A multi-copter wind field estimation method

By establishing dynamic equations based on the UAV's own state information, the airspeed direction and magnitude are decoupled and calculated, solving the problem that multi-rotor UAV wind field measurement relies on dedicated sensors. This enables low-cost, high-precision real-time wind field estimation and improves flight safety.

CN120688407BActive Publication Date: 2025-11-18杭州迅蚁网络科技有限公司
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Patent Information

Application Number
CN202511186856.7
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2025-08-25
Publication Date
2025-11-18
Estimated Expiration
2045-08-25

AI Technical Summary

Technical Problem

Existing wind field measurement methods for multi-rotor UAVs rely on dedicated wind speed sensors, which increases weight and cost and makes it difficult to achieve real-time airborne estimation.

Method used

By utilizing the UAV's own state information, including IMU, GPS, and motor controller, a dynamic equation is established to decouple the calculation of airspeed direction and magnitude, and a wind field estimation method is constructed. This avoids the direct use of complex motor models and relies solely on standard sensors for real-time wind field estimation.

Benefits of technology

It achieves low-cost, real-time, and high-precision wind field estimation, reduces hardware costs and power consumption, is suitable for embedded system implementation, and provides a basis for flight safety decision-making.

✦ Generated by Eureka AI based on patent content.

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Abstract

The application discloses a kind of multi-rotor wind field estimation method, it is related to unmanned vehicle technical field, including the following steps;S1, parameter definition and data acquisition;S2, real-time quality estimation of unmanned vehicle complete flight;S3, establish dynamics equation;S3-1, calculate horizontal thrust component;S3-2, calculate airspeed direction angle Δ a ;S3-3, calculate air resistance D;S3-4, calculate true airspeed V a ;S3-5, calculate wind speed component;S4, basic wind field estimation;S5, with take-off phase attitude error compensation estimation;S6, with resistance coefficient C D Calibration high-precision estimation.The multi-rotor wind field estimation method mainly relies on the sensors IMU, GPS, motor controller and the like already existing in the unmanned vehicle itself, the algorithm calculation steps are clear, suitable for embedded system implementation, and the estimation result has good reference value.
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Description

Technical Field

[0001] This invention relates to the field of unmanned aerial vehicle (UAV) technology, specifically to a method for estimating wind fields in multi-rotor aircraft. Background Technology

[0002] When multi-rotor drones perform tasks such as aerial photography, inspection, and logistics, their flight performance, trajectory tracking accuracy, and flight safety are significantly affected by the environmental wind field. Accurate wind field information is crucial for the autonomous control, navigation, and mission planning of drones.

[0003] Existing wind field measurement methods typically rely on mounting dedicated wind speed sensors (such as pitot tubes, anemometers, etc.) on drones. These sensors increase the weight, power consumption, and cost of the drones and may affect their aerodynamic characteristics. Furthermore, some methods based on complex airflow simulations are computationally intensive, making it difficult to achieve real-time airborne estimation.

[0004] Therefore, developing a method that does not rely on or relies less on dedicated wind speed sensors and can use the UAV's own state information for real-time and accurate wind field estimation has important practical application value. Summary of the Invention

[0005] The purpose of this invention is to provide a multi-rotor wind field estimation method to solve the problems in the background art where wind field measurement methods mainly rely on wind speed sensors, leading to increased weight and cost of UAVs and difficulty in achieving real-time airborne estimation.

[0006] To achieve the above objectives, the present invention provides the following technical solution: a multi-rotor wind field estimation method, including S1, parameter definition and data acquisition;

[0007] S2. Real-time quality estimation of the entire flight of the UAV;

[0008] S3. Establish the dynamic equations;

[0009] S3-1, Calculate the horizontal thrust components;

[0010] S3-2, Calculate the airspeed direction angle (Δ) a );

[0011] S3-3, Calculate air resistance (D);

[0012] S3-4. Calculate vacuum velocity (V) a );

[0013] S3-5, Calculate the wind speed components;

[0014] S4. Basic wind field estimation;

[0015] S5, with takeoff phase attitude error compensation estimation;

[0016] S6, with drag coefficient (C) D High-precision estimation calibrated.

[0017] Furthermore, the real-time state parameters in S1 include: aircraft mass m, gravitational acceleration g, total engine thrust T, the angle β (thrust tilt angle) between the thrust vector and the vertical direction, the angle Δ between the thrust projection on the horizontal plane and geographic north, and the acceleration components of the aircraft along the x, y, and z axes in the body coordinate system. x , y , z (or related terms calculated from velocity changes), the aircraft's ground speed vector (including the northward component of ground speed V) gN and the eastward component V gE (This information is provided by GPS or an inertial navigation system (INS).)

[0018] Furthermore, the complete flight of the UAV in S2 includes ground preparation, takeoff and climb, en route flight, landing and descent.

[0019] Furthermore, the real-time mass estimation in S2 includes estimating the total thrust T through the motor controller output during the UAV's uniform climb phase, and then calculating the aircraft mass m from the total thrust. By processing the "total rotor thrust T," model uncertainties are avoided. The total thrust T of a multi-rotor is a difficult quantity to model accurately, as it is related to various factors such as motor efficiency, battery voltage, air density, and propeller aging. Directly inferring the thrust from the motor PWM would introduce huge errors. Instead of directly using the motor model to calculate the thrust T, the algorithm indirectly and in real-time solves the total thrust T at the current moment using the vertical dynamic equation (T=(mg - maz) / cosβ). This solved T is a "result quantity" that already includes all factors affecting the thrust, making it more accurate than calculating it from the "cause" (motor speed). In this way, the algorithm successfully avoids the problem of complex and imprecise modeling of the rotor power system, greatly improving the algorithm's robustness and actual accuracy. This is a significant advantage over many other methods that rely on precise motor models.

[0020] Furthermore, the dynamic equations established in S3 include vertical dynamic equations, horizontal X-axis dynamic equations (body axis, along the horizontal projection direction of the thrust), and horizontal Y-axis dynamic equations (body axis, perpendicular to the horizontal projection direction of the thrust); direction first: by dividing the two horizontal dynamic equations, the unknown quantity "drag D" is cleverly eliminated, thereby directly and analytically (rather than iteratively) solving for the direction angle Δ of "airspeed". aThis is computationally extremely efficient and avoids the potential convergence problem of iterative algorithms; Magnitude post-calculation: After accurately obtaining the airspeed direction, it is substituted back into any horizontal dynamic equation to calculate the magnitude of "drag D", and finally, the magnitude of "airspeed" V is calculated using the drag formula. a The decoupling approach of "first determining the direction, then the magnitude" breaks down a complex multivariate coupled problem into two independent, simple problems that can be directly solved. This is the core ingenuity and creativity of this method in terms of mathematical processing.

[0021] Furthermore, in S3-2 and S3-3, the airspeed direction angle Δ a Let be the angle between the airspeed direction (the direction of the aircraft's velocity vector relative to the air) and the geographic north direction. Let D be the air resistance experienced by the aircraft.

[0022] Furthermore, the S4 basic wind field estimation includes the system acquiring attitude angles (roll, pitch, yaw) via IMU, acquiring three-axis acceleration via accelerometer, and acquiring ground speed and geographic heading via GPS. The calculation module calculates the thrust tilt angle and the angle Δ between the thrust horizontal projection and the north direction based on the attitude angle and heading angle. The core wind field estimation module executes this, outputting the total wind speed W and wind direction. This constructs a complete, low-cost, and practical system solution, forming a closed-loop technical solution.

[0023] The input is explicit: it only uses data from standard drone sensors such as IMU, GPS, and attitude calculation modules.

[0024] The process is clear: it follows a clear logical sequence of "acquiring data → calculating the direction and angle of the pull force → decoupling and calculating the airspeed vector → applying the wind triangle to calculate the wind speed".

[0025] Highly practical: It is a pure software algorithm that can be seamlessly integrated into existing flight control systems as a firmware module without any hardware modifications, resulting in extremely low implementation costs and high commercial promotion value.

[0026] Furthermore, the estimation with attitude error compensation in S5 includes adding attitude dead zone compensation during hovering / takeoff and landing. The attitude error compensation estimation during takeoff is not an estimation, but a logical constraint. In the absence of horizontal speed and with a small angle of the aircraft attitude, the output result of the wind estimation algorithm is directly made to be 0 (equivalent to the algorithm outputting 0 within the dead zone and the algorithm outputting the estimation result normally outside the dead zone).

[0027] Furthermore, S6 includes a drag coefficient C. D The high-precision estimation of the calibration includes optimizing the accuracy of the aerodynamic database through real-time flight data and through CFD simulation.

[0028] Compared with the prior art, the beneficial effects of the present invention are:

[0029] 1. Core Innovation: The most critical innovation is the decoupled analytical calculation method for the direction and magnitude of "airspeed". Existing technologies usually require iterative solutions to complex nonlinear equations or estimations using methods such as Kalman filtering.

[0030] Direction First: By dividing the two horizontal dynamic equations, the unknown "drag D" is cleverly eliminated, thus allowing the direction angle Δ of "airspeed" to be solved directly and analytically (rather than iteratively). a This is extremely efficient in terms of computation and avoids the problem of iterative algorithms not converging.

[0031] Post-calculation of magnitude: After accurately obtaining the airspeed direction, substitute it back into any horizontal dynamics equation to calculate the magnitude of "drag D", and finally calculate the magnitude of "airspeed V" using the drag formula. a The decoupling approach of "first determining the direction, then the magnitude" breaks down a complex multivariate coupled problem into two independent, simple problems that can be directly solved. This is the core ingenuity and creativity of this method in terms of mathematical processing.

[0032] 2. By processing the "total rotor thrust T," model uncertainties are avoided. The total thrust T of a multi-rotor is a difficult quantity to model accurately, as it is related to various factors such as motor efficiency, battery voltage, air density, and propeller aging. Directly inferring the thrust from the motor PWM would introduce significant errors. Instead of directly using the motor model to calculate the thrust T, the algorithm indirectly and in real-time solves the total thrust T at the current moment using the vertical dynamic equation (T = (mg - maz) / cosβ). This solved T is a "result quantity" that already includes all factors affecting the thrust, making it more accurate than calculating it from the "cause" (motor speed). In this way, the algorithm successfully avoids the problem of complex and imprecise modeling of the rotor power system, greatly improving the algorithm's robustness and practical accuracy. This is a significant advantage over many other methods that rely on precise motor models.

[0033] 3. A complete, low-cost, and practical systemic solution has been developed, forming a closed-loop technical solution:

[0034] The input is explicit: it only uses data from standard drone sensors such as IMU, GPS, and attitude calculation modules.

[0035] The process is clear: it follows a clear logical sequence of "acquiring data → calculating the direction and angle of the pull force → decoupling and calculating the airspeed vector → applying the wind triangle to calculate the wind speed".

[0036] Highly practical: It is a pure software algorithm that can be seamlessly integrated into existing flight control systems as a firmware module without any hardware modifications, resulting in extremely low implementation costs and high commercial promotion value.

[0037] This invention primarily relies on the UAV's existing sensors (IMU, GPS, motor controller, etc.), reducing additional hardware costs, weight, and power consumption. The algorithm's calculation steps are clear, making it suitable for embedded system implementation and enabling real-time wind field estimation. Especially during level flight or when wind speeds are high, the estimation results have good reference value and can provide a basis for flight safety decisions (such as determining whether to continue flying or whether wind speed limits have been exceeded). By introducing improvements such as real-time mass estimation, more accurate calibration, and modeling, the estimation accuracy and robustness can be further enhanced. Attached Figure Description

[0038] Figure 1 This is a schematic diagram of the workflow of the present invention. Detailed Implementation

[0039] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0040] Example 1: As Figure 1 The technical solution shown establishes the dynamic projection equation of the UAV in the body coordinate system in the plane perpendicular to the total thrust vector of the UAV (i.e., the "tangential normal direction"), and assumes that the wind mainly acts in the horizontal plane. The horizontal wind speed and wind direction are estimated by back-calculating the aerodynamic forces acting on the aircraft, including S1, parameter definition and data acquisition.

[0041] The real-time state parameters in S1 include: aircraft mass m, gravitational acceleration g, total engine thrust T, the angle β (thrust tilt angle) between the thrust vector and the vertical direction, the angle Δ between the thrust projection on the horizontal plane and geographic north, and the acceleration components of the aircraft along the x, y, and z axes in the body coordinate system. x , y , z (or related terms calculated from velocity changes), the aircraft's ground speed vector (including the northward component of ground speed V) gN and the eastward component V gE (This information is provided by GPS or an inertial navigation system (INS).)

[0042] By processing the "total rotor thrust T," model uncertainties are avoided. The total thrust T of a multi-rotor is a difficult quantity to model accurately, as it is related to various factors such as motor efficiency, battery voltage, air density, and propeller aging. Directly inferring the thrust from the motor PWM would introduce significant errors. Instead of directly using the motor model to calculate the thrust T, the algorithm indirectly and in real-time solves the total thrust T at the current moment using the vertical dynamic equation (T = (mg - maz) / cosβ). This inversely solved T is a "result quantity" that already includes all factors affecting the thrust, making it more accurate than calculating it from the "cause" (motor speed). In this way, the algorithm successfully avoids the problem of complex and imprecise modeling of the rotor power system, greatly improving the algorithm's robustness and practical accuracy. This is a significant advantage over many other methods that rely on precise motor models.

[0043] S2, Real-time quality estimation of the entire flight of the UAV.

[0044] The complete flight of the UAV in S2 includes ground preparation, takeoff and climb, en route flight, landing, and descent. Real-time quality estimation occurs during the takeoff and climb phase, during which the UAV typically undergoes accelerated climb (lifting off the ground), constant speed climb, and deceleration climb (approaching the en route). During the constant speed climb, the aircraft usually maintains a stable attitude and speed. The total thrust of the aircraft over a certain period during this phase is statistically analyzed, and then... This allows us to estimate the current mass of the aircraft. It should be noted that this is based on the assumption that the aircraft has a stable attitude and speed. In actual flight environments, various external disturbances often do not meet this assumption, so attitude and thrust compensation are required based on the actual situation.

[0045] S3. Establish the dynamic equations.

[0046] Based on the following assumptions: wind speed is mainly in the horizontal plane, and the aerodynamic lift generated by the aircraft is ignored (or its influence is considered small under certain conditions).

[0047] The dynamic equations, projected onto the fuselage axis in a plane perpendicular to the total thrust direction, are as follows:

[0048] 1. Dynamic equations in the vertical direction.

[0049]

[0050] 2. Horizontal X-direction dynamic equation (body axis, along the horizontal projection direction of the thrust).

[0051]

[0052] 3. Dynamic equations in the horizontal Y direction (body axis, perpendicular to the horizontal projection direction of the thrust).

[0053]

[0054] In S3-2 and S3-3, the airspeed direction angle Δa is the angle between the airspeed direction (the direction of the velocity vector of the aircraft relative to the air) and the geographic north direction, and the air resistance D is the air resistance experienced by the aircraft.

[0055] S3-1. Calculate the horizontal thrust component.

[0056] From the dynamic equations in the vertical direction, we can obtain

[0057] Then the horizontal thrust component Represented as:

[0058] S3-2. Calculate the airspeed direction angle (Δa).

[0059] Substitute the expression obtained from the above steps into the horizontal dynamics equation to calculate the tangent of the airspeed direction angle (Δa): Δa is obtained by calculating the arctangent function.

[0060] S3-3, Calculate air resistance (D).

[0061] Using the data obtained from the above steps and the horizontal dynamic equations (e.g., the X-direction equations), calculate the air resistance D:

[0062]

[0063] S3-4. Calculate the vacuum velocity (Va).

[0064] air resistance The physical equation is ,in air density, For reference area, The drag coefficient is the coefficient of performance in this invention. It is obtained from a pre-established aerodynamic database that stores different tension tilt angles. The corresponding value.

[0065] Based on the calculation and obtained from table lookup Inversely calculate the vacuum velocity :

[0066]

[0067] S3-5, Calculate the wind speed component.

[0068] According to the wind triangle relationship ( ), calculate the wind speed vector:

[0069] Northward component of airspeed .

[0070] Northward component of airspeed .

[0071] Northward component of wind speed .

[0072] Northward component of wind speed .

[0073] Total wind speed The wind direction angle can be determined by The calculation yields the following (in meteorology, the wind direction angle is usually defined as the direction from which the wind blows, but the actual algorithm outputs the direction from which the wind blows, so a conversion is needed).

[0074] Example 2: Figure 1 The technical solution shown in the figure estimates the total thrust of the UAV during the uniform climb phase by outputting the motor controller, and then calculates the mass of the aircraft from the total thrust. In order to avoid the influence of excessive error in a single or several back-calculation results, the average value of the back-calculation results over a continuous period of time can be used as the estimated mass.

[0075] S4. Basic wind field estimation;

[0076] i. The system obtains attitude angles (roll, pitch, yaw) through IMU, three-axis acceleration through accelerometer, and ground speed and geographic heading through GPS.

[0077] ii. The calculation module calculates the thrust tilt angle based on the attitude angle and heading angle. and the angle between the horizontal projection of the tension force and the north direction .

[0078] iii. Execution of technical solutions for the core module of wind field estimation, specifically iii~viii:

[0079] 1. Calculation .

[0080] 2. Calculation And thus obtain .

[0081] 3. Calculation .

[0082] 4. From the pre-stored pneumatic database, based on the current... Get from table Assuming air density and reference area Known.

[0083] 5. Calculation .

[0084] 6. Calculate the wind speed components and .

[0085] 7. Output total wind speed And wind direction.

[0086] S5, with takeoff phase attitude error compensation estimation;

[0087] Based on real-time quality estimation and S4 basic wind field estimation, attitude dead zone compensation is added for hovering / take-off and landing phases.

[0088] When the drone is under the above conditions, the tilt in a certain attitude direction is small. This tilt may be due to sensor zero-position error, installation offset, etc. The wind speed calculated from this small angle value has low reliability. It can be considered that the wind speed is 0 at this time, which increases the stability of the algorithm output under this condition. The attitude error compensation estimation during the takeoff stage is not an estimation, but a logical constraint. Under the condition of no horizontal speed and small body attitude, the output result of the wind estimation algorithm is directly made to be 0 (equivalent to the algorithm outputting 0 within the dead zone and the algorithm outputting the estimation result normally outside the dead zone).

[0089] S6, with drag coefficient (C) D High-precision estimation calibrated;

[0090] It can be calibrated in the following ways To improve the accuracy of the algorithm's estimation results:

[0091] Before leaving the factory or during routine maintenance, high-precision airspeed meters are temporarily installed on drones to optimize the accuracy of the aerodynamic database using real-time flight data and CFD simulation.

[0092] This invention primarily relies on the UAV's existing sensors (IMU, GPS, motor controller, etc.), reducing additional hardware costs, weight, and power consumption. The algorithm's calculation steps are clear, making it suitable for embedded system implementation and enabling real-time wind field estimation. Especially during level flight or when wind speeds are high, the estimation results have good reference value and can provide a basis for flight safety decisions (such as determining whether to continue flying or whether wind speed limits have been exceeded). By introducing improvements such as real-time mass estimation, more accurate calibration, and modeling, the estimation accuracy and robustness can be further enhanced.

[0093] Core innovation: The most critical innovation is the decoupled analytical calculation method for the direction and magnitude of "airspeed". Existing technologies usually require iterative solutions to complex nonlinear equations or estimation using methods such as Kalman filtering.

[0094] Direction First: By dividing the two horizontal dynamic equations, the unknown "drag D" is cleverly eliminated, thus allowing the direction angle Δ of "airspeed" to be solved directly and analytically (rather than iteratively). a This is extremely efficient in terms of computation and avoids the problem of iterative algorithms not converging.

[0095] Post-calculation of magnitude: After accurately obtaining the airspeed direction, substitute it back into any horizontal dynamics equation to calculate the magnitude of "drag D", and finally calculate the magnitude of "airspeed V" using the drag formula. a The decoupling approach of "first determining the direction, then the magnitude" breaks down a complex multivariate coupled problem into two independent, simple problems that can be directly solved. This is the core ingenuity and creativity of this method in terms of mathematical processing.

[0096] A complete, low-cost, and practical systemic solution has been developed, forming a closed-loop technical solution:

[0097] The input is explicit: it only uses data from standard drone sensors such as IMU, GPS, and attitude calculation modules.

[0098] The process is clear: it follows a clear logical sequence of "acquiring data → calculating the direction and angle of the pull force → decoupling and calculating the airspeed vector → applying the wind triangle to calculate the wind speed".

[0099] Highly practical: It is a pure software algorithm that can be seamlessly integrated into existing flight control systems as a firmware module without any hardware modifications, resulting in extremely low implementation costs and high commercial promotion value.

[0100] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for estimating wind field in a multi-rotor aircraft, characterized in that: Includes the following steps; S1. Parameter definition and data acquisition; S2. Real-time quality estimation of the entire flight of the UAV; S3. Establish the dynamic equations; S3-1. Calculate the horizontal thrust components: ; S3-2, Calculate the airspeed direction angle Δ a : And thus obtain ; S3-3, Calculate air resistance D: ; S3-4, Calculate vacuum velocity V a : From the pre-stored aerodynamic database, based on the current... Get from table Assuming air density and reference area Known; S3-5, Calculate the wind speed component: and ; S4. Basic wind field estimation, output total wind speed. Wind direction and total wind speed The wind direction angle is from Calculated; S5, with attitude error compensation estimation during takeoff, adds attitude dead zone compensation during hovering or takeoff and landing based on real-time mass estimation and S4 basic wind field estimation. S6, with drag coefficient C D The high-precision estimation is calibrated using the following methods. To improve the accuracy of algorithm estimation results: Before leaving the factory or during regular maintenance, high-precision airspeed meters are temporarily installed on the drones. The accuracy of the aerodynamic database is optimized by using real-time flight data and by using CFD simulation. The real-time state parameters in the above formula include: Aircraft mass m, gravitational acceleration g, total engine thrust T, angle β between the thrust vector and the vertical direction, angle Δ between the thrust projection onto the horizontal plane and geographic north, and acceleration components of the aircraft along the x, y, and z axes in the body coordinate system. x , y , z ; The aircraft's ground speed vector includes the northward component of the ground speed V. gN and the eastward component V gE Provided by GPS or Inertial Navigation System (INS).

2. The multi-rotor wind field estimation method according to claim 1, characterized in that: The complete flight of the UAV in S2 includes ground preparation, takeoff and climb, en route flight, landing and descent.

3. The multi-rotor wind field estimation method according to claim 2, characterized in that: The real-time mass estimation in S2 includes estimating the total thrust T through the output of the motor controller during the uniform climb phase of the UAV, and then calculating the mass m of the aircraft from the total thrust.

4. The multi-rotor wind field estimation method according to claim 3, characterized in that: The dynamic equations established in S3 include the dynamic equations in the vertical direction. Horizontal X-direction dynamic equation: body axis, along the horizontal projection direction of the thrust; and horizontal Y-direction dynamic equation: body axis, perpendicular to the horizontal projection direction of the thrust. Specifically, assuming the wind speed is primarily in the horizontal plane and the aerodynamic lift generated by the aircraft is neglected, the dynamic equations projected onto the aircraft axis in a plane perpendicular to the direction of total thrust are as follows: (1) Dynamic equations in the vertical direction ; (2) Dynamic equation in the horizontal X direction: Body axis, along the horizontal projection direction of the thrust. ; (3) Dynamic equation in the horizontal Y direction: body axis, perpendicular to the horizontal projection direction of the thrust. 。 5. The multi-rotor wind field estimation method according to claim 4, characterized in that: The airspeed direction angle Δ in S3-2 and S3-3 a The airspeed direction of an aircraft is specifically the angle between the direction of its velocity vector relative to the air and the geographic north direction. Air resistance D is the air resistance experienced by the aircraft.

6. The multi-rotor wind field estimation method according to claim 5, characterized in that: The S4 basic wind field estimation includes the system acquiring attitude angles (roll, pitch, yaw) through IMU, acquiring three-axis acceleration through accelerometer, and acquiring ground speed and geographical heading through GPS. The calculation module calculates the tension tilt angle and the angle Δ between the tension horizontal projection and the north direction based on the attitude angle and heading angle. The core module of wind field estimation is executed, and the total wind speed W and wind direction are output.

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