5) In the formula , is the Laplace operator, is the ship maneuverability index, and Equation (5) is written as the following differential equation (6) in, is the rudder angle, Represents the heading angle Taking the first derivative with respect to time, Represents the heading angle Taking the second derivative with respect to time, is the ship maneuverability index; Divide both sides of the equation (6) by have to: (7) in, is the rudder angle, Represents the heading angle Taking the first derivative with respect to time, Represents the heading angle Taking the second derivative with respect to time, is the ship maneuverability index; For a statically unstable ship, the second term on the left side of Equation (7) Need to be replaced by nonlinear terms Replace, and: (8) Thus, the nonlinear mathematical model of ship motion is established: (9) In formula (9): is the heading angle, is the rudder angle, Represents the heading angle Taking the first derivative with respect to time, Represents the heading angle Taking the second derivative with respect to time, is the ship maneuverability index, is a nonlinear parameter.

3. The great circle route guidance and control method based on the Mercator chart according to claim 1, characterized in that: The process of designing a great circle route guidance strategy to enable a ship to navigate along the great circle route according to a desired course is as follows: S31: Calculate the course and distance of a great circle route: Assume that the starting coordinates of the ship are , the end point coordinates are , use the spherical triangle formula to find the starting point coordinates The initial great circle heading at , the formula for calculating the great circle range and heading is: (10) (11) in, For great circle voyages, is the starting latitude, is the latitude of the arrival point, is the longitude difference between the starting point and the arrival point, is the initial great circle heading; S32: Calculate the coordinates of the point closest to the current ship position. The specific process is as follows: Assume the current ship position is , then the current ship position is The great circle distance between ,but: (12) (13) Where: is the longitude difference from the starting point to the current ship position; is the latitude of the current ship position; is the longitude of the current ship's position; is the great circle distance from the starting point to the current ship position; for Point relative to The great circle azimuth of the point, with the current position as Point, to the great circle route Draw a perpendicular line with the foot of , forming a spherical right triangle ,but: (14) (15) (16) Where: for arrive The vertical distance; is the great circle distance from the starting point to the current ship position; for arrive The great circle distance of a point; Two large arcs and The angle between for Point relative to The great circle azimuth of the point; is the initial great circle heading; The coordinates of a point and the great circle heading are calculated using the following formula: (17) (18) (19) (20) Where: is the initial great circle heading; for arrive The great circle distance of a point; Starting point to The longitude difference of a point; for The latitude of the point; yes The longitude of the point; is the latitude of the starting point; is the longitude of the starting point; is the latitude of the arrival point; is the longitude of the arrival point; for The great circle heading of the point; From the starting point to Great circle range of points; S33: Calculate the target point coordinates and the desired heading; Let the tracking target point be located at the perpendicular point Great circle distance ahead at point; (21) (22) (23) (24) Where: for point to The longitude difference of a point; for and The great circle distance between for The latitude of the point; yes The longitude of the point; for The latitude of the point; yes The longitude of the point; for The great circle heading of the point; for The great circle heading of the point, that is, the desired heading; is the latitude of the arrival point; is the longitude of the arrival point.

4. The great circle route guidance and control method based on the Mercator chart according to claim 1, characterized in that: The process of designing the final controller using a third-order closed-loop gain shaping algorithm based on the nonlinear mathematical model of ship motion is as follows: Assume that the bandwidth frequency of the closed-loop system of the ship is , then the compensation sensitivity function of the ship heading keeping control system is That is, the closed-loop transfer function of the control system is shown in formula (25): (25) The controller for: (26) From Equation (26), the controller is in the form of a typical PD controller connected in series with an oscillating link. Under the PD controller, a constant , the ship heading keeping control system will greatly reduce the adjustment time without changing the stability. Adding a positive constant to the proportional part of the PD link in Equation (26), the final controller is in the form of Equation (27): (27)。 5. The great circle route guidance and control method based on the Mercator chart according to claim 1, characterized in that: The process of converting the plan view of the great circle route during the ship's navigation process into a Mercator projection nautical chart to obtain the ship's navigation track along the great circle route under the Mercator projection is as follows: Plane coordinates of the planned route , based on the following formula, the plane coordinates of the planned great circle route are Convert to Mercator projection latitude and longitude using the following formula: (28) (29) in: is the radius of the Earth, is the horizontal coordinate of the rectangular coordinate on the plane, is the ordinate of the rectangular coordinate on the plane; is the Mercator projection coordinate longitude, is the Mercator projection coordinate latitude; The longitude and latitude obtained above are in radians and are converted according to the following formula: (30) (31) in, Represents Mercator projection coordinate longitude in radians, Represents the Mercator projection coordinate latitude in radians, represents the Mercator projection coordinate longitude in degrees, Represents Mercator projection coordinate latitude in degrees.

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