Dynamic orbit determination method based on integer ambiguity fixation

Abstract

The invention relates to the cross technical field of satellite navigation, satellite dynamics and the like, in particular to a dynamic orbit determination method based on integer ambiguity fixation, which can obviously improve orbit determination and extrapolation precision, and comprises the following steps: establishing a dynamic orbit determination model, establishing a GPS observation model, and then performing integer ambiguity fixation IAR, a single-difference IAR method adopting single receiver ambiguity correction specifically comprises the steps that firstly, GPS carrier phase observation is divided into continuous tracking segments for processing, for each tracking segment, an average Hatch-Melbourne-Wubbena (HMW) combination is constructed by means of double-frequency pseudo-range and carrier phase data, then cross-segment inconsistency is corrected, integers of wide lane ambiguity are achieved, and a single-receiver ambiguity correction result is obtained. And the corrected wide lane ambiguity is combined with an IF ambiguity floating point estimation value obtained by a least square method for use, and finally, the IF ambiguity difference is constrained as an integer value to realize the fixation of the single-difference ambiguity, and compared with the prior art, the orbit determination precision can be remarkably improved.

Description

Technical fields:

[0001] This invention relates to the interdisciplinary fields of satellite navigation and satellite dynamics, specifically a dynamic orbit determination method based on fixed integer ambiguity that can significantly improve the accuracy of orbit determination and extrapolation. Background technology:

[0002] Low Earth Orbit (LEO) satellite missions, such as the Gravity Recovery and Climate Experiment (GRACE) and its successor GRACE Follow-On (GFO), have played a crucial role in advancing Earth system science research. These missions not only support global gravity field inversion but also extend to inversion applications such as electron content in the ionosphere's top region. A prerequisite for these applications is high-precision satellite orbits, which is primarily due to the high-quality observations provided by onboard Global Positioning System (GPS) receivers.

[0003] Reduced-Dynamic Orbit Determination (RDOD) is the mainstream method for obtaining high-precision low-Earth orbit satellite orbits. Orbit dynamics modeling requires simultaneous consideration of conservative forces (such as Earth's gravity and various tides) and non-conservative forces (such as atmospheric drag, solar radiation pressure, and Earth's radiation pressure). Given the uncertainties inherent in the background model, et al. (2005) introduced pseudo-random acceleration (or pulses) in orbit determination to absorb the influence of unmodeled forces. Common practices include piecewise constants, linear accelerations, and velocity pulses, typically updated every 5–15 minutes, and jointly estimated with non-conservative force parameters (such as atmospheric drag factors) to compensate for unmodeled or less accurate conservative forces (such as time-varying gravity fields). The flexibility of these piecewise parameters, to some extent, bridges the gap between simplified and fully dynamical systems. Although RDOD can achieve excellent orbital accuracy, the pseudo-random parameters it introduces lack clear physical meaning, easily masking real dynamic characteristics and limiting orbital extrapolation capabilities. For example, Du et al. (2024) used empirical parameters containing constant and periodic terms for orbit determination and extrapolation, achieving a three-dimensional accuracy of 0.19 m within a 1-hour arc, but the real-time integration accuracy combined with accelerometer measured data still did not meet expectations.

[0004] To obtain more accurate dynamic orbits, it is essential to effectively mitigate the effects of unmodeled forces and model errors. In RDOD (Real-Time Orbital Analysis), these errors are often compensated for through pseudo-random parameters in the time domain to address insufficient modeling of high-frequency or short-period events. However, dominant error sources such as mass redistribution caused by hydrological, cryosphere, and ocean dynamic processes have global scales and long time scales, which cannot be adequately absorbed by time-domain adjustments alone.

[0005] Traditional Reduced-Dynamic Orbit Determination (RDOD) methods achieve centimeter-level accuracy by introducing pseudo-random parameters (such as velocity impulses and accelerations) to absorb the effects of unmodeled forces, but at the expense of physical interpretability and limit orbit extrapolation capabilities. In contrast, Dynamic Orbit Determination (DOD), by explicitly modeling time-varying perturbations, can simultaneously recover physically consistent orbits and models of Earth's time-varying gravity field over long arcs. However, due to the difficulty in effectively fixing GPS carrier phase ambiguities, DOD orbit accuracy remains lower than RDOD. Summary of the Invention:

[0006] This invention addresses the shortcomings and deficiencies of existing technologies by introducing IAR into the DOD framework, proposing a dynamic orbit determination method based on fixed integer ambiguity that can achieve orbital accuracy of 1cm while maintaining physical consistency.

[0007] This invention achieves its purpose through the following measures:

[0008] A dynamic orbit determination method based on fixed integer ambiguity is characterized by establishing a dynamic orbit determination model and a GPS observation model, followed by fixed integer ambiguity IAR. Specifically, a single-difference IAR method with single-receiver ambiguity correction is employed, comprising: first, dividing GPS carrier phase observations into continuous tracking segments for processing; for each tracking segment, constructing an average Hatch-Melbourne-Wübbena (HMW) combination using dual-frequency pseudorange and carrier phase data; subsequently, correcting cross-segment inconsistencies to integerize the wide-lane ambiguity; combining the corrected wide-lane ambiguity with a floating-point estimate of the ionospheric IF ambiguity obtained through least squares; and finally, constraining the IF ambiguity difference to integer values ​​to achieve fixed single-difference ambiguity.

[0009] Where i and j represent two different GPS satellites simultaneously tracked by the LEO receiver, σ N This represents the relevant standard deviation.

[0010] In this invention, σ N It is usually set to 10. -6 m.

[0011] This invention also includes carrier phase residual verification during integer ambiguity fixing. This invention introduces the monthly dynamic parameters estimated by the floating-point solution into the daily normal equation and constrains them with a corresponding diagonal weighting matrix. The diagonal weighting matrix of the monthly dynamic parameters is calculated as follows:

[0012]

[0013] Here, `diag(*)` extracts the diagonal elements of the matrix, N PP The monthly solution equation matrix is ​​represented by the equations. Once all ambiguities are corrected and verified, they will be incorporated into the corresponding celestial solution equations as high-weight constraints. Then, these celestial solution equations with ambiguity constraints are accumulated to construct a joint monthly solution equation system. Finally, the system is adjusted by least squares to jointly estimate the satellite's initial orbit, clock error, accelerometer calibration parameters, and spherical harmonic coefficients of the Earth's gravity field.

[0014] The establishment of the dynamic orbit determination model described in this invention specifically involves the following: In the inertial reference frame, the equation of motion for the satellite's center-of-mass position vector is:

[0015] Among them, f g Indicates conservative force; f ng To represent a non-conservative force, modeling the action of a non-conservative force using observational data from a spaceborne accelerometer requires calibration of the accelerometer data. The parameter S represents the calibration scaling factor, and b represents the bias factor. Let C be the satellite velocity vector. nm and S nm The spherical harmonic coefficients of the Earth's gravitational field, and the dynamic parameters S, b, and C. nm and S nm Collectively referred to as the dynamic parameter set p i ,…,p d Where d is the total number of parameters, the state of the satellite at any time t can be represented as the state vector y(t):

[0016]

[0017] Differentiating formula (2) yields:

[0018]

[0019] The dynamic force model is used to calculate the accelerations of conservative and non-conservative forces acting on the satellite. Due to the initial state vector y(t0) and the mechanical model parameters p i ,…,p d The estimated value is obtained during the orbit determination process, so it needs to be linearized to solve for the partial derivatives of the satellite state with respect to the estimated parameters at any time t. After determining the initial state vector, orbit integration is performed to advance the state forward. The partial derivatives of the satellite state with respect to the initial state vector y(t0) and dynamic parameters at any time t are represented by the state transition matrix Φ(t,t0) and the parameter sensitivity matrix S(t,t0), respectively.

[0020]

[0021] By differentiating formula (4) with respect to time t, we obtain:

[0022]

[0023] The initial conditions are:

[0024]

[0025] Given the satellite's orbital state, the orbit at any given time can be linearized using a truncated Taylor series expansion:

[0026]

[0027] Φ(t,t0) and S(t,t0) are determined by integrating equations (5) and (6), and equation (7) can be used to correct the nominal orbit using orbit observations.

[0028] In the GPS observation model established in this invention, the observation equations for low-orbit satellite dual-frequency pseudorange and carrier phase are as follows:

[0029]

[0030] in, and These represent pseudorange and carrier phase observations, respectively; i represents i th Frequency; ρ represents the geometric distance between the satellite s and the receiver r antenna reference point; c is the speed of light in a vacuum; dt r and dt s These are the clock offsets of the receiver and the satellite, respectively; μ r,Ci and μ r,Li It refers to the pseudorange and carrier phase hardware delay at the receiving end. and It refers to the pseudorange and carrier phase hardware delay at the satellite end; It is the ionospheric refractive index, f i This indicates the corresponding carrier phase frequency; λ represents the first-order ionospheric delay at the L1 frequency; i It is f i The carrier phase wavelength at a given frequency; It is an integer ambiguity within a unit period; systematic errors, such as relativistic effects, antenna phase entanglement effects, phase center offset and changes, have been corrected in advance by existing models, and residual errors can be ignored;

[0031] GPS satellite orbit and clock deviations, as well as hardware delays, are corrected using precise GPS products, and first-order ionospheric delays are eliminated through ionospheric free (IF) combination. The observation equations are simplified to the following form:

[0032]

[0033] in,

[0034]

[0035] In formula (9), λ if =c / (f1+f2) represents the IF carrier phase wavelength; To represent ambiguity, due to the rank deficiency of the observation equation, the actual observation equation used is as follows:

[0036]

[0037] Among them, dt r,Cif =dt r +μ r,Cif The pseudorange, ambiguity, is the receiver clock determination used in the pseudorange and carrier phase observation equations. It absorbs the deviations of pseudorange and carrier phase, and reorganizes them into floating-point form. Substituting equation (7) into equation (12) and linearizing it, the DOD observation model can be expressed as:

[0038] Among them, prefit * This represents the difference between the observed and calculated values ​​after linearization. This represents the initial orbital state; dp0 = [dS dbdC] nm dS nm [ ] represents the spherical harmonic coefficients of the accelerometer and gravitational field to be estimated.

[0039] This invention employs Variance Component Estimation (VCE), using solar arcs as the unit. It estimates the variance scaling factor based on the a priori residuals of the pseudorange and carrier phase, dynamically updating the observation weights to ensure that the relative contributions of different observation types achieve optimal balance according to their statistical characteristics. The prior weighting matrix for each observation type is defined in diagonal matrix form as follows:

[0040] in, Let f represent the variance of the i-th observation type, q represent the total number of observation types, and f represent the variance component of observation type i. i The calculation formula is: Among them, y i x represents the difference between the observed value and the computed value (Observable-Computed) vector. i Let A represent the parameter to be estimated. i Denotes the corresponding design matrix, m iThe updated weight matrix W represents the number of observations for each observation type.

[0041] To accelerate convergence, the initial standard deviation of pseudorange observations was set to 0.3m, and the initial standard deviation of carrier phase observations was set to 3mm.

[0042] In this invention, the SuperSTAR-FO accelerometer carried by the GFO satellite is inevitably affected by changes in the instrument's physical characteristics and the orbital environment during observations, manifesting as unknown scaling factors and time-varying deviations. Therefore, accelerometer observations must be calibrated. Within an inertial reference frame, the formula for calculating calibrated accelerometer measurements is as follows:

[0043] acc cal =R·(S·a obs +b bias (19), where a obs Represents the raw acceleration measurement in the Satellite Scientific Reference Frame (SRF), R represents the rotation matrix from the SRF to the inertial frame, S represents the scale factor matrix, and b bias The structure of the scaling factor matrix S, representing the deviation vector, is as follows:

[0044] The matrix consists of three components: (1) diagonal terms: s x s y and s z (2) Shear terms: α, β, and γ are used to account for the non-orthogonality between accelerometer axes; (3) Rotation terms ζ, ε, and δ represent the misalignment between the SRF and the accelerometer frame. The scale factor matrix S is usually estimated monthly, using the strategy employed by the GFO scientific data system. The deviation vector b bias Modeled as a second-order polynomial form for each accelerometer axis (x-axis, y-axis, and z-axis) with respect to time: b i =b i,0 +b i,1 (t-t0)+b i,2 (t-t0) 2 ,i=x,y,z(21), where,b i,0 b i,1 and b i,2 These are the constant term, linear term, and quadratic term for the corresponding axis, respectively. t is the current time, and t0 is the start time of the parameter estimation arc. The deviation parameters are estimated every 6 hours to account for short-term changes in instrument behavior.

[0045] This invention proposes a dynamic orbit determination (DOD) method based on integer ambiguity resolution (IAR). By simultaneously solving for the orbit and the spherical harmonic coefficients of the gravitational field, it significantly improves the accuracy of orbit determination and extrapolation. Verification results show that the difference from the precise scientific orbit of the Jet Propulsion Laboratory (JPL) is reduced to approximately 1 cm, reaching an internationally advanced level.

[0046] RDOD accuracy level. Independent verification shows that the residual of satellite laser ranging (SLR) is better than 1 cm, and the residual of K-band ranging (KBR) is better than 0.3 mm. Attached image description:

[0047] Appendix Figure 1 These are the root mean square values ​​of GPS pseudorange and carrier phase residuals of the GFO-C satellite under different gravity field model orders in this embodiment of the invention (September 2019). The highest orders of the spherical harmonic models are 2, 5, 10, 20, 30 and 60, respectively.

[0048] Appendix Figure 2 The root mean square difference between the GFO-C satellite and the JPL PSO orbit under different gravity field model orders in this embodiment of the invention (September 2019); the highest orders of the spherical harmonic models are 2, 5, 10, 20, 30 and 60 respectively.

[0049] Appendix Figure 3 This is the success rate of Integer Ambiguity Resolution (IAR) for GFO-C satellite wide-lane and narrow-lane ambiguity fixation in this embodiment of the invention (July-December 2019).

[0050] Appendix Figure 4 This is the root mean square difference between the GFO-C satellite dynamic orbit solution (floating-point solution and integer ambiguity fixed solution) and the JPL PSO orbit in this embodiment of the invention (July-December 2019).

[0051] Appendix Figure 5 This is the three-dimensional orbital difference distribution (cm) between the floating-point solution and integer ambiguity fixed solution (IAR) dynamic orbits of GFO-C and GFO-D satellites in single-satellite and dual-satellite modes and JPL PSO in this embodiment of the invention (July-December 2019).

[0052] Appendix Figure 6This is a comparison of the SLR residual histograms (cm) obtained from the floating-point solution and integer ambiguity fixed solution (IAR) dynamic orbits of GFO-C and GFO-D satellites in single-satellite and dual-satellite modes in this embodiment of the invention with the JPL PSO results (July-December 2019).

[0053] Appendix Figure 7 This is the daily standard deviation (STD) of the KBR residuals obtained from the dynamic orbits of GFO-C and GFO-D satellites in single-satellite and dual-satellite modes in this embodiment of the invention, based on the floating-point solution and the integer ambiguity fixed solution (IAR), and compared with the JPLPSO results (July-December 2019).

[0054] Appendix Figure 8 This embodiment of the invention describes the difference between the real-time integrated orbit and the JPL PSO of the GFO-C satellite in single-satellite mode on January 1, 2020, based on floating-point solutions and integer ambiguity fixed solutions (IAR). (Appendix) Figure 9 This is the geoid difference between the floating-point solution and the fixed solution of the lunar gravity field model obtained by inversion under single-star mode (GFO-C) and binary-star mode (GFO-C / D) in this embodiment of the invention (difference from EGM2008) (July 2019).

[0055] Appendix Figure 10 This is a schematic diagram of the results in an embodiment of the present invention, wherein... Figure 10 In the table, a represents the spherical harmonic coefficient error spectrum of the GFO-C floating-point solution inversion, b represents the spherical harmonic coefficient error spectrum of the GFO-C / D floating-point solution inversion, c represents the spherical harmonic coefficient error spectrum of the GFO-CIAR solution inversion, and d represents the spherical harmonic coefficient error spectrum of the GFO-C / D IAR solution inversion (Log10) (difference from EGM2008) (July 2019).

[0056] Appendix Figure 11 This is the power spectral density of the GPS carrier phase residual of the GFO-C satellite based on the floating-point solution and the fixed integer ambiguity solution in this embodiment of the invention (July 1, 2019).

[0057] Appendix Figure 12 The root mean square of the GPS carrier phase residuals (July-December 2019) of the floating-point solutions and integer ambiguity fixed (IAR) solutions of GFO-C and GFO-D satellites in this embodiment of the invention. Detailed implementation method:

[0058] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0059] This example relates to a method for improving orbit determination accuracy of low-Earth orbit (LEO) satellites based on fixed integer ambiguity (IAR). Since GPS carrier phase ambiguity is difficult to effectively fix, the accuracy of DOD (Orbit Determination) remains lower than that of RDOD (Orbit Determination Based on RDOD). To address this, this invention proposes a dynamic orbit determination (DOD) method based on fixed integer ambiguity (IAR). This method significantly improves the accuracy of orbit determination and extrapolation by simultaneously solving for the orbit and the spherical harmonic coefficients of the gravity field. Verification results using GRACE Follow-On (GFO) satellite data from July to December 2019 show that the difference from the precise scientific orbit of the Jet Propulsion Laboratory (JPL) is reduced to approximately 1 cm, reaching the internationally advanced RDOD accuracy level.

[0060] Independent verification in this example shows that the satellite laser ranging (SLR) residual is better than 1 cm, and the K-band ranging (KBR) residual is better than 0.3 mm. This invention further evaluates the 24-hour real-time orbit extrapolation performance based on onboard accelerometer observation data. The results show that the floating-point solution has radial, normal, and tangential errors of 1.9 cm, 4.4 cm, and 82.3 cm, respectively. After introducing IAR, the orbit extrapolation accuracy is improved to 1.3 cm, 3.3 cm, and 62.4 cm, respectively, demonstrating the potential of this method to achieve autonomous navigation without external observation. In the inversion of the lunar gravity field model, the IAR solution is basically consistent with the floating-point solution within the first 20 orders, but the differences are significant in higher orders, possibly due to the relatively large GPS phase residual of the IAR.

[0061] In the dynamic orbit determination model involved in this example, the equation of motion for the satellite's center of mass position vector in the inertial reference frame is: Among them, f g Indicates conservative force; f ng This represents a non-conservative force. In this example, spaceborne accelerometer observation data is used to model the action of a non-conservative force. Due to accuracy distortion caused by environmental changes and other factors, the accelerometer data needs to be calibrated. Parameter S represents the calibration scaling factor, and b represents the bias factor. Let C be the satellite velocity vector. nm and S nm Represents the spherical harmonic coefficients of the Earth's gravitational field; dynamic parameters such as S, b, and C. nm and S nm These are collectively referred to as the dynamic parameter set p. i ,…,p d Where d is the total number of parameters; the state of the satellite at any time t can be represented as the state vector y(t): Differentiating formula (2) yields:

[0062] The dynamic force model is used to calculate the accelerations of conservative and non-conservative forces acting on the satellite. Due to the initial state vector y(t0) and the mechanical model parameters p i ,…,p d The estimated value is obtained during the orbit determination process, therefore it needs to be linearized to solve for the partial derivatives of the satellite state with respect to the estimated parameters at any time t. After determining the initial state vector, orbit integration is performed to advance the state forward;

[0063] The partial derivatives of the satellite state at any time t with respect to the initial state vector y(t0) and the dynamic parameters are represented by the state transition matrix Φ(t,t0) and the parameter sensitivity matrix S(t,t0), respectively:

[0064] By differentiating formula (4) with respect to time t, we obtain:

[0065]

[0066] The initial conditions are:

[0067] Given the satellite's orbital state, the orbit at any given time can be linearized using a truncated Taylor series expansion:

[0068] Φ(t,t0) and S(t,t0) are determined by integrating equations (5) and (6), and equation (7) can be used to correct the nominal orbit using orbital observations. In addition, various orbital-related observation data can be incorporated into this set of equations to further constrain and refine the orbital parameters and mechanical model.

[0069] In this example, the GPS observation model uses the following equations for low-Earth orbit satellite dual-frequency pseudorange and carrier phase observations: in, and These represent pseudorange and carrier phase observations, respectively; i represents i th Frequency; ρ represents the geometric distance between the satellite s and the receiver r antenna reference point; c is the speed of light in a vacuum; dt r and dt s These are the clock offsets of the receiver and the satellite, respectively; μ r,Ci and μ r,Li It refers to the pseudorange and carrier phase hardware delay at the receiving end. and It refers to the pseudorange and carrier phase hardware delay at the satellite end; It is the ionospheric refractive index, f i This indicates the corresponding carrier phase frequency; λ represents the first-order ionospheric delay at the L1 frequency; i It is f i The carrier phase wavelength at a given frequency; It represents the integer ambiguity within a unit period. Systematic errors, such as relativistic effects, antenna phase entanglement, phase center shift and variation, have already been corrected in advance using existing models, and the residual errors are negligible.

[0070] GPS satellite orbit and clock deviations, as well as hardware delays, can be corrected using precise GPS products, and first-order ionospheric delays are eliminated through ionospheric free (IF) combination. The observation equations can be simplified to the following form:

[0071] in, In formula (9), λ if =c / (f1+f2) represents the IF carrier phase wavelength; This represents ambiguity. Due to the rank deficiency of the observation equation, the actual observation equation used is as follows:

[0072] Among them, dt r,Cif =dt r +μ r,Cif This represents the pseudorange determined by the receiver clock used in the pseudorange and carrier phase observation equations. Ambiguity. It absorbs the deviations of pseudorange and carrier phase, and reorganizes them into floating-point form. Substituting equation (7) into equation (12) and linearizing it, the DOD observation model can be expressed as:

[0073] Among them, prefit * This represents the difference between the observed and calculated values ​​after linearization. This is the initial orbital state;

[0074] dp0 = [dS db dC] nm dS nm [ ] represents the spherical harmonic coefficients of the accelerometer and gravitational field to be estimated.

[0075] This invention employs a single-difference IAR method for ambiguity correction using a single receiver. The specific process is as follows: First, GPS carrier phase observations are divided into continuous tracking segments for processing. For each tracking segment, an averaged Hatch-Melbourne-Wübbena (HMW) combination is constructed using dual-frequency pseudorange and carrier phase data. Subsequently, satellite-specific wide-lane (WL) bias products provided by Wuhan University (WHU) are used to correct cross-segment inconsistencies, thereby achieving integerization of the wide-lane ambiguity. The corrected wide-lane ambiguity is then combined with the ionosphere-free (IF) ambiguity floating-point estimate obtained through the least squares method. Finally, the IF ambiguity difference is constrained to an integer value to achieve fixation of the single-difference ambiguity. Where i and j represent two different GPS satellites simultaneously tracked by the LEO receiver, σ N This represents the relevant standard deviation, typically set to 10. -6 m.

[0076] In the process of fixing integer ambiguity, carrier phase residual verification is one of the key steps. This process verifies the correctness of ambiguity correction by evaluating whether the observation residuals of each candidate ambiguity solution are consistent with the standard deviation of the floating-point solution residuals. Ambiguity resolution and dynamic parameter estimation are both performed within a single daily arc to obtain the GPS carrier phase residuals. However, limited by the spatial coverage of low-Earth orbit (LEO) satellites within a single day, the daily solution equations exhibit rank deficiency in the joint estimation of the spherical harmonic coefficients (up to order 30) of the Earth's gravity field. To address this issue, this invention introduces the monthly solution dynamic parameters estimated by the floating-point solution into the daily normal equations and constrains them with a corresponding diagonal weighted matrix. Although theoretically, a fully filled covariance matrix can yield a better solution, this invention employs a diagonal approximation to achieve a balance between computational efficiency and accuracy. The diagonal weighted matrix of the monthly solution dynamic parameters is calculated as follows:

[0077] Here, `diag(*)` extracts the diagonal elements of the matrix, N PP This represents the monthly solution equation matrix. Once all ambiguities are corrected and verified, they are incorporated as high-weight constraints into the corresponding celestial solution equations. These celestial solution equations, incorporating ambiguity constraints, are then accumulated to construct a joint monthly solution equation system. Finally, least-squares adjustment is performed to jointly estimate the satellite's initial orbit, clock bias, accelerometer calibration parameters, and the spherical harmonic coefficients of the Earth's gravitational field.

[0078] This invention outlines the GPS data processing strategy employed in the experimental section. Dynamic orbit determination is based on a mathematical-physical model describing the forces acting on the satellite, primarily including the Earth's gravitational field, third-body perturbations, and tidal effects such as solid tides and ocean tides. The reference frame used for modeling is a non-rotating free-fall (inertial) reference frame, with its origin at the Earth's (system's) center of mass. The mechanical model implementation follows IERS Conventions 2010, and the static background gravity field uses GGM05C. No prior constraints or regularization are applied during the Earth's gravity field inversion process. This invention uses the observation dataset from the GFO binary satellites (GFO-C / GFO-D) from January 1st to December 31st, 2019. This dataset includes onboard GPS observation data (GPS1B), attitude data (SCA1B), mass data (MAS1B), accelerometer data (ACT1B for GFO-C, ACH1B for GFO-D), and related auxiliary data.

[0079] In GFO dynamic orbit determination, GPS pseudorange and carrier phase exhibit significant differences in noise level and parameter sensitivity. Therefore, rationally setting observation weights is crucial for achieving optimal estimation. To this end, this invention employs Variance Component Estimation (VCE), estimating the variance scaling factor of pseudorange and carrier phase based on their a priori residuals in daily arcs, and dynamically updating the observation weights. This daily update strategy ensures that the relative contributions of different observation types are optimally balanced according to their statistical characteristics. The prior weighting matrix for each observation type is defined in diagonal matrix form as follows: in, Let f represent the variance of the i-th observation type, and q represent the total number of observation types. The variance component f of observation type i. i The calculation formula is:

[0080] Among them, y i x represents the difference between the observed value and the computed value (Observable-Computed) vector. i Let A represent the parameter to be estimated. i Denotes the corresponding design matrix, m i This represents the number of observations for each observation type. The updated weight matrix W is: To accelerate convergence, the initial standard deviation of pseudorange observations was set to 0.3m, and the initial standard deviation of carrier phase observations was set to 3mm.

[0081] The SuperSTAR-FO accelerometer aboard the GFO satellite is inevitably affected by changes in instrument physical characteristics and the orbital environment during observations, resulting in unknown scaling factors and time-varying deviations. Therefore, accelerometer observations must be calibrated. Under the inertial reference frame, the formula for calculating calibrated accelerometer measurements is:

[0082] acc cal =R·(S·a obs +b bias (19), where a obs Represents the raw acceleration measurement in the Satellite Scientific Reference Frame (SRF), R represents the rotation matrix from the SRF to the inertial frame, S represents the scale factor matrix, and b bias This represents the deviation vector. The structure of the scaling factor matrix S is as follows:

[0083] The matrix consists of three components: (1) diagonal terms: s x s y and s z (1) Scale factor; (2) Shear terms: α, β, and γ are used to account for the non-orthogonality between the accelerometer axes; (3) Rotation terms ζ, ε, and δ represent the misalignment between the SRF and the accelerometer frame. The scale factor matrix S is typically estimated monthly, using the strategy employed by the GFO scientific data system. Deviation vector b bias Modeled as a second-order polynomial form for each accelerometer axis (x-axis, y-axis, and z-axis) with respect to time: b i =b i,0 +b i,1 (t-t0)+b i,2 (t-t0) 2 ,i=x,y,z(21), where,b i,0 b i,1 and b i,2 These represent the constant, linear, and quadratic terms for the corresponding axes, respectively. t is the current time, and t0 is the start time of the parameter estimation arc. A set of deviation parameters is estimated every 6 hours to account for short-term variations in instrument behavior.

[0084] This example provides a systematic evaluation of the dynamic orbit determination results proposed in this invention, focusing on the influence of two factors: the degree / order (d / o) of the gravity field estimation and the integer ambiguity resolution (IAR). Orbit accuracy was verified by comparing it with the Precise Science Orbit (PSO) provided by the US JPL, combined with K-band ranging and satellite laser ranging observations. For the influence of the gravity field order, this invention selected GFO-C satellite data from September 2019 for analysis; while the IAR accuracy evaluation used single-satellite and binary-mode observation data from the GFO-C and GFO-D binary satellites from July to December 2019.

[0085] In dynamic orbit determination, the order of the gravity field model to be estimated plays a crucial role. At an orbital altitude of approximately 500 km, the GFO satellite is affected by Earth's gravity field up to order 180. Previous studies have attempted to recover the time-varying gravity field using GPS observations from CHAMP, GRACE, GOCE, Swarm, and low-Earth orbit satellite constellations, with a spatial resolution of approximately 2000 km, corresponding to an Earth gravity field of approximately order 20. To comprehensively assess the impact of gravity field resolution on orbital quality, this invention estimates gravity field models of orders 2-60 while maintaining other processing parameters consistent. Since the residuals and orbital accuracy are unsatisfactory without gravity field estimation, this scenario was not included in the analysis.

[0086] Figure 1 The root-mean-square (RMS) results for the GPS pseudorange and carrier phase residuals of the GFO-C satellite are presented. It can be seen that the pseudorange observations have limited sensitivity to the gravity field model: when the estimation order increases from 2 d / o to 5 d / o, the pseudorange residuals only decrease by about 2.5%, mainly due to the relatively high noise in the pseudorange observations themselves. In contrast, the carrier phase residuals are more sensitive to the gravity field order: when the gravity field model is estimated to 5 d / o, the residuals decrease significantly and approach the noise level of the carrier phase observations; beyond 5 d / o, the residual changes for both types of observations tend to level off, indicating that the solution results have basically converged in terms of gravity field resolution.

[0087] Figure 2The impact of the gravity field model order on orbit accuracy was demonstrated, including differences in the radial, tangential, and cross directions, as well as the 3D RMS. As the model order increased from 2 to 20 d / o, the accuracy of each component improved significantly, with the 3D RMS decreasing from 4.4 cm to below 1.7 cm. The improvement was most pronounced in the 2-10 order range, particularly in the radial and tangential directions. Beyond order 20, orbit accuracy tended to stabilize, with limited improvement from further increases in order. When the model was extended to order 60, orbit accuracy decreased slightly, possibly due to noise amplification or overfitting. Overall, the results indicate that the 20-order model achieves a better balance between orbit accuracy and model complexity.

[0088] This invention evaluates the impact of integer ambiguity fixation on the accuracy of dynamic orbit determination, focusing on the ambiguity fixation success rate (SR) and its contribution to improving orbit accuracy. Experiments were conducted in both single-satellite and binary configurations; the binary configuration enhances the predictability of spherical harmonic coefficients by sharing gravity field parameters. Simultaneously, the additional benefits of the binary configuration in orbit determination and its potential introduced biases are analyzed.

[0089] Figure 3 The success rates of ambiguity fixation for wide-lane (WL) and narrow-lane (NL) orbit determination are presented. The results show that the average success rate for WL ambiguity is approximately 98%, while the success rate for NL ambiguity is approximately 92%, slightly lower than the >95% typically reported in simplified dynamic orbit determination. This decrease is mainly attributed to modeling errors in the background field model during dynamic orbit determination. To ensure reliable results, ambiguity fixation is only performed when both the mean residual and variance of a single arc segment are less than 1 cm. On some observation days, the NL success rate drops to approximately 85%, primarily due to frequent attitude maneuvers and poor GPS observation data quality.

[0090] Figure 4The root mean square (RMS) differences in orbits relative to JPL PSO are presented, covering the radial, tangential, and normal directions, with results compiled over a six-month period. For the floating-point solution, the orbital differences in the three directions are 0.7 cm, 1.1 cm, and 1.3 cm, respectively, with the normal direction showing the lowest accuracy, and some days showing errors approaching 2 cm. This is mainly attributed to the fact that the Y-axis of the onboard accelerometer is a non-sensitive axis, resulting in larger measurement errors. After introducing integer ambiguity fixation, the orbital residuals in all three directions are reduced to approximately 0.7 cm. Notably, the improvement in the radial component is limited. During the period from DOY 182 to 305, the accuracy of both the floating-point solution and the IAR solution remained at approximately 0.7 cm; however, the improvement in the radial component only became apparent after DOY 305. This may be related to the frequent maneuvers of the GFO satellite during this period, which led to a decrease in the accuracy of the floating-point solution, while IAR compensated for this deficiency to some extent. Overall, the introduction of IAR enhances the geometric constraints of the solution and effectively reduces the impact of accelerometer errors.

[0091] Figure 5 The differences in three-dimensional orbits at various epochs before and after IAR processing are presented in single-star and binary-star modes. Whether using floating-point or IAR solutions, the orbital difference for GFO-D is slightly greater than that for GFO-C. This difference may stem from the quality degradation of GFO-D accelerometer data after June 21, 2018, with all six components (roll, cross, yaw, radial, pitch, and tangential) exhibiting significant common-mode errors. After introducing IAR, the orbital differences between single-star and binary-star modes statistically converge, and orbital accuracy is significantly improved. The accuracy of radial, tangential, and normal orbits improves by approximately 10%, 40%, and 40%, respectively. However, according to... Figure 5 It is evident that, regardless of whether floating-point or IAR solutions are used, the binary satellite model does not exhibit higher orbital accuracy compared to the single-satellite model. This result is not only attributed to the high fidelity of the GFO satellite dynamics model and the rigorous handling of observation errors, but also closely related to the almost identical orbital configurations of GFO-C and GFO-D. Under these conditions, the accuracy gain of joint processing of binary satellite data compared to single-satellite solutions is quite limited.

[0092] Satellite Laser Ranging (SLR) is commonly used to verify the orbital accuracy of low-Earth orbit (LEO) satellites, particularly sensitive to the radial component. Conventional methods estimate ranging bias and other corrections during orbit determination to improve the consistency between SLR and orbit. However, because instrument system errors are difficult to completely separate from orbital errors, the resulting residuals may underestimate the true error. To avoid this bias, this invention employs an Observation Minus Calculation (OMC) strategy for verification. Under this strategy, the SLR residual values ​​may be larger, but they maintain good comparability when using a Precise Scientific Orbit (PSO) as a reference. Figure 6The SLR residuals for dynamic orbits (floating-point and IAR solutions) are presented; samples exceeding 10 cm were removed to eliminate outliers. Given that both the orbit integration step size and GPS sampling interval are 30 s, the PSO provided by JPL was downsampled to 30 s during SLR verification to ensure consistency and fairness with the SLR epoch interpolation. Results show that in both single-star and binary-star modes, the SLR residuals of GFO-D decreased from 1.3 cm to 1.1 cm after introducing IAR, demonstrating some improvement. This result is consistent with the SLR residual level of JPL PSO, thus indirectly confirming the effectiveness of the SLR verification method of this invention and the positive effect of IAR on dynamic orbits. In contrast, the floating-point and IAR solutions of GFO-C do not show significant differences in SLR residuals, suggesting that SLR has limited sensitivity to its improvement, and the residuals may still be affected by other factors.

[0093] In addition to external SLR checks, GFO orbital accuracy can also be assessed through inter-satellite K-band ranging (KBR) observations. Figure 7 The empirical cumulative distribution function (ECDF) of KBR residuals for dynamic orbits based on floating-point solutions and integer ambiguity fixed solutions (IAR) in single-satellite and binary modes is presented and compared with the results of JPL PSO. Overall, the residuals in the single-satellite mode are slightly larger than those in the binary mode, reflecting the advantage of jointly estimating spherical harmonic coefficients in improving relative orbit consistency. IAR further improves inter-satellite accuracy: in the single-satellite mode, the standard deviation (STD) of the KBR residuals decreases from 0.68 cm to 0.30 cm; in the binary mode, it decreases from 0.60 cm to 0.30 cm. This result is consistent with the simplified dynamic ambiguity fixed solutions reported by Zhang et al. (2021) and Zhou et al. (2021), further demonstrating the effectiveness of IAR. JPL PSO, by using double-difference ambiguity fixing, achieves a residual standard deviation of 0.16 cm. In summary, the dynamic orbit determination method combined with IAR can achieve high-precision inter-satellite distance estimation.

[0094] Real-time orbit integration of low-Earth orbit satellites driven by onboard accelerometer observations provides a rigorous internal verification method for testing the consistency of dynamic mechanical models and their parameterization. Following the experimental scheme designed by Du et al. (2024), this invention conducted a 24-hour real-time orbit integration on January 1, 2020. This integration used the 30th-order time-varying gravity field model and accelerometer scaling factor obtained in December 2019; the accelerometer bias parameter was taken from the time mean of the estimation results on December 31, 2019; and the initial orbital state vector was the integration result from the previous day. Figure 8The differences between the floating-point solution and the IAR solution in orbit integration results and the JPL Precision Scientific Orbit (PSO) are shown. The radial and normal errors of the 24-hour integration are less than 4 cm and 12 cm, respectively, exhibiting a clear 1-cycle-per-revolution (1-CPR) oscillation characteristic. In contrast, the tangential error increases to approximately 1.2 m, mainly attributed to the unmodeled thruster pulse effect, which is particularly significant in the tangential direction. As can be seen from the figure, the introduction of IAR significantly suppresses the error accumulation in the tangential direction compared to the floating-point solution. The three-dimensional RMS errors of the 3-hour arc segment are all less than 7 cm, indicating that this dynamic method has strong robustness in short-arc orbit prediction. In the 24-hour integration, the radial, normal, and tangential RMS errors of the floating-point solution are 1.9 cm, 4.4 cm, and 82.3 cm, respectively; after introducing IAR, the errors are further reduced to 1.3 cm, 3.3 cm, and 62.4 cm. The results show that IAR can effectively improve accuracy in long-arc orbit integration, especially in mitigating the accumulation of tangential errors.

[0095] Within the framework of dynamic orbit determination, the spherical harmonic coefficients of the Earth's gravity field are simultaneously estimated along with the satellite orbit. This invention evaluates the inverted gravity field model by comparing ambiguity-floating-point solutions with integer ambiguity-fixed solutions, aiming to examine the impact of ambiguity fixation on the quality of gravity field reconstruction. Furthermore, it compares...

[0096] The results of GFO-C, GFO-D and their joint solution are used to evaluate the effectiveness of single-satellite receiver gravitational field recovery and to explore its implications for future mission design. Figure 9 The geoid elevation differences for different gravity field models are presented. Since the gravity field model derived from GFO-D has similar characteristics and performance to GFO-C, the results from GFO-D are omitted. As shown in the figure, the geoid elevation difference curves illustrate the overall similarity and significant differences between the different schemes. Before proceeding with a detailed comparison, it is important to note that the estimation of low-order coefficients is highly sensitive to various systematic errors—such as variations in antenna phase center due to inaccurate modeling. These systematic effects often dominate the biases in low-order coefficients, thus masking the influence of IAR or satellite configuration.

[0097] Compared to the single-satellite (GFO-C) configuration, the double-satellite (GFO-C / D) configuration exhibits a lower geoid elevation difference under both floating-point and integer ambiguity-fixed solutions. This is mainly attributed to the increased number of observations and better spatiotemporal sampling brought about by the nadir trajectory in the double-satellite mode, thereby enhancing the rank and conditionality of the normal equations and improving the estimability of the spherical harmonic coefficients. Regarding the impact of IAR, within the first 20 orders, the Float and IAR error curves in both single-satellite and double-satellite configurations are almost parallel and the difference is negligible; considering that the effective resolution for gravity field reconstruction using only GPS observations is approximately 20 orders (about 2000 km), this result is physically consistent. Above the 20th order, the error curves begin to separate significantly, suggesting that although IAR improves orbital accuracy, the residual bias introduced by the fixed ambiguity may increase the sensitivity of higher-order coefficients to unmodeled errors. Error spectrum analysis of gravity field spherical harmonic coefficients ( Figure 10 Furthermore, it is shown that the IAR solution exhibits more significant error amplification near the resonance order (especially the 0th order band term and the 16th order orbital resonance term), indicating that IAR may exacerbate the sensitivity of the spherical harmonic coefficients to the orbital resonance effect.

[0098] Although IAR significantly improves orbital accuracy, noise is amplified in higher-order spherical harmonic coefficients derived from the gravity field (especially those related to orbital resonance). To further analyze this phenomenon, this invention performs amplitude spectral density (ASD) analysis on the post-fit GPS carrier phase residuals. Figure 11 Considering that the continuous tracking arc of a single GPS satellite by a GFO satellite is typically about 40 minutes, to ensure temporal coverage of the entire solar arc, this invention selects the GPS satellite carrier phase residual with the highest elevation angle as the representative value at each epoch. The results show that the carrier phase residual of the dynamic solution is not white noise, but exhibits obvious non-stationary characteristics. Its main sources include uncertainties in the background force model, aliasing signals caused by atmospheric and oceanic variability, and high-frequency quality variations that are difficult to characterize in the lunar solution's gravity field.

[0099] Comparing the ASD of the floating-point solution and the IAR solution reveals that the noise in the low-frequency (<5-CPR) range of the IAR solution is noticeably increased, indicating that some disturbances or modeling uncertainties are not fully absorbed within the IAR-based DOD framework and are instead reflected in the GPS carrier phase residuals. Figure 12The post-fit phase residual RMS of two types of solutions during DOY 182–365 in 2019 are presented. The average RMS for both is below 10 mm. For the GFO-C satellite, the average floating-point solution is 5.5 mm, while the IAR solution increases to 6.8 mm (an increase of 19%). For the GFO-D satellite, the average floating-point solution is 5.9 mm, while the IAR solution increases to 7.1 mm (an increase of 17%). This indicates that while IAR can improve orbit accuracy, it may introduce residual biases during the fixation process due to unmodeled effects. The increased carrier phase residual RMS of IAR implies a weakening of the effective contribution of carrier phase observations to gravity field recovery. Although the specific mechanism of noise amplification still needs further clarification, the above results suggest that caution should be exercised when applying IAR in gravity field estimation. Future work should optimize the IAR strategy to better accommodate background model uncertainties, preserving orbit accuracy gains while avoiding adverse effects on higher-order gravity field coefficients.

Claims

1. A kinematic orbit determination method based on integer ambiguity fixing, characterized in that, The dynamic orbit determination model is established, and a GPS observation model is established, and then the integer ambiguity fixing IAR is carried out, wherein a single receiver ambiguity correction single difference IAR method is adopted, specifically including: first, the GPS carrier phase observation is divided into continuous tracking segments for processing, for each tracking segment, the HMW combination is constructed by using the dual-frequency pseudo-range and carrier phase data, then, the cross-segment inconsistency is corrected, the integer of the wide-lane ambiguity is realized, the corrected wide-lane ambiguity is combined with the ionosphere-free IF ambiguity float estimation value obtained by the least square method, and finally, the IF ambiguity difference is constrained to an integer value, so that the single difference ambiguity is fixed: where i and j represent two different GPS satellites being simultaneously tracked by the LEO receiver, σ N denotes the associated standard deviation.

2. The dynamic orbit determination method based on integer ambiguity fixing according to claim 1, characterized in that, σ N set to 10 -6 m.

3. The dynamic orbit determination method based on integer ambiguity fixing as claimed in claim 1, characterized in that, The carrier phase residual test in the integer ambiguity fixing process is also included, the monthly solution dynamic parameter estimated by the float solution is introduced into the daily normal equation, and the corresponding diagonal weight matrix is constrained, and the diagonal weight matrix of the monthly solution dynamic parameter is calculated as follows: where diag(*) extracts the diagonal elements of a matrix, N PP denotes the monthly solution equation matrix, once all ambiguities are fixed and verified, they will be included in the corresponding daily solution equation as high-weight constraints, then, these daily solution equations with introduced ambiguity constraints are cumulated to build a joint monthly solution equation system, finally, least squares adjustment is performed on it to jointly estimate the satellite initial orbit, clock error, accelerometer calibration parameters and the spherical harmonic coefficients of the Earth's gravity field.

4. The dynamic orbit determination method based on integer ambiguity fixing according to claim 1, wherein, The dynamic orbit determination model is established, and a GPS observation model is established, and then the integer ambiguity fixing IAR is carried out, wherein a single receiver ambiguity correction single difference IAR method is adopted, specifically including: first, the GPS carrier phase observation is divided into continuous tracking segments for processing, for each tracking segment, the HMW combination is constructed by using the dual-frequency pseudo-range and carrier phase data, then, the cross-segment inconsistency is corrected, the integer of the wide-lane ambiguity is realized, the corrected wide-lane ambiguity is combined with the ionosphere-free IF ambiguity float estimation value obtained by the least square method, and finally, the IF ambiguity difference is constrained to an integer value, so that the single difference ambiguity is fixed: where f g represents conservative forces; f ng represents non-conservative forces, which are modeled using on-board accelerometer observations, which need to be calibrated, with parameters S representing a calibration scaling factor and b representing a bias factor, is the satellite velocity vector, C nm and S nm represent spherical harmonic coefficients of the Earth's gravitational field, dynamic parameters S, b, C nm and S nm collectively referred to as the set of dynamic parameters p i ,…,p d where d is the total number of parameters, and the state of the satellite at any time t can be represented as a state vector y(t): The derivative of formula (2) is obtained as follows: where the dynamic force model is used to compute the conservative and non-conservative force accelerations acting on the satellite Since the initial state vector y(t0) and the dynamic model parameters p i ,…,p d are the estimated values obtained in the orbit determination process, they need to be linearized to solve the partial derivatives of the satellite state with respect to the estimated parameters at any time t. After the initial state vector is determined, an orbit integration is performed to propagate the state forward in time. The partial derivatives of the satellite state with respect to the initial state vector y(t0) and the dynamic parameters at any time t are represented by the state transition matrix Φ(t, t0) and the parameter sensitivity matrix S(t, t0), respectively: The derivative of formula (4) with respect to time t is obtained as follows: The initial condition is: Given the satellite orbit state, the orbit is linearized by truncating the Taylor series expansion at any time: Φ(t, t0) and S(t, t0) are determined by integrating formula (5) and (6), and formula (7) can be used to correct the nominal orbit by using the orbit observation.

5. The dynamic orbit determination method based on integer ambiguity fixing as claimed in claim 1, characterized in that, In the GPS observation model, the low-orbit satellite dual-frequency pseudo-range and carrier phase observation equation is: wherein, and respectively represent the pseudo-range and carrier phase observations; i represents the i th frequency; p represents the geometric distance between satellite s and the antenna reference point of receiver r; c is the speed of light in vacuum; dt r and dt s are the clock biases of the receiver and satellite respectively; m r,Ci and m r,Li are the pseudo-range and carrier phase hardware delays at the receiving end, and are the pseudo-range and carrier phase hardware delays at the satellite end; is the ionospheric refraction factor, f i represents the corresponding carrier phase frequency; represents the first-order ionospheric delay on L1 frequency; l i is the carrier phase wavelength on f i frequency; is the integer ambiguity in a unit cycle; system errors, such as relativistic effects, antenna phase winding effects, phase center offsets and variations, have been pre-corrected by existing models, and the residual errors can be ignored; The GPS satellite orbit and clock bias and hardware delay are corrected by accurate GPS products, and the first-order ionospheric delay is eliminated by the ionosphere-free (IF) combination, and the observation equation is simplified as follows: Wherein, In equation (9), λ if = c / (f1+f2) represents the IF carrier phase wavelength; represents the ambiguity, due to the rank deficiency of the observation equation, the actual observation equation used is as follows: where dt r,Cif = dt r + μ r,Cif denotes the pseudo-range for the receiver clock determination of the pseudo-range and carrier phase observation equations, ambiguity absorbing the bias of the pseudo-range and carrier phase, reorganized as a floating-point number Substituting equation (7) into equation (12) and linearizing, the DOD observation model can be expressed as: where prefit * is the difference between the linearized observations and computed values; is the initial orbit state; dp0= [dS db dC nm dS nm is the accelerometer to be estimated and the gravity field spherical harmonic coefficients.

6. The dynamic orbit determination method based on integer ambiguity fixing as claimed in claim 1, characterized in that, Variance component estimation (VCE) is adopted, the variance scale factor of the pseudo-range and carrier phase is estimated according to the posteriori residual respectively, and the observation weight is dynamically updated, so that the relative contribution of different observation types is balanced according to the statistical characteristics, and the prior weighting matrix of the observation type is defined in the form of a diagonal matrix as follows: where, denotes the variance of the ith observation type, q denotes the total number of observation types, the variance component f i The calculation formula is: where, y i denotes the Observable-Computed vector, x i denotes the parameter to be estimated, A i denotes the corresponding design matrix, m i denotes the number of observations of each observation type, and the updated weight matrix W is: To accelerate the convergence, the initial standard deviation of pseudorange observation is set to 0.3 m and the initial standard deviation of carrier phase observation is set to 3 mm.

7. The dynamic orbit determination method based on integer ambiguity fixing as claimed in claim 1, characterized in that, The SuperSTAR-FO accelerometer carried by the GFO satellite is inevitably affected by the instrument physical characteristics and the change of the orbit environment in observation, which is manifested as an unknown scale factor and a time-varying bias, etc., therefore, the accelerometer observation must be calibrated, and the formula for calculating the calibrated accelerometer measurement value in the inertial reference frame is as follows: acc cal = R · (S · a obs + b bias )(19), where a obs denotes the raw acceleration measurements in the satellite science reference frame (SRF), R denotes the rotation matrix from the SRF to the inertial frame, S denotes the scale factor matrix, and b bias denotes the bias vector, and the scale factor matrix S has the structure: The matrix includes three components: (1) diagonal terms: s x , s y , and s z represent scale factors; (2) shear terms: a, b, and g for considering non-orthogonality between accelerometer axes; (3) rotation terms z, e, and d represent SRF misalignment with accelerometer frame. The scale factor matrix S is usually estimated once a month, the bias vector b bias is modeled as a second-order polynomial in time for each accelerometer axis (x, y, and z): b i = b i,0 + b i,1 (t - t0) + b i,2 (t - t0) 2 , i = x, y, z (21), where b i,0 , b i,1 , and b i,2 are constant, linear, and quadratic terms for the corresponding axis, t is the current time, and t0 is the start time of the parameter estimation arc. The bias parameters are estimated in sets of every 6 hours to consider short-term changes in instrument behavior.

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