A rocket engine thrust vector swing angle measurement solving method, device and medium

Abstract

The present application belongs to the field of space vehicle and missile flight control technology, and particularly relates to a rocket engine thrust vector swing angle measurement solving method, device and medium. The method comprises: constructing a space vector model containing an axial bias parameter, converting equation solving into an unconstrained nonlinear least squares optimization problem, and solving by using a numerical differential assisted gradient descent method to obtain mathematical model parameters. The present application quickly and accurately inversely solves the real swing angle of the rocket engine by using a numerical optimization method with low calculation power consumption.

Description

Technical Field

[0001] This invention belongs to the field of aerospace launch vehicle and missile flight control technology, and particularly relates to a method, equipment and medium for measuring and calculating the thrust vector swing angle of a rocket engine. Background Technology

[0002] In the flight control systems of launch vehicles, strategic missiles, and hypersonic vehicles (hereinafter, we will only use rockets as an example), rocket engines typically employ bidirectional oscillating thrust vector calculation technology to precisely adjust their flight attitude. Two servo actuators, orthogonally mounted on the engine, are usually used as actuators to drive the engine oscillation. However, during engine operation, nonlinear factors such as deformation of the mounting points and mounting gaps cause discrepancies between the actual engine nozzle oscillation angle and the servo actuator commands. During the development of the control system, it is necessary to obtain the actual deflection angle of the engine nozzle relative to the rocket body (i.e., the thrust vector oscillation angle) to analyze the time-domain and frequency-domain dynamic performance of the actuators.

[0003] Current pitch angle measurement technologies are mainly divided into two categories: "direct measurement methods" and "indirect measurement methods." Direct measurement methods typically involve directly mounting angular displacement sensors (such as rotary potentiometers, rotary transformers, or photoelectric encoders) on the rotating shaft system of the engine universal joint (such as a cross shaft or ball joint). While this method can directly read the angle value, the universal joint is located near the engine throat, operating in an extremely harsh environment (high-temperature radiation, high-frequency severe vibration, structural deformation). Furthermore, the compact structure of the universal joint often makes it difficult to reserve installation space for the sensor, and it also introduces significant measurement errors. Currently, indirect measurement methods are more widely used in engineering. These methods utilize linear displacement sensors (such as LVDTs or magnetostrictive sensors) mounted on the opposite side of the servo actuator to measure the extension or retraction length of the actuator or sensor, and then use approximate algorithms to inversely solve for the corresponding engine nozzle pitch angle. Although indirect measurement methods avoid harsh environments, existing calculation algorithms face significant challenges when dealing with the complex layouts of new types of vehicles. Traditional measurement and calculation algorithms are typically based on simplified assumptions of "planar trigonometric geometry" or "coplanar mounting." This means assuming that the upper fulcrum of the linear sensor (on the frame side) and the rotation center of the engine universal joint lie in the same cross-sectional plane (with an axial height difference of 0). Under this assumption, the motion relationship between the sensor and the nozzle is simplified to a solution using the cosine theorem for a triangle in a two-dimensional plane.

[0004] However, due to the limited space in the engine compartment and the layout of the load-bearing structure, the upper support point of the sensor often cannot be arranged within the cross-section of the rocket body passing through the center of the universal joint (perpendicular to the longitudinal axis of the rocket body, hereinafter referred to as the cross-section of the universal joint center or the pivot cross-section), but rather there is a significant axial offset (i.e., the distance between the upper support point and the cross-section passing through the center of the universal joint). At this point, the sensor, upper fulcrum, lower fulcrum, and universal joint center form a complex "spatial quadrilateral" motion mechanism. If the traditional algorithm based on the coplanar assumption is still used, the nonlinear effect of the axial height difference on the extension will be ignored, resulting in a significant static geometric error in the calculated swing angle, which seriously affects the measurement and solution accuracy.

[0005] When the engine swings bidirectionally simultaneously, due to the constraints of the spatial mechanism, the swing of one channel will cause a change in the length of the sensor in the other channel (i.e., geometric cross-coupling effect). Let the actual swing angles corresponding to the two servo actuators be... The length of the two sensors There are two swing angles. Complex nonlinear bivariate functions .

[0006] To find the inverse solution from the measured length is essentially to solve a set of nonlinear binary coupled equations. Existing solutions typically fall into two categories, but both have shortcomings: Lookup table method: This method involves pre-storing a large number of discrete points in a data table. However, this requires a significant amount of storage space, and its accuracy is limited by the interpolation algorithm, making it difficult to achieve high-precision solutions.

[0007] The traditional Newton-Raphson iterative method solves the problem by calculating the inverse of the Jacobian matrix. However, this method involves a large number of matrix derivatives, inversions, and complex trigonometric function calculations, resulting in a huge computational burden and the risk of iterative non-convergence.

[0008] Existing algorithms often derive analytical solutions for specific geometric configurations. Once the engine installation parameters (such as offset distance and initial length) are slightly adjusted, it is necessary to re-derive the cumbersome mathematical formulas and modify the code, which lacks versatility and increases the software maintenance cost.

[0009] In summary, existing thrust vector measurement and calculation methods often impose a strict constraint that the upper support point of the sensor must be located within the cross-section of the rocket body passing through the center of the universal joint, which brings difficulties to structural design and layout.

[0010] Therefore, there is an urgent need for a technical solution that is based on modeling in a universal Cartesian coordinate system and can quickly and accurately solve for the true engine pitch angle using a numerical optimization method with low computational cost. Summary of the Invention

[0011] The purpose of this invention is to provide a method, equipment, and medium for measuring and calculating the thrust vector angle of a rocket engine, so as to solve the above-mentioned defects in existing thrust vector angle measurement and calculation methods.

[0012] The first aspect of the present invention provides a method for measuring and calculating the thrust vector swing angle of a rocket engine, comprising: constructing a spatial vector model including axial offset parameters, transforming the solution of the equation system into an unconstrained nonlinear least squares optimization problem, and solving it using a gradient descent method assisted by numerical differentiation to obtain mathematical model parameters.

[0013] In some embodiments of the first aspect of the present invention, after obtaining the mathematical model parameters, a geometric decoupling and restoration step is introduced to output the true physical swing angle in the fixed coordinate system of the rocket body.

[0014] According to some embodiments of the first aspect of the present invention, a fixed coordinate system for the rocket body is constructed with the universal joint rotation center of the rocket engine as the origin, and the geometric installation characteristics of two linear displacement sensors are obtained.

[0015] According to some embodiments of the first aspect of the present invention, it further includes: obtaining real-time length data of the corresponding two-channel linear displacement sensors based on the geometric installation characteristics of the two-channel linear displacement sensors and the acquired voltages of the two sensors; Based on the real-time length data of the two-channel linear displacement sensor and the generalized Euler angle of the rocket engine rotating around the gimbal, a set of geometric constraint equations for rotation transformation is established. Based on the geometric constraint equations of the rotation transformation, a nonlinear least squares objective function is constructed to obtain a binary scalar loss function; Based on the binary scalar loss function, the gradient descent method is used to search for the optimal angle solution in real time within the solution cycle. Based on the optimal angle solution, geometric decoupling and restoration are performed and the physical swing angle is output, thereby completing the measurement and calculation of the rocket engine thrust vector swing angle.

[0016] A second aspect of the present invention provides another method for measuring and calculating the thrust vector swing angle of a rocket engine, comprising: Construct a fixed coordinate system for the rocket body with the universal joint rotation center of the rocket engine as the origin, and obtain the geometric installation characteristics of two linear displacement sensors. Based on the geometric installation characteristics of the two linear displacement sensors and the collected voltages of the two sensors, the real-time length data of the corresponding channel linear displacement sensor is obtained. Based on the real-time length data of the two-channel linear displacement sensor and the generalized Euler angle of the rocket engine rotating around the gimbal, a set of geometric constraint equations for rotation transformation is established. Based on the geometric constraint equations of the rotation transformation, a nonlinear least squares objective function is constructed to obtain a binary scalar loss function; Based on the binary scalar loss function, the gradient descent method is used to search for the optimal angle solution in real time within the solution cycle. Based on the optimal angle solution, geometric decoupling and restoration are performed and the physical swing angle is output, thereby completing the measurement and calculation of the rocket engine thrust vector swing angle.

[0017] In some embodiments of the second aspect of the present invention, the geometric mounting characteristics of the two-line linear displacement sensors include a fixed-point vector and a moving-point initial vector; The fixed-point vector is the position vector of the upper support point of the two linear displacement sensors on the frame side; The initial vector of the moving point is the position vector of the lower support point of the two linear displacement sensors on the nozzle side when the rocket engine is in the zero position.

[0018] In some embodiments of the second aspect of the present invention, the establishment of a set of geometric constraint equations for rotational transformation based on the real-time length data of the two-channel linear displacement sensor and the generalized Euler angles of the rocket engine rotating around the gimbal includes: Based on the generalized Euler angles of the rocket engine rotating around the gimbal, a rotational transformation matrix is ​​constructed from the nozzle coordinate system to the fixed coordinate system of the rocket body. Based on the rotation transformation matrix, the upper support point position vectors of the two linear displacement sensors on the frame side, the lower support point position vectors of the two linear displacement sensors on the nozzle side, and the real-time length data of the two-channel linear displacement sensors, a set of geometric constraint equations for the rotation transformation is established.

[0019] In some embodiments of the second aspect of the present invention, a nonlinear least-squares objective function is constructed based on the geometric constraint equations of the rotation transformation to obtain a binary scalar loss function, including: The higher-order terms of the geometric constraint equations of the rotation transformation are eliminated by using the properties of orthogonal matrices, thereby transforming the geometric constraint equations into a scalar inner product form constraint equations. Based on the scalar inner product form constraint equations, the residual function equations are obtained. Based on the residual function equations, a nonlinear least squares objective function is constructed to obtain the binary scalar loss function.

[0020] In some embodiments of the second aspect of the present invention, the step of searching for the optimal angle solution in real time within the solution period using gradient descent based on the binary scalar loss function includes the following steps: After setting the initial value of the angle iteration and the maximum number of iterations, and calculating the current loss function value based on the initial value of the angle iteration and the binary scalar loss function, the iteration loop is entered. In each iteration, the convergence of the current iteration is determined based on the current loss function value to obtain the optimal angle solution. : If the current loss function value is less than a preset threshold, the current iteration is determined to have converged, and the iteration loop is exited to obtain the optimal angle solution. ; Otherwise, if the current iteration has not converged, the central difference method is used to calculate the numerical gradient and update the angle parameter along the negative gradient direction. This updated angle value is then used as the angle iteration value for the next iteration to calculate the current loss function value. This process continues until the current loss function value is less than a preset threshold or the maximum number of iterations is reached, thus obtaining the optimal angle solution. ; in, For generalized Euler angles in Cartesian coordinates The optimal solution for the principal rotation angle of the axis. For generalized Euler angles in Cartesian coordinates The optimal solution for the rotation angle of the axis.

[0021] In some embodiments of the second aspect of the present invention, setting the initial value for angle iteration includes: using the solution result of the previous solution cycle as the initial value for angle iteration in the current cycle. Alternatively, the initial value for the angle iteration in the current cycle can be set to 0.

[0022] In some embodiments of the second aspect of the present invention, the step of calculating the numerical gradient using the central difference method and updating the angle parameter along the negative gradient direction includes: The partial derivatives of the binary scalar loss function with respect to the generalized Euler angles are approximated using the central difference method to obtain the numerical gradient, as shown in the following formula: ; The angle parameter is updated based on the numerical gradient along the negative gradient direction, and the formula is as follows: ; in, A constant representing a small perturbation. For adaptive or fixed learning rate, Let be the binary scalar loss function. For generalized Euler angles in Cartesian coordinates The principal rotation angle of the axis, For generalized Euler angles in Cartesian coordinates The rotation angle of the axis The numerical gradient of the principal rotation angle. The numerical gradient is the value derived from the rotation angle.

[0023] In some embodiments of the second aspect of the present invention, the step of performing geometric decoupling and restoration based on the optimal angle solution and outputting the physical swing angle includes: The generalized Euler angle is rotated in the Cartesian coordinate system. The optimal solution for the principal rotation angle of the axis is directly mapped through the principal rotation axis, outputting the physical swing angle of the principal rotation axis in the fixed coordinate system of the rocket body: The generalized Euler angle is rotated in the Cartesian coordinate system. The optimal solution for the rotation angle of the axis is corrected by projection of the rotation axis, outputting the physical swing angle of the rotation axis in the fixed coordinate system of the rocket body. The projection correction formula is as follows: ; in, , This represents the actual physical swing angle driven by the servo actuator in the fixed coordinate system of the rocket body. The generalized Euler angles are oriented around the Cartesian coordinate system. The optimal solution for the principal rotation angle of the axis. The generalized Euler angles are oriented around the Cartesian coordinate system. The optimal solution for the rotation angle of the axis.

[0024] A third aspect of the present invention provides a computer device, including a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the steps of the method described in the first or second aspect.

[0025] A fourth aspect of the present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the method described in the first or second aspect of the present invention.

[0026] This invention provides a method, equipment, and medium for measuring and calculating the thrust vector angle of a rocket engine. The method establishes a spatial vector model including axial offset parameters, transforming the complex inverse kinematics problem into an unconstrained optimization problem of a low-dimensional scalar function. Specifically, it constructs a bivariate scalar loss function, converting the complex equation system into an unconstrained nonlinear least-squares optimization problem, and employs a numerical differential-assisted gradient descent method for solution. After optimizing and solving the mathematical model parameters, a geometric decoupling and restoration step is introduced to output the true physical angle of the rocket body in a fixed coordinate system. This invention can quickly and accurately solve for the true angle of the engine using a low-computational-consumption numerical optimization method. Attached Figure Description

[0027] Figure 1 This is a flowchart of the rocket engine thrust vector swing angle measurement and calculation method according to an embodiment of the present invention; Figure 2 This is a schematic diagram of the spatial coordinate system and the installation of non-coplanar sensors in the rocket engine oscillation measurement system according to an embodiment of the present invention. Detailed Implementation

[0028] Various embodiments and features of this application are described herein with reference to the accompanying drawings.

[0029] It should be understood that various modifications can be made to the embodiments described herein. Therefore, the above description should not be considered as limiting, but merely as an example of embodiments. Other modifications within the scope and spirit of this application will be apparent to those skilled in the art.

[0030] The accompanying drawings, which are included in and form part of this specification, illustrate embodiments of the present application and, together with the general description of the present application given above and the detailed description of the embodiments given below, serve to explain the principles of the present application.

[0031] These and other features of this application will become apparent from the following description of preferred forms of embodiments given as non-limiting examples, with reference to the accompanying drawings.

[0032] It should also be understood that although this application has been described with reference to some specific examples, those skilled in the art can certainly implement many other equivalent forms of this application.

[0033] This invention addresses a series of measurement and calculation challenges arising from structural layout limitations and computational resource constraints in existing rocket engine thrust vector control systems, primarily solving the following three technical problems: (1) Solve the problem of principle geometric measurement error caused by “non-coplanar installation”.

[0034] Existing technologies are generally based on the "coplanar assumption" (i.e., assuming the sensor's fulcrum is within the central cross-section of the gimbal), simplifying complex spatial motion into planar trigonometric relationships. Due to the compact spatial layout of rockets, relaxing the installation constraints of the sensor's fulcrum would greatly facilitate rocket structural design and installation. However, this would result in a significant axial offset between the linear displacement sensor's fulcrum and the gimbal's central cross-section, forming a nonlinear "spatial quadrilateral" motion mechanism. If a planar approximation model is still used, it will introduce fundamental geometric errors that cannot be eliminated through calibration. This invention solves the problem of decreased swing angle measurement accuracy caused by installation position offset by establishing a Cartesian spatial model that explicitly includes axial offset parameters.

[0035] (2) Solve the problem of "high computing power consumption" in nonlinear inverse calculation under bidirectional swing conditions.

[0036] When the engine oscillates in both directions (pitch and yaw) simultaneously, the length changes of the two sensors exhibit complex nonlinear spatial coupling. Determining the angle from the length is a "reverse kinematics" solution. Traditional methods (such as the Newton-Raphson method) typically require calculating the Jacobian matrix and its inverse, involving numerous high-order matrix operations and floating-point processing. This invention avoids complex matrix inversion operations by constructing a scalar loss function and employing gradient descent, thus resolving the contradiction between high-precision inverse kinematics and low-computing-power hardware resources.

[0037] (3) Solve the problem of “geometric decoupling and restoration” from intermediate process variables to the actual physical swing angle.

[0038] In three-dimensional rotation, different rotation sequences (such as rotating the Z-axis first and then the Y-axis) can lead to inconsistencies between Euler angle values ​​and physical projection angle values. Traditional calculation methods tend to overlook this deviation (i.e., projection distortion) between the mathematical parameters defined by the rotation sequence and the physical angles defined by the guidance system. This invention not only calculates the optimal angle in the mathematical model but also introduces a geometric decoupling and restoration step to solve the problem of distorted swing angle measurements caused by spatial projection effects, ensuring that the output swing angle is strictly consistent with the physical meaning defined by the guidance command.

[0039] Figure 1 A flowchart of a method for measuring and calculating the thrust vector swing angle of a rocket engine, as provided in an embodiment of the present invention, is shown below. Figure 1 As shown, a method for measuring and calculating the thrust vector swing angle of a rocket engine includes the following steps: S101, construct a Cartesian space coordinate system with the universal joint rotation center of the rocket engine as the origin, and obtain the geometric installation characteristics of the two linear displacement sensors. Specifically, establishing such Figure 2 The diagram shows the rotation center of the engine universal joint. Cartesian coordinate system with origin This serves as a fixed coordinate system for the rocket body. This coordinate system is defined when the engine is at zero position and remains constant relative to the rocket body. The axis is parallel to the rocket body axis and points in the direction of the nozzle exit. shaft and The axes are located within the cross-section of the universal joint and satisfy the right-hand rule (the Y-axis lies on the intersection of the plane formed by servo actuators A and X-axis and the cross-section of the universal joint, and the Z-axis lies on the intersection of the plane formed by servo actuators B and X-axis and the cross-section of the universal joint). In this coordinate system, two linear displacement sensors are mounted opposite the corresponding servo actuators (symmetric about the X-axis), and the geometric mounting characteristics of the two linear displacement sensors (defined as channel A and channel B) are parametrically described.

[0040] S102, based on the geometric installation characteristics of the two linear displacement sensors and the collected voltages of the two sensors, the real-time length data of the corresponding two-channel linear displacement sensors is obtained. S103, Based on the real-time length data of the two-channel linear displacement sensor and the generalized Euler angle of the rocket engine rotating around the gimbal, establish a set of geometric constraint equations for rotation transformation; S104, Based on the geometric constraint equations of the rotation transformation, construct a nonlinear least squares objective function to obtain a binary scalar loss function; S105, based on the binary scalar loss function, the gradient descent method is used to search for the optimal angle solution in real time within the solution cycle, and the geometric decoupling restoration is performed based on the optimal angle solution to output the physical swing angle, thereby completing the measurement and calculation of the rocket engine thrust vector swing angle.

[0041] This invention constructs a binary scalar loss function, transforming the solution of complex equation systems into an unconstrained nonlinear least squares optimization problem, and employs a gradient descent method aided by numerical differentiation for solution. This algorithm involves only basic floating-point addition, subtraction, multiplication, division, and trigonometric function operations, without matrix inversion, and therefore avoids the risk of singularity.

[0042] Compared with existing technologies, the present invention provides a method for calculating the thrust vector swing angle of a rocket engine based on inverse kinematics of measurement data from a non-coplanar linear displacement sensor. This method, based on the principles of rigid body kinematics, establishes a spatial vector model including axial offset parameters, transforming the complex inverse kinematics solution into an unconstrained optimization problem of a low-dimensional scalar function. After optimizing and solving the mathematical model parameters, a geometric decoupling and restoration step is specifically introduced, using the principles of spherical trigonometry to accurately map intermediate variables (from rotation angle) to the true physical swing angle in the fixed coordinate system of the rocket body. This mechanism eliminates measurement distortion caused by projection distortion, ensuring that the measurement results are strictly consistent with the physical definition of the guidance commands.

[0043] It should be noted that the method for inversely solving the rocket engine thrust vector swing angle measurement based on the measurement data of a non-coplanar mounted linear displacement sensor includes the following steps: Step S1: Construct a spatial vector motion model of the non-coplanar sensor mounting configuration, i.e., establish a model such as... Figure 2 The Cartesian coordinate system shown; Step S2: Establish a set of geometric constraint equations based on rotational transformation, that is, establish the mapping relationship between the real-time length of the sensor and the engine attitude angle based on rigid body rotation theory; Step S3: Construct a nonlinear least squares objective function, that is, transform the inverse kinematics solution into a parameter optimization problem, avoiding the solution of complex nonlinear equation system analytical inverses; Step S4: Fast iterative optimization based on numerical differential gradient, that is, using the gradient descent method with extremely low computational cost to search for the optimal solution in real time within the solution cycle; Step S5: Geometric decoupling and restoration of physical swing angle and output.

[0044] In this embodiment of the invention, within a real-time calculation cycle, the complete algorithm flow of the onboard computer or measurement computer using linear displacement sensor data to inversely solve the engine thrust vector swing angle is mainly divided into three stages: data preprocessing, iterative optimization, and geometric decoupling. The specific descriptions of steps S2 to S5 are as follows: Steps S2 and S3 above constitute the data acquisition and constraint construction phase. First, real-time length data from the A and B channel linear displacement sensors are acquired. And perform filtering. Based on the sensor's geometric installation parameters (including axial offset) ), calculate the geometric constraint constants that are only related to length. This step transforms complex spatial geometric operations into scalar forms, reducing the computational load in subsequent iterations.

[0045] Step S4 above is the gradient descent iterative optimization stage, including warm start, loop calculation, convergence determination, and parameter update. Details are as follows: Hot Start: Uses the solution result from the previous cycle as the initial value for the current cycle. By utilizing the continuity of engine motion, the number of iterations can be significantly reduced.

[0046] Iterative calculation: In each iteration, construct the attitude rotation matrix. Calculate the residual between the theoretical projection and the measurement constant. and loss function .

[0047] Convergence criterion: If the loss function Less than the preset threshold (like If the loop converges, the loop will exit.

[0048] Parameter update: If convergence fails, calculate the numerical gradient using the central difference method. And update the angle parameters along the negative gradient direction. The entire process does not require calculating the inverse of the Jacobian matrix, ensuring the numerical stability of the algorithm.

[0049] Step S5 above is the geometric decoupling, restoration, and output stage. The optimal solution obtained through iteration... These are parameters for the mathematical model, influenced by the order of spatial rotation. This is not the actual yaw angle. The algorithm finally performs a geometric decoupling step, using the formula... Eliminate projection distortion, It restores the true physical swing angle.

[0050] Based on the above embodiments, the geometric installation characteristics of the two-line linear displacement sensors include fixed-point vector and moving-point initial vector; The fixed-point vector is the position vector of the upper support point of the two linear displacement sensors on the frame side; The initial vector of the moving point is the position vector of the lower support point of the two linear displacement sensors on the nozzle side when the rocket engine is in the zero position.

[0051] Specifically, such as Figure 2 In the fixed coordinate system of the rocket body shown, the geometric mounting characteristics of two linear displacement sensors (defined as channel A and channel B) are parametrically described by defining fixed-point vectors and initial vectors of moving points: Define fixed-point vector: Set the position vector of the sensor's upper support point on the frame side as follows: Explicitly introduce a non-zero axial offset. That is, the vector of the upper pivot point. Axial components This allows for a precise description of the non-coplanar characteristics of the sensor's fulcrum not being located in the gimbal normal plane, thus enabling a mathematical model to accurately describe these characteristics. Define the initial vector of the moving point: When the engine is at zero position, the position vectors of the sensor's lower pivot point on the nozzle side are respectively... At this point, the entire measurement system consists of two systems sharing the same center of rotation. A spatial quadrilateral closed-loop mechanism.

[0052] It should be noted that, Figure 2 A schematic diagram of the spatial coordinate system and non-coplanar sensor installation for a rocket engine oscillation measurement system (shown) ).

[0053] When the engine is in the zero position, the fixed coordinate system of the rocket body is defined as follows: origin Defined as the rotation center of the engine universal joint (ball joint).

[0054] Axis: The positive direction is along the initial axis of the engine nozzle, pointing towards the nozzle (below).

[0055] Axis: Located within the mounting plane of sensor A, perpendicular to... The axis points outwards.

[0056] Axis: Located within the mounting plane of sensor B, perpendicular to... The axis is determined by the right-hand rule (pointing outwards).

[0057] The fixed coordinate system of the rocket body remains unchanged relative to the rocket body. In this coordinate system, sensor A is initially located at... In the plane ( Sensor B is initially located at In the plane ( ).

[0058] At the same time, an engine nozzle coordinate system is established, which coincides with the fixed coordinate system of the rocket body when the engine is in the zero position. This coordinate system moves with the engine swing.

[0059] Key geometric points: 1) These are the upper support points (rack-side mounting points) for sensors A and B, respectively. Figure 2 Display in and Located at the origin of Above the plane in the negative direction of the axis (i.e.) This reflects the installation characteristic of the upper support point and the center of the universal joint being "non-coplanar" (axial offset).

[0060] 2) : These are the initial positions of the lower support points (nozzle-side mounting points) of sensors A and B, respectively.

[0061] Sensor A is initially located at In the plane ( Sensor B is initially located at In the plane ( ).

[0062] In this embodiment of the invention, the specific implementation scheme for parameter initialization and data reading is as follows: (1) Read the geometric parameters stored in memory: Sensor A (Pitch) pivot vector Initial vector of the lower pivot point .

[0063] Sensor B (Yaw) pivot vector Initial vector of the lower pivot point .

[0064] To clarify, This demonstrates the characteristics of non-coplanar installation.

[0065] (2) Read real-time sensor data: The voltage value of sensor A is collected and converted into physical length. .

[0066] The voltage value of sensor B is collected and converted into physical length. .

[0067] Based on the above embodiments, the step of establishing a set of geometric constraint equations for rotational transformation based on the real-time length data of the two-channel linear displacement sensor and the generalized Euler angles of the rocket engine's rotation around the gimbal includes: Based on the generalized Euler angles of the rocket engine rotating around the gimbal, a rotational transformation matrix is ​​constructed from the nozzle coordinate system to the fixed coordinate system of the rocket body. Specifically, first, define the rotation sequence, and let the generalized Euler angles of the engine's rotation around the universal joint be . The fixed coordinate system of the rocket body is rotated in this order to coincide with the nozzle coordinate system. Among these, To bypass The principal rotation angle of the axis, To bypass The rotation angle of the shaft (the rotation sequence can be defined according to the specific cross shaft or ball joint structure of the universal joint; in this embodiment of the invention, the rotation order is taken first) back Secondly, construct the attitude transfer matrix: construct the rotation transformation matrix from the nozzle coordinate system to the fixed coordinate system of the rocket body. : .

[0068] Based on the rotation transformation matrix, the upper support point position vectors of the two linear displacement sensors on the frame side, the lower support point position vectors of the two linear displacement sensors on the nozzle side, and the real-time length data of the two-channel linear displacement sensors, a set of geometric constraint equations for the rotation transformation is established.

[0069] Specifically, the length constraint equation is derived: based on the invariance of spatial Euclidean distance, the real-time measured length of the sensor... It should be strictly equal to the straight-line distance between the upper pivot and the lower pivot after rotation. Establish constraint equations for channels A and B respectively: .

[0070] Based on the above embodiments, a nonlinear least squares objective function is constructed based on the geometric constraint equations of the rotation transformation to obtain a binary scalar loss function, including: The higher-order terms of the geometric constraint equations of the rotation transformation are eliminated by using the properties of orthogonal matrices, thereby transforming the geometric constraint equations into a scalar inner product form constraint equations. Specifically, the constraint equations established for channels A and B are simplified by linearization, the above modulus formula is expanded, and the properties of orthogonal matrices are utilized ( Eliminating higher-order terms transforms the constraint equation into a scalar inner product form: ; in, As a measurement scalar that is only related to the real-time measured length, it greatly simplifies the subsequent computational complexity.

[0071] It should be noted that in this embodiment... The physical meaning of represents the inner product of the rotated vector projections, and this value depends only on the rod length. The above formula... This is a simplified representation of the calculation of geometric constraint constants. Based on the transformed formula of the cosine theorem, the current constraint target value is calculated. It is expressed as follows: ; .

[0072] Based on the scalar inner product form constraint equations, the residual function equations are obtained. Specifically, define the residual function The difference between the theoretical projected value and the actual measured scalar: ; .

[0073] Based on the residual function equations, a nonlinear least squares objective function is constructed to obtain the binary scalar loss function.

[0074] Specifically, construct a binary scalar loss function (Objective Function). : ; The global minimum point of the loss function This corresponds to the engine's actual current geometric attitude.

[0075] Based on the above embodiments, the step of using gradient descent based on the binary scalar loss function to search for the optimal angle solution in real time within the solution cycle includes the following steps: After setting the initial value of the angle iteration and the maximum number of iterations, and calculating the current loss function value based on the initial value of the angle iteration and the binary scalar loss function, the iteration loop is entered. In each iteration, the convergence of the current iteration is determined based on the current loss function value to obtain the optimal angle solution. : If the current loss function value is less than a preset threshold, the current iteration is determined to have converged, and the iteration loop is exited to obtain the optimal angle solution. ; Otherwise, if the current iteration has not converged, the central difference method is used to calculate the numerical gradient and update the angle parameter along the negative gradient direction. This updated angle value is then used as the angle iteration value for the next iteration to calculate the current loss function value. This process continues until the current loss function value is less than a preset threshold or the maximum number of iterations is reached, thus obtaining the optimal angle solution. ; in, For generalized Euler angles in Cartesian coordinates The optimal solution for the principal rotation angle of the axis. For generalized Euler angles in Cartesian coordinates The optimal solution for the rotation angle of the axis.

[0076] Specifically, the core algorithm for iteratively solving the swing angle is implemented in the following steps: Set initial guess value Use the solution result from the previous solution cycle (to accelerate convergence using continuity), or initially set to 0. Set the maximum number of iterations. Learning rate Enter the following iterative loop ( ).

[0077] 1) Utilize the current angle Calculate the sine and cosine values ​​and construct the rotation matrix. : ; 2) Calculate the residual using the inner product of the rotated vector projections and the geometric constraint target value: ; ; 3) To avoid tedious formula derivation and to utilize the parallel capabilities of computers, the central difference method is used to calculate the gradient: ; Assume a small perturbation .

[0078] 4) Calculation Directional gradient: ;in, ; Similarly, calculate Directional gradient .

[0079] 5) Update angle: ; .

[0080] 6) Convergence judgment: If If so, the loop will exit.

[0081] Based on the above embodiments, setting the initial value for angle iteration includes: using the solution result of the previous solution cycle as the initial value for angle iteration in the current cycle. Alternatively, the initial value for the angle iteration in the current cycle can be set to 0.

[0082] Specifically, in step S4 above, a warm start initialization is first performed, utilizing the solution results from the previous solution cycle. As the initial value for the current cycle of iteration Due to the extremely short solution cycle (usually...) Since the engine attitude changes continuously, this strategy can significantly reduce the number of iterations.

[0083] Based on the above embodiments, the step of calculating the numerical gradient using the central difference method and updating the angle parameters along the negative gradient direction includes: The partial derivatives of the binary scalar loss function with respect to the generalized Euler angles are approximated using the central difference method to obtain the numerical gradient, as shown in the following formula: ; The angle parameter is updated based on the numerical gradient along the negative gradient direction, and the formula is as follows: ; in, A constant representing a small perturbation. For adaptive or fixed learning rate, Let be the binary scalar loss function. For generalized Euler angles in Cartesian coordinates The principal rotation angle of the axis, For generalized Euler angles in Cartesian coordinates The rotation angle of the axis The numerical gradient of the principal rotation angle. The numerical gradient is the value derived from the rotation angle.

[0084] It should be noted that the loss function is approximated using the central difference method. about and The partial derivatives of the rotation matrix are used to avoid complex analytical differentiation of the rotation matrix (i.e., there is no need to calculate the Jacobian matrix). For small perturbations (e.g.) ).

[0085] When the loss function value Stop iteration and output the optimal solution when the preset threshold or the maximum number of iterations is reached. .

[0086] Based on the above embodiments, the step of performing geometric decoupling and restoration based on the optimal angle solution and outputting the physical swing angle includes: The generalized Euler angle is rotated in the Cartesian coordinate system. The optimal solution for the principal rotation angle of the axis is directly mapped through the principal rotation axis, outputting the physical swing angle of the principal rotation axis in the fixed coordinate system of the rocket body. The generalized Euler angle is rotated in the Cartesian coordinate system. The optimal solution for the rotation angle of the axis is corrected by projection onto the rotation axis, outputting the physical swing angle of the rotation axis in the fixed coordinate system of the rocket body. The projection correction formula is as follows: ; in, , This represents the actual physical swing angle driven by the servo actuator in the fixed coordinate system of the rocket body. The generalized Euler angles are oriented around the Cartesian coordinate system. The optimal solution for the principal rotation angle of the axis. The generalized Euler angles are oriented around the Cartesian coordinate system. The optimal solution for the rotation angle of the axis.

[0087] It should be noted here that, due to the order dependency of the rotation matrix defined in step S2, the calculated... In reality, it's the non-orthogonal projection component of channel B after rotation over channel A, not the physical swing angle corresponding to the servo actuator's movement. The following geometric decoupling and restoration is required: 1) Direct mapping of channel A (main rotation axis): ; 2) Projection correction of channel B (from the rotation axis): Using spherical trigonometry, the coupling effect caused by the deflection of channel A is eliminated, and the true projection angle of channel B in the fixed coordinate system of the rocket body is calculated: ; 3) Data output: The decoupled actual physical swing angle The data is sent to a measurement computer for swing characteristic analysis.

[0088] In some specific embodiments, after the above convergence judgment exits the loop, the mathematical model parameters after the iteration is completed are obtained. and Convert them into engineering parameters with clear physical meaning: First, determine the swing angle of channel A: since the rotation sequence is defined as rotating first... (Z-axis) This axis is not affected by subsequent rotation, so it is directly assigned a value: ; Then restore the B channel swing angle: It has already deflected in channel A. A secondary rotation projection is performed based on this. To obtain the true command angle of channel B in the fixed coordinate system of the rocket body, the following inverse transformation is performed: ; It should be noted that in the actual operating conditions of a rocket, Usually less than 15° This division operation is safe and reliable.

[0089] The final output is the calculated truth value. The output is sent to a measurement computer for feature analysis and evaluation.

[0090] The technical solution of this invention focuses on protecting the following innovative aspects: 1) Cartesian space closed-chain vector modeling method based on non-zero axial offset parameters This invention overcomes the theoretical limitation of traditional thrust vector measurement and calculation, which relies on the requirement that "the fulcrum on the sensor must be on the cross-section of the rocket body where the gimbal center is located." It innovatively constructs an explicit method in Cartesian coordinate system that includes axial offset parameters. This paper presents a sensor-universal joint spatial quadrilateral closed-chain motion model. This model utilizes rigid body transformation matrices to accurately describe the spatial geometric constraints under non-coplanar mounting configurations, mathematically eliminating nonlinear geometric errors caused by neglecting installation height differences, and achieving a precise description of the engine's oscillation state under arbitrary spatial layouts.

[0091] 2) Fast iterative algorithm for solving the Jacobian-free matrix based on a binary scalar loss function To address the technical challenge of excessive computational cost associated with solving bidirectional coupled inverse kinematics problems, which typically requires high-dimensional matrix inversions (such as the Newton-Raphson method), this invention proposes a lightweight solution strategy combining geometrically constrained scalarization and numerical gradient descent. This invention transforms the complex inverse kinematics problem into an unconstrained bivariate scalar least squares optimization problem and employs the central difference method to calculate the numerical gradient, thus avoiding the complex analytical differentiation of the Jacobian matrix and its inverse matrix operations. This innovation significantly reduces the algorithm's time and space complexity while maintaining strong numerical stability.

[0092] 3) Integrating the physical swing angle geometric decoupling and restoration mechanism of rotational order projection correction To address the projection distortion problem caused by different definitions of Euler angles in three-dimensional rotation, this invention establishes a precise mapping mechanism from "mathematical model parameters (Euler angles)" to "engineering physical parameters (projection angles)." After optimizing and solving for intermediate variables, this invention utilizes the principles of spherical trigonometry to introduce a mechanism based on... The decoupling and restoration steps precisely isolate the spatial projection coupling effect of the main rotating channel on the slave rotating channel. This innovation ensures that the measured swing angle value is strictly consistent with the physical meaning defined by the guidance and control system.

[0093] It should be noted that the present invention has the following advantages over the prior art: 1) It completely eliminates the fundamental geometric errors caused by non-coplanar layouts, breaking through the spatial constraints of the overall design. Existing technologies generally rely on the fact that "the fulcrum of the sensor and the universal joint are coplanar" The idealized assumption of "( )" means that when there is axial offset in actual engineering, the traditional plane approximation algorithm will produce static measurement deviations that are difficult to eliminate.

[0094] This invention, based on the fundamental principles of spatial mechanisms, establishes a spatial quadrilateral closed-chain vector model that includes axial offset parameters. Regardless of whether the fulcrum on the sensor is located in front of, behind, or at any spatial position on the gimbal plane, this method can achieve a zero-error mathematical description of the actual physical length through precise analytical constraint equations.

[0095] This breakthrough not only improved measurement accuracy but also liberated the overall structural design of launch vehicles. Structural designers are no longer limited by the stringent requirement of "coplanar installation" and can flexibly choose the installation location of sensors based on the load-bearing structure of the compartment, the thermal protection layout, and the routing of pipelines, significantly optimizing the space utilization of the engine compartment.

[0096] 2) Reduced computational requirements for inverse kinematics solutions, making it compatible with rocket-borne embedded hardware environments. To address the challenging problem of solving bidirectional coupled inverse kinematics, this invention abandons the traditional Newton-Raphson iterative method, thus avoiding high-dimensional operations involving the Jacobian matrix and its inverse.

[0097] This invention innovatively constructs a binary scalar loss function, transforming the solution of complex equation systems into an unconstrained nonlinear least squares optimization problem, and employs a gradient descent method aided by numerical differentiation for solution. This algorithm involves only basic floating-point addition, subtraction, multiplication, division, and trigonometric function operations, without matrix inversion, and therefore avoids the risk of singularity.

[0098] 3) It accurately compensates for the spatial projection distortion of bidirectional swaying, improving the quality of attitude control. Traditional single-channel independent solution methods ignore the spatial projection effect caused by the Euler angle rotation order. When a channel is deflected at a large angle, it will cause false measurement coupling (i.e., "cross-coupling effect") in another channel.

[0099] After optimizing and solving the mathematical model parameters, this invention specifically introduces a geometric decoupling and restoration step, using the principles of spherical trigonometry to accurately map the intermediate variable (from the rotation angle) to the true physical swing angle in the fixed coordinate system of the rocket body. This mechanism eliminates measurement distortion caused by projection distortion, ensuring that the measurement results are strictly consistent with the physical definition of the guidance commands.

[0100] Further improvements and modifications to the above technical solutions are also within the scope of protection of this invention, such as, but not limited to: 1) Quaternion-based rotation model alternative To further improve computational efficiency and avoid repeated calls to trigonometric functions in the iterative loop, the Euler angle rotation matrix in step S2... Can be derived from unit quaternions Replacement.

[0101] Specific variation: The optimization variables are changed from a scalar perspective. Replace with the imaginary part vector of a quaternion (assuming a small angle) Alternatively, it can be solved using normalization constraints.

[0102] Advantages: Quaternion rotation transformations involve only algebraic multiplication and addition operations, completely avoiding transcendental function (sin / cos) calculations and eliminating gimbal lock issues. Although the final output still needs to be converted to Euler angles, the computation speed is faster in high-frequency iterative cores. This type of substitution of mathematical expression is an equivalent transformation of the technical solution of this invention.

[0103] 2) Adjustment of rotation sequence to adapt to different universal joint configurations The embodiments of this invention employ a rotation sequence of "Z first, then Y". For engine universal joints with different physical structures (such as different orthogonal relationships of universal joint shaft systems, or the use of soft bearing structures), their physical motion laws may correspond to different rotation matrix construction methods.

[0104] Specific variation: Adjust the order of matrix multiplication in step S2 (e.g., change it to...). (or definition method). Regardless of how the rotation matrix is ​​defined, as long as it preserves the non-coplanar mounting parameters... The inverse solution obtained by the least squares method falls within the protection scope of this invention.

[0105] Based on the above embodiments, this embodiment of the invention provides a computer device, including a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the steps of the method described in the above embodiments.

[0106] In some embodiments of the present invention, a computer-readable storage medium is provided, the computer-readable storage medium storing a computer program, which, when executed by a processor, implements the steps of the method described in the above embodiments.

[0107] The processor may include, but is not limited to, one or more processors or microprocessors. Each processor may be implemented as an Application Specific Integrated Circuit (ASIC), Digital Signal Processor (DSP), Digital Signal Processing Device (DSPD), Programmable Logic Device (PLD), Field Programmable Gate Array (FPGA), controller, microcontroller, microprocessor, or other electronic component, for executing the methods in the above embodiments.

[0108] Computer-readable storage media can be implemented by any type of volatile or non-volatile storage device or a combination thereof. Computer-readable storage media may include, but are not limited to, random access memory (RAM), read-only memory (ROM), flash memory, EPROM memory, EEPROM memory, registers, and computer storage media (e.g., hard disks, floppy disks, solid-state drives, removable disks, CD-ROMs, DVD-ROMs, Blu-ray discs, etc.).

[0109] Computer-readable storage media may also store at least one computer-executable program / instruction, such as computer-readable instructions. Computer-readable storage media include, but are not limited to, volatile memory and / or non-volatile memory. Volatile memory may include, for example, random access memory (RAM) and / or cache memory. Computer-readable storage media may include, for example, read-only memory (ROM), hard disk, flash memory, etc. For example, a non-transitory computer-readable storage medium may be connected to a computing device such as a computer, and then, when the computing device executes the computer-readable instructions stored on the computer-readable storage medium, the various methods described above can be performed.

[0110] In addition, the computer device may include (but is not limited to) a data bus, an input / output (I / O) bus, a display, and input / output devices (e.g., keyboard, mouse, speakers, etc.).

[0111] The processor can communicate with external devices via the I / O bus through wired or wireless networks.

[0112] Those skilled in the art will understand that all or part of the steps of the methods described above can be implemented by a program instructing related hardware. The program can be stored in a readable storage medium, and when executed, the program includes one or a combination of the steps of the method implementation.

[0113] In the various embodiments of this application, the functional units can be integrated into a single processing module, or each unit can exist physically separately, or two or more units can be integrated into a single module. The integrated module can be implemented in hardware or as a software functional module. If the integrated module is implemented as a software functional module and sold or used as an independent product, it can also be stored in a readable storage medium. The storage medium can be a read-only memory, a disk, or an optical disk, etc.

[0114] In the description of this specification, the references to terms such as "one embodiment / mode," "some embodiments / modes," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment / mode or example is included in at least one embodiment / mode or example of this application. Furthermore, the described specific features, structures, materials, or characteristics may be combined in any suitable manner in one or more embodiments / modes or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments / modes or examples described in this specification, as well as the features of different embodiments / modes or examples.

[0115] Furthermore, 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 technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this application, "multiple" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0116] Those skilled in the art should understand that the above embodiments are merely for illustrative purposes and are not intended to limit the scope of this application. Those skilled in the art can make other changes or modifications based on the above disclosure, and these changes or modifications still fall within the scope of this application.

Claims

1. A method for measuring and calculating the thrust vector swing angle of a rocket engine, characterized in that, include: A spatial vector model including axial offset parameters is constructed, and the solution of the equation system is transformed into an unconstrained nonlinear least squares optimization problem. The gradient descent method with numerical differentiation is used to solve the problem and obtain the mathematical model parameters.

2. The method for measuring and calculating the thrust vector swing angle of a rocket engine according to claim 1, characterized in that, After obtaining the mathematical model parameters, a geometric decoupling and restoration process was introduced to output the true physical swing angle of the rocket body in the fixed coordinate system.

3. The method for measuring and calculating the thrust vector swing angle of a rocket engine according to claim 2, characterized in that, A fixed coordinate system for the rocket body was constructed with the universal joint rotation center of the rocket engine as the origin, and the geometric installation characteristics of two linear displacement sensors were obtained.

4. The method for measuring and calculating the thrust vector swing angle of a rocket engine according to claim 3, characterized in that, Further includes: Based on the geometric installation characteristics of the two linear displacement sensors and the collected voltages of the two sensors, the real-time length data of the corresponding two-channel linear displacement sensors are obtained. Based on the real-time length data of the two-channel linear displacement sensor and the generalized Euler angle of the rocket engine rotating around the gimbal, a set of geometric constraint equations for rotation transformation is established. Based on the geometric constraint equations of the rotation transformation, a nonlinear least squares objective function is constructed to obtain a binary scalar loss function; Based on the binary scalar loss function, the gradient descent method is used to search for the optimal angle solution in real time within the solution cycle. Based on the optimal angle solution, geometric decoupling and restoration are performed and the physical swing angle is output, thereby completing the measurement and calculation of the rocket engine thrust vector swing angle.

5. The method according to claim 4, characterized in that, The geometric mounting characteristics of the two linear displacement sensors include the fixed-point vector and the moving-point initial vector; The fixed-point vector is the position vector of the upper support point of the two linear displacement sensors on the frame side; The initial vector of the moving point is the position vector of the lower support point of the two linear displacement sensors on the engine nozzle side when the rocket engine is in the zero position.

6. The method according to claim 4, characterized in that, The geometric constraint equations for establishing the rotational transformation based on the real-time length data from the two-channel linear displacement sensor and the generalized Euler angles of the rocket engine's rotation around the gimbal include: Based on the generalized Euler angles of the rocket engine rotating around the gimbal, a rotational transformation matrix is ​​constructed from the engine nozzle coordinate system to the fixed coordinate system of the rocket body. Based on the rotation transformation matrix, the upper support point position vectors of the two linear displacement sensors on the frame side, the lower support point position vectors of the two linear displacement sensors on the engine nozzle side, and the real-time length data of the two-channel linear displacement sensors, a set of geometric constraint equations for the rotation transformation is established.

7. The method according to claim 4, characterized in that, Based on the geometric constraint equations of the rotation transformation, a nonlinear least squares objective function is constructed to obtain a binary scalar loss function, including: The higher-order terms of the geometric constraint equations of the rotation transformation are eliminated by using the properties of orthogonal matrices, thereby transforming the geometric constraint equations into a scalar inner product form constraint equations. Based on the scalar inner product form constraint equations, the residual function equations are obtained. Based on the residual function equations, a nonlinear least squares objective function is constructed to obtain the binary scalar loss function.

8. The method according to claim 4, characterized in that, The method of using gradient descent based on the binary scalar loss function to search for the optimal angle in real time within the solution cycle includes the following steps: After setting the initial value of the angle iteration and the maximum number of iterations, and calculating the current loss function value based on the initial value of the angle iteration and the binary scalar loss function, the iteration loop is entered. In each iteration, the convergence of the current iteration is determined based on the current loss function value to obtain the optimal angle solution. : If the current loss function value is less than a preset threshold, the current iteration is determined to have converged, and the iteration loop is exited to obtain the optimal angle solution. ; Otherwise, if the current iteration has not converged, the central difference method is used to calculate the numerical gradient and update the angle parameter along the negative gradient direction. This updated angle value is then used as the angle iteration value for the next iteration to calculate the current loss function value. This process continues until the current loss function value is less than a preset threshold or the maximum number of iterations is reached, thus obtaining the optimal angle solution. ; in, For generalized Euler angles in Cartesian coordinates The optimal solution for the principal rotation angle of the axis. For generalized Euler angles in Cartesian coordinates The optimal solution for the rotation angle of the axis.

9. The method according to claim 8, characterized in that, The process of calculating the numerical gradient using the central difference method and updating the angle parameters along the negative gradient direction includes: The partial derivatives of the binary scalar loss function with respect to the generalized Euler angles are approximated using the central difference method to obtain the numerical gradient, as shown in the following formula: ； The angle parameter is updated based on the numerical gradient along the negative gradient direction, and the formula is as follows: ； in, A constant representing a small perturbation. For adaptive or fixed learning rate, Let be the binary scalar loss function. For generalized Euler angles in Cartesian coordinates The principal rotation angle of the axis, For generalized Euler angles in Cartesian coordinates The rotation angle of the axis The numerical gradient of the principal rotation angle. The numerical gradient is the value derived from the rotation angle.

10. The method according to claim 8, characterized in that, The process of geometric decoupling and reconstruction based on the optimal solution of the angle and outputting the physical swing angle includes: The generalized Euler angle is rotated in the Cartesian coordinate system. The optimal solution for the principal rotation angle of the axis is directly mapped through the principal rotation axis, outputting the physical swing angle of the principal rotation axis in the fixed coordinate system of the rocket body: ； The generalized Euler angle is rotated in the Cartesian coordinate system. The optimal solution for the rotation angle of the axis is corrected by projection of the rotation axis, outputting the physical swing angle of the rotation axis in the fixed coordinate system of the rocket body. The projection correction formula is as follows: ； in, This represents the actual physical swing angle driven by the servo actuator in the fixed coordinate system of the rocket body. The generalized Euler angles are oriented around the Cartesian coordinate system. The optimal solution for the principal rotation angle of the axis. The generalized Euler angles are oriented around the Cartesian coordinate system. The optimal solution for the rotation angle of the axis.

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