A center of mass measurement method based on gravitational potential energy multipole expansion

Abstract

This invention discloses a method for measuring the center of mass based on the multipole expansion of gravitational potential energy. This invention improves accuracy through joint calibration of multiple physical quantities by simultaneously measuring the changes in inertial tensor and gravitational potential energy. The specific process involves simultaneously measuring the multipole moment of gravitational potential energy and the inertial tensor. Using the multipole expansion formula of gravitational potential energy, the potential energy is expressed as a function of mass, center of mass position, and inertial tensor. The center of mass coordinates are solved by fitting multiple sets of displacement data using the least squares method. The least squares fitting method eliminates systematic errors in the calibration of single physical quantities and dynamically corrects center of mass offset errors. This method can handle asymmetric and complex shapes and is suitable for measuring the center of mass position of precision devices such as spacecraft. This invention has wide applications in the field of precision measurement technology.

Description

Technical Field

[0001] This invention relates to the field of precision measurement technology, and in particular to a method for measuring the center of mass based on the multipole moment expansion of gravitational potential energy. Background Technology

[0002] Center of mass measurement is an important fundamental measurement project in scientific research, production, and engineering development. Currently available methods for center of mass measurement mainly employ static methods such as the suspension method, four-point balance method, inclined platform method, and knife-edge balance method, as well as dynamic methods such as the moment of inertia method. These methods have some limitations in practical applications, such as insufficient measurement accuracy, complex equipment, high cost, and poor adaptability to complex shapes. Summary of the Invention

[0003] In view of the technical problems of insufficient measurement accuracy, complex equipment, high cost and poor adaptability to complex-shaped objects in current center of mass measurement technology, the purpose of this invention is to provide a center of mass measurement method based on the multipole moment expansion of gravitational potential energy.

[0004] This invention includes a method for measuring the center of mass based on the expansion of gravitational potential energy multipole moments. The method comprises:

[0005] Determine multiple coordinate axes in the space where the object being measured is located;

[0006] Traverse all the coordinate axes. For any of the coordinate axes that are traversed, translate the object under test step by step along the coordinate axis. For any position point that the object under test is translated to, detect the change in inertial tensor and the change in gravitational potential energy of the object under test at the position point. Based on the change in inertial tensor and the change in gravitational potential energy corresponding to each position point, determine the centroid position component of the object under test on the coordinate axis.

[0007] The centroid position of the measured object is determined based on the centroid position components corresponding to each of the coordinate axes.

[0008] Further, the stepwise translation of the measured object along the coordinate axis includes:

[0009] Keep the object under test stationary along the other coordinate axes;

[0010] Set the translation spacing;

[0011] The object under test is translated multiple times at equal intervals along the coordinate axis according to the translation interval.

[0012] Furthermore, after gradually translating the object under test along the coordinate axis and before detecting the change in inertial tensor and the change in gravitational potential energy, a static waiting process is performed.

[0013] Further, detecting the change in inertial tensor and the change in gravitational potential energy of the object under test at the specified location includes:

[0014] At least 6 different rotation axes are determined; the angles between any one of the rotation axes and each coordinate axis form the angle combinations corresponding to the rotation axes.

[0015] For any of the rotation axes, detect the moment of inertia of the object under test rotating about the rotation axis;

[0016] By combining the aforementioned moments of inertia, the inertia tensor of the measured object at the specified position point is calculated; the inertia tensor includes the moment of inertia and the product of inertia.

[0017] The change in the inertia tensor is determined based on the inertia tensor corresponding to the given location point and the inertia tensor corresponding to the previous location point.

[0018] Further, the calculation of the inertial tensor of the measured object at the specified position point by simultaneously solving the various moments of inertia includes:

[0019] Establish equations

[0020]

[0021] in, This represents the moment of inertia of the measured object as it rotates about the axis of rotation. This represents the moment of inertia of the measured object about the X-axis. This represents the moment of inertia of the measured object about the Y-axis. This represents the moment of inertia of the measured object about the Z-axis. , and Represents the product of inertia. The angle between the rotation axis and the X-axis is given. The angle between the rotation axis and the Y-axis is given. The angle between the rotation axis and the Z-axis;

[0022] Will , and The values ​​for each combination of included angles. Solve the equations simultaneously to obtain the moment of inertia. , , With inertial product , and .

[0023] Further, detecting the change in inertial tensor and the change in gravitational potential energy of the object under test at the specified location includes:

[0024] The object under test is suspended at the lower end of the optical lever system of the second-stage pendulum;

[0025] The rotating platform located below the object under test in the second-order pendulum is activated, driving the source mass block installed on the rotating platform to perform periodic rotational motion, so that the source mass block generates a time-varying gravitational potential;

[0026] The optical lever system is used to detect the deflection angle of the object under test caused by the time-varying gravitational potential.

[0027] The gravitational potential energy of the object under test at the specified position point is calculated based on the deflection angle.

[0028] The change in gravitational potential energy is determined based on the gravitational potential energy corresponding to the given location point and the gravitational potential energy corresponding to the previous location point.

[0029] Further, calculating the gravitational potential energy of the object at the specified position based on the deflection angle includes:

[0030] According to the formula

[0031]

[0032]

[0033] Perform calculations; among which, The gravitational potential energy of the object being measured at the specified location point is... The fiber torque of the optical lever system. The fiber torsional stiffness of the optical lever system. The deflection angle is denoted as .

[0034] Further, determining the mass center position component of the measured object on the coordinate axis based on the change in inertial tensor and the change in gravitational potential energy corresponding to each position point includes:

[0035] Obtain the formula relating the gravitational potential energy to the inertial tensor of the object under test;

[0036] Based on the formula relating gravitational potential energy and inertial tensor, the change in inertial tensor, and the change in gravitational potential energy, a nonlinear relationship between the change in gravitational potential energy and the position of the measured object is established using the least squares method.

[0037] Based on the nonlinear relationship, the centroid position component of the measured object on the coordinate axis is obtained by fitting.

[0038] Furthermore, the formula relating the gravitational potential energy to the inertial tensor of the measured object includes:

[0039]

[0040] in, Indicates that the measured object is relative to The gravitational potential energy at the point. The gravitational constant is... The mass of the object being measured. Indicates that the measured object and Distance between points The density of the object being measured. This indicates that the volume of the space occupied by the measured object is divided. , , , , and This represents the inertial tensor of the object being measured.

[0041] Further, the step of fitting the centroid position components of the measured object on the coordinate axis according to the nonlinear relationship includes:

[0042] Set the objective to minimize the sum of squared residuals;

[0043] The Gauss-Newton method is used to solve for the parameter vector to be estimated in the nonlinear relationship through local linearization iteration.

[0044] Based on the solved parameter vector, the centroid position component of the measured object on the coordinate axis is determined.

[0045] The beneficial effects of this invention are as follows: The centroid measurement method based on the multipole expansion of gravitational potential energy in the embodiments improves accuracy by simultaneously measuring the change in inertial tensor and the change in gravitational potential energy. The specific process is as follows: simultaneously measuring the multipole moment of gravitational potential energy and the inertial tensor, combining the multipole expansion formula of gravitational potential energy, expressing the potential energy as a function of mass, centroid position, and inertial tensor, and solving for the centroid coordinates by fitting multiple sets of displacement data using the least squares method. The least squares fitting method can eliminate systematic errors when calibrating a single physical quantity, dynamically correct centroid offset errors, handle asymmetric complex shapes of the measured object, and is suitable for measuring the centroid position of precision devices such as spacecraft. Attached Figure Description

[0046] Figure 1This is a schematic diagram illustrating the use of a two-stage pendulum device to measure the object in this embodiment.

[0047] Figure 2 This is a schematic diagram of the coordinate system in the embodiment;

[0048] Figure 3 This is a schematic diagram illustrating the steps of the centroid measurement method based on the multipole expansion of gravitational potential energy in the embodiment. Detailed Implementation

[0049] For a system of objects, its center of mass is the center of mass distribution. The calculation of gravitational potential energy is usually based on the position of the center of mass, meaning that gravitational potential energy is related to the position of the system's center of mass. Therefore, the position of the center of mass can be derived by indirectly measuring the gravitational potential energy of the object at a point, which provides a theoretical basis for measuring the position of the center of mass using the multipole moment expansion of gravitational potential energy.

[0050] Specifically, setting up points (a, b, c) are the objects being measured. A point outside the object being measured. for The general formula for the gravitational potential energy at a point is:

[0051]

[0052] in (P) represents the gravitational potential energy. The value of the point, The gravitational constant is... For the object being measured density, , = In Cartesian coordinate system Point to object The distance to a point on the surface.

[0053] Based on the above potential energy formula, the object to be measured At one point The potential energy generated above can be expressed using a Taylor series.

[0054]

[0055] Among them, the object being tested Regarding the center of mass The mass moment is:

[0056]

[0057] Summation Is it satisfying = + + All non-negative integers , , It is performed on the set.

[0058] In this embodiment, take =2, at this time the Taylor series expansion of the potential energy formula is:

[0059]

[0060] in ; ; ; ; ; ; ; ; ; ;

[0061] For test objects of arbitrary shapes (e.g., irregular shapes) Based on the integral definition of moment of inertia, the relationship between the mass moment and the moment of inertia can be established:

[0062] ; ;

[0063] ;

[0064] ;

[0065] achievable

[0066] ; ; ;

[0067] ; ; ;

[0068] Therefore, the object being measured for The formula for the gravitational potential energy at point A can be written as:

[0069]

[0070] in, The object being measured quality The density of the object, This represents the volume integral of the space occupied by the measured object, other than integral expressions. , , Indicates the object being measured center of mass Coordinates in the Cartesian coordinate system.

[0071] If the object being measured If the density distribution is uniform, then the object being measured... for The formula for the gravitational potential energy at a point can be further written as:

[0072]

[0073] The measured object obtained from the above derivation for formula for gravitational potential energy at a point

[0074]

[0075] This indicates the object being measured. gravitational potential energy With inertial tensor , , , , and The relationship between these parameters can serve as the theoretical basis for a method of measuring the center of mass based on the multipole expansion of gravitational potential energy.

[0076] In this embodiment, the object whose center of mass is to be measured using the center-of-mass measurement method based on the expansion of gravitational potential energy multipole moments can be a precision device such as a spacecraft, specifically an artificial satellite or similar equipment. When performing the center-of-mass measurement method based on the expansion of gravitational potential energy multipole moments, the following methods can be used: Figure 1 The two-stage pendulum device shown performs relevant measurements on the object being measured.

[0077] Reference Figure 1 The two-stage pendulum device used includes a magnetic damper 1, a quartz wire 2, a reflector 3, a crossbar 4, a counterweight 5, a source mass 7, and a rotating platform 8. The magnetic damper 1, quartz wire 2, reflector 3, crossbar 4, and counterweight 5 constitute an optical lever system. (Refer to...) Figure 1A counterweight 5 is installed at one end of the crossbar 4. During measurement, the object to be measured 6 is suspended from the other end of the crossbar 4 by a fiber. The two-stage pendulum device can achieve multi-degree-of-freedom measurement by changing the position of the crossbar 4, thereby measuring parameters such as the deflection angle and translation displacement of the object to be measured 6. For example, a laser beam is emitted towards the reflector 3, which reflects the laser beam in a certain direction, forming a spot that can be detected by the detector. When the object to be measured 6 is subjected to force and deflects or displaces, the object to be measured 6 will transmit the force through the fiber suspending it and the crossbar 4, causing the reflector 3 to deflect. The deflection of the reflector 3 causes the laser beam to displace the spot, which is then detected by the detector, thereby calculating the corresponding deflection angle.

[0078] In this embodiment, refer to Figure 3 The method for measuring the center of mass based on the multipole expansion of gravitational potential energy includes the following steps:

[0079] S1. Determine multiple coordinate axes in the space where the object being measured is located;

[0080] S2. Traverse all coordinate axes:

[0081] For any coordinate axis that is traversed, the object under test is translated step by step along the coordinate axis. For any position point that the object under test is translated to, the change in inertial tensor and the change in gravitational potential energy of the object under test at the position point are detected. Based on the change in inertial tensor and the change in gravitational potential energy corresponding to each position point, the mass center position component of the object under test on the coordinate axis is determined.

[0082] S3. Determine the position of the centroid of the object being measured based on the corresponding centroid position components of each coordinate axis.

[0083] In step S1, a Cartesian coordinate system can be constructed using the rotation axis and pendulum platform of the secondary pendulum device as a reference, serving as a reference frame for subsequent position control and data calculation. For example, referring to... Figure 2 The location of the magnetic damper 1 can be taken as the origin O, and an XY plane can be established with a plane horizontal to the ground to establish the X coordinate axis and Y coordinate axis. A Z coordinate axis perpendicular to the X coordinate axis and Y coordinate axis can be established to obtain multiple coordinate axes such as the X coordinate axis, Y coordinate axis and Z coordinate axis. Figure 2 In addition, an inertial principal axis coordinate system can also be established. .

[0084] In step S2, the same measurement and calculation are performed on the X-axis, Y-axis, and Z-axis, respectively. The explanation can also be based on one of the coordinate axes, such as the X-axis.

[0085] Reference Figure 3 For the X-axis, the translation spacing can be set. Starting from the starting point on the X-axis Start, each time according to the translation interval The object being measured is translated along the X-axis, and the total translation is... This results in the measured object reaching the X-coordinate axis at each translation. ( =1, 2, ..., ( ) multiple location points.

[0086] In this embodiment, when the object being measured is translated on the X-axis, it remains stationary on the other coordinate axes (Y-axis and Z-axis), meaning its coordinates on the Y-axis and Z-axis remain unchanged. Similarly, when the object is translated on the Y-axis, it remains stationary on the other coordinate axes (X-axis and Z-axis), and when the object is translated on the Z-axis, it remains stationary on the other coordinate axes (X-axis and Y-axis).

[0087] Assume the object being measured is from position point Translate to position point , to the location point The following explanation uses a measurement as an example. Specifically, the object being measured can be moved to a position point. Subsequently, a static waiting period is performed before detecting the changes in inertial tensor and gravitational potential energy at that location. Specifically, the static waiting period can be 3-5 data acquisition cycles, during which the changes in inertial tensor and gravitational potential energy are not detected. This static waiting period allows the vibrations caused by the translation of the object under test to decay before detection, thereby reducing detection errors caused by vibration. When measuring the change in inertial tensor at each location, multiple changes in inertial tensor can be measured using the same method, and their average value can be calculated as the final measured change in inertial tensor, thus reducing measurement errors. Similarly, when measuring the change in gravitational potential energy at each location, multiple changes in gravitational potential energy can be measured using the same method, and their average value can be calculated as the final measured change in gravitational potential energy.

[0088] In this embodiment, when at the location point To measure the change in inertia tensor and the change in gravitational potential energy, the following steps can be performed:

[0089] S201. Identify at least 6 different rotation axes;

[0090] S202. For any rotation axis, detect the moment of inertia of the object being measured as it rotates around the rotation axis;

[0091] S203. Combine the moments of inertia to calculate the inertia tensor of the measured object at its position point; the inertia tensor includes the moment of inertia and the product of inertia;

[0092] S204. Determine the change in inertia tensor based on the inertia tensor corresponding to the current position and the inertia tensor corresponding to the previous position.

[0093] Steps S201-S204 involve measuring at the location point. The steps for determining the change in the inertial tensor.

[0094] The principle of steps S201-S204 is as follows: Figure 2 As shown.

[0095] Reference Figure 2 When the object being measured is translated to the position point Then, the axis of rotation can be selected. Rotating shaft For passing point Any axis, that is, the selected axis of rotation. Must pass through the origin ,and The direction can be arbitrarily determined. For a given axis of rotation... Its relationship with the coordinate axes The angle between the (X-axis) and the coordinate axis is , and coordinate axes The angle between the (Y-axis) and the coordinate axis is , and coordinate axes The angle between the (Z-axis) and the coordinate axis is Thus forming a rotating shaft Corresponding angle combinations , , And the rotating shaft can be obtained. Corresponding direction cosine , , If the object being measured is considered a rigid body, then the object being measured about its axis of rotation... Moment of inertia It can be represented as:

[0096]

[0097] in The moment of inertia of the measured object about the X-axis, The moment of inertia of the measured object about the Y-axis, Let be the moment of inertia of the measured object about the Z-axis. Product of inertia. Moment of inertia. , , and inertial product The measured object rotates around its axis. The inertia tensor corresponding to rotation.

[0098] On the other hand, the object being measured rotates around its axis of rotation Moment of inertia corresponding to rotation This can be obtained through actual measurement. Specifically, when performing step S202, the following steps can be performed:

[0099] S20201. Apply an external force to twist the object under test to the initial angle. ;

[0100] S20202. After the external force is released, the measured object undergoes a single-degree-of-freedom torsional oscillation.

[0101] Neglecting friction and air resistance, according to the formula:

[0102]

[0103] In the formula: Let the moment of inertia of the torsion bar be denoted as . For torsional stiffness; For the twist angle, The damping coefficient is a constant. When friction and air resistance are negligible, the damping coefficient is approximately zero. The free oscillation period of the torsional pendulum system is T = Then the moment of inertia of the torsion table = Assume the period under no-load (torsional table) is... The period after loading the test object is The moment of inertia of the object being measured can be obtained through... Calculation, where = This is a system constant. Once the torsion table testing equipment is determined, and Given a known quantity, the period after loading is measured. The moment of inertia of the object being measured can then be calculated. .

[0104] Based on the above principle, it can be concluded that for any given axis of rotation... Its moment of inertia On the one hand, it can be obtained through actual measurement; on the other hand, it satisfies:

[0105]

[0106] in , , It is with this axis of rotation The relevant combinations of included angles. That is... There exists a moment of inertia , , and inertial product The six unknowns can be obtained by combining six linear equations.

[0107] Therefore, in step S201, at least 6 different rotation axes are determined; in step S202, the moment of inertia of the object being tested rotating about each rotation axis is detected. This results in 6 linearly uncorrelated pairs. The system of equations; in step S203, the moment of inertia can be obtained by solving the system of equations. , , and inertial product Iso-inertial tensor.

[0108] The location point of the measured object can be calculated through steps S201-S203. Moment of inertia , , and inertial product Based on the same principle, the position of the measured object at its previous position can also be obtained. These parameters. In step S204, the object to be measured can be placed at the position point. Moment of inertia Subtract at position point Moment of inertia Thus, the position of the measured object is obtained. The change in moment of inertia Based on the same principle, we can also obtain , , This constitutes the change in rotational inertia.

[0109] In this embodiment, when at the location point To measure the change in inertia tensor and the change in gravitational potential energy, the following steps can be performed:

[0110] S205. The object to be measured is suspended at the lower end of the optical lever system of the second-stage pendulum;

[0111] S206. Start the rotating platform located below the object being measured in the second-stage pendulum, drive the source mass block installed on the rotating platform to perform periodic rotational motion, so that the source mass block generates a time-varying gravitational potential;

[0112] S207. The deflection angle of the object under test caused by the time-varying gravitational potential is detected by an optical lever system;

[0113] S208. Calculate the gravitational potential energy of the object at the position point based on the deflection angle;

[0114] S209. Determine the change in gravitational potential energy based on the gravitational potential energy corresponding to the current position and the gravitational potential energy corresponding to the previous position.

[0115] Steps S205-S209 involve measuring at the location point. The steps for determining the change in gravitational potential energy.

[0116] In step S205, such as Figure 1 The object to be measured is suspended as shown.

[0117] In step S206, the rotating platform located below the object under test in the second-order pendulum is activated, driving the source mass block mounted on the rotating platform to perform periodic rotational motion, causing the source mass block to generate a time-varying gravitational potential. Specifically, the rotating platform can alternately change its rotation direction, for example, rotating forward and then stopping, then rotating in the opposite direction and then stopping, then rotating forward again and then stopping, and so on, thus causing the source mass block to generate an alternating time-varying gravitational potential. The alternating time-varying gravitational potential exerts a periodically changing gravitational force on the object under test, thereby causing the object to be subjected to alternating torque.

[0118] In step S207, the alternating torque acting on the object being measured is transmitted to the reflector 3, thereby enabling the detector to detect the deflection angle of the object being measured. .

[0119] In step S208, the formula can be used.

[0120]

[0121]

[0122] Calculate the position of the measured object at point gravitational potential energy on .in For the torsional stiffness of the fiber (quartz wire 2) in the optical lever system, The torque generated by the fiber in the optical lever system. The integration limit can be from 0 to 1. .

[0123] The location point of the measured object can be calculated through steps S205-S208. gravitational potential energy Based on the same principle, the position of the measured object at its previous position can also be obtained. The gravitational potential energy. In step S209, the object to be measured can be placed at position point. gravitational potential energy Subtract at position point The gravitational potential energy of the object being measured is obtained at the position point. change in gravitational potential energy .

[0124] For the object being measured at position point change in gravitational potential energy and the change in inertia tensor , , , According to the formula relating the gravitational potential energy and the inertial tensor of the measured object:

[0125]

[0126] Can Considered one of them The change in amount, , , , Each is considered as one of them , , , , , The change in gravitational potential energy is used to establish a nonlinear relationship between the change in gravitational potential energy and the position of the measured object using the least squares method:

[0127]

[0128] in It is the vector of parameters to be estimated. For error, Indicates will Mapped to the change in gravitational potential energy The error between them is Nonlinear functions.

[0129] For the aforementioned nonlinear relationship, the Gauss-Newton method can be used to solve for the estimated parameter vector in the nonlinear relationship through local linearization iteration:

[0130]

[0131] in For Jacobian matrices, For the residual vector, Let be the number of iteration rounds. The objective is to minimize the sum of squared residuals, i.e., the goal of the iteration is:

[0132]

[0133] in This represents the number of times the object being measured is translated along the X-axis. This is achieved by making... Minimize, thereby determining the vector of parameters to be estimated. The specific values ​​are used to determine the nonlinear relationship. .

[0134] In this embodiment, it can be based on nonlinear relationships. The centroid position component of the measured object on the X-axis is obtained by fitting. For example, it can be determined The position corresponding to a value of 0 (or closest to 0) is taken as the centroid position component of the measured object on the X-axis. .

[0135] Reference Figure 3 After performing step S2, the centroid position component of the measured object on the X-axis is determined. Next, the centroid position components of the measured object on the Y-axis are measured and calculated. Then, measure and calculate the centroid position component of the measured object on the Z-axis. At this point, all coordinate axes have been calculated, thus obtaining the coordinates of the centroid of the measured object. , , .

[0136] The centroid measurement method based on the multipole expansion of gravitational potential energy in this embodiment improves accuracy through joint calibration of multiple physical quantities. It simultaneously measures the multipole moment of gravitational potential energy and the inertial tensor. Combining the multipole expansion formula of gravitational potential energy, the potential energy is expressed as a function of mass, centroid position, and inertial tensor. The centroid coordinates are solved by fitting multiple sets of displacement data using the least squares method. The least squares fitting method eliminates systematic errors when calibrating a single physical quantity. A double-stage pendulum can achieve multi-degree-of-freedom measurement by changing the position of the pendulum rod, and its kinetic and potential energy models are sensitive to position changes. The centroid measurement method based on the multipole expansion of gravitational potential energy in this embodiment utilizes this characteristic to dynamically correct centroid offset errors through multiple displacement measurements. Traditional centroid measurement relies on geometric symmetry, while the centroid measurement method based on the multipole expansion of gravitational potential energy in this embodiment can handle asymmetric and complex shapes of the measured object, making it suitable for precision devices such as spacecraft.

[0137] It should be noted that, unless otherwise specified, when a feature is referred to as "fixed" or "connected" to another feature, it can be directly fixed or connected to the other feature, or indirectly fixed or connected to the other feature. Furthermore, the descriptions of "upper," "lower," "left," and "right" used in this disclosure are only relative to the relative positional relationships of the components of this disclosure in the accompanying drawings. The singular forms "a," "an," and "the" used in this disclosure are also intended to include the plural forms, unless the context clearly indicates otherwise. Moreover, unless otherwise defined, all technical and scientific terms used in this embodiment have the same meaning as commonly understood by one of ordinary skill in the art. The terminology used in this embodiment specification is only for describing particular embodiments and is not intended to limit the invention. The term "and / or" as used in this embodiment includes any combination of one or more of the associated listed items.

[0138] It should be understood that although the terms first, second, third, etc., may be used to describe various elements in this disclosure, these elements should not be limited to these terms. These terms are only used to distinguish elements of the same type from each other. For example, a first element may also be referred to as a second element without departing from the scope of this disclosure, and similarly, a second element may also be referred to as a first element. The use of any and all instances or exemplary language (“e.g.,” “such as,” etc.) provided in this embodiment is intended only to better illustrate embodiments of the invention and, unless otherwise required, does not impose a limitation on the scope of the invention.

[0139] It should be recognized that embodiments of the present invention can be implemented or carried out by computer hardware, a combination of hardware and software, or by computer instructions stored in a non-transitory computer-readable storage medium. The method can be implemented using standard programming techniques—including a non-transitory computer-readable storage medium configured with a computer program, wherein such a storage medium causes the computer to operate in a specific and predefined manner—according to the methods and drawings described in the specific embodiments. Each program can be implemented in a high-level procedural or object-oriented programming language to communicate with the computer system. However, if desired, the program can be implemented in assembly or machine language. In any case, the language can be a compiled or interpreted language. Furthermore, for this purpose, the program can run on a programmed application-specific integrated circuit (ASIC).

[0140] Furthermore, the procedures described in this embodiment can be performed in any suitable order unless otherwise indicated by this embodiment or otherwise obviously contradict the context. The procedures (or variations and / or combinations thereof) described in this embodiment can be executed under the control of one or more computer systems configured with executable instructions, and can be implemented by hardware or a combination thereof as code (e.g., executable instructions, one or more computer programs, or one or more applications) that commonly executes on one or more processors. A computer program includes a plurality of instructions executable by one or more processors.

[0141] Furthermore, the method can be implemented in any suitable type of computing platform, including but not limited to personal computers, minicomputers, mainframes, workstations, networked or distributed computing environments, standalone or integrated computer platforms, or in communication with charged particle tools or other imaging devices, etc. Aspects of the invention can be implemented as machine-readable code stored on a non-transitory storage medium or device, whether removable or integrated into a computing platform, such as a hard disk, optical read and / or write storage medium, RAM, ROM, etc., such that it is readable by a programmable computer, and when the storage medium or device is read by the computer, it can be used to configure and operate the computer to perform the processes described herein. Furthermore, the machine-readable code, or portions thereof, can be transmitted via wired or wireless networks. The invention of this embodiment includes these and other different types of non-transitory computer-readable storage media when such media comprises instructions or programs that implement the steps above in conjunction with a microprocessor or other data processor. When programmed according to the methods and techniques of the invention, the invention also includes the computer itself.

[0142] A computer program can be applied to input data to perform the functions of this embodiment, thereby transforming the input data to generate output data stored in non-volatile memory. The output information can also be applied to one or more output devices, such as a display. In a preferred embodiment of the invention, the transformed data represents physical and tangible objects, including specific visual depictions of physical and tangible objects generated on the display.

[0143] The above are merely preferred embodiments of the present invention. The present invention is not limited to the above-described embodiments. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention, as long as they achieve the technical effects of the present invention by the same means, should be included within the scope of protection of the present invention. Within the scope of protection of the present invention, the technical solutions and / or implementation methods can have various modifications and variations.

Claims

1. A method for measuring the center of mass based on the multipole expansion of gravitational potential energy, characterized in that, The method for measuring the center of mass based on the multipole expansion of gravitational potential energy includes: Determine multiple coordinate axes in the space where the object being measured is located; Traverse all the coordinate axes. For any of the coordinate axes that are traversed, translate the object under test step by step along the coordinate axis. For any position point that the object under test is translated to, detect the change in inertial tensor and the change in gravitational potential energy of the object under test at the position point. Based on the change in inertial tensor and the change in gravitational potential energy corresponding to each position point, determine the centroid position component of the object under test on the coordinate axis. The position of the centroid of the object under test is determined based on the centroid position components corresponding to each of the coordinate axes. The step of determining the mass center position component of the measured object on the coordinate axis based on the change in inertial tensor and the change in gravitational potential energy corresponding to each position point includes: Obtain the formula relating the gravitational potential energy to the inertial tensor of the object under test; Based on the formula relating gravitational potential energy and inertial tensor, the change in inertial tensor, and the change in gravitational potential energy, a nonlinear relationship between the change in gravitational potential energy and the position of the measured object is established using the least squares method. Based on the nonlinear relationship, the centroid position component of the measured object on the coordinate axis is obtained by fitting. The formula relating the gravitational potential energy and the inertial tensor of the object under test includes: in, Indicates that the measured object is relative to The gravitational potential energy at the point. The gravitational constant is... The mass of the object being measured. Indicates that the measured object and Distance between points The density of the object being measured. This indicates that the volume of the space occupied by the measured object is divided. , , , , and This represents the inertial tensor of the object being measured. The step of fitting the centroid position components of the measured object on the coordinate axis according to the nonlinear relationship includes: Set the objective to minimize the sum of squared residuals; The Gauss-Newton method is used to solve for the parameter vector to be estimated in the nonlinear relationship through local linearization iteration. Based on the solved parameter vector, the centroid position component of the measured object on the coordinate axis is determined.

2. The method for measuring the center of mass based on the multipole expansion of gravitational potential energy according to claim 1, characterized in that, The stepwise translation of the measured object along the coordinate axis includes: Keep the object under test stationary along the other coordinate axes; Set the translation spacing; The object under test is translated multiple times at equal intervals along the coordinate axis according to the translation interval.

3. The method for measuring the center of mass based on the multipole expansion of gravitational potential energy according to claim 2, characterized in that: After the object under test is gradually translated along the coordinate axis and before the change in inertial tensor and the change in gravitational potential energy are detected, a static waiting process is performed.

4. The method for measuring the center of mass based on the multipole expansion of gravitational potential energy according to claim 1, characterized in that, The detection of the change in inertial tensor and the change in gravitational potential energy of the object under test at the specified location includes: At least 6 different rotation axes are determined; the angles between any one of the rotation axes and each coordinate axis form the angle combinations corresponding to the rotation axes. For any of the rotation axes, detect the moment of inertia of the object under test rotating about the rotation axis; By combining the aforementioned moments of inertia, the inertia tensor of the measured object at the specified position point is calculated; the inertia tensor includes the moment of inertia and the product of inertia. The change in the inertia tensor is determined based on the inertia tensor corresponding to the given location point and the inertia tensor corresponding to the previous location point.

5. The method for measuring the center of mass based on the multipole expansion of gravitational potential energy according to claim 4, characterized in that, The calculation of the inertia tensor of the measured object at the specified position point by simultaneously combining the aforementioned moments of inertia includes: Establish equations in, This represents the moment of inertia of the measured object as it rotates about the axis of rotation. This represents the moment of inertia of the measured object about the X-axis. This represents the moment of inertia of the measured object about the Y-axis. This represents the moment of inertia of the measured object about the Z-axis. , and Represents the product of inertia. The angle between the rotation axis and the X-axis is given. The angle between the rotation axis and the Y-axis is given. The angle between the rotation axis and the Z-axis; Will , and The values ​​for each combination of included angles. Solve the equations simultaneously to obtain the moment of inertia. , , With inertial product , and .

6. The method for measuring the center of mass based on the multipole expansion of gravitational potential energy according to claim 1, characterized in that, The detection of the change in inertial tensor and the change in gravitational potential energy of the object under test at the specified location includes: The object under test is suspended at the lower end of the optical lever system of the second-stage pendulum; The rotating platform located below the object under test in the second-order pendulum is activated, driving the source mass block installed on the rotating platform to perform periodic rotational motion, so that the source mass block generates a time-varying gravitational potential; The optical lever system is used to detect the deflection angle of the object under test caused by the time-varying gravitational potential. The gravitational potential energy of the object under test at the specified position point is calculated based on the deflection angle. The change in gravitational potential energy is determined based on the gravitational potential energy corresponding to the given location point and the gravitational potential energy corresponding to the previous location point.

7. The method for measuring the center of mass based on the multipole expansion of gravitational potential energy according to claim 6, characterized in that, The calculation of the gravitational potential energy of the object at the specified position based on the deflection angle includes: According to the formula Perform calculations; among which, The gravitational potential energy of the object being measured at the specified location point is... The fiber torque of the optical lever system. The fiber torsional stiffness of the optical lever system. The deflection angle is denoted as .

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