Gyroscope accelerometer dynamic error calibration method considering angular motion of base

Abstract

The invention discloses a gyro accelerometer dynamic error modeling and calibration method considering angular motion of a base, and belongs to the field of dynamics and control. The implementation method comprises the following steps: establishing kinematics and dynamics equations of the PIGA under the triaxial angular motion of the base, and deducing a complete error model comprising a static error term, a dynamic error term and a nonlinear error term; constructing a kinetic equation including inertia tensor, a moment balance equation and servo loop influence; quantitatively analyzing the influence of the angular motion parameters of the base on PIGA output errors; based on three-axis turntable experimental design, dynamic response of each axis of the PIGA is excited through single-axis, double-axis and three-axis angular velocity input, and error model coefficients are separated and calibrated; error model parameters are calibrated by adopting a mode of combining a least square method and other data fitting algorithms; angular velocity and angular acceleration terms are separated based on direction cosine matrix transformation, and error coefficient aliasing is avoided; dynamic response data are fitted through a least square method, and all error coefficients are calibrated at a time.

Description

Technical Field

[0001] This invention relates to a novel method for calibrating the error model of a pendulum integral gyroscope accelerometer (PIGA), and more particularly to a method for calibrating the coefficients of dynamic error in the error model based on the precession angles α and β of the PIGA under a moving base condition, belonging to the field of dynamics and control. Background Technology

[0002] Since the late 20th century, with the development of information technology, materials research and development, and even artificial intelligence, the application of inertial technology has become more widespread. It has formed close connections and intersections with multiple disciplines such as control science, microelectronics, precision instruments, and computer science, playing an increasingly significant role in fields such as aviation, aerospace, navigation, and medicine. It can be said that the level of development of inertial technology is closely related to the overall scientific and technological level of a country. Currently, the main types of accelerometers include pendulum integrating gyroscope accelerometers, flexible pendulum accelerometers, quartz vibrating beam accelerometers, silicon micromechanical accelerometers, micro-optical accelerometers, atomic accelerometers, and optomechanical accelerometers, among which the pendulum integrating gyroscope accelerometer (PIGA) has an output accuracy of 10. -8 g~10 -5 g is a technologically mature and highly accurate accelerometer.

[0003] As an important inertial measurement device, the error model of PIGA has evolved from simple to complex, from static to dynamic, and from single-source to multi-source coupling. By establishing a comprehensive error model, the error performance of PIGA in complex environments can be more comprehensively described and analyzed. This process has deepened with the advancement of inertial navigation technology, computer technology, and sensor technology. PIGA applications often face complex mechanical environments, such as vibration, tilt, and rotation. Under high-speed motion or severe vibration, the output of PIGA may be affected by the angular velocity and instability of the base, thus affecting the accuracy of the measurement results. Since the angular motion of the base causes dynamic errors in the gyro accelerometer, current research mainly focuses on the calibration of the error model of the gyro accelerometer under static base conditions, while research on the calibration of the dynamic error model of the gyro accelerometer is insufficient. Moreover, traditional calibration methods based on fixed base turntables require high-precision turntable equipment, while dynamic base calibration methods can reduce the requirements for equipment, improve the feasibility of on-site identification and calibration, and help to better understand and compensate for the impact of these environmental factors on system performance, thereby improving the accuracy and reliability of inertial navigation systems.

[0004] Therefore, studying the influence of base angular motion on the output of gyro accelerometers and its compensation methods is of great significance for improving the accuracy of navigation systems in practical applications. Summary of the Invention

[0005] To address the issue of low measurement accuracy of PIGA due to inaccurate calibration of relevant error model coefficients under dynamic environments, this invention aims to provide a dynamic error modeling and calibration method for gyro accelerometers considering base angular motion. This method establishes a complete error model for PIGA under moving base conditions, including dynamic and static errors. Each error parameter is analyzed individually, clarifying the impact of static, dynamic, and nonlinear errors of the gyro accelerometer on the accelerometer's output accuracy, ultimately identifying the parameters requiring calibration. Each parameter is calibrated to make the simulation of PIGA under experimental conditions more consistent with the actual working environment of PIGA, thereby improving the output accuracy of PIGA under dynamic environments.

[0006] This invention is achieved through the following technical solution:

[0007] This invention discloses a method for dynamic error modeling and calibration of a gyro accelerometer considering the angular motion of the base. It employs Euler's dynamic equations to establish the differential equations of motion of the PIGA (piga accelerometer) about the inner frame axis, and simplifies and transforms these equations to obtain... The expression is used to derive the expression for each error coefficient. Each error is analyzed, and the negligible errors are eliminated by controlling the PIGA's integer precession. The errors that need to be retained are calibrated to obtain a simplified error model. A transformation matrix for the PIGA's moving base three-axis coordinate system is established. The optimal angular motion projection method is obtained by controlling the angular velocity input of the base's three axes. This method is used to input the base's three-axis angular velocities into the PIGA's three axes and substitute them into the simplified error model. The base's angular velocity is applied as a harmonic variation to calibrate the angular acceleration error term. The differential equations of motion of the PIGA around the inner frame axis and the PIGA around the outer frame axis are solved simultaneously. The trend of the α angle is observed, and the time point where the α angle is 2π under different angular velocity inputs and the corresponding angular acceleration are found. Based on the number of calibration parameters n, 2n simulations are performed. To prevent singularity phenomena in the matrix during data fitting that would result in no solution, the PIGA is upright in the first n simulations, so that its sensitive axis is opposite to the direction of gravitational acceleration g. In the last n simulations, the PIGA is inverted, so that its sensitive axis is in the same direction as gravitational acceleration g. The PIGA is calibrated using the least squares method to obtain the output matrix Y, the coefficient matrix A, and the error coefficient matrix b. This achieves the dynamic error modeling and calibration of the gyro accelerometer considering the angular motion of the base.

[0008] The present invention discloses a method for dynamic error modeling and calibration of a gyro accelerometer considering base angular motion, comprising the following steps:

[0009] Step 1: Establish the kinematic and dynamic equations of PIGA under the triaxial angular motion of the base, and derive the complete error model including static error terms, dynamic error terms and nonlinear error terms;

[0010] Step 2: Analyze the angular velocity transmission relationship from the base coordinate system to the outer frame, inner frame and rotor coordinate system through coordinate transformation, and construct dynamic equations that include inertia tensor, torque balance equation and servo loop influence;

[0011] Step 3: Quantitatively analyze the impact of base angular motion parameters on PIGA output error; base angular motion parameters include the order of magnitude of angular velocity, angular acceleration, and cross-coupling terms;

[0012] Step 4: Based on the three-axis turntable experimental design, the dynamic response of each axis of PIGA is excited by single-axis, dual-axis and three-axis angular velocity inputs, and the error model coefficients are separated and calibrated.

[0013] Step 5: The error model parameters are calibrated by combining the least squares method with other data fitting algorithms, and the accuracy of the calibration results is verified by numerical simulation.

[0014] The angular velocity and angular acceleration terms are separated by direction cosine matrix transformation to avoid the aliasing of error coefficients; the dynamic response data is fitted by least squares method to calibrate all error coefficients at once.

[0015] Furthermore, it also includes step 6: using the EV model method to cross-validate the least squares calibration results to improve calibration accuracy.

[0016] Furthermore, the method for implementing step 1 is as follows:

[0017] The differential equations of motion for PIGA's rotation about the outer frame axis and the inner frame axis are obtained using Euler's equations of motion; the equations of motion about the outer frame axis are:

[0018]

[0019] Where: α and β are the precession angles of the outer frame and the inner frame, respectively. The moment of inertia of the outer frame system about the OY1 axis, The moment of inertia of the inner frame system about the OZ2 axis, ω is the inertia product of the inner frame system. x0 ω y0 ω z0 ω is the angular velocity of the base relative to inertial space along the corresponding axis; x1 ω y1 ω z1 Let be the absolute angular velocity of the outer frame along the corresponding axis; H is the rotational angular momentum of the rotor. This is the disturbance torque of the outer frame axis. The expression is as follows:

[0020]

[0021] Among them, J e The moment of inertia of the rotor shaft

[0022] Equations of motion about the inner frame axis:

[0023]

[0024] Where α and β are the precession angles of the outer frame and the inner frame, respectively. ω is the inertia product of the inner frame system. x0 ω y0 ω z0 ω is the angular velocity of the base relative to inertial space along the corresponding axis; x1 ω y1 ω z1 Let be the absolute angular velocity of the outer frame along the corresponding axis; H is the rotational angular momentum of the rotor. This is the disturbance torque of the outer frame axis.

[0025] Equation (2) yields the complete error model for PIGA:

[0026]

[0027] By eliminating the relevant error terms through integer precession, the complete error model is obtained in the following form:

[0028]

[0029] Where K0 is a constant error term, K x This is the input axis scaling factor. Δ xx Δ yy Δ zz δ represents the coefficients of the squared terms of angular acceleration and angular velocity. x The coefficients for the acceleration and angular velocity coupling terms.

[0030] Furthermore, in step 3,

[0031] The term represents the sensitive axis angular acceleration error, on the order of 10. -5 Priority calibration is required;

[0032] Δ xx Δ yy Δ zz The items are the centrifugal force error of the sensitive shaft, the centrifugal force error of the output shaft, and the centrifugal force error of the rotor shaft, with an order of magnitude of 10. -6It requires separation calibration via dual-axis angular velocity input;

[0033] δ x The term represents the sensitive axis angular velocity-acceleration coupling error, on the order of 10. -6 It needs to be calibrated by combining acceleration and angular velocity coupling experiments.

[0034] Furthermore, in step 4,

[0035] Establish the transformation matrices for the outer, middle, and inner rings of the moving base in a three-axis coordinate system; let the platform coordinate system of the three-axis rotary table be OX. T Y T Z T The outer ring coordinate system is OX. O Y O Z O The middle ring coordinate system is -OX. M Y M Z M The inner ring axis coordinate system is OX. I Y I Z I The coordinate system of the inertial measurement unit housing, i.e., the PIGA base, is O-X0Y0Z0; in the initial state, all corresponding axes coincide, OY... I Pointing towards the inner ring axis, OX M Pointing towards the central axis, OZ O Pointing towards the outer ring axis; PIGA is fixed to the inner ring; the initial orientation is taken as the local geographic coordinate system East (E)-North (N)-Sky (ξ);

[0036] The angular velocities of the three-axis rotary table are ω TX ω TY ω TZ After a period of time t, the angles they have rotated through are θ. X θ Y θ Z , (θ X =ω TX t, θ Y =ω TY t, θ Z =ω TZ t), θ1, θ2, and θ3 are the initial phases of the outer ring, middle ring, and inner ring, respectively, from OX T Y T Z T To OX O Y O Z O The rotation transformation matrix between the two coordinate systems is as follows:

[0037]

[0038] From OXO Y O Z O To OX M Y M Z M The rotation transformation matrix between the two coordinate systems is as follows:

[0039]

[0040] From OX M Y M Z M To OX I Y I Z I The rotation transformation matrix between the two coordinate systems is as follows:

[0041]

[0042] Based on the final form of the error model given by equation (4), by controlling the single-axis angular velocity input in the three rings, the dual-axis angular velocity input in the three rings, and the three-axis angular velocity input in the three rings, the base motion mode that can calibrate all error coefficients at once is analyzed and selected.

[0043] Furthermore, in step 4,

[0044] ① X-axis single-axis input angular velocity mode

[0045] Input the angular velocity ω along the X-axis of the turntable. Tx The direction cosine matrix between the coordinate systems during rotation along each axis After projection transformation, the expressions for PIGA's three-axis angular velocity and angular acceleration are obtained as follows:

[0046]

[0047] The angular acceleration is obtained from equation (9):

[0048]

[0049] Substituting the expressions of equations (9) and (10) into equation (5), we get:

[0050]

[0051] Get

[0052]

[0053] From equation (12), we know that, since ω y0 ω z0 It is 0, therefore Δyy Δ zz The three terms are not reflected in the expression. The method of applying angular velocity to the X-axis alone cannot calibrate these three errors; it can only calibrate the other coefficients.

[0054] ② Y-axis single-axis input angular velocity mode

[0055] Apply an angular velocity ω separately to the Y-axis of the turntable Ty The direction cosine matrix between the coordinate systems when rotating along each axis After projection transformation, the expressions for PIGA's three-axis angular velocity and angular acceleration are obtained as follows:

[0056]

[0057] The angular acceleration is obtained from equation (13):

[0058]

[0059] Substituting equations (13) and (14) into equation (5) yields

[0060]

[0061] Get

[0062]

[0063] From equation (16), the simplified error model can only be applied to K0 and K x Δ yy Two coefficients were calibrated, but the remaining coefficients were not reflected in the expression because the relevant shaft angular velocity was 0, so they could not be calibrated.

[0064] ③Z-axis single-axis input angular velocity mode

[0065] Input the angular velocity ω separately along the Z-axis of the turntable. Tz Through the direction cosine matrix between coordinate systems ω Tz The expressions for the angular velocities and angular accelerations along the X, Y, and Z axes, projected onto the three axes of the PIGA model, are as follows:

[0066]

[0067] The angular acceleration is obtained from equation (17):

[0068]

[0069] Substituting equations (17) and (18) into equation (5) yields

[0070]

[0071] That is, we get:

[0072]

[0073] According to equation (20), when an angular velocity is applied to the outer ring of the base and projected onto the three axes of the PIGA, all three axes of the PIGA can obtain angular velocities and undergo angular motion, which can excite angular acceleration on the three axes of the PIGA; by analyzing equation (3-15), it is known that Unable to One item is calibrated, and the remaining error coefficients are calibrated;

[0074] ④ Dual-axis input angular velocity mode (X-axis and Y-axis)

[0075] Input angular velocities ω on the X and Y axes of the turntable respectively. Tx ω Ty The direction cosine transformation matrix given by rotation along each axis ω Tx ω Ty Projecting onto the PIGA three-axis coordinate system, the angular velocities and angular accelerations of the three axes are obtained as follows:

[0076]

[0077] The angular acceleration is obtained from equation (21):

[0078]

[0079] Substituting equations (21) and (22) into equation (5), we get:

[0080]

[0081] That is, we get:

[0082]

[0083] Analysis of equation (24), where the angular velocity ω is applied to the middle ring and the inner ring. Tx ω Ty When the PIGA triaxial angular velocity is excited, the final expression can be obtained as shown in equation (24), and there is no aliasing in the coefficients of the error model. The method of applying angular velocity can calibrate all coefficients at once.

[0084] ⑤ Dual-axis input angular velocity mode (X-axis and Z-axis)

[0085] Apply angular velocities ω to the X and Z axes of the turntable respectively. Tx ω Tz Through coordinate transformation matrix The following can be obtained regarding the three-axis angular velocities and angular accelerations of the PIGA:

[0086]

[0087] The angular acceleration is obtained from equation (25):

[0088]

[0089] Substituting equations (25) and (26) into equation (5) yields

[0090]

[0091] That is, we get:

[0092]

[0093] Analysis of equation (28) shows that there is no overlap between the errors, and all error model coefficients are calibrated.

[0094] ⑥ Y-axis and Z-axis dual-axis input angular velocity mode

[0095] Apply angular velocities ω to the Y and Z axes of the turntable respectively. Ty ω Tz Through coordinate transformation matrix The following can be obtained regarding the three-axis angular velocities and angular accelerations of the PIGA:

[0096]

[0097] The angular acceleration is obtained from equation (29):

[0098]

[0099] Substituting equations (29) and (30) into equation (5) yields:

[0100]

[0101] That is, we get:

[0102]

[0103] Analyzing equation (32), when there is angular velocity input on the Y and Z axes, the PIGA triaxial angular velocity is excited, and the error model coefficients do not overlap, so all error model coefficients can be calibrated at once;

[0104] ⑦ Simultaneous input of angular velocity along the X, Y, and Z axes

[0105] When angular velocities are simultaneously input on the X, Y, and Z axes of the turntable, the coordinate transformation matrix is ​​used... The angular velocities and angular accelerations obtained along the three axes of the PIGA are as follows:

[0106]

[0107] The angular acceleration is obtained from equation (33):

[0108]

[0109] Substituting equations (33) and (34) into equation (5) yields:

[0110]

[0111] That is, we get:

[0112]

[0113] According to equations (35) and (36), simultaneously inputting uniform angular velocity on the three axes of the turntable can excite all dynamic error terms related to angular acceleration and angular velocity, and there is no aliasing phenomenon. All error coefficients of equation (5) can be calibrated.

[0114] When there is angular velocity input on two axes, it can be seen from the expression analysis that all error model coefficients of PIGA can be calibrated at one time when angular velocity is input on three axes. Similarly, when angular velocity is input on three axes, all error models will also be calibrated at one time without aliasing.

[0115] When angular velocities are input individually for the X, Y, and Z axes of the base, only the error model coefficients for the corresponding axes can be calibrated. However, when angular velocities are input in pairs for the X and Y axes, X and Z axes, and Y and Z axes of the base, all error coefficients can be calibrated at once. Similarly, when angular velocities are input for the X, Y, and Z axes of the base, all error coefficients can also be calibrated at once.

[0116] When angular velocities are input along the X and Y axes, the corresponding equation (24) adopts the fourth angular velocity input method, namely, inputting angular velocities along the X and Y axes, to calibrate the final error model of PIGA compared to other angular velocity input methods; angular velocities ω are simultaneously applied to the X and Y axes of the base. Tx ω Ty The PIGA triaxial dynamic response is excited by coordinate system projection, and the projection relationship of the dual-axis input angular velocity is as follows:

[0117]

[0118] The angular velocity of the base input is set to harmonic transformation ω. Tx =ω D sin(ω0t), ω TyFor a constant value, the corresponding angular acceleration Calibrate the coefficients The value;

[0119] By combining equations (1) and (2), the PIGA dynamic equations under different angular velocity inputs are solved to obtain the trend of the change of angle α. The corresponding time t is taken at the integer precession of angle α. n Through time t n Find the angular velocity at the corresponding moment

[0120] Furthermore, in step 4, the PIGA should be upright and inverted during measurement; the direction of the gravitational acceleration g should be controlled to be the same as and opposite to the direction of the PIGA's sensitive axis, respectively.

[0121] The gyro accelerometer is tested using the whole precession period method: let its whole precession period N. t The total measurement time is T m The PIGA precesses for 16384 pulses per revolution, therefore its precession N t The number of pulses bound by Zhou was 16384N. t After the vibration table has stabilized and the test has started, the precession N... t After the weekly measurement was completed, the average precession output of the gyro accelerometer was:

[0122] Furthermore, in step 5,

[0123] The parameters that need to be calibrated are The output matrix is: The coefficient matrix is

[0124]

[0125] Where: 1 is the coefficient used to calibrate K0 in the least squares method, g is the gravitational acceleration, The angular acceleration, ω obtained for the sensitive axis of PIGA xn ω yn ω zn Angular velocities obtained from the PIGA triaxial axis; K0, K x , Δ xx Δ yy Δ zz δ x These are constant error, proportional factor error, sensitive shaft angular acceleration error, sensitive shaft centrifugal force error, output shaft centrifugal force error, rotor shaft centrifugal force error, and sensitive shaft angular velocity-acceleration coupling error, respectively. The output value of PIGA at integer intervals

[0126] Furthermore, step 6 is implemented as follows:

[0127] Step 6.1: Construct a coefficient matrix containing angular velocity input, acceleration input, and output response;

[0128] Step 6.2: Construct the covariance matrix and regularization parameter matrix;

[0129] Step 6.3: Compare the results of the EV model method and the least squares method, and select the calibration parameters with the best consistency.

[0130] Furthermore, it also includes step 7: based on steps 1 to 6, establish the output error model of the gyro accelerometer considering the angular motion of the base and calibrate the dynamic error, and use the calibrated dynamic error for spacecraft guidance, strategic missile inertial navigation, and high-dynamic launch vehicle attitude control to improve the accuracy and efficiency of spacecraft guidance, strategic missile inertial navigation, and high-dynamic launch vehicle attitude control.

[0131] Beneficial effects:

[0132] 1. This invention discloses a dynamic error modeling and calibration method for gyro accelerometers that considers base angular motion. Addressing the deficiency of traditional error models that do not consider base angular motion coupling, it establishes a complete mathematical model including static error, dynamic linear terms, nonlinear angular velocity terms, and acceleration-angular velocity cross terms. Through integer precession elimination analysis, the coupling mechanism of base angular motion on the error is analyzed and obtained. Errors are eliminated based on this coupling mechanism, significantly improving the integrity and accuracy of the error model under dynamic conditions, and effectively solving the model distortion problem caused by multi-source error coupling under high-maneuverability conditions.

[0133] 2. The dynamic error modeling and calibration method for gyro accelerometers considering base angular motion disclosed in this invention breaks through the limitations of traditional single-axis calibration parameters, constructs a dual-axis harmonic angular velocity collaborative excitation strategy, and decouples angular acceleration, angular velocity squares, and cross-coupling terms by combining orthogonal projection matrices. Through an upright-inverted dual-mode gravity field switching mechanism, parameter aliasing interference during the calibration process is eliminated, achieving synchronous calibration of all error terms, significantly improving calibration efficiency and dynamic error separation capability, and meeting the high-precision calibration requirements under complex motion environments.

[0134] 3. This invention discloses a dynamic error modeling and calibration method for gyro accelerometers considering base angular motion. It constructs a cross-validation framework combining least squares and neural networks, and through dynamic dataset training and anti-interference optimization design, significantly improves the stability and convergence of the calibration system under vibration and rotation conditions. In practical engineering applications, this technology effectively reduces reliance on high-precision turntable equipment, shortens the calibration cycle, and improves system reliability, thereby enhancing the accuracy and efficiency of spacecraft guidance, strategic missile inertial navigation, and high-dynamic launch vehicle attitude control. Attached Figure Description

[0135] Figure 1 This is a schematic diagram of the basic structure of PIGA;

[0136] Figure 2 This is a schematic diagram of the PIGA coordinate system;

[0137] Figure 3 This is a flowchart of the dynamic error modeling and calibration method for a gyro accelerometer considering the angular motion of the base disclosed in this invention;

[0138] Figure 4 A simplified schematic diagram of a three-axis rotary table;

[0139] Figure 5 This involves the coordinate system transformation between the base and the outer ring.

[0140] Figure 6 This involves the coordinate system transformation between the outer and middle rings.

[0141] Figure 7 This represents the coordinate system transformation between the middle ring and the inner ring. Detailed Implementation

[0142] Considering the integer precession of the PIGA, it can be seen from equation (4) that a total of 7 parameters need to be calibrated, including K0, K... x , Δ xx Δ yy Δ zz δ x Calibration is performed. First, the sensitivity axis error of the PIGA is calibrated by controlling the input of the base angular velocity. According to the analysis in Chapter 3, the X-axis of the PIGA is the sensitivity axis of the PIGA. Under the condition of applying angular velocity on a single axis, the X-axis angular velocity of the three-axis turntable is given priority.

[0143] like Figure 3 As shown in the figure, the specific implementation steps of the dynamic error modeling and calibration method for a gyro accelerometer considering the angular motion of the base disclosed in this embodiment are as follows:

[0144] For the simulation experiment setup, the PIGA was first upright in the first n sets of experiments, so that the sensitive axis direction was opposite to the direction of gravitational acceleration. In the last n sets of experiments, the PIGA was inverted, so that the sensitive axis direction coincided with the direction of gravitational acceleration. This was done to prevent matrix singularities during the calibration of the least squares method and the EV model method. Then, the PIGA error model was calibrated. Based on the analysis in Chapter 3, when applying angular velocity along the single X-axis, the angular velocity needs to be controlled as a harmonic input to prevent the X-axis angular acceleration from being zero. The item cannot be specified; the corresponding expression is as follows:

[0145]

[0146] Among them, a x =g, where g is the local gravitational acceleration, ω Tx =ω D sin(ω0t), According to the least squares method, equation (4-4) can be written in the following form:

[0147] Y 2n×1 =φ 2n×5 k 5×1 (39)

[0148] When applying angular velocity along a single axis, and the angle α is at 2π during integer precession, five parameters need to be calibrated, the rank N of matrix φ must be ≥ 5, and the output vector must be... φ 2n×5 It can be obtained from equation (4-4), which will not be given in detail here.

[0149] According to the least squares method, we get

[0150] k=(φ T φ) -1 φ T Y (40)

[0151] Applying the harmonic transformation angular velocity given above to the X-axis of the turntable, and performing 2n simulations, the calibration results obtained using the least squares method are as follows:

[0152] Table 4-1 Least Squares Method for Calibration of Single-Axis Error

[0153]

[0154] Using the EV model method to analyze K0 and K x , Δ xx δ x During calibration, under ideal simulation conditions, the input angular velocity has no error, but the output value Y will have an error due to the time step. Therefore, it is necessary to set the observation vector Y. mean and ideal vector Y true The simulation settings are the same as the least squares method, and 2n simulations are performed. In the first n groups, the PIGA is upright, so that the sensitive axis direction is opposite to the direction of gravitational acceleration. In the last n groups of experiments, the PIGA is inverted, so that the sensitive axis direction coincides with the direction of gravitational acceleration. The simulation is still performed with the single-axis angular velocity set as the harmonic input ω. Tx =ω D sin(ω0t), We obtained 2n sets of data on the variation of angle α with time, recorded the time it took for the PIGA to precess to an integer point, and obtained the observation vector Y. meanY mean The expression is as follows:

[0155]

[0156] Observing the values ​​of α at integer points in the above equation, and setting the absolute value of the theoretical value of α to 2π, we obtain the ideal vector Y. true Y true The expression is as follows:

[0157]

[0158] The ±2π is because solving the PIGA's dynamic equations revealed that the PIGA may precess in the opposite direction during its precession. In this case, the PIGA will precess in the opposite direction to a whole circle. The time and value of the angle α located at the whole circle ±2π are recorded to obtain the corresponding angular velocity and angular acceleration. After integrating each term, the average is taken, and the resulting values ​​are combined to form a coefficient matrix. Due to its complexity, it will not be listed here. Regarding the coefficient matrix... We integrate each term and then take the average. Let the parameter vector be θ. The matrix form is as follows:

[0159]

[0160] In the formula, W is the covariance matrix, σ i =Y true (i,1)-Y mean (i,1), where λ is the regularization parameter, the EV model method is used to calibrate the three-axis rotary table under the condition of applying angular velocity to the single-axis X-axis rotary table. The calibration results are as follows:

[0161] Table 4-2 Single-axis error calibration using the EV model method

[0162]

[0163] In summary, based on the theoretical analysis of the PIGA dynamic error model presented above, and combined with the specific expression of the error term, it can be seen that due to the inherent physical limitations of the PIGA, its angular acceleration term under dynamic operating conditions... The actual contribution of this term is very limited, usually on a very small order of magnitude, or even close to zero. Therefore, in the modeling and simulation process of this study, it is reasonable to assume that the actual impact of this term on the PIGA output can be ignored, and treat it as 0 or a very small amount. Such simplification will not have a substantial impact on the overall recognition accuracy of the model.

[0164] On the other hand, the β-angle response is weak, which is due to the nutation drift phenomenon that occurs in PIGA during actual operation. In actual operation, PIGA internally incorporates relevant servo loops for control, and the amplitude is typically less than 10. -3 The order of magnitude. Therefore, in the operating condition settings of this study, the theoretical order of magnitude of the β angle is controlled at 1×10. -3 The following approach aligns with actual engineering practices and helps simplify the model, thereby improving the stability of numerical solutions.

[0165] In summary, under dynamic testing conditions, especially under well-optimized experimental conditions, the impact of these disturbances is typically manifested as minute perturbations, far smaller than the order of magnitude of the dynamic error term. Particularly in simulation environments using high-precision turntables, effective vibration isolation supports, and stable power supply control, these minute perturbations can be further suppressed, making the influence of the disturbance torque on model identification relatively limited. Therefore, in this study, the order of magnitude of the influence of the disturbance torque on the dynamic error term is controlled to be within 10. -7 The following is based on both actual experimental conditions and modeling accuracy, and has sufficient physical basis and engineering rationality.

[0166] Table 4-3 Comparison of two parameter calibration methods under uniaxial working conditions

[0167]

[0168]

[0169] When applying angular velocity along a single axis, comparing the performance of the least squares method and the EV model method in PIGA single-axis dynamic error model calibration reveals that the EV model method exhibits higher accuracy and lower error across multiple parameters compared to the least squares method. For example, in the angular acceleration error term... In the calibration, the calibration results of the EV model method are better than those of the least squares method, while the squared term of the sensitive axis angular velocity Δ xx During calibration, the difference from the set values ​​decreased significantly, by approximately 77%, demonstrating that the EV model method has stronger resolution and smaller fitting bias when handling small parameters. Simultaneously, the EV model method maintained a high degree of consistency between the calibrated and set values ​​for each parameter, without any physically significant deviation or overfitting. Its error control performance was more focused, with smaller error fluctuations, reflecting good model stability and physical consistency.

[0170] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for modeling and calibrating dynamic errors of a gyro accelerometer considering base angular motion, characterized in that: Includes the following steps: Step 1: Establish the kinematic and dynamic equations of PIGA under the triaxial angular motion of the base, and derive the complete error model including static error terms, dynamic error terms and nonlinear error terms; Step 2: Analyze the angular velocity transmission relationship from the base coordinate system to the outer frame, inner frame and rotor coordinate system through coordinate transformation, and construct dynamic equations that include inertia tensor, torque balance equation and servo loop influence; Step 3: Quantitatively analyze the impact of base angular motion parameters on PIGA output error; base angular motion parameters include the order of magnitude of angular velocity, angular acceleration, and cross-coupling terms; Step 4: Based on the three-axis turntable experimental design, the dynamic response of each axis of PIGA is excited by single-axis, dual-axis and three-axis angular velocity inputs, and the error model coefficients are separated and calibrated. Step 5: The error model parameters are calibrated by combining the least squares method with other data fitting algorithms, and the accuracy of the calibration results is verified by numerical simulation. The angular velocity and angular acceleration terms are separated by direction cosine matrix transformation to avoid the aliasing of error coefficients; the dynamic response data is fitted by least squares method to calibrate all error coefficients at once.

2. The method for dynamic error modeling and calibration of a gyro accelerometer considering base angular motion as described in claim 1, characterized in that: It also includes step 6: using the EV model method to cross-validate the least squares calibration results to improve calibration accuracy.

3. The method for dynamic error modeling and calibration of a gyro accelerometer considering base angular motion as described in claim 1 or 2, characterized in that: Step 1 is implemented as follows: The differential equations of motion for PIGA's rotation about the outer frame axis and the inner frame axis are obtained using Euler's equations of motion; the equations of motion about the outer frame axis are: Where: α and β are the precession angles of the outer frame and the inner frame, respectively. The moment of inertia of the outer frame system about the OY1 axis, The moment of inertia of the inner frame system about the OZ2 axis, ω is the inertia product of the inner frame system. x0 ω y0 ω z0 ω is the angular velocity of the base relative to inertial space along the corresponding axis. x1 ω y1 ω z1 Let be the absolute angular velocity of the outer frame along the corresponding axis; H is the rotational angular momentum of the rotor. This is the disturbance torque of the outer frame axis. The expression is as follows: Among them, J e The moment of inertia of the rotor shaft Equations of motion about the inner frame axis: Where α and β are the precession angles of the outer frame and the inner frame, respectively. ω is the inertia product of the inner frame system. x0 ω y0 ω z0 ω is the angular velocity of the base relative to inertial space along the corresponding axis. x1 ω y1 ω z1 Let be the absolute angular velocity of the outer frame along the corresponding axis; H is the rotational angular momentum of the rotor. This is the disturbance torque of the outer frame axis. Equation (2) yields the complete error model for PIGA: By eliminating the relevant error terms through integer precession, the complete error model is obtained in the following form: Where K0 is a constant error term, K x This is the input axis scaling factor. Δ xx Δ yy Δ zz δ represents the coefficients of the squared terms of angular acceleration and angular velocity. x The coefficients for the acceleration and angular velocity coupling terms.

4. The method for dynamic error modeling and calibration of a gyro accelerometer considering base angular motion as described in claim 3, characterized in that: In step 3, The term represents the sensitive axis angular acceleration error, on the order of 10. -5 Priority calibration is required; Δ xx Δ yy Δ zz The items are the centrifugal force error of the sensitive shaft, the centrifugal force error of the output shaft, and the centrifugal force error of the rotor shaft, with an order of magnitude of 10. -6 It requires separation calibration via dual-axis angular velocity input; δ x The term represents the sensitive axis angular velocity-acceleration coupling error, on the order of 10. -6 It needs to be calibrated by combining acceleration and angular velocity coupling experiments.

5. The method for dynamic error modeling and calibration of a gyro accelerometer considering base angular motion as described in claim 4, characterized in that: In step 4, Establish the transformation matrices for the outer, middle, and inner rings of the moving base in a three-axis coordinate system; let the platform coordinate system of the three-axis rotary table be OX. T Y T Z T The outer ring coordinate system is OX. O Y O Z O The middle ring coordinate system is -OX. M Y M Z M The inner ring axis coordinate system is OX. I Y I Z I The coordinate system of the inertial measurement unit housing, i.e., the PIGA base, is O-X0Y0Z0; in the initial state, all corresponding axes coincide, OY... I Pointing towards the inner ring axis, OX M Pointing towards the central axis, OZ O Pointing towards the outer ring axis; PIGA is fixed to the inner ring; the initial orientation is taken as the local geographic coordinate system East (E)-North (N)-Sky (ξ); The angular velocities of the three-axis rotary table are ω TX ω TY ω TZ After a period of time t, the angles they have rotated through are θ. X θ Y θ Z , (θ X =ω TX t, θ Y =ω TY t, θ Z =ω TZ t), θ1, θ2, and θ3 are the initial phases of the outer ring, middle ring, and inner ring, respectively, from OX T Y T Z T To OX O Y O Z O The rotation transformation matrix between the two coordinate systems is as follows: From OX O Y O Z O To OX M Y M Z M The rotation transformation matrix between the two coordinate systems is as follows: From OX M Y M Z M To OX I Y I Z I The rotation transformation matrix between the two coordinate systems is as follows: Based on the final form of the error model given by equation (4), by controlling the single-axis angular velocity input in the three rings, the dual-axis angular velocity input in the three rings, and the three-axis angular velocity input in the three rings, the base motion mode that can calibrate all error coefficients at once is analyzed and selected.

6. The method for dynamic error modeling and calibration of a gyro accelerometer considering base angular motion as described in claim 5, characterized in that: In step 4, ①X-axis single-axis input angular velocity mode Input the angular velocity ω along the X-axis of the turntable. Tx The direction cosine matrix between the coordinate systems during rotation along each axis After projection transformation, the expressions for PIGA's three-axis angular velocity and angular acceleration are obtained as follows: The angular acceleration is obtained from equation (9): Substituting the expressions of equations (9) and (10) into equation (5), we get: Get From equation (12), we know that, since ω y0 ω z0 It is 0, therefore Δ yy Δ zz The three terms are not reflected in the expression. The method of applying angular velocity to the X-axis alone cannot calibrate these three errors; it can only calibrate the other coefficients. ② Y-axis single-axis input angular velocity mode Apply an angular velocity ω separately to the Y-axis of the turntable Ty The direction cosine matrix between the coordinate systems when rotating along each axis After projection transformation, the expressions for PIGA's three-axis angular velocity and angular acceleration are obtained as follows: The angular acceleration is obtained from equation (13): Substituting equations (13) and (14) into equation (5) yields Get From equation (16), the simplified error model can only be applied to K0 and K x Δ yy Two coefficients were calibrated, but the remaining coefficients were not reflected in the expression because the relevant shaft angular velocity was 0, so they could not be calibrated. ③Z-axis single-axis input angular velocity mode Input the angular velocity ω separately along the Z-axis of the turntable. Tz Through the direction cosine matrix between coordinate systems ω Tz The expressions for the angular velocities and angular accelerations along the X, Y, and Z axes, projected onto the three axes of the PIGA model, are as follows: The angular acceleration is obtained from equation (17): Substituting equations (17) and (18) into equation (5) yields That is, we get: According to equation (20), when an angular velocity is applied to the outer ring of the base and projected onto the three axes of the PIGA, all three axes of the PIGA can obtain angular velocities and undergo angular motion, which can excite angular acceleration on the three axes of the PIGA; by analyzing equation (3-15), it is known that Unable to One item is calibrated, and the remaining error coefficients are calibrated; ④ Dual-axis input angular velocity mode (X-axis and Y-axis) Input angular velocities ω on the X and Y axes of the turntable respectively. Tx ω Ty The direction cosine transformation matrix given by rotation along each axis ω Tx ω Ty Projecting onto the PIGA three-axis coordinate system, the angular velocities and angular accelerations of the three axes are obtained as follows: The angular acceleration is obtained from equation (21): Substituting equations (21) and (22) into equation (5), we get: That is, we get: Analysis of equation (24), where the angular velocity ω is applied to the middle ring and the inner ring. Tx ω Ty When the PIGA triaxial angular velocity is excited, the final expression can be obtained as shown in equation (24), and there is no aliasing in the coefficients of the error model. The method of applying angular velocity can calibrate all coefficients at once. ⑤ Dual-axis input angular velocity mode (X-axis and Z-axis) Apply angular velocities ω to the X and Z axes of the turntable respectively. Tx ω Tz Through coordinate transformation matrix The following can be obtained regarding the three-axis angular velocities and angular accelerations of the PIGA: The angular acceleration is obtained from equation (25): Substituting equations (25) and (26) into equation (5) yields That is, we get: Analysis of equation (28) shows that there is no overlap between the errors, and all error model coefficients are calibrated. ⑥ Y-axis and Z-axis dual-axis input angular velocity mode Apply angular velocities ω to the Y and Z axes of the turntable respectively. Ty ω Tz Through coordinate transformation matrix The following can be obtained regarding the three-axis angular velocities and angular accelerations of the PIGA: The angular acceleration is obtained from equation (29): Substituting equations (29) and (30) into equation (5) yields: That is, we get: Analyzing equation (32), when there is angular velocity input on the Y and Z axes, the PIGA triaxial angular velocity is excited, and the error model coefficients do not overlap, so all error model coefficients can be calibrated at once; ⑦ Simultaneous input of angular velocity along the X, Y, and Z axes When angular velocities are simultaneously input on the X, Y, and Z axes of the turntable, the coordinate transformation matrix is ​​used... The angular velocities and angular accelerations obtained along the three axes of the PIGA are as follows: The angular acceleration is obtained from equation (33): Substituting equations (33) and (34) into equation (5) yields: That is, we get: According to equations (35) and (36), simultaneously inputting uniform angular velocity on the three axes of the turntable can excite all dynamic error terms related to angular acceleration and angular velocity, and there is no aliasing phenomenon. All error coefficients of equation (5) can be calibrated. When there is angular velocity input on two axes, it can be seen from the expression analysis that all error model coefficients of PIGA can be calibrated at one time when angular velocity is input on three axes. Similarly, when angular velocity is input on three axes, all error models will also be calibrated at one time without aliasing. When angular velocities are input individually for the X, Y, and Z axes of the base, only the error model coefficients for the corresponding axes can be calibrated. However, when angular velocities are input in pairs for the X and Y axes, X and Z axes, and Y and Z axes of the base, all error coefficients can be calibrated at once. Similarly, when angular velocities are input for the X, Y, and Z axes of the base, all error coefficients can also be calibrated at once. When angular velocities are input along the X and Y axes, the corresponding equation (24) adopts the fourth angular velocity input method, namely, inputting angular velocities along the X and Y axes, to calibrate the final error model of PIGA compared to other angular velocity input methods; angular velocities ω are simultaneously applied to the X and Y axes of the base. Tx ω Ty The PIGA triaxial dynamic response is excited by coordinate system projection, and the projection relationship of the dual-axis input angular velocity is as follows: The angular velocity of the base input is set to harmonic transformation ω. Tx =ω D sin(ω0t), ω Ty For a constant value, the corresponding angular acceleration Calibrate the coefficients The value; By combining equations (1) and (2), the PIGA dynamic equations under different angular velocity inputs are solved to obtain the trend of the change of angle α. The corresponding time t is taken at the integer precession of angle α. n Through time t n Find the angular velocity at the corresponding moment 7. The method for dynamic error modeling and calibration of a gyro accelerometer considering base angular motion as described in claim 6, characterized in that: In step 4, the PIGA should be upright and inverted during measurement; the direction of the gravitational acceleration g should be controlled to be the same as and opposite to the direction of the PIGA's sensitive axis, respectively. The gyro accelerometer is tested using the whole precession period method: let its whole precession period N. t The total measurement time is T m The PIGA precesses for 16384 pulses per revolution, therefore its precession N t The number of pulses bound by Zhou was 16384N. t After the vibration table has stabilized and the test has started, the precession N... t After the weekly measurement was completed, the average precession output of the gyro accelerometer was:

8. The method for dynamic error modeling and calibration of a gyro accelerometer considering base angular motion as described in claim 7, characterized in that: In step 5, The parameters that need to be calibrated are The output matrix is: The coefficient matrix is Where: 1 is the coefficient used to calibrate K0 in the least squares method, g is the gravitational acceleration, The angular acceleration, ω obtained for the sensitive axis of PIGA xn ω yn ω zn Angular velocities obtained from the PIGA triaxial axis; K0, K x , Δ xx Δ yy Δ zz δ x These are constant error, proportional factor error, sensitive shaft angular acceleration error, sensitive shaft centrifugal force error, output shaft centrifugal force error, rotor shaft centrifugal force error, and sensitive shaft angular velocity-acceleration coupling error, respectively. This is the output value of PIGA at integer intervals.

9. The method for dynamic error modeling and calibration of a gyro accelerometer considering base angular motion as described in claim 8, characterized in that: Step 6 is implemented as follows: Step 6.1: Construct a training dataset containing angular velocity input, acceleration input, and output response; Step 6.2: Optimize the error model coefficients using the backpropagation algorithm to suppress calibration bias caused by singular matrices; Step 6.3: Compare the results of the neural network and the least squares method, and select the calibration parameters with the best consistency.

10. The method for dynamic error modeling and calibration of a gyro accelerometer considering base angular motion as described in claim 9, characterized in that: It also includes step 7: Based on steps 1 to 6, establish the output error model of the gyro accelerometer considering the angular motion of the base and calibrate the dynamic error, and use the calibrated dynamic error for spacecraft guidance, strategic missile inertial navigation, and high-dynamic launch vehicle attitude control to improve the accuracy and efficiency of spacecraft guidance, strategic missile inertial navigation, and high-dynamic launch vehicle attitude control.

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