Inertia tensor measurement optimal attitude angle solving method considering centroid offset

Abstract

The invention provides an inertia tensor measurement optimal attitude angle solving method considering centroid offset, and belongs to the technical field of inertia tensor measurement. In order to solve the problem that when the true centroid of a measured rigid body does not coincide with a clamping reference point in traditional inertia tensor measurement, tiny deviation of an attitude causes a significant error, the method comprises the following steps: determining a rotation matrix of a rigid body coordinate system relative to a measurement coordinate system; according to the parallel axis theorem, obtaining an inertia tensor related to a measurement principle O; placing a measured object at a specified attitude angle, and obtaining an inertia tensor of the measured object in a measurement coordinate system; in any posture, a measured object is measured along three axes through the multi-point weighing platform and the torsional pendulum mechanism, and a linear model is established; on the basis of the Fisher information matrix, the optimal solution of the pitch angle of the measured body is obtained through D-optimal accurate measurement; and determining a value taking rule of a yaw angle and a final enforceable attitude set, and taking a fixed constant of the roll angle to obtain an optimal attitude angle.

Description

TECHNICAL FIELD

[0001] The present application relates to a method for solving optimal attitude angles in inertia tensor measurement considering centroid offset, and belongs to the technical field of inertia tensor measurement. BACKGROUND

[0002] Traditional inertia tensor measurement generally faces the problem of decreased accuracy and reliability of measurement results caused by insufficient modeling of centroid offset and lack of optimization in attitude selection in practical application. Existing processes mostly use empirical formula or equally spaced attitude transformation, and are supplemented by simple denoising and repeated measurement to improve measurement accuracy. When the real centroid of the measured rigid body does not coincide with the clamping reference point, a slight deviation of the attitude can cause significant error. Therefore, a new technical path is needed to improve information utilization and parameter estimation stability through optimal attitude planning under the condition of high real-time requirement or limited measurement times. SUMMARY

[0003] The present application is to solve the problem that a slight deviation of the attitude causes significant error when the real centroid of the measured rigid body does not coincide with the clamping reference point in traditional inertia tensor measurement, and further proposes a method for solving optimal attitude angles in inertia tensor measurement considering centroid offset.

[0004] The technical solution adopted by the present application to solve the above problem is as follows: Step 1: Establish a measurement coordinate system and a coordinate system of the measured rigid body, and obtain the rotation matrix of the rigid body coordinate system relative to the measurement coordinate system by combining Z-Y-X Euler angles; Step 2: Set the mass, offset and inertia tensor at the centroid of the measured body, and obtain the inertia tensor about the measurement principle O according to the parallel axis theorem; Step 3: Obtain the optimal solution of the pitch angle of the measured body based on the Fisher information matrix and D-optimal criterion; Place the measured body to obtain the inertia tensor of the measured body in the measurement coordinate system; Step 4: Measure the measured body along three axes by a multi-point weighing platform and a torsion pendulum mechanism under an arbitrary attitude, and establish a linear model based on the measurement results; Step 5: Obtain the optimal solution of the pitch angle of the measured body based on the Fisher information matrix and D-optimal criterion; Step 6: Apply the transpose of the rotation matrix of the rigid body coordinate system relative to the measurement coordinate system to the measurement coordinate system basis vector to determine the value rule of the yaw angle and the final implementable attitude set, take the roll angle as a fixed constant, and obtain the optimal attitude angle.

[0005] Further, step 1 specifically includes: the measurement coordinate system is defined as the coordinate system of the measured rigid body is the Z-Y-X Euler angle is used to describe the attitude relationship, wherein the Z-Y-X Euler angle 、 、 respectively represent the yaw angle, the pitch angle and the roll angle, the yaw angle is the rotation angle of the rigid body around the z axis, the pitch angle is the rotation angle of the rigid body around the y axis, and the roll angle is the rotation angle of the rigid body around the x axis; the rotation matrix of the rigid body coordinate system relative to the measurement coordinate system is obtained based on the defined measurement coordinate system, the coordinate system of the measured rigid body and the Z-Y-X Euler angle ; the rotation matrix of the rigid body coordinate system relative to the measurement coordinate system is ; (1).

[0006] Further, the expression of the inertia tensor of the measurement principle O in step 2 is: (2); In formula (2), E is a unit matrix, T is a transpose, is an additional inertia term caused by the center of mass offset, which is an important error source not explicitly considered in the traditional measurement method, is the inertia tensor relative to the center of mass of the object.

[0007] Further, the expression of the inertia tensor of the measured body in the measurement coordinate system in step 3 is: (3); In formula (3), T is a transpose, .

[0008] Further, step 4 specifically comprises: measuring the measured body along three axes by a multi-point weighing platform and a torsion mechanism at any attitude to obtain observation data of the measured body ; organizing the observation data of the measured body at multiple attitudes to obtain a linear model ; The expression of the observation data of the measured body is: (4); In formula (4), is a unit directional vector in the measurement coordinate system, T is a transpose, is a noise term; The expression of the linear model is: (5); In formula (5), is the inertia parameter vector to be solved, is the sensitivity matrix, and b(d) is the correction introduced by the mass center offset term.

[0009] Further, step 5 specifically includes: Based on the Fisher information matrix , the D-optimal criterion is used to optimize the attitude angle, and the optimal solution of the pitch angle is obtained. The expression of the optimal solution of the pitch angle is: (6); (7); In formulas (6) and (7), represents the objective function, which is used to calculate the pitch angle value, is the optimal pitch angle, is the main inertia component of the information intensity of the three parameter subspaces, is the inertial integral component of the information intensity of the three parameter subspaces.

[0010] Further, step 6 specifically includes: Step 6.1: Apply the transpose of the rotation matrix of the rigid body coordinate system relative to the measurement coordinate system to the measurement coordinate system base vector to obtain the three-axis direction cosine , wherein after applying the transpose to the measurement coordinate system base vector, any observation behavior , is the direction cosine of the reference coordinate system in the measurement coordinate system, indicating the rotation from the measurement coordinate system to the reference coordinate system, is the correction introduced by the mass center offset term, is the inertia parameter vector, is the error term, which represents the error of a certain direction in the measurement process, and is usually used to describe the uncertainty or noise of the measurement; Step 6.2: According to any observation behavior, the Fisher information matrix under isotropic noise is , the components of are expanded according to Z-Y-X, wherein F is the product of the components of about α, and the yaw angle is equally sampled within a circle, and the discrete orthogonality is used to strictly cancel all non-zero harmonic waves in the summation of H and isotropic noise F, is the yaw rotation angle of the system, leaving only the dependence The symmetric average term maximizes detF and minimizes isotropic noise F under a given attitude angle β; detF is the determinant of the Fisher information matrix, used to quantify the identifiability of the parameters. Step 6.3: Considering the feasibility of the tooling, the roll angle is taken as a fixed constant and substituted into Step 6.2 to obtain the value pattern of the yaw angle. Combined with the calculated optimal solution for the pitch angle, the range of the optimal attitude angles is obtained. ; The expression for yaw angle sampling within one revolution is: (8).

[0011] The beneficial effects of this invention are: This invention introduces a centroid offset compensation model into inertial tensor measurement, constructing an offset term based on the parallel axis theorem and incorporating it into the observation equation, thereby reducing the systematic deviation caused by the misalignment of the centroid and the clamping datum. This invention employs Z–Y–X Euler angle parameterization and D-optimal experimental design, using a minimum set of attitudes with equally spaced yaw angles αk (k=0…5), an optimal pitch angle β≈45°, and a roll angle γ≈0°, maximizing the determinant and minimizing the condition number of the Fisher information matrix, significantly reducing parameter coupling and ill-conditioned phenomena. Compared to empirical angle selection, parameter identifiability and numerical stability are significantly enhanced. This invention requires only a few attitudes (6 equally spaced yaw attitudes are sufficient to cover full parameter identification), and is compatible with multi-axis observation methods such as weighing platforms and torsion mechanisms, reducing the number of attitude adjustments and changes, lowering the requirements for tooling travel, improving measurement efficiency, and shortening the testing cycle. It is suitable for engineering applications in large rotors and confined space scenarios. Attached Figure Description

[0012] Figure 1 A flowchart illustrating the method for solving the optimal attitude angle using inertial tensor measurement that takes into account centroid offset; Figure 2 Optimize the graph for the Fisher information matrix; Figure 3 This is a schematic diagram of six feasible optimal poses. Detailed Implementation

[0013] like Figure 1 As shown, the steps of the method for solving the optimal attitude angle of inertial tensor measurement considering centroid offset described in this embodiment include: S1: Establish the measurement coordinate system and the coordinate system of the rigid body under test, and determine the rotation matrix of the rigid body coordinate system relative to the measurement coordinate system; First, establish a coordinate system, let the measurement coordinate system be... The coordinate system of the rigid body being measured is The attitude relationship is described using Z–Y–X Euler angles. , , respectively represent: yaw angle (rotation angle of rigid body around z axis), pitch angle (rotation angle of rigid body around y axis), roll angle (rotation angle of rigid body around x axis). Based on the above coordinate system and definition, the rotation matrix of the rigid body coordinate system relative to the measurement coordinate system is: (1).

[0014] S2: Set the mass, offset and inertia tensor at the center of mass of the measured body, and obtain the inertia tensor about the measurement principle O according to the parallel axis theorem; Suppose the mass of the measured body is m, the offset of the center of mass relative to the measurement origin is , and the inertia tensor at the center of mass is I c . According to the parallel axis theorem, the inertia tensor about the measurement principle O is: (2); In formula (2), E is the unit matrix, T is the transpose, is the additional inertia term caused by the offset of the center of mass, which is an important error source not explicitly considered in traditional measurement methods, is the inertia tensor relative to the center of mass of the object.

[0015] S3: Place the measured body at a specified attitude angle to obtain the inertia tensor of the measured body in the measurement coordinate system; When the measured rigid body is placed at an attitude angle , the inertia tensor of the rigid body in the measurement coordinate system is: (3); In formula (3), T is the transpose, .

[0016] S4: In any attitude, measure the measured body along three axes through a multi-point weighing platform and a torsion mechanism, and establish a linear model based on the measurement results; In any attitude, the observation obtained by measuring along three axes through a multi-point weighing platform and a torsion mechanism can be expressed as: (4); In formula (4), is the unit direction vector in the measurement coordinate system, T is the transpose, is the noise term; The measurement data under multiple attitudes can be sorted to obtain a linear model: (5); In formula (5), Let be the vector of inertia parameters to be solved. Let b(d) be the sensitivity matrix, and b(d) be the correction introduced by the centroid offset term.

[0017] S5: Based on the Fisher information matrix, the optimal solution for the pitch angle of the measured object is obtained by using the D-optimal criterion. To achieve high-precision measurement of inertia parameters under finite attitudes, this invention uses the Fisher information matrix. Based on, such as Figure 2 As shown, the attitude angle is optimized using D-optimal quasi-measurement: (6); (7); In formulas (6) and (7), This represents the objective function used to calculate the pitch angle value. To achieve the optimal pitch angle, Principal inertia components Information intensity of the three parameter subspaces Inertial integral quantity Information intensity of the three parameter subspaces.

[0018] S6: Apply the transpose of the rotation matrix of the rigid body coordinate system relative to the measurement coordinate system to the basis vector of the measurement coordinate system to determine the value of the yaw angle and the final feasible attitude set. Set the roll angle to a fixed constant to obtain the optimal attitude angle.

[0019] After obtaining the optimal pitch angle solution, it is necessary to further determine the value pattern of the yaw angle and the final feasible attitude set. To this end, the rotation matrix... The transpose of the vector is applied to the basis vectors of the measurement coordinate system to obtain the cosines of the three-axis directions. An arbitrary observation row can be denoted as: (8); In formula (8), It is the direction cosine of the reference coordinate system in the measurement coordinate system, representing the rotation from the measurement coordinate system to the reference coordinate system. The correction amount introduced for the centroid offset term. For inertia parameter vectors, This is the error term, representing the error in a specific direction during the measurement process. It is usually used to describe the uncertainty or noise of the measurement.

[0020] Therefore, the Fisher information matrix under isotropic noise is: .Will Expanding the components according to ZYX, we can see that the terms concerning α appear at most up to the second harmonic ( ). ), while F is the product of these terms, up to the fourth harmonic. Let the yaw angle be equally sampled over a circle: (9).

[0021] Utilizing discrete orthogonality All non-zero harmonics can be strictly canceled in the summation of H and F, for the yaw rotation angle of the system, leaving only the symmetric average term that depends on the attitude angle β, maximizing detF and minimizing cond(F) under the given attitude angle β; detF is the determinant of the Fisher information matrix, which is used to quantify the identifiability of the parameters; leaving only the symmetric average term that depends on the attitude angle β. This symmetrization significantly attenuates the off-focus elements of F and reduces the column correlation, maximizing detF and minimizing cond(F) under the given β, which maximizes the identifiability of the measurement system, i.e., minimizes the mutual influence between different parameters, thereby obtaining the most accurate estimation.

[0022] Based on the above derivation and combined with the implementability of the tooling, the roll angle can be taken as a fixed constant. The range of the optimal attitude angle is . The final calculation of the six implementable optimal attitudes is shown in Figure 3 . This configuration theoretically maximizes the determinant of the Fisher information matrix, making the inertial tensor measurement reach the globally optimal precision under the conditions of mass center offset, noise, and assembly constraints.

[0023] In summary, the present application introduces a mass center offset compensation model in inertial tensor measurement, constructs a bias term based on the parallel axis theorem and incorporates it into the observation equation, thereby reducing the system bias caused by the non-coincidence of the mass center and the clamping reference. The present application uses Z-Y-X Euler angle parameterization and D-optimal experimental design, uses an equally spaced yaw angle αk(k=0…5), an optimal pitch angle β≈45°, and a roll angle γ≈0° of the minimum attitude set, maximizes the determinant of the Fisher information matrix, and minimizes the condition number, significantly reducing parameter coupling and ill-conditioning. Compared with the empirical angle selection, the parameter identifiability and numerical stability are significantly enhanced. The present application only needs a small number of attitudes (6 equally spaced yaw attitudes can cover the full parameter identification), and is compatible with weighing platforms and torsional pendulum mechanisms and other multi-axis observation methods, reducing the number of attitude adjustments and changes, reducing the requirements for tooling travel, improving measurement efficiency, shortening the test period, and being suitable for engineering applications of large rotors and limited space scenarios.

[0024] The above merely describes preferred embodiments of the present application, and is not intended to limit the present application in any form. Although the present application has been disclosed with preferred embodiments as above, it is not intended to limit the present application. Any person skilled in the art can make some changes or modifications to the above disclosed technical contents without departing from the technical solution of the present application, and can make equivalent embodiments with equivalent changes, as long as they do not depart from the technical solution of the present application and are within the spirit and principle of the present application. Any simple modification, equivalent replacement and improvement of the above embodiments, as long as they are within the protection scope of the present application, are still within the protection scope of the present application.

Claims

1. A method for solving the optimal attitude angle using inertial tensor measurement considering center-of-mass offset, characterized in that, include: Step 1: Establish the measurement coordinate system and the coordinate system of the rigid body under test, and obtain the rotation matrix of the rigid body coordinate system relative to the measurement coordinate system by combining the Z-Y-X Euler angles; Step 2: Set the mass, offset, and inertia tensor at the center of mass of the measured object. According to the parallel axis theorem, obtain the inertia tensor with respect to measurement principle O. Step 3: Using attitude angle Place the object under test and obtain the inertia tensor of the object under test in the measurement coordinate system; Step 4: Under any posture, measure the object along the three axes using a multi-point weighing platform and a torsion pendulum mechanism, and establish a linear model based on the measurement results; Step 5: Based on the Fisher information matrix, use the D-optimal criterion to obtain the optimal solution for the pitch angle of the measured object; Step 6: Apply the transpose of the rotation matrix of the rigid body coordinate system relative to the measurement coordinate system to the basis vector of the measurement coordinate system to determine the value of the yaw angle and the final feasible attitude set. Set the roll angle to a fixed constant to obtain the optimal attitude angle.

2. The method for solving the optimal attitude angle of inertial tensor measurement considering centroid offset according to claim 1, characterized in that, Step 1 specifically includes: Define the measurement coordinate system as The coordinate system of the rigid body being measured is The attitude relationship is described using Z–Y–X Euler angles, where Z–Y–X Euler angles are... , , These represent the yaw angle, pitch angle, and roll angle, respectively. The yaw angle is the angle around the rigid body. z The rotation and pitch angles of the axis are for rigid body rotation around the axis. y The rotation angle and roll angle of the shaft are for rigid body about the axis. x The rotation angle of the axis; The rotation matrix of the rigid body coordinate system relative to the measurement coordinate system is obtained based on the defined measurement coordinate system, the coordinate system of the rigid body under test, and the Z–Y–X Euler angles. ; Rotation matrix of rigid body coordinate system relative to measurement coordinate system for: （1）。 3. The method for solving the optimal attitude angle of inertial tensor measurement considering centroid offset according to claim 1, characterized in that, The expression for the inertia tensor of measurement principle O in step 2 is: （2）； In formula (2), E is the identity matrix, used to represent the adjustment of the inertia tensor in this coordinate system. T For transpose, This is the additional inertial term caused by the shift of the center of mass, and it is an important source of error that has not been explicitly considered in traditional measurement methods. Let be the inertial tensor relative to the object's center of mass.

4. The method for solving the optimal attitude angle by considering the centroid offset in inertial tensor measurement according to claim 2, characterized in that, The expression for the inertia tensor of the measured object in the measurement coordinate system in step 3 is: （3）； In formula (3), T For transpose, .

5. The method for solving the optimal attitude angle of inertial tensor measurement considering centroid offset according to claim 1, characterized in that, Step 4 specifically includes: Under any orientation, the measured object is measured along three axes using a multi-point weighing platform and a torsion pendulum mechanism to obtain the observed data of the measured object. ; The observation data of the object under multiple poses are processed to obtain a linear model. ; Observational data of the tested object The expression is: （4）； In formula (4), The unit direction vector in the measurement coordinate system, where T is the transpose. Noise term; linear model The expression is: （5）； In formula (5), Let be the vector of inertia parameters to be solved. Let b(d) be the sensitivity matrix, and b(d) be the correction introduced by the centroid offset term.

6. The method for solving the optimal attitude angle of inertial tensor measurement considering centroid offset according to claim 1, characterized in that, Step 5 specifically includes: Fisher's Information Matrix Based on this, the attitude angles are optimized using D-optimal criterion to obtain the optimal solution for the pitch angle. ; Optimal solution for pitch angle The expression is: （6）； （7）； In formulas (6) and (7), This represents the objective function used to calculate the pitch angle value. To achieve the optimal pitch angle, Principal inertia components Information intensity of the three parameter subspaces Inertial integral quantity Information intensity of the three parameter subspaces.

7. The method for solving the optimal attitude angle of inertial tensor measurement considering centroid offset according to claim 6, characterized in that, Step 6 specifically includes: Step 6.1: Rotate the rigid body coordinate system relative to the measurement coordinate system using the rotation matrix. The transpose of the vector is applied to the basis vectors of the measurement coordinate system to obtain the cosines of the three-axis directions. Among them, after the transpose is applied to the basis vectors of the measurement coordinate system, any observation behavior , It is the direction cosine of the reference coordinate system in the measurement coordinate system, representing the rotation from the measurement coordinate system to the reference coordinate system. The correction amount introduced for the centroid offset term. For inertia parameter vectors, This is the error term, representing the error in a specific direction during the measurement process. It is usually used to describe the uncertainty or noise in the measurement. Step 6.2: Obtain the Fisher information matrix under isotropic noise based on arbitrary observation behavior. ,Will The components are expanded according to ZYX, where F is the expanded component. The product of the components with respect to α, and the yaw angle is sampled at equal angles within one circle, utilizing discrete orthogonality. All non-zero harmonics are strictly canceled out in the summation of H and isotropic noise F. For the system's yaw rotation angle, only leave the dependency. The symmetric average term maximizes detF and minimizes isotropic noise F under a given attitude angle β; detF is the determinant of the Fisher information matrix, used to quantify the identifiability of the parameters. Step 6.3: Considering the feasibility of the tooling, the roll angle is taken as a fixed constant and substituted into Step 6.2 to obtain the value pattern of the yaw angle. Combined with the calculated optimal solution for the pitch angle, the range of the optimal attitude angles is obtained. ; The expression for yaw angle sampling within one revolution is: （8）。

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