Solid sounding rocket drop point control guidance method

Abstract

The invention provides a solid sounding rocket landing point control guidance method which comprises the following steps: calculating a geocentric angle between a rocket depletion point and a target landing point, the speed of the rocket depletion point and the to-be-increased speed of the target landing point of the rocket according to current real-time parameters of the rocket; taking the target drop point of the rocket as a constraint, and calculating a theoretical program angle of the rocket according to the depletion point speed of the rocket and the to-be-accelerated speed of the target drop point of the rocket; the actual pneumatic influence and the earth rotation influence are considered, the theoretical program angle of the rocket is corrected, and the actual program angle of the rocket is obtained; and the actual program angle is issued, and guidance calculation is completed. According to the method, the theoretical program angle is iteratively calculated in real time on the rocket computer to complete flight control of the rocket, it is guaranteed that the drop point precision meets the requirement, the problems that a traditional perturbation guidance method is large in calculation amount and poor in adaptability are solved, and high-precision and low-calculation-complexity real-time guidance is achieved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of flight control of solid sounding rocket, in particular to a solid sounding rocket landing point control guidance method. BACKGROUND

[0002] Solid sounding rocket is a kind of rocket for flight test in near-earth space. In order to recover the scientific experimental equipment, the landing point accuracy of the rocket is generally better than 1km. In order to achieve high landing point control, real-time guidance calculation needs to be performed according to the estimated energy of the rocket and the landing point constraint in the active stage (when the solid rocket engine is working), so as to estimate the program angle that needs to be met at the moment of engine shutdown to control the landing point accuracy. After the engine is shut down, the free flight in the passive stage is performed to land at the specified landing point (nearby).

[0003] At present, the guidance method of the sounding rocket is mainly perturbation guidance method, which needs a large amount of calculation before the rocket is launched to cope with various disturbance factors encountered in the actual flight process.

[0004] At present, the guidance method of the rocket in the atmosphere is mainly perturbation guidance method, which needs a large amount of calculation before the launch to determine the perturbation guidance parameters to cope with various disturbance factors encountered in the actual flight process. However, due to the dramatic characteristics of the solid rocket engine and the influence of the aerodynamics of the rocket in the atmosphere, the parameter disturbance in the actual flight is large, so the parameter wide perturbation may occur, and the perturbation guidance mainly copes with small factor disturbance, so it is difficult to meet the landing point accuracy requirement by using the traditional perturbation guidance. SUMMARY

[0005] The purpose of the present application is to provide a solid sounding rocket landing point control guidance method, which iteratively calculates the theoretical program angle on the rocket computer in real time to complete the flight control of the rocket, ensures that the landing point accuracy meets the requirement, overcomes the problems of large calculation amount and poor adaptability of the traditional perturbation guidance method, and realizes real-time guidance with high accuracy and low calculation complexity.

[0006] In order to achieve the above purpose, the present application provides a solid sounding rocket landing point control guidance method, which comprises the following steps: calculating the geocentric angle between the rocket depletion point and the target landing point, the rocket depletion point speed and the to-be-increased speed of the rocket target landing point according to the current real-time parameters of the rocket; taking the target landing point of the rocket as a constraint, calculating the theoretical program angle of the rocket according to the rocket depletion point speed and the to-be-increased speed of the rocket target landing point; considering the actual aerodynamic influence and the influence of the earth rotation, correcting the theoretical program angle of the rocket to obtain the actual program angle of the rocket; and issuing the actual program angle to complete the guidance calculation.

[0007] The solid sounding rocket landing point control guidance method as claimed in claim 1, wherein the calculation of the geocentric angle between the rocket depletion point and the target landing point, the rocket depletion point speed and the to-be-increased speed of the rocket target landing point according to the current real-time parameters of the rocket comprises:

[0008] According to the current real-time parameters of the rocket, the estimated parameters of the rocket engine depletion shutdown moment are obtained by using a linear interpolation algorithm;

[0009] According to the estimated parameters of the rocket engine depletion shutdown moment and the current attitude angle data of the rocket, the position of the rocket engine shutdown moment and the rocket depletion point speed are calculated;

[0010] The remaining flight time obtained by the last iteration, and the position of the rocket engine shutdown moment and the target landing point position are used to calculate the geocentric angle between the rocket depletion point and the target landing point;

[0011] According to the geocentric angle between the rocket depletion point and the target landing point, and the rocket depletion point speed, the to-be-increased speed of the rocket target landing point is calculated.

[0012] The solid sounding rocket landing point control guidance method as claimed in claim 1, wherein the calculation formula of the position of the rocket engine shutdown moment is:

[0013]

[0014] Wherein, R off represents the position of the rocket engine shutdown moment; R1 represents the current position of the rocket; V1 represents the current speed of the rocket; tg represents the remaining working time of the rocket engine shutdown moment; ws represents the remaining visual position of the rocket engine shutdown moment; dws represents the remaining visual speed of the rocket engine shutdown moment; represents the pitch angle; ψ represents the yaw angle; g represents the gravity acceleration, and t represents the time.

[0015] The solid sounding rocket landing point control guidance method as claimed in claim 1, wherein the calculation formula of the rocket depletion point speed is:

[0016]

[0017] Wherein, V off represents the rocket depletion point speed; represents the pitch angle; ψ represents the yaw angle; dws represents the remaining visual speed of the rocket engine shutdown moment.

[0018] The solid sounding rocket landing point control guidance method as claimed in claim 1, wherein the calculation formula of the to-be-increased speed of the rocket target landing point is:

[0019]

[0020] Where δV represents the velocity to be increased at the rocket's target landing point; β represents the geocentric angle between the rocket's exhaustion point and the target landing point; V off θ represents the velocity at the point of rocket exhaustion; θ represents the angle between the local velocity vector and the local horizontal plane.

[0021] In the solid-propellant sounding rocket impact point control guidance method described above, the formula for calculating the angle θ between the local velocity vector and the local horizontal plane is as follows:

[0022]

[0023] Among them, R off Indicates the position of the rocket engine at the moment of shutdown; R c β represents the Earth's reference radius; β represents the geocentric angle between the rocket's exhaustion point and the target impact point.

[0024] The solid-propellant sounding rocket impact point control guidance method described above, wherein the target impact point of the rocket is used as a constraint, and the theoretical program angle of the rocket is calculated based on the rocket's exhaustion point velocity and the required acceleration at the target impact point, includes:

[0025] Calculate the remaining flight time for the current iteration based on the rocket's exhaustion point velocity;

[0026] The difference in remaining flight time obtained from two adjacent iterations is used as the convergence criterion to calculate the theoretical program angle.

[0027] The solid sounding rocket impact point control guidance method described above includes, in part, the theoretical program angles: theoretical pitch program angle and theoretical yaw program angle.

[0028] The formula for calculating the theoretical pitch program angle is as follows:

[0029]

[0030] in, δV represents the theoretical pitch program angle; δV represents the velocity increment required for the rocket to reach the target point; δV x The X-axis component representing the velocity to be increased at the rocket target's impact point; δV y The Y-axis component representing the velocity to be increased at the rocket target's impact point;

[0031] In the solid-propellant sounding rocket impact point control guidance method described above, the formula for calculating the theoretical yaw program angle is as follows:

[0032]

[0033] Where, ψ 0c Indicates the theoretical yaw program angle; δV z The Z-axis component represents the velocity to be increased at the rocket target's landing point.

[0034] In the solid-propellant sounding rocket impact point control guidance method described above, the formula for calculating the actual program angle is as follows:

[0035]

[0036] Ψ c =Ψ 0c +Ψ b ;

[0037] in, Indicates the actual pitch program angle; ψ c Indicates the actual yaw program angle; Represents the theoretical pitch program angle; ψ 0c Indicates the theoretical yaw program angle; and ψ b These are the pitch compensation angle and the yaw compensation angle, respectively.

[0038] The beneficial effects achieved by this application are as follows:

[0039] (1) This application calculates parameters such as the rocket's geocentric angle and velocity increment in real time, and combines them with a theoretical elliptical trajectory model to quickly generate program angle commands, thereby achieving high-precision landing point control.

[0040] (2) The entire guidance calculation process of this application is simple, requires less data for pre-launch calculation, and is less demanding on calculation.

[0041] There are no high requirements for stride length, and the requirements for onboard equipment are not high. Attached Figure Description

[0042] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this application. For those skilled in the art, other drawings can be obtained based on these drawings.

[0043] Figure 1 The flowchart of a solid sounding rocket impact point control and guidance method according to an embodiment of this application is as follows. Figure 1 .

[0044] Figure 2 The flowchart of a solid sounding rocket impact point control and guidance method according to an embodiment of this application is as follows. Figure 2 . Detailed Implementation

[0045] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without creative effort are within the scope of protection of this application.

[0046] like Figure 1 and 2 As shown, this application provides a solid sounding rocket impact point control guidance method, the method comprising:

[0047] Step S100: Based on the rocket's current real-time parameters, calculate the geocentric angle between the rocket's exhaustion point and the target landing point, the rocket's exhaustion point velocity, and the rocket's target landing point velocity.

[0048] like Figure 2 As shown, step S100 includes:

[0049] Step S110: Based on the current real-time parameters of the rocket, use a linear interpolation algorithm to obtain the estimated parameters of the rocket engine exhaustion shutdown time.

[0050] The estimated parameters for the moment when the rocket engines exhaust and shut down include: the rocket's remaining flight time, remaining apparent velocity, and apparent position. The rocket's current real-time parameters include its real-time position, velocity, and remaining flight time.

[0051] Based on the rocket's current real-time parameters, the remaining flight time, remaining apparent velocity, and apparent position are calculated using a linear interpolation algorithm. The linear interpolation algorithm is an existing algorithm and will not be described in detail here.

[0052] Specifically, during rocket flight, based on the rocket engine's apparent position, apparent velocity, and remaining operating time, interpolation calculations are performed on the real-time position, velocity, and remaining flight time obtained from rocket navigation calculations to obtain the remaining flight time, remaining apparent velocity, and remaining apparent position when the rocket engine shuts down. In other words, linear interpolation is performed using the rocket engine's apparent position, apparent velocity, and remaining operating time to calculate the real-time position, velocity, and flight time obtained from rocket navigation calculations, yielding the remaining flight time, apparent velocity increment, and remaining apparent position when the engine is exhausted and shut down.

[0053] Step S120: Calculate the position of the rocket engine shutdown moment and the rocket exhaustion point velocity based on the estimated parameters of the rocket engine exhaustion shutdown moment and the current attitude angle data of the rocket.

[0054] The rocket's current attitude angle data includes pitch, yaw, and roll. Pitch, yaw, and roll angles are respectively... ψ, γ.

[0055] The formula for calculating the position of the rocket engine at shutdown time is as follows:

[0056]

[0057] Among them, R off The rocket engine shuts down at its current position; R1 represents the rocket's current position; V1 represents the rocket's current velocity; tg represents the remaining operating time when the rocket engine shuts down; ws represents the remaining apparent position when the rocket engine shuts down; dws represents the remaining apparent velocity when the rocket engine shuts down. ψ represents the pitch angle; y represents the yaw angle. g represents gravitational acceleration, and t represents time, in seconds.

[0058] The formula for calculating the rocket's exhaustion point velocity is as follows:

[0059]

[0060] Among them, V off This indicates the point at which the rocket reaches its exhaustion velocity; ψ represents the pitch angle; y represents the yaw angle; dws represents the remaining apparent velocity when the rocket engine is shut down.

[0061] Step S130: Using the remaining flight time obtained from the previous iteration, the position of the rocket engine shutdown, and the target landing position, calculate the geocentric angle between the rocket exhaustion point and the target landing point.

[0062] The remaining flight time calculated in the first iteration is the initial value ts0, which is obtained by simulation calculation in the computer according to step S110.

[0063] The formula for calculating the geocentric angle between the rocket's exhaustion point and the target impact point is as follows:

[0064]

[0065] Where β represents the geocentric angle between the rocket exhaustion point and the target landing point; R1 represents the geocentric vector of the rocket at the moment the rocket engine shuts down (exhausts) (in the launch inertial coordinate system); and Rt represents the geocentric vector of the target landing point (in the launch inertial coordinate system).

[0066] Step S140: Calculate the required acceleration to meet the target landing point of the rocket based on the geocentric angle between the rocket exhaustion point and the target landing point, and the rocket exhaustion point velocity.

[0067] Based on the elliptical ballistics theory, and combining the geocentric angle β between the rocket exhaustion point and the target landing point with the rocket engine exhaustion point velocity, the magnitude and direction of the required acceleration of the rocket from the shutdown point to the target landing point are calculated to meet the landing point constraint. The elliptical ballistics theory is an ideal ballistics model that only considers the two-body motion condition.

[0068] The formula for calculating the velocity increment required for the rocket target impact point is as follows:

[0069]

[0070] Where δV represents the velocity to be increased at the rocket's target landing point; β represents the geocentric angle between the rocket's exhaustion point and the target landing point; V off θ represents the velocity at the point of rocket exhaustion; θ represents the angle between the local velocity vector and the local horizontal plane.

[0071] The formula for calculating the angle θ between the local velocity vector and the local horizontal plane is as follows:

[0072]

[0073] Among them, R off Indicates the position of the rocket engine at the moment of shutdown; R c β represents the Earth's reference radius; β represents the geocentric angle between the rocket's exhaustion point and the target impact point.

[0074] Step S200: Using the target landing point of the rocket as a constraint, calculate the theoretical program angle of the rocket based on the rocket exhaustion point velocity and the expected increase velocity at the target landing point.

[0075] The theoretical program angles of a rocket include: pitch program angle and yaw program angle.

[0076] like Figure 2 As shown, step S200 includes:

[0077] Step S210: Calculate the remaining flight time for the current iteration based on the rocket's exhaustion point velocity.

[0078] The remaining flight time in the current iteration is the remaining flight time ts from exhaustion and shutdown to landing. The formula for calculating ts is as follows:

[0079]

[0080] Where a, e, and μ are the semi-major axis, eccentricity, and Earth constant, respectively. V off The asterisk (*) represents the rocket's exhaust velocity; the asterisk (*) represents the multiplication symbol.

[0081] Step S220: Use the difference in remaining flight time obtained from two adjacent iterations as the convergence criterion to calculate the theoretical program angle.

[0082] Understandably, once the difference between the remaining flight times obtained from two consecutive iterations is less than or equal to a set threshold, the iteration stops, and the theoretical program angle is calculated (or "extracted") using the time interval (ts) obtained from the last iteration. The theoretical program angle is still derived from the inverse solution of the landing point constraints under the shutdown point state, but now the converged ts is used to determine this set of states. The final converged ts is used to inversely solve the velocity increment at the shutdown point, and then the theoretical program angle is calculated.

[0083] As a specific embodiment of the present invention, it is determined whether the difference between the remaining flight time calculated in two adjacent iterations meets the requirements. If so, the theoretical program angle is calculated; otherwise, the iterative calculation process is re-executed and the remaining flight time is recalculated until the difference between the remaining flight time calculated in two adjacent iterations meets the requirements.

[0084] Specifically, the ts used in step S130 is compared with the remaining flight time ts calculated in step S210. If the difference between the two is within the set range, the theoretical program angle of the acceleration to be increased obtained in step S140 can be calculated. Otherwise, the remaining flight time ts calculated in step S210 is substituted into step S130, and the calculations from step S130 to step S210 are iteratively calculated until the set range is met.

[0085] As a specific embodiment of the present invention, the remaining flight time calculated in two adjacent iterations is defined as the first remaining flight time and the second remaining flight time, respectively. The second remaining flight time is compared with the first remaining flight time. If the difference between the two exceeds a set range, iterative calculation is performed until the difference meets the set range.

[0086] As a specific embodiment of the present invention, the iterative calculation process is completed in real time in the onboard computer, and the calculation time for each iteration does not exceed 50ms.

[0087] The theoretical program angles include the theoretical pitch program angle and the theoretical yaw program angle. Within the launch inertial coordinate system, there are the X, Y, and Z axes. The launch inertial coordinate system is established at launch time, with its origin typically located at the rocket's launch point. The directions of the X, Y, and Z axes remain unchanged in inertial space after being determined at launch time.

[0088] The formula for calculating the theoretical pitch program angle is as follows:

[0089]

[0090] in, δV represents the theoretical pitch program angle; δV represents the velocity increment required for the rocket to reach the target point; δV x The X-axis component representing the velocity to be increased at the rocket target's impact point; δV yThe Y-axis component represents the velocity to be increased at the rocket target's landing point.

[0091] The formula for calculating the theoretical yaw procedure angle is as follows:

[0092]

[0093] Where, ψ 0c Indicates the theoretical yaw program angle; δV z The Z-axis component represents the velocity to be increased at the rocket target's landing point.

[0094] In step S300, considering the actual aerodynamic effects and the Earth's rotation, the theoretical program angle of the rocket is corrected to obtain the actual program angle of the rocket.

[0095] As a specific embodiment of the present invention, the theoretical program angle is compensated and corrected to obtain the actual program angle.

[0096] As a specific embodiment of the present invention, the theoretical program angle is compensated for by atmospheric disturbances and the influence of Earth's rotation based on empirical data to obtain the actual program angle, which is used for rocket attitude control.

[0097] As a specific embodiment of the present invention, the actual program angle includes the pitch program angle and the yaw program angle, which are used to control the pitch and yaw attitudes of the rocket, respectively.

[0098] Define pitch compensation angle and yaw compensation angle as follows: and ψ b .

[0099] The formula for calculating the actual program angle is as follows:

[0100]

[0101] ψ c =ψ 0c +ψ b ;

[0102] in, Indicates the actual pitch program angle; ψ c Indicates the actual yaw program angle; Represents the theoretical pitch program angle; ψ 0c This represents the theoretical yaw program angle.

[0103] Step S400: The actual program angle is sent out to complete the guidance calculation.

[0104] As a specific embodiment of the present invention, the actual program angle is output as a guidance command to the rocket control system to achieve flight control.

[0105] As a specific embodiment of the present invention, when the engine is about to run out of power, the real-time program angle update is stopped and the fixed axis control mode is entered.

[0106] As a specific embodiment of the present invention, after the engine is shut down, the rocket enters the passive phase of free flight and eventually lands near the target landing point.

[0107] The beneficial effects achieved by this application are as follows:

[0108] (1) This application calculates parameters such as the rocket's geocentric angle and velocity increment in real time, and combines them with a theoretical elliptical trajectory model to quickly generate program angle commands, thereby achieving high-precision landing point control.

[0109] (2) The entire guidance calculation process of this application is simple, the amount of data to be calculated before launch is small, and there are no high requirements for the calculation step size and the requirements for the on-rocket equipment are not high.

[0110] In the description of this application, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of the stated features. In the description of this application, "multiple" means two or more, unless otherwise explicitly specified.

[0111] In the description of this application, the word "for example" is used to mean "used as an example, illustration, or description." Any embodiment described as "for example" in this application is not necessarily to be construed as being more preferred or advantageous than other embodiments. The following description is provided to enable any person skilled in the art to make and use the invention. Details are set forth in the following description for purposes of explanation. It should be understood that those skilled in the art will recognize that the invention can be made without using these specific details. In other instances, well-known structures and processes will not be described in detail to avoid obscuring the description of the invention with unnecessary detail. Therefore, the invention is not intended to be limited to the embodiments shown, but is consistent with the broadest scope of the principles and features disclosed in this application.

[0112] The above description is merely an embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principle of the present invention should be included within the scope of the claims of the present invention.

Claims

1. A method for controlling the landing point of a solid-propellant sounding rocket, characterized in that, The method includes the following steps: Based on the rocket's current real-time parameters, calculate the geocentric angle between the rocket's exhaustion point and the target landing point, the rocket's exhaustion point velocity, and the rocket's expected velocity at the target landing point; Using the rocket's target landing point as a constraint, the theoretical program angle of the rocket is calculated based on the rocket's exhaustion point velocity and the expected acceleration at the target landing point. Taking into account the actual aerodynamic effects and the Earth's rotation, the theoretical program angle of the rocket is corrected to obtain the actual program angle of the rocket; The actual program angle is issued to complete the guidance calculation.

2. The solid-propellant sounding rocket impact point control and guidance method according to claim 1, characterized in that, Based on the rocket's current real-time parameters, the calculation of the geocentric angle between the rocket's exhaustion point and the target impact point, the rocket's exhaustion point velocity, and the rocket's expected velocity at the target impact point includes: Based on the rocket's current real-time parameters, the estimated parameters for the rocket engine's exhaustion and shutdown time are obtained using a linear interpolation algorithm; Based on the estimated parameters of the rocket engine's exhaustion shutdown moment and the rocket's current attitude angle data, calculate the location of the rocket engine's shutdown moment and the rocket's exhaustion point velocity; Using the remaining flight time obtained from the previous iteration, as well as the location of the rocket engine shutdown and the target landing point, calculate the geocentric angle between the rocket exhaustion point and the target landing point; Based on the geocentric angle between the rocket's exhaustion point and the target landing point, and the rocket's velocity at the exhaustion point, calculate the required acceleration to meet the target landing point.

3. The solid-propellant sounding rocket impact point control and guidance method according to claim 2, characterized in that, in, The formula for calculating the position at which the rocket engine shuts down is: Among them, R off The rocket engine shuts down at its current position; R1 represents the rocket's current position; V1 represents the rocket's current velocity; tg represents the remaining operating time when the rocket engine shuts down; ws represents the remaining apparent position when the rocket engine shuts down; dws represents the remaining apparent velocity when the rocket engine shuts down. ψ represents the pitch angle; g represents the gravitational acceleration; and t represents time.

4. The solid-propellant sounding rocket impact point control and guidance method according to claim 2, characterized in that, The formula for calculating the rocket's exhaustion point velocity is: Among them, V off This indicates the rocket's exhaust velocity. ψ represents the pitch angle; y represents the yaw angle; dws represents the remaining apparent velocity when the rocket engine is shut down.

5. The solid-propellant sounding rocket impact point control and guidance method according to claim 2, characterized in that, The formula for calculating the velocity increment required for the rocket target's impact point is: Where δV represents the velocity to be increased at the rocket's target landing point; β represents the geocentric angle between the rocket's exhaustion point and the target landing point; V off θ represents the velocity at the point of rocket exhaustion; θ represents the angle between the local velocity vector and the local horizontal plane.

6. The solid-propellant sounding rocket impact point control and guidance method according to claim 5, characterized in that, The formula for calculating the angle θ between the local velocity vector and the local horizontal plane is: Among them, R off Indicates the position of the rocket engine at the moment of shutdown; R c β represents the Earth's reference radius; β represents the geocentric angle between the rocket's exhaustion point and the target impact point.

7. The solid-propellant sounding rocket impact point control and guidance method according to claim 1, characterized in that, Using the rocket's target landing point as a constraint, and based on the rocket's exhaustion velocity and the expected velocity increase at the target landing point, the theoretical program angle of the rocket is calculated, including: Calculate the remaining flight time for the current iteration based on the rocket's exhaustion point velocity; The difference in remaining flight time obtained from two adjacent iterations is used as the convergence criterion to calculate the theoretical program angle.

8. The solid-propellant sounding rocket impact point control and guidance method according to claim 7, characterized in that, Theoretical program angles include: theoretical pitch program angle and theoretical yaw program angle; The formula for calculating the theoretical pitch program angle is as follows: in, δV represents the theoretical pitch program angle; δV represents the velocity increment required for the rocket to reach the target point; δV x The X-axis component representing the velocity to be increased at the rocket target's impact point; δV y The Y-axis component represents the velocity to be increased at the rocket target's landing point.

9. The solid-propellant sounding rocket impact point control and guidance method according to claim 8, characterized in that, in, The formula for calculating the theoretical yaw procedure angle is: Among them, Ψ 0c Indicates the theoretical yaw program angle; δV z The Z-axis component represents the velocity to be increased at the rocket target's landing point.

10. The solid-propellant sounding rocket impact point control and guidance method according to claim 9, characterized in that, in, The formula for calculating the actual program angle is: P c =Ψ 0c +Ψ b ; in, Indicates the actual pitch program angle; Ψ c Indicates the actual yaw program angle; Indicates the theoretical pitch program angle; Ψ 0c Indicates the theoretical yaw program angle; and Ψ b These are the pitch compensation angle and the yaw compensation angle, respectively.

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