Mass center measuring method based on gravitational potential energy multi-pole moment expansion

Abstract

The invention discloses a centroid measurement method based on gravitational potential energy multi-pole moment expansion, and the method achieves the combined calibration of multiple physical quantities through the simultaneous measurement of an inertia tensor variable quantity and a gravitational potential energy variable quantity, and improves the precision. The specific process is as follows: simultaneously measuring the multi-pole moment and inertia tensor of the gravitational potential energy, expressing the potential energy as a function of mass, centroid position and inertia tensor in combination with a multi-pole moment expansion formula of the gravitational potential energy, fitting multiple groups of displacement data through a least square method to solve centroid coordinates, and calculating the multi-pole moment of the gravitational potential energy. The system error during calibration of a single physical quantity can be eliminated by means of least square fitting, the mass center offset error is dynamically corrected, the measured object of an asymmetric complex shape can be processed, and the method is suitable for mass center position measurement of precise devices such as spacecrafts and the like. The method is widely applied to the technical field of precision measurement.

Description

Technical Field

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

[0002] Center of mass measurement is an important fundamental measurement method in scientific research, production, and engineering development. Existing methods for center of mass measurement primarily utilize static measurement methods such as the suspension method, four-point counterweight method, tilting platform method, and knife-edge balance method, as well as dynamic measurement methods such as the moment of inertia method. These methods have limitations in practical applications, such as insufficient measurement accuracy, complex equipment, high costs, and poor adaptability to complex objects. Summary of the Invention

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

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

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

[0006] Traversing all the coordinate axes, for any of the traversed coordinate axes, gradually translating the object under test along the coordinate axis, for any position point to which the object under test is translated, detecting the change in the inertia tensor and the change in the gravitational potential energy of the object under test at the position point, and determining the center of mass position component of the object under test on the coordinate axis based on the change in the inertia tensor and the change in the gravitational potential energy corresponding to each position point;

[0007] The center of mass position of the object to be measured is determined according to the center of mass position components corresponding to each of the coordinate axes.

[0008] Furthermore, the stepwise translation of the object to be measured along the coordinate axis includes:

[0009] Keeping the measured object stationary along other coordinate axes;

[0010] Set the translation spacing;

[0011] According to the translation interval, the measured object is translated multiple times at equal intervals along the coordinate axis.

[0012] Furthermore, after the measured object is gradually translated along the coordinate axis and before the change in the inertia tensor and the change in the gravitational potential energy are detected, a standby waiting process is performed.

[0013] Furthermore, detecting the change in the inertia tensor and the change in the gravitational potential energy of the object at the position point includes:

[0014] Determine at least 6 different rotation axes; the angles between any of the rotation axes and each coordinate axis form angle combinations corresponding to the rotation axes;

[0015] For any of the rotation axes, detecting the moment of inertia of the measured object corresponding to the rotation of the rotation axis;

[0016] Combining the moments of inertia, the inertia tensor of the measured object at the 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 according to the inertia tensor corresponding to the position point and the inertia tensor corresponding to the previous position point.

[0018] Furthermore, the combining of the moments of inertia to calculate the inertia tensor of the measured object at the position point includes:

[0019] Establishing the equation

[0020] I H =I xx cos 2 α+I yy cos 2 β+I zz cos 2 γ-2I xy cosαcosβ-2I xz cosαcosγ-2I yz cosβcosγ

[0021] Among them, I H represents the moment of inertia of the measured object rotating around the rotation axis, I xs Indicates the moment of inertia of the object being measured around the X-axis, I yy Indicates the moment of inertia of the object being measured around the Y coordinate axis, I zz Indicates the moment of inertia of the object being measured around the Z coordinate axis, I xy , I xz and I yz represents the product of inertia, α is the angle between the rotation axis and the X-axis, β is the angle between the rotation axis and the Y-axis, and γ is the angle between the rotation axis and the Z-axis;

[0022] When α, β and γ are the values ​​of each angle combination, I H Solve the equations of the moment of inertia I xx , I yy , I zz and product of inertia I xy , I xz and I yz .

[0023] Furthermore, detecting the change in the inertia tensor and the change in the gravitational potential energy of the object at the position point includes:

[0024] The object to be measured is suspended at the lower end of the optical lever system of the secondary pendulum;

[0025] Starting a rotating platform located below the measured object in the secondary pendulum to drive a source mass block mounted on the rotating platform to perform periodic rotational motion, so that the source mass block generates a time-varying gravitational potential;

[0026] detecting, by means of the optical lever system, a deflection angle of the measured object caused by the time-varying gravitational potential;

[0027] Calculating the gravitational potential energy of the measured object at the position point according to the deflection angle;

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

[0029] Furthermore, the calculating of the gravitational potential energy of the measured object at the position point according to the deflection angle includes:

[0030] According to the formula

[0031] U=∫τdθ

[0032] τ=κθ

[0033] Calculation is performed; wherein, U is the gravitational potential energy of the object to be measured at the position point, τ is the fiber torque of the optical lever system, κ is the fiber torsional stiffness of the optical lever system, and θ is the deflection angle.

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

[0035] Obtaining a formula for the relationship between the gravitational potential energy and the inertia tensor of the measured object;

[0036] According to the relationship formula between the gravitational potential energy and the inertia tensor, the change in the inertia tensor and the change in the gravitational potential energy, a nonlinear relationship between the change in the gravitational potential energy and the position of the measured object is established by the least squares method;

[0037] According to the nonlinear relationship, the center of mass position component of the measured object on the coordinate axis is obtained by fitting.

[0038] Furthermore, the relationship formula between the gravitational potential energy and the inertia tensor of the measured object includes:

[0039]

[0040] Wherein, U(P) represents the gravitational potential energy of the object under test relative to point P, G is the gravitational constant, M is the mass of the object under test, r represents the distance between the object under test and point P, σ c is the density of the object being measured, ∫∫∫ V Indicates the volume integration of the space occupied by the measured object, and represents the inertia tensor of the object being measured.

[0041] Furthermore, fitting the center of mass position component of the object under test on the coordinate axis according to the nonlinear relationship includes:

[0042] Set the minimization of the residual sum of squares as the objective;

[0043] Using the Gauss-Newton method, solving the parameter vector to be estimated in the nonlinear relationship through local linearization iteration;

[0044] The center of mass position component of the measured object on the coordinate axis is determined based on the solved parameter vector to be estimated.

[0045] The beneficial effects of the present invention are as follows: the center of mass measurement method based on the expansion of the gravitational potential energy multipole moment in the embodiment realizes the joint calibration of multiple physical quantities to improve the accuracy by simultaneously measuring the change of the inertia tensor and the change of the gravitational potential energy; the specific process is: simultaneously measuring the gravitational potential energy multipole moment and the inertia tensor, combining the multipole moment expansion formula of the gravitational potential energy, expressing the potential energy as a function of the mass, the center of mass position and the inertia tensor, fitting multiple sets of displacement data by the least squares method to solve the center of mass coordinates, using the least squares method fitting to eliminate the systematic error in the calibration of a single physical quantity, dynamically correcting the center of mass offset error, being able to handle the measured objects with asymmetric and complex shapes, and being suitable for the center of mass position measurement of precision devices such as spacecraft. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 Schematic diagram of measuring an object using a two-stage pendulum device in an embodiment;

[0047] Figure 2 Schematic diagram of the coordinate system in the embodiment;

[0048] Figure 3 Schematic diagram of the steps of the center of mass measurement method based on gravitational potential energy multipole moment expansion in an embodiment. DETAILED DESCRIPTION

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

[0050] Specifically, let point P(a, b, c) be a point outside the measured object C. The general formula for the gravitational potential energy of the measured object C at point P is:

[0051]

[0052] Where U(P) is the value of gravitational potential energy at point P, G is the universal gravitational constant, σ C is the density of the object C being measured, is the distance from point P to a point on object C in the Cartesian coordinate system.

[0053] Based on the above potential energy formula, the potential energy generated by the measured object C at point P can be expressed as Taylor series:

[0054]

[0055] The object being measured C is about the center of mass C c The mass moment of is:

[0056]

[0057] Sum It is performed on the set of all non-negative integers N1, N2, N3 that satisfy N=N1+N2+N3.

[0058] In this embodiment, N=2, and the Taylor series expansion of the potential energy formula is:

[0059]

[0060] in

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

[0062]

[0063] Available

[0064]

[0065] Therefore, the gravitational potential energy formula of the measured object C at point P can be written as:

[0066]

[0067] Where M is the mass of the object C being measured, σ c is the density of the object, ∫∫∫ V It represents the volume integral of the space occupied by the measured object. The x, y, and z outside the integral represent the center of mass C of the measured object C. c Coordinates in the Cartesian coordinate system.

[0068] If the density distribution of the measured object C is uniform, then the gravitational potential energy formula of the measured object C at point P can be further written as:

[0069]

[0070] The formula for the gravitational potential energy of the object C at point P derived above is:

[0071]

[0072] It represents the gravitational potential energy U(P) and inertia tensor of the measured object C and The relationship between them can serve as the theoretical basis for the center of mass measurement method based on the multipole moment expansion of gravitational potential energy.

[0073] In this embodiment, the object to be measured for the center of mass measurement method based on the expansion of gravitational potential energy multipole moment can be a precision device such as a spacecraft, and specifically can be a device such as an artificial satellite. When executing the center of mass measurement method based on the expansion of gravitational potential energy multipole moment, you can use Figure 1 The secondary pendulum device shown performs relevant measurements on the object being measured.

[0074] Reference Figure 1 The secondary pendulum device used includes a magnetic damper 1, a quartz wire 2, a reflector 3, a crossbar 4, a counterweight mass block 5, a source mass block 7 and a rotating platform 8, wherein the magnetic damper 1, the quartz wire 2, the reflector 3, the crossbar 4 and the counterweight mass block 5 constitute an optical lever system. Figure 1, the counterweight mass 5 is installed at one end of the crossbar 4, and when measuring, the object to be measured 6 is suspended at the other end of the crossbar 4 by a fiber. The secondary pendulum device can achieve multi-degree-of-freedom measurement by changing the position of the crossbar 4, etc., thereby measuring parameters such as the deflection angle and translational displacement of the object to be measured 6. For example, a laser beam is emitted to the reflector 3, and the reflector 3 will reflect the laser beam in a certain direction. The reflected laser beam forms a light spot that can be detected by the detector; when the object to be measured 6 is deflected or displaced by force, the object to be measured 6 will transmit the force through the fiber and crossbar 4 on which it is suspended, causing the reflector 3 to deflect. The deflection of the reflector 3 causes the laser beam to cause the light spot to displace, which is detected by the detector, thereby calculating the corresponding deflection angle.

[0075] In this embodiment, refer to Figure 3 The center of mass measurement method based on the gravitational potential energy multipole moment expansion includes the following steps:

[0076] S1. Determine multiple coordinate axes of the space where the object to be measured is located;

[0077] S2. Traverse all coordinate axes:

[0078] For any coordinate axis that is traversed, the object under test is gradually translated along the coordinate axis. For any position point to which the object under test is translated, the change in the inertia tensor and the change in the gravitational potential energy of the object under test at the position point are detected. Based on the change in the inertia tensor and the change in the gravitational potential energy corresponding to each position point, the center of mass position component of the object under test on the coordinate axis is determined;

[0079] S3. Determine the center of mass position of the object being measured according to the corresponding center of mass position components of each coordinate axis.

[0080] In step S1, the rotation axis and the pendulum platform of the secondary pendulum device can be used as a reference to construct a Cartesian coordinate system as a reference frame for subsequent position control and data calculation. Figure 2 , the position of the magnetic damper 1 can be used as the origin O, and an XY plane can be established with a plane horizontal to the ground to establish an X coordinate axis and a Y coordinate axis, and a Z coordinate axis perpendicular to the X coordinate axis and the Y coordinate axis can be established to obtain multiple coordinate axes such as the X coordinate axis, the Y coordinate axis and the Z coordinate axis. Figure 2 In the , you can also establish the inertial principal axis coordinate system (Ox p y p z p ).

[0081] In step S2, the same measurement and calculation are performed on the X-axis, the Y-axis, and the Z-axis, respectively. One of the axes, such as the X-axis, is used as an example for explanation.

[0082] Reference Figure 3For the X-axis, the translation interval Δx can be set. Starting from the starting point x0 on the X-axis, the object to be measured is translated along the X-axis according to the translation interval Δx each time, for a total of n translations, so that the object to be measured reaches the coordinate x on the X-axis after each translation. i Multiple location points (i=1, 2, ..., n).

[0083] In this embodiment, when the object being measured is translated on the X-axis, the object being measured can be kept stationary on the other coordinate axes (Y-axis and Z-axis). That is, the coordinates of the object being measured on the Y-axis and Z-axis remain unchanged. Similarly, when the object being measured is translated on the Y-axis, the object being measured can be kept stationary on the other coordinate axes (X-axis and Z-axis). When the object being measured is translated on the Z-axis, the object being measured can be kept stationary on the other coordinate axes (X-axis and Y-axis).

[0084] Assume that the object to be measured is moved from position x i-1 Translate to position x i , at position x i Specifically, the object to be measured can be moved to the position point x i Afterwards, before detecting the change in the inertia tensor and the change in the gravitational potential energy at that position, a static waiting process is performed. Specifically, the duration of the static waiting can be 3-5 data acquisition cycles, and the detection of the change in the inertia tensor and the change in the gravitational potential energy is not performed during the static waiting period. Through the static waiting process, the detection can be performed after the vibration generated by the translation of the measured object is attenuated, thereby reducing the detection error caused by the vibration. When measuring the change in the inertia tensor at each position point, the same method can be used to measure multiple changes in the inertia tensor, and their average value can be calculated as the final measured change in the inertia tensor, thereby reducing the measurement error; similarly, when measuring the change in the gravitational potential energy at each position point, the same method can be used to measure multiple changes in the gravitational potential energy, and their average value can be calculated as the final measured change in the gravitational potential energy.

[0085] In this embodiment, when at position x i To measure the change in inertia tensor and gravitational potential energy, you can perform the following steps:

[0086] S201. Determine at least 6 different rotation axes;

[0087] S202. For any rotation axis, detecting the moment of inertia of the object being measured corresponding to the rotation around the rotation axis;

[0088] S203. Combine the moments of inertia to calculate the inertia tensor of the object being measured at the position point; the inertia tensor includes the moment of inertia and the product of inertia;

[0089] S204. Determine the change in the inertia tensor based on the inertia tensor corresponding to the position point and the inertia tensor corresponding to the previous position point.

[0090] Steps S201-S204 are to measure the position x i The step of the change in the inertia tensor.

[0091] The principles of steps S201-S204 are as follows: Figure 2 shown.

[0092] Reference Figure 2 , when the object is translated to position x i After that, the rotation axis OH can be selected. The rotation axis OH is an arbitrary axis passing through the point O, that is, the selected rotation axis OH must pass through the origin O, and its direction can be determined arbitrarily. For a determined rotation axis OH, its angle with the coordinate axis OX (X coordinate axis) is α, the angle with the coordinate axis OY (Y coordinate axis) is β, and the angle with the coordinate axis OZ (Z coordinate axis) is γ, thereby forming the angle combination α, β, γ corresponding to the rotation axis OH, and the direction cosines cosα, cosβ, and cosγ corresponding to the rotation axis OH can be obtained. If the object to be measured is regarded as a rigid body, then the moment of inertia I of the object to be measured about the rotation axis OH H It can be expressed as:

[0093]

[0094] Among them I xx is the moment of inertia of the object being measured around the X-axis, I yy is the moment of inertia of the object being measured around the Y coordinate axis, I zz is the moment of inertia of the object being measured around the Z coordinate axis, I xy , I xz , I yz is the product of inertia. Moment of inertia I xx , I yy , I zz and product of inertia I xy , I xz , I yz It constitutes the inertia tensor corresponding to the rotation of the measured object around the rotation axis OH.

[0095] On the other hand, the moment of inertia I corresponding to the rotation of the measured object around the rotation axis OH H Specifically, when executing step S202, the following steps may be performed:

[0096] S20201. Apply external force to twist the object to an initial angle θ;

[0097] S2202. After releasing the external force, the object under test vibrates in a single-degree-of-freedom torsional pendulum.

[0098] When friction and air resistance are neglected, the formula is:

[0099]

[0100] Where: I is the moment of inertia of the torsion bar; k is the torsional stiffness; θ is the torsion angle, and c is a constant. When friction and air resistance are negligible, the damping coefficient is approximately zero. The free vibration period of the torsion pendulum system The moment of inertia of the torsion pendulum is I = kT 2 / (4π 2 ). Assume that the period when no load (torsion pendulum) is T0, and the period after loading the object is T x , the moment of inertia of the object being measured can be obtained by Calculation, where m = k / (4π 2 ) is the system constant. After the torsion pendulum test equipment is determined, k and T0 are known quantities, and the moment of inertia I of the object under test can be calculated by measuring the period T1 after loading. H .

[0101] According to the above principle, for any given rotation axis OH, its moment of inertia I H On the one hand, it can be obtained through actual measurement, and on the other hand, it satisfies:

[0102] I H =I xx cos 2 α+I yy cos 2 β+I zz cos 2 γ-2I xy cosαcosβ-2I xz cosαcosγ-2I yz cosβcosγ

[0103] Among them, α, β, and γ are the angle combinations related to the rotation axis OH. H There is a moment of inertia I xx , I yy , I zz and product of inertia I xy , I xz , I yz The six unknown quantities can be obtained by combining six linear equations.

[0104] Therefore, in step S201, at least 6 different rotation axes are determined; in step S202, the moment of inertia I corresponding to the rotation of the object under test around the rotation axis is detected for each rotation axis. H , so we get 6 linearly unrelated I HIn step S203, by solving the equation group, the moment of inertia I can be obtained. xx , I yy , I zz and product of inertia I xy , I xz , I yz Isoinertia tensor.

[0105] Through steps S201-S203, it can be calculated that the object being measured is at position x i The moment of inertia I xx , I yy , I zz and product of inertia I xy , I xz , I yz , and based on the same principle, we can also get the position of the object being measured at the previous point, that is, position x i-1 In step S204, the object to be measured can be placed at position x i The moment of inertia I xx Subtract the value at position x i-1 The moment of inertia I xx , thus obtaining the object being measured at position x i ΔI in the change of moment of inertia xx Based on the same principle, we can also get ΔI yy , ΔI zz , ΔI xy , ΔI xz , ΔI yz , thus constituting the change in moment of inertia.

[0106] In this embodiment, when at position x i To measure the change in inertia tensor and gravitational potential energy, you can perform the following steps:

[0107] S205. The object to be measured is suspended at the lower end of the optical lever system of the secondary pendulum;

[0108] S206. Start the rotating platform of the secondary pendulum located below the object being measured, driving the source mass block mounted on the rotating platform to perform periodic rotational motion, so that the source mass block generates a time-varying gravitational potential;

[0109] S207. Detecting the deflection angle of the object under test due to the time-varying gravitational potential through an optical lever system;

[0110] S208. Calculate the gravitational potential energy of the object at the position according to the deflection angle;

[0111] S209. Determine the change in gravitational potential energy based on the inertia tensor corresponding to the position point and the inertia tensor corresponding to the previous position point.

[0112] Steps S205-S209 are to measure the position x i The step of the change in gravitational potential energy.

[0113] In step S205, Figure 1 Hang the object to be measured as shown.

[0114] In step S206, the rotating platform of the secondary pendulum, located below the object under test, is activated to drive the source mass mounted on the rotating platform to perform periodic rotational motion, causing the source mass 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 reverse and then stopping, then rotating forward again and then stopping, thereby driving the source mass to generate an alternating time-varying gravitational potential. This alternating time-varying gravitational potential exerts a periodically changing gravitational force on the object under test, thereby subjecting the object to an alternating torque.

[0115] In step S207 , the alternating torque applied to the object under test is transmitted to the reflector 3 , so that the detector detects the deflection angle θ of the object under test.

[0116] In step S208, the formula

[0117] U=∫τdθ

[0118] τ=κθ

[0119] Calculate the position x of the object being measured i The gravitational potential energy U on the optical lever is given by κ, where κ is the torsional stiffness of the optical lever system fiber (quartz wire 2), and τ is the torque generated by the optical lever system fiber. The integration limit of ∫ can be from 0 to θ.

[0120] Through steps S205-S208, it can be calculated that the object being measured is at position x i The gravitational potential energy Y of the object under test can also be obtained based on the same principle at the previous position point x. i-1 In step S209, the object to be measured can be placed at position x i The gravitational potential energy Y minus the gravitational potential energy at position x i-1 The gravitational potential energy of the object being measured is obtained at the position point x i The change in gravitational potential energy ΔU i .

[0121] For the object to be measured, the i The change in gravitational potential energy ΔU i and the change in inertia tensor ΔI xx , ΔI yy , ΔI zz, ΔI xy , ΔI xz , Δi yz Etc., according to the relationship formula between the gravitational potential energy and inertia tensor of the measured object:

[0122]

[0123] ΔU can be i As the change of U(p), ΔI xx , ΔI yy , Δi zz , ΔI xy , ΔI xz , ΔI yz Considered as The nonlinear relationship between the change in gravitational potential energy and the position of the measured object is established by the least squares method:

[0124] ΔU i =f(x i ,β)+∈ i

[0125] Where β is the parameter vector to be estimated, ∈ i is the error, f() means x i ,β is mapped to the change in gravitational potential energy ΔU i The error between i nonlinear function.

[0126] For the above nonlinear relationship, the Gauss-Newton method can be used to solve the parameter vector to be estimated in the nonlinear relationship through local linearization iteration:

[0127] β (k+1) =β (k) +(J T J) -1 J T r

[0128] Where J is the Jacobian matrix, r is the residual vector, and k is the number of iterations. Set the goal to minimize the residual sum of squares, that is, the iteration goal is:

[0129]

[0130] Where n is the number of times the object is translated on the X-axis. By minimizing S(β), the specific value of the estimated parameter vector β is determined, and the nonlinear relationship ΔU is determined. i =f(x i ,β)+∈ i .

[0131] In this embodiment, according to the nonlinear relationship ΔU i=f(x i ,β)+∈ i The center of mass position component x of the object being measured on the X-axis is obtained by fitting C For example, ΔU can be determined i = 0 (or closest to 0) as the position of the center of mass of the object being measured on the X-axis. C .

[0132] Reference Figure 3 After executing step S2, determine the center of mass position component x of the object to be measured on the X coordinate axis C After that, the center of mass position component y of the object being measured on the Y coordinate axis is measured and calculated. C , then measure and calculate the center of mass position component z of the object being measured on the Z coordinate axis C At this point, all coordinate axes have been calculated, and the coordinates of the center of mass of the object being measured are obtained as x C ,y C , z C .

[0133] The center of mass measurement method based on the expansion of gravitational potential energy multipole moments in this embodiment improves accuracy through the joint calibration of multiple physical quantities. It simultaneously measures the gravitational potential energy multipole moment and the inertia tensor. Combining the multipole moment expansion formula for gravitational potential energy, it expresses potential energy as a function of mass, center of mass position, and inertia tensor. The center of mass coordinates are solved by fitting multiple sets of displacement data using the least squares method. This least squares fitting eliminates systematic errors in the calibration of a single physical quantity. A secondary pendulum can achieve multi-degree-of-freedom measurement by changing the position of the pendulum rod. Its kinetic and potential energy models are sensitive to position changes. The center of mass measurement method based on the expansion of gravitational potential energy multipole moments in this embodiment leverages this characteristic to dynamically correct center of mass offset errors through multiple displacement measurements. Traditional center of mass measurement relies on geometric symmetry, while the center of mass measurement method based on the expansion of gravitational potential energy multipole moments in this embodiment can handle asymmetric and complex objects through the expansion of gravitational potential energy multipole moments, making it suitable for precision devices such as spacecraft.

[0134] It should be noted that, unless otherwise specified, when a feature is referred to as being "fixed" or "connected" to another feature, it may be directly fixed or connected to the other feature, or it may be indirectly fixed or connected to the other feature. In addition, the descriptions of up, down, left, right, etc. used in this disclosure are only relative to the relative positional relationships of the components of the present disclosure in the accompanying drawings. The singular forms of "a", "" and "the" used in this disclosure are also intended to include the plural forms, unless the context clearly indicates otherwise. In addition, unless otherwise defined, all technical and scientific terms used in this embodiment have the same meaning as those generally understood by those skilled in the art. The terms used in the specification of this embodiment are only for describing specific embodiments and are not intended to limit the invention. The term "and / or" used in this embodiment includes any combination of one or more related listed items.

[0135] It should be understood that, although the present disclosure may adopt the term first, second, third etc. to describe various elements, these elements should not be limited to these terms.These terms are only used to distinguish the elements of the same type from each other.For example, without departing from the scope of the present disclosure, the first element may also be referred to as the second element, and similarly, the second element may also be referred to as the first element.The use of any and all examples or exemplary language ("for example", "such as" etc.) provided by the present embodiment is only intended to better illustrate embodiments of the present invention, and unless otherwise required, the scope of the present invention will not be limited.

[0136] It should be appreciated that embodiments of the present invention can be implemented or practiced by computer hardware, a combination of hardware and software, or by computer instructions stored in a non-transitory computer-readable memory. The methods can be implemented in a computer program using standard programming techniques - including a non-transitory computer-readable storage medium configured with a computer program, wherein the storage medium so configured causes the computer to operate in a specific and predefined manner - according to the methods and figures 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. In addition, the program can be run on a programmed application-specific integrated circuit for this purpose.

[0137] In addition, the operations of the process described in this embodiment may be performed in any suitable order, unless otherwise indicated in this embodiment or otherwise clearly contradicted by the context. The process described in this embodiment (or variations and / or combinations thereof) may be performed under the control of one or more computer systems configured with executable instructions, and may be implemented as a code (e.g., executable instructions, one or more computer programs, or one or more applications) executed on one or more processors, by hardware or a combination thereof. A computer program includes a plurality of instructions that may be executed by one or more processors.

[0138] Furthermore, the method can be implemented in any type of computing platform that is operably connected to a suitable computer, including but not limited to a personal computer, a minicomputer, a mainframe, a workstation, a network or distributed computing environment, a separate or integrated computer platform, or in communication with a charged particle tool or other imaging device, etc. Various aspects of the present 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, an optical read and / or write storage medium, RAM, ROM, etc., so that it can be read 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 process described herein. In addition, the machine-readable code, or portions thereof, can be transmitted over a wired or wireless network. When such media includes instructions or programs that implement the above steps in conjunction with a microprocessor or other data processor, the invention of this embodiment includes these and other different types of non-transitory computer-readable storage media. When programmed according to the methods and techniques of the present invention, the present invention also includes the computer itself.

[0139] The computer program can be applied to input data to perform the functions of the present embodiment, thereby converting the input data to generate output data that is stored in a 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 present invention, the converted data represents a physical and tangible object, including a specific visual depiction of the physical and tangible object produced on the display.

[0140] The above are merely preferred embodiments of the present invention. The present invention is not limited to the aforementioned embodiments. As long as the technical effects of the present invention are achieved by the same means, any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention shall be included within the scope of protection of the present invention. Within the scope of protection of the present invention, various modifications and variations of the technical solutions and / or implementation methods may be made.

Claims

1. A method for measuring the center of mass based on the expansion of gravitational potential energy multipole moment, characterized in that: The center of mass measurement method based on gravitational potential energy multipole moment expansion includes: Determine multiple coordinate axes of the space where the object being measured is located; Traversing all the coordinate axes, for any of the traversed coordinate axes, gradually translating the object under test along the coordinate axis, for any position point to which the object under test is translated, detecting the change in the inertia tensor and the change in the gravitational potential energy of the object under test at the position point, and determining the center of mass position component of the object under test on the coordinate axis based on the change in the inertia tensor and the change in the gravitational potential energy corresponding to each position point; The center of mass position of the object to be measured is determined according to the center of mass position components corresponding to each of the coordinate axes.

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

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

4. The method for measuring the center of mass based on the expansion of gravitational potential energy multipole moments according to claim 1, characterized in that: The detecting of the change in the inertia tensor and the change in the gravitational potential energy of the object at the position point includes: Determine at least 6 different rotation axes; the angles between any of the rotation axes and each coordinate axis form angle combinations corresponding to the rotation axes; For any of the rotation axes, detecting the moment of inertia of the measured object corresponding to the rotation of the rotation axis; Combining the moments of inertia, the inertia tensor of the measured object at the 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 according to the inertia tensor corresponding to the position point and the inertia tensor corresponding to the previous position point.

5. The method for measuring the center of mass based on the expansion of gravitational potential energy multipole moments according to claim 4, characterized in that: The combining of the moments of inertia to calculate the inertia tensor of the measured object at the position point includes: Establishing the equation I H =I xx cos 2 α+I yy cos 2 β+I zz cos 2 γ-2I xy cosαcosβ-2I xz cosαcosγ-2I yz cosβcosγwhere I H represents the moment of inertia of the measured object rotating around the rotation axis, I xx Indicates the moment of inertia of the object being measured around the X-axis, I yy Indicates the moment of inertia of the object being measured around the Y coordinate axis, I zz Indicates the moment of inertia of the object being measured around the Z coordinate axis, I xy , I xz and I yz represents the product of inertia, α is the angle between the rotation axis and the X-axis, β is the angle between the rotation axis and the Y-axis, and γ is the angle between the rotation axis and the Z-axis; When α, β and γ are the values ​​of each angle combination, I H Solve the equations of the moment of inertia I xx , I yy , I zz and product of inertia I xy , I xz and I yz .

6. The method for measuring the center of mass based on the expansion of gravitational potential energy multipole moments according to claim 1, characterized in that: The detecting of the change in the inertia tensor and the change in the gravitational potential energy of the object at the position point includes: The object to be measured is suspended at the lower end of the optical lever system of the secondary pendulum; Starting a rotating platform located below the measured object in the secondary pendulum to drive a source mass block mounted on the rotating platform to perform periodic rotational motion, so that the source mass block generates a time-varying gravitational potential; detecting, by means of the optical lever system, a deflection angle of the measured object caused by the time-varying gravitational potential; Calculating the gravitational potential energy of the measured object at the position point according to the deflection angle; The change in gravitational potential energy is determined based on the inertia tensor corresponding to the position point and the inertia tensor corresponding to the previous position point.

7. The method for measuring the center of mass based on the expansion of gravitational potential energy multipole moments according to claim 6, characterized in that: Calculating the gravitational potential energy of the measured object at the position point according to the deflection angle includes: According to the formula U=∫τdθ τ=kθ Calculation is performed; wherein, U is the gravitational potential energy of the object to be measured at the position point, τ is the fiber torque of the optical lever system, κ is the fiber torsional stiffness of the optical lever system, and θ is the deflection angle.

8. The method for measuring the center of mass based on the expansion of gravitational potential energy multipole moments according to any one of claims 1 to 7, characterized in that: Determining the mass center position component of the measured object on the coordinate axis according to the inertia tensor change and the gravitational potential energy change corresponding to each position point includes: Obtaining a formula for the relationship between the gravitational potential energy and the inertia tensor of the measured object; According to the relationship formula between the gravitational potential energy and the inertia tensor, the change in the inertia tensor and the change in the gravitational potential energy, a nonlinear relationship between the change in the gravitational potential energy and the position of the measured object is established by the least squares method; According to the nonlinear relationship, the center of mass position component of the measured object on the coordinate axis is obtained by fitting.

9. The method for measuring the center of mass based on the expansion of gravitational potential energy multipole moments according to claim 8, characterized in that: The relationship formula between the gravitational potential energy and the inertia tensor of the measured object includes: Wherein, U(P) represents the gravitational potential energy of the object under test relative to point P, G is the gravitational constant, M is the mass of the object under test, r represents the distance between the object under test and point P, σ c is the density of the object being measured, ∫∫∫ V Indicates the volume integration of the space occupied by the measured object, and represents the inertia tensor of the object being measured.

10. The method for measuring the center of mass based on the expansion of gravitational potential energy multipole moments according to claim 8, characterized in that: The step of fitting the center of mass position component of the object under test on the coordinate axis according to the nonlinear relationship includes: Set the minimization of the residual sum of squares as the objective; Using the Gauss-Newton method, solving the parameter vector to be estimated in the nonlinear relationship through local linearization iteration; The center of mass position component of the measured object on the coordinate axis is determined based on the solved parameter vector to be estimated.

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