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19 pages, 6733 KiB  
Article
Real-Time Orbit Determination of Micro–Nano Satellite Using Robust Adaptive Filtering
by Jing Chen, Xiaojun Jin, Cong Hou, Likai Zhu, Zhaobin Xu and Zhonghe Jin
Sensors 2024, 24(24), 7988; https://doi.org/10.3390/s24247988 (registering DOI) - 14 Dec 2024
Viewed by 181
Abstract
Low-performing GPS receivers, often used in challenging scenarios such as attitude maneuver and attitude rotation, are frequently encountered for micro–nano satellites. To address these challenges, this paper proposes a modified robust adaptive hierarchical filtering algorithm (named IARKF). This algorithm leverages robust adaptive filtering [...] Read more.
Low-performing GPS receivers, often used in challenging scenarios such as attitude maneuver and attitude rotation, are frequently encountered for micro–nano satellites. To address these challenges, this paper proposes a modified robust adaptive hierarchical filtering algorithm (named IARKF). This algorithm leverages robust adaptive filtering to dynamically adjust the distribution of innovation vectors and employs a fading memory weighted method to estimate measurement noise in real time, thereby enhancing the filter’s adaptability to dynamic environments. A segmented adaptive filtering strategy is introduced, allowing for flexible parameter adjustment in different dynamic scenarios. A micro–nano satellite equipped with a miniaturized dual-frequency GPS receiver is employed to demonstrate precise orbit determination capabilities. On-orbit GPS data from the satellite, collected in two specific scenarios—slow rotation and Earth-pointing stabilization—are analyzed to evaluate the proposed algorithm’s ability to cope with weak GPS signals and satellite attitude instability as well as to assess the achievable orbit determination accuracy. The results show that, compared to traditional Extended Kalman Filters (EKF) and other improved filtering algorithms, the IARKF performs better in reducing post-fit residuals and improving orbit prediction accuracy, demonstrating its superior robustness. The three-axes orbit determination internal consistency precision can reach the millimeter level. This work explores a feasible approach for achieving high-performance orbit determination in micro–nano satellites. Full article
(This article belongs to the Section Navigation and Positioning)
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Figure 1
<p>Flowchart of IARKF strategy.</p>
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<p>Comparison of posterior residuals, number of positioning satellites, and GDOP.</p>
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<p>Comparison of posterior residuals in the sky vision.</p>
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<p>Comparison of posterior residual scatter.</p>
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<p>Filtering level variation curve and distribution map.</p>
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<p>Comparison of errors for overlapping arc.</p>
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<p>Comparison of posterior residuals, number of positioning satellites, and GDOP.</p>
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<p>Comparison of posterior residuals in the sky vision.</p>
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<p>Comparison of posterior residual scatter.</p>
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<p>Filtering level variation curve and distribution map.</p>
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<p>Comparison of errors for overlapping arc.</p>
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18 pages, 6204 KiB  
Article
An Integrity Monitoring Method for Navigation Satellites Based on Multi-Source Observation Links
by Jie Xin, Dongxia Wang and Kai Li
Remote Sens. 2024, 16(23), 4574; https://doi.org/10.3390/rs16234574 - 6 Dec 2024
Viewed by 313
Abstract
The BeiDou-3 navigation satellite system (BDS-3) has officially provided positioning, navigation, and timing (PNT) services to global users since 31 July 2020. With the application of inter-satellite link technology, global integrity monitoring becomes possible. Nevertheless, the content of integrity monitoring is still limited [...] Read more.
The BeiDou-3 navigation satellite system (BDS-3) has officially provided positioning, navigation, and timing (PNT) services to global users since 31 July 2020. With the application of inter-satellite link technology, global integrity monitoring becomes possible. Nevertheless, the content of integrity monitoring is still limited by the communication capacity of inter-satellite links and the layout of ground monitoring stations. Low earth orbit (LEO) satellites have advantages in information-carrying rate and kinematic velocity and can be used as satellite-based monitoring stations for navigation satellites. Large numbers of LEO satellites can provide more monitoring data than ground monitoring stations and make it easier to obtain full-arc observation data. A new challenge of redundant data also arises. This study constructs multi-source observation links with satellite-to-ground, inter-satellite, and satellite-based observation data, proposes an integrity monitoring method with optimization of observation links, and verifies the performance of integrity monitoring with different observation links. The experimental results show four findings. (1) Based on the integrity status of BDS-3, the proposed system-level integrity mode can realize full-arc anomaly diagnosis in information and signals according to the observation conditions of the target satellite. Apart from basic navigation messages and satellite-based augmentation messages, autonomous messages and inter-satellite ranging data can be used to evaluate the state of the target satellite. (2) For a giant LEO constellation, only a small number of LEO satellites need to be selected to construct a minimum satellite-based observation unit that can realize multiple returns of navigation messages and reduce the redundancy of observation data. With the support of 12 and 30 LEO satellites, the minimum number of satellite-based observation links is 1 and 4, respectively, verifying that a small amount of LEO satellites could be used to construct a minimum satellite-based observation unit. (3) A small number of LEO satellites can effectively improve the observation geometry of the target satellite. An orbit determination observation unit, which consists of chosen satellite-to-ground and/or satellite-based observation links based on observation geometry, is proposed to carry out fast calculations of satellite orbit. If the orbit determination observation unit contains 6 satellite-to-ground monitoring links and 6/12/60 LEO satellites, the value of satellite position dilution of precision (SPDOP) is 38.37, 24.60, and 15.71, respectively, with a 92.95%, 95.49%, and 97.12% improvement than the results using 6 satellite-to-ground monitoring links only. (4) LEO satellites could not only expand the resolution of integrity parameters in real time but also augment the service accuracy of the navigation satellite system. As the number of LEO satellites increases, the area where UDRE parameters can be solved in real time is constantly expanding to a global area. The service accuracy is 0.93 m, 0.88 m, and 0.65 m, respectively, with augmentation of 6, 12, and 60 LEO satellites, which is an 8.9%, 13.7%, and 36.3% improvement compared with the results of regional service. LEO satellites have practical application values by improving the integrity monitoring of navigation satellites. Full article
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<p>Design of integrity monitoring system for the BDS-3 satellite.</p>
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<p>Design of integrity monitoring system with support of multi-source observation links.</p>
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<p>The chosen valid satellite-based observation links (<math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>≤</mo> <msub> <mi>γ</mi> <mn>0</mn> </msub> </mrow> </semantics></math>).</p>
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<p>The chosen valid satellite-based observation links (<math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>&gt;</mo> <msub> <mi>γ</mi> <mn>0</mn> </msub> </mrow> </semantics></math>).</p>
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<p>The chosen valid satellite-to-ground observation links.</p>
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<p>Processing of integrity monitoring system.</p>
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<p>Divided grid points and chosen ground monitoring stations.</p>
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<p>Multiple numbers in scenario 3.</p>
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<p>Coverage rate of four-multiple numbers in scenario 3.</p>
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<p>Coverage rate of more than four-multiple numbers in scenario 3. (The grids in the blue area can be monitored by no less than four LEO satellites; the grids in the yellow area can be monitored by less than four LEO satellites).</p>
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<p>Number of available links and SPDOP values in scenario 1.</p>
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<p>Number of available links and SPDOP values in scenario 2.</p>
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<p>Number of available links and SPDOP values in scenario 3.</p>
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<p>Number of available links and SPDOP values in scenario 4.</p>
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<p>Number of available links and PDOP values of BJFS in scenario 4.</p>
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19 pages, 5444 KiB  
Article
Two-Dimensional Directions Determination for GNSS Spoofing Source Based on MEMS-Based Dual-GNSS/INS Integration
by Chengzhong Zhang, Dingjie Wang and Jie Wu
Remote Sens. 2024, 16(23), 4568; https://doi.org/10.3390/rs16234568 - 5 Dec 2024
Viewed by 425
Abstract
Satellite navigation spoofing is a major challenge in the field of satellite/inertial integrated navigation security. To effectively enhance the anti-spoofing capability of a low-cost GNSS/MEMS-SINS integrated navigation system, this paper proposes a method integrating a dual-antenna global navigation satellite system (GNSS) and a [...] Read more.
Satellite navigation spoofing is a major challenge in the field of satellite/inertial integrated navigation security. To effectively enhance the anti-spoofing capability of a low-cost GNSS/MEMS-SINS integrated navigation system, this paper proposes a method integrating a dual-antenna global navigation satellite system (GNSS) and a micro-inertial measurement unit (MIMU) to determine the two-dimensional directions of spoofing signal sources. The proposed method evaluates whether the single-difference carrier-phase measurements conform to the corresponding directions given in ephemeris files and employs micro-inertial navigation technology to determine the two-dimensional directions of the signal source. Based on a set of short-baseline dual-station measurements, the accuracy of the proposed method in determining the two-dimensional azimuths of satellites in synchronous orbits is verified, and the deviation from the real value is evaluated. The experimental results show that the proposed method can effectively identify the spoofed satellite signals while providing high-precision direction information at three different distances: 100 m, 10 km, and 36,000 km. The two-dimensional angle errors do not exceed 0.2 rad, 0.05 rad, and 0.01 rad, respectively. Full article
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Figure 1
<p>Typical application scenario for the proposed method. The dotted line represents the trajectory of the unmanned vehicle during satellite signal deception.</p>
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<p>Integrated navigation model.</p>
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<p>Single-differenced carrier phase between two antennas on the baseline vector-source plane.</p>
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<p>Signal source position in the body coordinate system.</p>
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<p>Domains and standardised points of inter-station single-difference integer ambiguities on B<sub>1</sub> and B<sub>3.</sub> The square area represents the range of values for carrier phase ambiguities N<sub>1</sub> and N<sub>3</sub>. Due to the integer constraints for N<sub>1</sub> and N<sub>3</sub>, the feasible candidates for (N<sub>1</sub>, N<sub>3</sub>) can be illustrated as the dots in the square area.</p>
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<p>Experimental device.</p>
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<p>BDS real signal two-dimensional signal source determining errors. (<b>a</b>) Two-dimensional direction-finding error of BeiDou-2 in this experiment; (<b>b</b>) two-dimensional direction-finding error of BeiDou-3 in this experiment; (<b>c</b>) yaw angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line of BeiDou-2 in this experiment; (<b>d</b>) yaw angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line of BeiDou-3 in this experiment; (<b>e</b>) pitch angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line of BeiDou-2 in this experiment; and (<b>f</b>) pitch angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line of BeiDou-3 in this experiment.</p>
Full article ">Figure 7 Cont.
<p>BDS real signal two-dimensional signal source determining errors. (<b>a</b>) Two-dimensional direction-finding error of BeiDou-2 in this experiment; (<b>b</b>) two-dimensional direction-finding error of BeiDou-3 in this experiment; (<b>c</b>) yaw angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line of BeiDou-2 in this experiment; (<b>d</b>) yaw angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line of BeiDou-3 in this experiment; (<b>e</b>) pitch angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line of BeiDou-2 in this experiment; and (<b>f</b>) pitch angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line of BeiDou-3 in this experiment.</p>
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<p>Horizontal reference trajectories of the carrier.</p>
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<p>Near-range spoofing signal two-dimensional signal source determining errors. (<b>a</b>) Two-dimensional direction-finding error for the short-distance spoofing signal source; (<b>b</b>) yaw angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line for the short-distance spoofing signal source; and (<b>c</b>) pitch angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line for the short-distance spoofing signal source.</p>
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<p>MIMU attitude error curves.</p>
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<p>Simulation of spoofing signal two-dimensional signal source determining errors. (<b>a</b>) Simulation of two-dimensional direction-finding errors for the spoofing signal source; (<b>b</b>) simulation of the yaw angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line for the spoofing signal source; and (<b>c</b>) simulation of the pitch angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line for the spoofing signal source.</p>
Full article ">Figure 11 Cont.
<p>Simulation of spoofing signal two-dimensional signal source determining errors. (<b>a</b>) Simulation of two-dimensional direction-finding errors for the spoofing signal source; (<b>b</b>) simulation of the yaw angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line for the spoofing signal source; and (<b>c</b>) simulation of the pitch angle error and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math> line for the spoofing signal source.</p>
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36 pages, 14204 KiB  
Article
A Novel Algorithm for Precise Orbit Determination Using a Single Satellite Laser Ranging System Within a Single Arc for Space Surveillance and Tracking
by Dong-Gu Kim, Sang-Young Park and Eunji Lee
Aerospace 2024, 11(12), 989; https://doi.org/10.3390/aerospace11120989 - 29 Nov 2024
Viewed by 418
Abstract
A satellite laser ranging (SLR) system uses lasers to measure the range from ground stations to space objects with millimeter-level precision. Recent advances in SLR systems have increased their use in space surveillance and tracking (SST). The problem we are addressing, the precise [...] Read more.
A satellite laser ranging (SLR) system uses lasers to measure the range from ground stations to space objects with millimeter-level precision. Recent advances in SLR systems have increased their use in space surveillance and tracking (SST). The problem we are addressing, the precise orbit determination (POD) using one-dimensional range observations within a single arc, is challenging owing to infinite solutions because of limited observability. Therefore, general orbit determination algorithms struggle to achieve reasonable accuracy. The proposed algorithm redefines the cost value for orbit determination by leveraging residual tendencies in the POD process. The tendencies of residuals are quantified as R-squared values using Fourier series fitting to determine velocity vector information. The algorithm corrects velocity vector errors through the grid search method and least squares (LS) with a priori information. This approach corrects all six dimensions of the state vectors, comprising position and velocity vectors, utilizing only one dimension of the range observations. Simulations of three satellites using real data validate the algorithm. In all cases, the errors of the two-line element data (three-dimensional position error of 1 km and velocity error of 1 m/s, approximately) used as the initial values were reduced by tens of meters and the cm/s level, respectively. The algorithm outperformed the general POD algorithm using only the LS method, which does not effectively reduce errors. This study offers a more efficient and accurate orbit determination method, which improves the safety, cost efficiency, and effectiveness of space operations. Full article
(This article belongs to the Section Astronautics & Space Science)
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Figure 1
<p>An example of a geometrically infinite number of orbits with range values observed at a single station within a single arc. (<math display="inline"><semantics> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mi>o</mi> <mo>,</mo> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mrow> </semantics></math> is the actual observation value and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mi>c</mi> <msub> <mrow> <mo>,</mo> <mi>t</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mrow> </semantics></math> is the calculated observation value at observation time <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">X</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> is the given initial value with error, and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">X</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> is the true state vector at <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">X</mi> </mrow> <mo>^</mo> </mover> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> is the estimated state vector by POD using <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mi>o</mi> <mo>,</mo> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mrow> </semantics></math>).</p>
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<p>Simplified flowchart of RSR method.</p>
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<p>Example of unwanted overshoot effect: approximation of square wave by 30-term Fourier series [<a href="#B34-aerospace-11-00989" class="html-bibr">34</a>].</p>
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<p>Correlation of LS and position error (solid line) with dynamics (dotted line) and measurement error (dashed line).</p>
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<p>Examples of the bow-tie effect depending on the inadequacies of the LS dominated by the influence of the velocity vector error in the initial state.</p>
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<p>Meaning of SST and SSE for the R-squared value.</p>
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<p>Comparison of random search, grid search, and proposed grid search methods.</p>
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<p>Algorithm process of proposed grid search method for velocity vector searching.</p>
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<p>Flowchart of RSR POD algorithm.</p>
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<p>Range observations, which are the laser photon’s two-way travel time divided in half and multiplied by the speed of light, of Cryosat-2, Sentinel-3A, and Saral satellites utilized in the simulations.</p>
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<p>The comparison of errors between the SP-3 and TLE data in the GCRF for approximately 1 month around the time of observation in each case; (<b>a</b>) 3D position and velocity errors in all cases, (<b>b</b>) velocity vector errors from the Cryosat-2 satellite’s TLE near December 2018, (<b>c</b>) velocity vector errors from the Sentinel-3A satellite’s TLE near June 2018, (<b>d</b>) velocity vector errors from the Saral satellite’s TLE near August 2018.</p>
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<p>Results of the RSR POD for Case 1 (<math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> is the velocity vector error computed from SP-3, assumed to be the true error value); (<b>a</b>) optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> among the gridded values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math> on the x-axis, (<b>b</b>) optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> among the gridded values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math> on the y-axis, (<b>c</b>) optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> among the gridded values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math> on the z-axis, (<b>d</b>) all cost values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>J</mi> </mrow> <mrow> <mi>R</mi> <mi>S</mi> <mi>R</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> <mo>(</mo> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math>) and minimum cost <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>J</mi> </mrow> <mrow> <mi>R</mi> <mi>S</mi> <mi>R</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> <mo>(</mo> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>) in the k-th loop.</p>
Full article ">Figure 13
<p>Calculated residuals <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>19</mn> </mrow> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> with the first given initial state from TLE (left y-axis). <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>19</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> using the optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> of the first, second, and third loops (right y-axis), the fitted curve <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow> <mn mathvariant="bold">6</mn> </mrow> </msub> <mo>(</mo> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>19</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math> using the sixth Fourier series function for residuals <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>19</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> in the RSR POD algorithm process for Case 1, and the difference <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> between the dependent variable of the fitted curve at observation time <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> and the residual <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mi>i</mi> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> as data used in the fitting.</p>
Full article ">Figure 14
<p>POD results using only LS compared to that using RSR in the same condition for Case 1; (<b>a</b>) 3D position and velocity errors, (<b>b</b>) residuals (only LS: left y-axis, RSR: right y-axis).</p>
Full article ">Figure 15
<p>Results of the RSR POD for Case 2 (<math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> is the velocity vector error computed from SP-3, assumed to be the true error value); (<b>a</b>) optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> among the gridded values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math> on the x-axis, (<b>b</b>) optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> among the gridded values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math> on the y-axis, (<b>c</b>) optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> among the gridded values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math> on the z-axis, (<b>d</b>) all cost values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>J</mi> </mrow> <mrow> <mi>R</mi> <mi>S</mi> <mi>R</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> <mo>(</mo> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math>) and minimum cost <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>J</mi> </mrow> <mrow> <mi>R</mi> <mi>S</mi> <mi>R</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> <mo>(</mo> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>) in the first, third, and fifth loops.</p>
Full article ">Figure 16
<p>Calculated residuals <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>18</mn> </mrow> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> with the first given initial state from TLE (left y-axis). <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>18</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> using the optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> of the first, third, and fifth loops (right y-axis), the fitted curve <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow> <mn mathvariant="bold">6</mn> </mrow> </msub> <mo>(</mo> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>18</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math> using the sixth Fourier series function for residuals <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>18</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> in the RSR POD algorithm process for Case 2, and the difference <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> between the dependent variable of the fitted curve at observation time <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> and the residual <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mi>i</mi> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> as data used in the fitting.</p>
Full article ">Figure 17
<p>POD results of only LS compared to RSR in the same condition for Case 2; (<b>a</b>) 3D position and velocity errors, (<b>b</b>) residuals (only LS: left y-axis, RSR: right y-axis).</p>
Full article ">Figure 18
<p>Results of the RSR POD for Case 3 (<math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> is the velocity vector error computed from SP-3, which is assumed to be the true error value); (<b>a</b>) optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> among the gridded values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math> on the x-axis, (<b>b</b>) optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> among the gridded values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math> on the y-axis, (<b>c</b>) optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> among the gridded values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math> on the z-axis, (<b>d</b>) all cost values <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>J</mi> </mrow> <mrow> <mi>R</mi> <mi>S</mi> <mi>R</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> <mo>(</mo> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </mrow> </semantics></math>) and minimum cost <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>J</mi> </mrow> <mrow> <mi>R</mi> <mi>S</mi> <mi>R</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> <mo>(</mo> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math>) in the first, second, third, fourth, and fifth loops.</p>
Full article ">Figure 19
<p>Calculated residuals <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>23</mn> </mrow> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> with the first given initial state from TLE (left y-axis). <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>23</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> using the optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> of the first, third, and fifth loops (right y-axis), the fitted curve <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow> <mn mathvariant="bold">6</mn> </mrow> </msub> <mo>(</mo> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>23</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math> using the sixth Fourier series function for residuals <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>23</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> in the RSR POD algorithm process for Case 3, and the difference <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> between the dependent variable of the fitted curve at observation time <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> and the residual <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">ϵ</mi> </mrow> <mrow> <mi>i</mi> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> as data used in the fitting.</p>
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<p>POD results of only LS compared to RSR in the same condition for Case 3; (<b>a</b>) 3D position and velocity errors, (<b>b</b>) residuals (only LS: left y-axis, RSR: right y-axis).</p>
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<p>Results compared to SP-3 by propagating the state vector <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">X</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>0</mn> </mrow> </msubsup> </mrow> </semantics></math> given initially through the TLE, the estimated state vector <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">X</mi> </mrow> <mo>^</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mn>0</mn> </mrow> </msubsup> </mrow> </semantics></math> through the only LS method, and the estimated state vector <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi mathvariant="bold-italic">X</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> calculated through the RSR POD algorithm for 24 h; (<b>a</b>) Case 1: Cryosat-2 satellite, (<b>b</b>) Case 2: Sentinel-3A satellite, (<b>c</b>) Case 3: Saral satellite.</p>
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<p>RMS of residuals per gridded vector of the first, second, and third loops for Case 1.</p>
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<p>Calculated residuals <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>11</mn> </mrow> <mrow> <msubsup> <mrow> <mi>e</mi> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> with the first given initial state from TLE (left y-axis). <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>11</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> using the optimal solution <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>k</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </semantics></math> of the first, third, and fifth loops (right y-axis), the fitted curve <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msub> <mo>(</mo> <msubsup> <mrow> <mi>ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>11</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math> using the fourth Fourier series function for residuals <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>ϵ</mi> </mrow> <mrow> <mn>1</mn> <mo>:</mo> <mn>11</mn> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> in the RSR POD algorithm process for 1/2 arc length of Case 3, and the difference <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> between the dependent variable of the fitted curve at observation time <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> and the residual <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>ϵ</mi> </mrow> <mrow> <mi>i</mi> </mrow> <mrow> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mo>*</mo> </mrow> </msubsup> </mrow> </msubsup> </mrow> </semantics></math> as data used in the fitting.</p>
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<p>All cost values, R-squared, <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi>J</mi> </mrow> <mrow> <mi>R</mi> <mi>S</mi> <mi>R</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msubsup> <mo>(</mo> <msubsup> <mrow> <mover accent="true"> <mrow> <mi>e</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>v</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msubsup> </mrow> </semantics></math>), in the first loop of the RSR POD for 1/4 arc length of Case 3.</p>
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<p>Bow-tie effect following velocity error for MEO satellite, LAGEOS-1; (<b>a</b>) not-corrected velocity error case (error level: meter), (<b>b</b>) corrected velocity error case (error level: centimeter).</p>
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<p>Results of POD by EKF and UKF for Cases 1, 2, and 3; (<b>a</b>) 3D position and velocity errors for Case 1, (<b>b</b>) 3D position and velocity errors for Case 2, (<b>c</b>) 3D position and velocity errors for Case 3, and (<b>d</b>) residuals for three cases.</p>
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18 pages, 6209 KiB  
Article
Impact of Latency and Continuity of GNSS Products on Filter-Based Real-Time LEO Satellite Clock Determination
by Meifang Wu, Kan Wang, Jinqian Wang, Wei Xie, Jiawei Liu, Beixi Chen, Yulong Ge, Ahmed El-Mowafy and Xuhai Yang
Remote Sens. 2024, 16(22), 4315; https://doi.org/10.3390/rs16224315 - 19 Nov 2024
Viewed by 504
Abstract
High-precision Low Earth Orbit (LEO) satellite clocks are essential for LEO-augmented Positioning, Navigation, and Timing (PNT) services. Nowadays, high-precision LEO satellite clocks can be determined in real-time using a Kalman filter either onboard or on the ground, as long as the GNSS observations [...] Read more.
High-precision Low Earth Orbit (LEO) satellite clocks are essential for LEO-augmented Positioning, Navigation, and Timing (PNT) services. Nowadays, high-precision LEO satellite clocks can be determined in real-time using a Kalman filter either onboard or on the ground, as long as the GNSS observations collected onboard LEO satellites can be transmitted to the ground in real-time. While various real-time and high-precision GNSS products are available nowadays in the latter case, their continuity and latencies in engineering reality are not as perfect as expected and will lead to unignorable impacts on the precision of the real-time LEO satellite clocks. In this study, based on real observations of Sentinel-3B, the impacts of different latencies and continuity of the real-time GNSS products on LEO real-time clocks are determined and discussed for two scenarios, namely the “epoch estimation” and “arc estimation” scenarios. The former case refers to the traditional filter-based processing epoch-by-epoch, and the latter case connects LEO satellite clocks from different rounds of filter-based processing under a certain arc length. The two scenarios lead to the “end-loss” and “mid-gap” situations. Latencies of the real-time GNSS products are discussed for the cases of orbit-only latency, clock-only latency, and combined forms, and different handling methods for the missing GNSS satellite clocks are discussed and compared. Results show that the real-time LEO satellite clock precision is very sensitive to the precision of real-time GNSS satellite clocks, and prediction of the latter becomes essential in case of their latencies. For the “end-loss” situation, with a latency of 30 to 120 s for the GNSS real-time clocks, the LEO satellite clock precision is reduced from about 0.2 to 0.28–0.57 ns. Waiting for the GNSS products in case of their short latencies and predicting the LEO satellite clocks instead could be a better option. For “arc-estimation”, when the gap of GNSS real-time products increases from 5 to 60 min, the real-time LEO clock precision decreases from 0.26 to 0.32 ns. Full article
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Graphical abstract

Graphical abstract
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<p>The continuity of the received real-time GPS satellite clocks from four analysis centers.</p>
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<p>Epoch-processing (<b>left</b>) and arc-processing (<b>right</b>) of real-time LEO satellite clock determination based on the kinematic model using the Kalman filter.</p>
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<p>Processing timelines of the two methods for real-time LEO satellite clocks determination based on the kinematic model and using the Kalman filter.</p>
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<p>(<b>Top</b>): Sentinel-3B satellite clocks determined using the Kalman filter-based kinematic model with the CNES real-time products without latency (blue line) and using the BLS-based reduced-dynamic model with the CODE final products (red line); (<b>Bottom</b>): differences between the two sets of clocks (real-time and final) in the top panel after unifying the time reference.</p>
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<p>Sentinel-3B clocks determined using the Kalman filter-based kinematic model and CNES real-time products with different latencies for Scenario A. The clocks using CNES real-time products without loss at the end are used as the reference.</p>
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<p>Sentinel-3B clocks determined using the Kalman filter-based kinematic model and CNES real-time products with different latencies for Scenario A. The BLS clocks using the CODE final products are used as the reference.</p>
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<p>Sentinel-3B clocks in the five scenarios (see <a href="#remotesensing-16-04315-t003" class="html-table">Table 3</a>) with a latency of 120 s compared with the clocks estimated using the CNES real-time products without latencies.</p>
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<p>Sentinel-3B clocks in the five scenarios (with a latency of 120 s, see <a href="#remotesensing-16-04315-t003" class="html-table">Table 3</a>) and the clocks without latencies in the CNES products, compared with the BLS clocks using the CODE final products.</p>
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<p>The MDEVs of the satellite clocks (<b>left</b>) and the clock estimation errors and prediction errors over 120 s (<b>right</b>) for G16 and G30.</p>
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<p>Sentinel-3B satellite clocks determined using the Kalman filter-based kinematic model with the CNES real-time products without latency (black line); using the BLS-based reduced-dynamic model with the CODE final products (red line); the connected near-real-time clocks estimated in a Kalman filter using the CNES real-time products without gaps (blue line).</p>
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<p>Differences between the C and K clocks (<b>top</b>) and C and R clocks (<b>bottom</b>) for Sentinel-3B. The explanations of C, K, and R clocks are given in <a href="#remotesensing-16-04315-f010" class="html-fig">Figure 10</a>.</p>
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<p>Near-real-time Sentinel-3B clocks based on a Kalman filter-based kinematic model, using CNES products with different “mid-gap” durations. (<b>Top</b>): Sentinel-3B clocks for the entire day; (<b>Bottom</b>): Sentinel-3B clocks from the first epoch affected to the last epoch affected.</p>
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<p>Near-real-time Sentinel-3B clock errors using the Kalman filter-based kinematic model with the CNES real-time products suffering from different “mid-gap” durations, compared with those using continuous CNES real-time products. (<b>Top</b>): Sentinel-3B clocks for the entire day; (<b>Bottom</b>): Sentinel-3B clocks from the first epoch affected to the last epoch affected.</p>
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<p>Near-real-time Sentinel-3B clock errors determined using the Kalman filter-based kinematic model with CNES products of different “mid-gap” durations, compared with the BLS clocks determined using the CODE final products. (<b>Top</b>): Sentinel-3B clocks for the entire day; (<b>Bottom</b>): Sentinel-3B clocks from the first epoch affected to the last epoch affected.</p>
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17 pages, 5464 KiB  
Article
Geographically-Informed Modeling and Analysis of Platform Attitude Jitter in GF-7 Sub-Meter Stereo Mapping Satellite
by Haoran Xia, Xinming Tang, Fan Mo, Junfeng Xie and Xiang Li
ISPRS Int. J. Geo-Inf. 2024, 13(11), 413; https://doi.org/10.3390/ijgi13110413 - 15 Nov 2024
Viewed by 590
Abstract
The GF-7 satellite, China’s inaugural sub-meter-level stereoscopic mapping satellite, has been deployed for a wide range of applications, including natural resource investigation, environmental monitoring, fundamental surveying, and the development of global geospatial information resources. The satellite’s stable platform and reliable imaging systems are [...] Read more.
The GF-7 satellite, China’s inaugural sub-meter-level stereoscopic mapping satellite, has been deployed for a wide range of applications, including natural resource investigation, environmental monitoring, fundamental surveying, and the development of global geospatial information resources. The satellite’s stable platform and reliable imaging systems are crucial for achieving high-quality imaging and precise attitude measurements. However, the satellite’s operation is affected by both internal and external factors, which induce vibrations in the satellite platform, thereby affecting image quality and mapping accuracy. To address this challenge, this paper proposes a novel method for constructing a satellite platform vibration model based on geographic location information. The model is developed by integrating composite data from star sensors and gyroscopes (gyro) with subsatellite point location data. The experimental methodology involves the composite processing of gyro data and star sensor optical axis angles, integration of the processed data through time-matching and normalization, and denoising of the integrated data, followed by trigonometric fitting to capture the periodic characteristics of platform vibrations. The positions of the satellite substellar points are determined from the satellite orbit data. A rigorous geometric imaging model is then used to construct a vibration model with geographic location correlation in combination with the satellite subsatellite point positions. The experimental results demonstrate the following: (1) Over the same temporal range, there is a significant convergence in the waveform similarities between the gyro data and the star sensor optical axis angles, indicating a strong correlation in the jitter information; (2) The platform vibration exhibits a robust correlation with the satellite’s geographic location along its orbit. Specifically, the model reveals that the GF-7 satellite experiences the maximum vibration amplitude between 5° S and 20° S latitude during its ascending phase, and the minimum vibration amplitude between 5° N and 20° N latitude during the descending phase. The model established in this study offers theoretical support for optimizing satellite attitude and mitigating platform vibrations. Full article
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Figure 1

Figure 1
<p>Comparative analysis of real and ideal vibration information from the star sensor and the gyro.</p>
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<p>Flow chart for vibration modeling method of the GF-7 satellite.</p>
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<p>Time consistency analysis of gyro data and star sensor data.</p>
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<p>Time offset analysis in composite data from the gyro and star sensor. Time Offset is the temporal difference between the two datasets. T1 is a specific time point.</p>
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<p>Schematic of the moving average filter.</p>
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<p>Satellite orbital data for calculating geographic locations of subsatellite points.</p>
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<p>Filtered gyro data across multiple satellite tracks: (<b>a</b>) Track 016396; (<b>b</b>) Track 016416; (<b>c</b>) Track 016431; (<b>d</b>) Track 016446.</p>
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<p>Optical axis clamping angle measurements from star sensor.</p>
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<p>Composite analysis of gyro and star sensor pinch angle data.</p>
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<p>Denoising results using moving average filter.</p>
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<p>Results of trigonometric function fitting for vibration data analysis. The blue color indicates composite data. The red curve represents the fitted result.</p>
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<p>Geographic distribution of satellite orbital paths.</p>
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<p>Variations in satellite flutter amplitude relative to geographic location.</p>
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26 pages, 11851 KiB  
Article
Development and Evaluation of Machine Learning Models for Air-to-Land Temperature Conversion Using the Newly Established Kunlun Mountain Gradient Observation System
by Yongkang Li, Qing He, Yongqiang Liu, Amina Maituerdi, Yang Yan and Jiao Tan
Land 2024, 13(11), 1903; https://doi.org/10.3390/land13111903 - 13 Nov 2024
Viewed by 499
Abstract
Mountainous land types are characterized by a scarcity of observational data, particularly in remote areas such as the Kunlun Mountains, where conventional Automatic Weather Stations (AWSs) typically do not record land surface temperature (LST) data. This study aims to develop and evaluate models [...] Read more.
Mountainous land types are characterized by a scarcity of observational data, particularly in remote areas such as the Kunlun Mountains, where conventional Automatic Weather Stations (AWSs) typically do not record land surface temperature (LST) data. This study aims to develop and evaluate models for converting air temperature (TA) to LST using newly established meteorological station data from the Kunlun Mountain Gradient Observation System, thereby providing time-continuous LST data for AWSs. We constructed a conceptual model to explore the relationship between 1.5 m TA and LST and instantiated it using three machine learning algorithms: Support Vector Machine (SVR), Convolutional Neural Network (CNN), and CatBoost. The results demonstrated that the CatBoost algorithm outperformed the others under complex terrain and climatic conditions, achieving a coefficient of determination (R2) of 0.997 and the lowest root mean square error (RMSE) of 0.627 °C, indicating superior robustness and accuracy. Consequently, CatBoost was selected as the optimal model. Additionally, this study analyzed the spatiotemporal distribution characteristics of cloud cover in the Kunlun Mountain region using the MOD11A1 product and assessed the uncertainties introduced by the 8-day average compositing method of the MOD11A2 product. The results revealed significant discrepancies between the monthly average LST derived from polar-orbiting satellites and the hourly composite monthly LST measured on-site or under ideal cloud-free conditions. These differences were particularly pronounced in high-altitude regions (4000 m and above), with the greatest differences occurring in winter, reaching up to 10.2 °C. These findings emphasize the importance of hourly LST calculations based on AWSs for accurately assessing the spatiotemporal characteristics of LST in the Kunlun Mountains, thus providing more precise spatiotemporal support for remote sensing applications in high-altitude regions. Full article
(This article belongs to the Section Land Innovations – Data and Machine Learning)
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<p>Overview of study area.</p>
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<p>Flowchart of data processing steps. Note: The red text section represents the Derivative Data constructed based on the conceptual model proposed in this article.</p>
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<p>Comparison of 10-fold cross-validation of three machine learning algorithms for whether <span class="html-italic">at_diff_up</span> participates or not. (<b>a</b>) is a box plot of RMSE against the Pearson correlation coefficient, (<b>b</b>) is the box plot of significance test probability p against the Pearson correlation coefficient. Note: The feature variable denoted by <b><span class="html-italic">at_diff_up</span></b> represents the difference between the current hourly 1.5 m atmospheric temperature and the corresponding value from the preceding hour.</p>
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<p>Importance box graph of Catboost, the corresponding optimal parameters based on the grid search, and the model validation accuracy metric. Note: ** represents significance through a <span class="html-italic">p</span> &lt; 0.01 correlation test.</p>
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<p>Comparison of Supersite-based measured LST with Catboost-simulated LST across the Kunlun Mountain Vertical Gradient. Note: The elevations of Yeyike, Kalasai, Akesusai, Khunjerab, and Wolonggang are 2275 m, 3013 m, 3934.7 m, 4700 m, and 5896 m, respectively.</p>
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<p>Comparison of Supersite-based measured LST with Catboost-simulated LST across the Kunlun Mountain Vertical Gradient. Note: The elevations of Yeyike, Kalasai, Akesusai, Khunjerab, and Wolonggang are 2275 m, 3013 m, 3934.7 m, 4700 m, and 5896 m, respectively.</p>
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<p>Comparison of Supersite-based measured LST with Catboost-simulated LST across the Kunlun Mountain Vertical Gradient. Note: The elevations of Yeyike, Kalasai, Akesusai, Khunjerab, and Wolonggang are 2275 m, 3013 m, 3934.7 m, 4700 m, and 5896 m, respectively.</p>
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<p>Accuracy evaluation of the CatBoost model across different seasons at five supersites. Panels (<b>a</b>,<b>b</b>) represent the coefficient of determination and the root mean square error of the CatBoost model, respectively.</p>
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<p>Cross-validated Taylor diagrams of Suomi NPP’s VIIRS LST versus Catboost-simulated LST for spring, summer, autumn, winter, daytime, nighttime, and yearly comparisons.</p>
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<p>Cross-validated Taylor diagrams of Terra and Aqua MODIS LST versus Catboost-simulated LST for spring, summer, autumn, winter, daytime, nighttime, and yearly comparisons.</p>
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<p>Spatial distribution of MOD11A1 data availability under cloud influence from 2000 to 2023. Note: (<b>a</b>–<b>g</b>) represent the spatial distribution of data availability for spring, summer, autumn, winter, daytime, nighttime, and the entire year, respectively. (<b>h</b>) represents the spatial distribution of elevation.</p>
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<p>Spatial distribution of MOD11A1 data availability under cloud influence from 2000 to 2023. Note: (<b>a</b>–<b>g</b>) represent the spatial distribution of data availability for spring, summer, autumn, winter, daytime, nighttime, and the entire year, respectively. (<b>h</b>) represents the spatial distribution of elevation.</p>
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<p>Time series comparison of four datasets—8-Day Composite MOD11A2 LST, Site LST in Terra Scan Time, Site LST No Clouds, and Site LST 24h—based on observations from Kunlun Mountain Gradient Stations. Note: (<b>a</b>) Yeyike station represents the 2000 m gradient, (<b>b</b>) Kalasai represents the 3000 m gradient, (<b>c</b>) Akesuaai represents the 4000 m gradient, (<b>d</b>) Khunjurab represents the 5000 m gradient, and (<b>e</b>) Wolonggang represents the 6000 m gradient.</p>
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<p>Time series comparison of four datasets—8-Day Composite MOD11A2 LST, Site LST in Terra Scan Time, Site LST No Clouds, and Site LST 24h—based on observations from Kunlun Mountain Gradient Stations. Note: (<b>a</b>) Yeyike station represents the 2000 m gradient, (<b>b</b>) Kalasai represents the 3000 m gradient, (<b>c</b>) Akesuaai represents the 4000 m gradient, (<b>d</b>) Khunjurab represents the 5000 m gradient, and (<b>e</b>) Wolonggang represents the 6000 m gradient.</p>
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<p>Comparative analysis of time series data from four distinct datasets—derived from monthly composite MOD11A2 LST and site-specific LST measurements at the Kunlun Mountain Gradient LST Observatory. The datasets include MOD11A2 LST, Site LST in Terra Scan Time, Site LST No Clouds, and Site LST 24h. Note: (<b>a</b>) Yeyike station represents the 2000 m gradient, (<b>b</b>) Kalasai represents the 3000 m gradient, (<b>c</b>) Akesuaai represents the 4000 m gradient, (<b>d</b>) Khunjurab represents the 5000 m gradient, and (<b>e</b>) Wolonggang represents the 6000 m gradient.</p>
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24 pages, 3810 KiB  
Article
Study on the Feasibility and Performance Evaluation of High-Orbit Spacecraft Orbit Determination Based on GNSS/SLR/VLBI
by Zhengcheng Wu, Shaojie Ni, Wei Xiao, Zongnan Li and Huicui Liu
Remote Sens. 2024, 16(22), 4214; https://doi.org/10.3390/rs16224214 - 12 Nov 2024
Viewed by 837
Abstract
Deep space exploration utilizing high-orbit vehicles is a vital approach for extending beyond near-Earth space, with orbit information serving as the foundation for all functional capabilities. The performance of orbit determination is primarily influenced by observation types, errors, geometrical structures, and physical perturbations. [...] Read more.
Deep space exploration utilizing high-orbit vehicles is a vital approach for extending beyond near-Earth space, with orbit information serving as the foundation for all functional capabilities. The performance of orbit determination is primarily influenced by observation types, errors, geometrical structures, and physical perturbations. Currently, research on orbit determination for high-orbit spacecraft predominantly focuses on single observation methods, error characteristics, multi-source fusion techniques, and algorithms. However, these approaches often suffer from low observation accuracy and increased costs. This paper advocates for the comprehensive utilization of existing multi-source observation methods, such as GNSS (Global Navigation Satellite System), SLR (Satellite Laser Ranging), and VLBI (Very Long Baseline Interferometry), in research. The decoupled Kalman filter reveals a positive correlation between measurement positioning accuracy and orbit determination accuracy, and it derives a simple orbit performance evaluation model that considers the influence of observation value types and geometric configurations, without the need to introduce complex dynamic models. Simulations are then employed to verify and analyze antenna gain, observation values, and performance evaluation. The results indicate the following: (1) Under simulated conditions, the optimal strategy involves employing the SLR/VLBI dual system during periods when VLBI orbit determination is feasible, yielding an average Weighted Position Dilution of Precision (WPDOP) of 26.79. (2) For periods when VLBI orbit determination is not feasible, the optimal approach is to utilize the GNSS/SLR/VLBI triple system, resulting in an average WPDOP of 16.32. (3) The orbit determination performance of the triple system is not significantly impacted by the use of global SLR stations compared to using only Chinese SLR stations. However, the global network enables continuous, round-the-clock orbit determination capabilities. Full article
(This article belongs to the Special Issue GNSS Positioning and Navigation in Remote Sensing Applications)
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<p>In the Kalman filter probability distribution diagram, the blue dotted line area is the estimated probability density distribution at the previous time, the blue solid line area is the predicted probability density distribution at the later time, the red solid line area is the observed probability density distribution at the later time, and the green solid line area is the recursive Kalman filter probability density distribution at the later time.</p>
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<p>Position and track map of the high-orbit vehicle during the period from 20:00:00 on 1 January 2024 to 20:00:00 on 6 January 2024.</p>
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<p>Distribution map of SLR stations in China.</p>
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<p>Distribution map of VLBI stations in China.</p>
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<p>Radio signal link analysis flow chart.</p>
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<p>An elevation map of the Earth.</p>
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<p>Geometric diagram of GNSS, SLR, and VLBI stations for high-orbit vehicle detection.</p>
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<p>Directions of different types of antennas. (<b>a</b>) Block IIR. (<b>b</b>) Block IIR-M. (<b>c</b>) Block IIF. (<b>d</b>) Block III. (<b>e</b>) BDS-M. (<b>f</b>) BDS-I/G. (<b>g</b>) GLO. (<b>h</b>) GAL.</p>
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<p>A 2D orientation diagram of different types of antennas after processing.</p>
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<p>(<b>a</b>) GPS/BDS/GLO/GAL digital maps are available from 0:00 on 1 January 2024 to 23:30 on 1 January 2024. (<b>b</b>) Visible number of SLR stations from 0:00 on 1 January 2024 to 23:30 on 1 January 2024. (<b>c</b>) Visible number of VLBI stations from 0:00 on 1 January 2024 to 23:30 on 1 January 2024.</p>
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<p>WPDOP values of single GNSS, single SLR and single VLBI systems for high-orbit vehicles. (<b>a</b>) WPDOP value of single GNSS system for high-orbit vehicles. (<b>b</b>) WPDOP value of single SLR system for high-orbit vehicles. (<b>c</b>) WPDOP value of single VLBI system for high-orbit vehicles.</p>
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<p>WPDOP value of GNSS and VLBI dual system for high-orbit vehicle.</p>
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<p>WPDOP value of GNSS and SLR dual system for high-orbit vehicle.</p>
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<p>The WPDOP value of the SLR and VLBI dual system for a high-orbit vehicle.</p>
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<p>WPDOP values of GNSS, SLR, and VLBI systems for high-orbit spacecraft.</p>
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<p>The <math display="inline"><semantics> <mi>δ</mi> </semantics></math>WPDOP values of the three systems and the single SLR vary with the measurement accuracy of the single SLR.</p>
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<p>Orbit determination performance of different combinations: V (VLBI), C (Chinese SLR), G (GNSS), S (Global SLR).</p>
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17 pages, 528 KiB  
Article
Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System
by Dmitry Pavlov and Ivan Dolgakov
Universe 2024, 10(11), 413; https://doi.org/10.3390/universe10110413 - 3 Nov 2024
Viewed by 599
Abstract
Physical properties of the Sun (orientation of rotation axis, oblateness coefficient J2, and change rate of the gravitational parameter μ˙) are determined using a dynamical model describing the motion of the Sun, planets, the Moon, asteroids, and [...] Read more.
Physical properties of the Sun (orientation of rotation axis, oblateness coefficient J2, and change rate of the gravitational parameter μ˙) are determined using a dynamical model describing the motion of the Sun, planets, the Moon, asteroids, and Trans-Neptunian objects (TNOs). Among the many kinds of observations used to determine the orbits and physical properties of the bodies, the most important for our study are precise interplanetary ranging data: Earth–Mercury ranges from MESSENGER spacecraft and Earth–Mars ranges from Odyssey and MRO. The findings allow us to improve the model of the Sun in modern planetary ephemerides. First, the dynamically determined direction of the Sun’s pole is ≈2° off the visible axis of rotation of the Sun’s surface, which is corroborated by present knowledge of the Sun’s interior. Second, the change rate of the Sun’s gravitational parameter is found to be smaller (in absolute value) than the nominal value derived from the estimate of mass loss through radiation and solar wind. Possible interpretations are discussed. Full article
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<p>Difference between model values of MESSENGER ranges (<math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>II</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi mathvariant="normal">I</mi> </msub> </mrow> </semantics></math>).</p>
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<p>Difference between model values of MESSENGER ranges (<math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>III</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi>II</mi> </msub> </mrow> </semantics></math>).</p>
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<p>Difference between model values of MESSENGER ranges (<math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>III</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi mathvariant="normal">I</mi> </msub> </mrow> </semantics></math>).</p>
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<p>MESSENGER residuals <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>−</mo> <msub> <mi>C</mi> <mi>III</mi> </msub> </mrow> </semantics></math>.</p>
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20 pages, 747 KiB  
Article
Neural Network-Based Optimization of LEO Transfers
by Andrea Forestieri and Lorenzo Casalino
Aerospace 2024, 11(11), 879; https://doi.org/10.3390/aerospace11110879 - 25 Oct 2024
Viewed by 616
Abstract
This study investigates the application of neural networks to the evaluation of minimum-time low-thrust transfers in low Earth orbit. The findings demonstrate the effectiveness of utilizing costates to regularize the training loss, significantly enhancing the accuracy of the predictions of the neural networks, [...] Read more.
This study investigates the application of neural networks to the evaluation of minimum-time low-thrust transfers in low Earth orbit. The findings demonstrate the effectiveness of utilizing costates to regularize the training loss, significantly enhancing the accuracy of the predictions of the neural networks, even when working with limited datasets. Remarkably precise estimates of transfer times are achieved by training the regularized networks on datasets comprising one million samples. The incorporation of a warm-started guess strategy, involving simpler neural networks to provide transfer time and costates predictions for new transfers, accelerates the data collection process, making this approach highly practical for real-world applications. Overall, the methodology employed in this research study holds significant promise for low-thrust space missions, particularly when the evaluation of multiple minimum-time transfers is necessary in mission planning. In fact, the trained neural networks significantly speed up convergence when solving optimal control problems with indirect optimization methods. Furthermore, the remarkable accuracy in estimating both minimum transfer times and costates provides the flexibility of relying entirely on neural networks for determining minimum time. Full article
(This article belongs to the Section Astronautics & Space Science)
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<p>Neural network with two hidden layers. The number of layers and neurons per layer define <math display="inline"><semantics> <mo mathvariant="bold-italic">θ</mo> </semantics></math>.</p>
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<p>Warm-started guess strategy adopted for the collection of the datasets.</p>
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<p>Performance comparison between the DNN and the other ML algorithms on the sequence identified by the DNN during the beam search for the OOS mission.</p>
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<p>Performance comparison between the DNN and the other ML algorithms on the sequence identified by random forest during the beam search for the OOS mission.</p>
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14 pages, 1809 KiB  
Article
Dual-Arm Space Robot On-Orbit Operation of Auxiliary Docking Prescribed Performance Impedance Control
by Dongbo Liu and Li Chen
Aerospace 2024, 11(11), 867; https://doi.org/10.3390/aerospace11110867 - 23 Oct 2024
Viewed by 552
Abstract
The impedance control of a dual-arm space robot in orbit auxiliary docking operation is studied. First, for the closed-chain hybrid system formed by the dual-arm space robot after capture operation, the dynamic equation of position uncontrolled and attitude controlled is established. The second-order [...] Read more.
The impedance control of a dual-arm space robot in orbit auxiliary docking operation is studied. First, for the closed-chain hybrid system formed by the dual-arm space robot after capture operation, the dynamic equation of position uncontrolled and attitude controlled is established. The second-order linear impedance model and second-order approximate environment model are established for the problem of simultaneous output force/pose control of the end of the manipulator. Then, aiming at the transient performance control requirements of the dual-arm space robot auxiliary docking operation in orbit, a sliding mode controller with equivalent replacement of tracking errors is designed by introducing Prescribed Performance Control (PPC) theory. Next, Radial Basis Function Neural Networks (RBFNN) are used to accurately compensate for the modeling uncertainties of the system. Finally, the stability of the system is verified by Lyapunov stability determination. The simulation results show that the attitude control accuracy is better than 0.5°, the position control accuracy is better than 103 m, and the output force control accuracy is better than 0.5 N when it reaches 30 N. It also indicated that the proposed control algorithm can limit the transient performance of the controlled system within the preset range and achieve high-precision force/pose control, which ensures a more stable on-orbit auxiliary docking operation of the dual-arm space robot. Full article
(This article belongs to the Section Astronautics & Space Science)
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<p>Dual-arm space robot system and auxiliary docking target spacecraft system.</p>
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<p>Dual-arm space robot end trajectory tracking curves of on-orbit auxiliary docking operation. (<b>a</b>) The carrier attitude. (<b>b</b>) The X-direction position. (<b>c</b>) The Y-direction position. (<b>d</b>) The space robot end attitude.</p>
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<p>Dual-arm space robot joint trajectory tracking curves of on-orbit auxiliary docking operation. (<b>a</b>) The left arm. (<b>b</b>) The right arm.</p>
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<p>Dual-arm space robot end output force, base, and joints control torques of on-orbit docking operation. (<b>a</b>) The end. (<b>b</b>) The base. (<b>c</b>) The left-arm joints. (<b>d</b>) The right-arm joints.</p>
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<p>Tracking errors of on-orbit insertion operation. (<b>a</b>) PPC method. (<b>b</b>) PD method.</p>
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18 pages, 9065 KiB  
Article
Modeling of Solar Radiation Pressure for BDS-3 MEO Satellites with Inter-Satellite Link Measurements
by Yifei Lv, Zihao Liu, Rui Jiang and Xin Xie
Remote Sens. 2024, 16(20), 3900; https://doi.org/10.3390/rs16203900 - 20 Oct 2024
Viewed by 787
Abstract
As the largest non-gravitational force, solar radiation pressure (SRP) causes significant errors in precise orbit determination (POD) of the BeiDou global navigation satellite system (BDS-3) medium Earth orbit (MEO) satellite. This is mainly due to the imperfect modeling of the satellite’s cuboid body. [...] Read more.
As the largest non-gravitational force, solar radiation pressure (SRP) causes significant errors in precise orbit determination (POD) of the BeiDou global navigation satellite system (BDS-3) medium Earth orbit (MEO) satellite. This is mainly due to the imperfect modeling of the satellite’s cuboid body. Since the BDS-3’s inter-satellite link (ISL) can enhance the orbit estimation of BDS-3 satellites, the aim of this study is to establish an a priori SRP model for the satellite body using 281-day ISL observations to reduce the systematic errors in the final orbits. The adjustable box wind (ABW) model is employed to refine the optical parameters for the satellite buses. The self-shadow effect caused by the search and rescue (SAR) antenna is considered. Satellite laser ranging (SLR), day-boundary discontinuity (DBD), and overlapping Allan deviation (OADEV) are utilized as indicators to assess the performance of the a priori model. With the a priori model developed by both ISL and ground observation, the slopes of SLR residual for the China Academy of Space Technology (CAST) and Shanghai Engineering Center for Microsatellites (SECM) satellites decrease from −0.097 cm/deg and 0.067 cm/deg to −0.004 cm/deg and −0.009 cm/deg, respectively. The standard deviation decreased by 21.8% and 26.6%, respectively. There are slight enhancements in the average values of DBD and OADEV, and a reduced β-dependent variation is observed in the OADEV of the corresponding clock offset. We also found that considering the SAR antenna only slightly improves the orbit accuracy. These results demonstrate that an a priori model established for the BDS-3 MEO satellite body can reduce the systematic errors in orbits, and the parameters estimated using both ISL and ground observation are superior to those estimated using only ground observation. Full article
(This article belongs to the Special Issue GNSS Positioning and Navigation in Remote Sensing Applications)
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<p>Dimensions of the satellite body and its shadow on the panel (unit: m).</p>
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<p>Flowchart for the refinement of SRP models of BDS-3 MEO satellite.</p>
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<p>Correlation between the parameters for CAST (C20) satellite and SECM (C29) satellite obtained by the G1 and J1.</p>
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<p>Optical parameter estimated by strategy G1 (blue), G2 (purple), J1 (red), and J2 (green).</p>
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<p>SLR residual dependent on ε angle for satellite without SAR antenna. The first row represents G1 estimates, and the second row represents J1 estimates.</p>
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<p>OADEV at 9000 s of CAST-A (C20) and SECM-A (C30) with respect to the <span class="html-italic">β</span> angle.</p>
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<p>OADEV at 9000 s of CAST-B (C33) and SECM-B (C43) with respect to the <span class="html-italic">β</span> angle.</p>
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<p>SLR residual estimated by ECOM (left column), ECOM+G1 (middle column), and ECOM+J1 (right column) dependent on <math display="inline"><semantics> <mrow> <mi>ε</mi> </mrow> </semantics></math> angle. The red line represents the linear trend line, with the slope value (red font) denoted in cm/deg.</p>
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<p>SLR residual distribution based on ECOM (blue), ECOM+G1 (green), and ECOM+J1 (red).</p>
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<p>OADEV at 9000 s of CAST-A (C20), SECM-A (C30), and SECM-B (C43) satellites with respect to the <span class="html-italic">β</span> angle.</p>
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<p>OADEV at 9000 s of CAST-B (C33) satellite with respect to the <span class="html-italic">β</span> angle.</p>
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23 pages, 6144 KiB  
Article
Advancing CubeSats Capabilities: Ground-Based Calibration of Uvsq-Sat NG Satellite’s NIR Spectrometer and Determination of the Extraterrestrial Solar Spectrum
by Mustapha Meftah, Christophe Dufour, David Bolsée, Lionel Van Laeken, Cannelle Clavier, Amal Chandran, Loren Chang, Alain Sarkissian, Patrick Galopeau, Alain Hauchecorne, Pierre-Richard Dahoo, Luc Damé, André-Jean Vieau, Emmanuel Bertran, Pierre Gilbert, Fréderic Ferreira, Jean-Luc Engler, Christophe Montaron, Antoine Mangin, Odile Hembise Fanton d’Andon, Nicolas Caignard, Angèle Minet, Pierre Maso, Nuno Pereira, Étienne Brodu, Slimane Bekki, Catherine Billard and Philippe Keckhutadd Show full author list remove Hide full author list
Remote Sens. 2024, 16(19), 3655; https://doi.org/10.3390/rs16193655 - 30 Sep 2024
Viewed by 1179
Abstract
Uvsq-Sat NG is a French 6U CubeSat (10 × 20 × 30 cm) of the International Satellite Program in Research and Education (INSPIRE) designed primarily for observing greenhouse gases (GHG) such as CO2 and CH4, measuring the Earth’s radiation budget [...] Read more.
Uvsq-Sat NG is a French 6U CubeSat (10 × 20 × 30 cm) of the International Satellite Program in Research and Education (INSPIRE) designed primarily for observing greenhouse gases (GHG) such as CO2 and CH4, measuring the Earth’s radiation budget (ERB), and monitoring solar spectral irradiance (SSI) at the top-of-atmosphere (TOA). It epitomizes an advancement in CubeSat technology, showcasing its enhanced capabilities for comprehensive Earth observation. Scheduled for launch in 2025, the satellite carries a compact and miniaturized near-infrared (NIR) spectrometer capable of performing observations in both nadir and solar directions within the wavelength range of 1100 to 2000 nm, with a spectral resolution of 7 nm and a 0.15° field of view. This study outlines the preflight calibration process of the Uvsq-Sat NG NIR spectrometer (UNIS), with a focus on the spectral response function and the absolute calibration of the instrument. The absolute scale of the UNIS spectrometer was accurately calibrated with a quartz-halogen lamp featuring a coiled-coil tungsten filament, certified by the National Institute of Standards and Technology (NIST) as a standard of spectral irradiance. Furthermore, this study details the ground-based measurements of direct SSI through atmospheric NIR windows conducted with the UNIS spectrometer. The measurements were obtained at the Pommier site (45.54°N, 0.83°W) in Charentes–Maritimes (France) on 9 May 2024. The objective of these measurements was to verify the absolute calibration of the UNIS spectrometer conducted in the laboratory and to provide an extraterrestrial solar spectrum using the Langley-plot technique. By extrapolating the data to AirMass Zero (AM0), we obtained high-precision results that show excellent agreement with SOLAR-HRS and TSIS-1 HSRS solar spectra. At 1.6 μm, the SSI was determined to be 238.59 ± 3.39 mW.m−2.nm−1 (k = 2). These results demonstrate the accuracy and reliability of the UNIS spectrometer for both SSI observations and GHG measurements, providing a solid foundation for future orbital data collection and analysis. Full article
(This article belongs to the Special Issue Advances in CubeSats for Earth Observation)
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<p>(<b>left</b>) Location of the Uvsq-Sat NG NIR spectrometer (UNIS) inside the satellite. (<b>right</b>) Location of the UNIS NIR spectrometer aperture on the Uvsq-Sat NG satellite.</p>
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<p>Spectra of the various lamps (Hg, Xe, Kr) and laser (He-Ne) used for wavelength calibration of the UNIS NIR spectrometer of the Uvsq-Sat NG space-based mission.</p>
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<p>(<b>top</b>) Signal measured by the Uvsq-Sat NG NIR spectrometer (UNIS) when facing the Hg reference lamp. (<b>bottom</b>) The spectrum of the Hg lamp with the location of strong lines (nominal signal and signal convolved with a spectral resolution of 7 nm).</p>
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<p>Pixel-wavelength relationship of the spectrometer obtained from various calibrations (lamps, laser, light source, and monochromators).</p>
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<p>(<b>top</b>) Signal observed by the Uvsq-Sat NG NIR spectrometer (UNIS), with data corrected for dark current. (<b>bottom</b>) SSI of the F-546 (FEL 546) reference lamp used under standard conditions (8.2 A, distance of the device under test at 50 cm).</p>
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<p>Absolute response (TF) of the Uvsq-Sat NG NIR spectrometer (UNIS) from 1100 to 2000 nm.</p>
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<p>FWHM of the slit function of the Uq-Sat NG NIR spectrometer (UNIS) at 1523 nm, determined using a laser.</p>
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<p>(<b>top</b>) Distribution of relative spectral responses (slit functions) of the Uvsq-Sat NG NIR spectrometer (UNIS). (<b>bottom</b>) FWHM values of the slit functions of the spectrometer.</p>
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<p>Temperature response (dark current) of the Uvsq-Sat NG NIR spectrometer (UNIS) during a test in a climatic chamber (26 April 2024). The raw data (ADU) is plotted against temperature (°C) for various exposure times.</p>
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<p>Image of the Sun taken in H-alpha (656.28 nm) with our telescope during the test campaign of the spectrometer. This period of observations was marked by intense solar activity characterized by powerful solar storms, extreme solar flares, and geomagnetic storm components.</p>
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<p>(<b>left</b>) Evolution of AirMass from 11:00 UTC to 18:00 UTC on 9 May 2024, at the Pommier site (Charente-Maritime). (<b>right</b>) Temperature variation during the observations.</p>
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<p>(<b>top</b>) Raw data intensities acquired by the Uvsq-Sat NG NIR spectrometer (UNIS) over time (Frame Number of the measurements). (<b>bottom</b>) SSI was measured by the spectrometer at air masses of 1.17 (red line) and 1.63 (blue line).</p>
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<p>(<b>left</b>) Solar spectrum obtained with the Uvsq-Sat NG NIR spectrometer (UNIS) for the two methods implemented. SOLAR-HRS and TSIS-1 HSRS solar spectra are shown for comparison within a wavelength range between 1100 and 2000 nm. (<b>right</b>) The ratio of the different spectra convolved to 7 nm compared to the spectrum obtained with the UNIS NIR spectrometer.</p>
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<p>Optical layout of the Uvsq-Sat NG NIR spectrometer (UNIS). UNIS is a miniaturized spectrometer with an aperture of 15 mm, a spectral range from 1100 to 2000 nm, a spectral resolution of 7 nm, and a field of view of 0.15°.</p>
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<p>Measured irradiance and corresponding Langley-plot fits for four wavelengths (1179.0, 1271.2, 1601.0, and 1749.5 nm) observed by the Uvsq-Sat NG NIR spectrometer (UNIS) on 9 May 2024, based on the use of Method 1.</p>
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<p>Measured irradiance and corresponding Langley-plot fits for four wavelengths (1179.0, 1271.2, 1601.0, and 1749.5 nm) observed by the Uvsq-Sat NG NIR spectrometer (UNIS) on 9 May 2024, based on the use of Method 2.</p>
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<p>Brightness temperature of the Sun obtained from the Uvsq-Sat NG NIR spectrometer (UNIS) in the wavelength bands where the Langley-plot technique is applicable. The brightness temperatures from HRS and TSIS are also shown for comparison.</p>
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19 pages, 15677 KiB  
Article
Automatic Correction of Time-Varying Orbit Errors for Single-Baseline Single-Polarization InSAR Data Based on Block Adjustment Model
by Huacan Hu, Haiqiang Fu, Jianjun Zhu, Zhiwei Liu, Kefu Wu, Dong Zeng, Afang Wan and Feng Wang
Remote Sens. 2024, 16(19), 3578; https://doi.org/10.3390/rs16193578 - 26 Sep 2024
Viewed by 775
Abstract
Orbit error is one of the primary error sources of interferometric synthetic aperture radar (InSAR) and differential InSAR (D-InSAR) measurements, arising from inaccurate orbit determination of SAR platforms. Typically, orbit error in the interferogram can be estimated using polynomial models. However, correcting for [...] Read more.
Orbit error is one of the primary error sources of interferometric synthetic aperture radar (InSAR) and differential InSAR (D-InSAR) measurements, arising from inaccurate orbit determination of SAR platforms. Typically, orbit error in the interferogram can be estimated using polynomial models. However, correcting for orbit errors with significant time-varying characteristics presents two main challenges: (1) the complexity and variability of the azimuth time-varying orbit errors make it difficult to accurately model them using a set of polynomial coefficients; (2) existing patch-based polynomial models rely on empirical segmentation and overlook the time-varying characteristics, resulting in residual orbital error phase. To overcome these problems, this study proposes an automated block adjustment framework for estimating time-varying orbit errors, incorporating the following innovations: (1) the differential interferogram is divided into several blocks along the azimuth direction to model orbit error separately; (2) automated segmentation is achieved by extracting morphological features (i.e., peaks and troughs) from the azimuthal profile; (3) a block adjustment method combining control points and connection points is proposed to determine the model coefficients of each block for the orbital error phase estimation. The feasibility of the proposed method was verified by repeat-pass L-band spaceborne and P-band airborne InSAR data, and finally, the InSAR digital elevation model (DEM) was generated for performance evaluation. Compared with the high-precision light detection and ranging (LiDAR) elevation, the root mean square error (RMSE) of InSAR DEM was reduced from 18.27 m to 7.04 m in the spaceborne dataset and from 7.83~14.97 m to 3.36~6.02 m in the airborne dataset. Then, further analysis demonstrated that the proposed method outperforms existing algorithms under single-baseline and single-polarization conditions. Moreover, the proposed method is applicable to both spaceborne and airborne InSAR data, demonstrating strong versatility and potential for broader applications. Full article
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<p>Flowchart of the proposed method for estimating the time-varying orbital error phase.</p>
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<p>Schematic diagram of differential interferogram division by the proposed method. (<b>a</b>) is the differential interferogram, (<b>b</b>) is the profile of the time-varying orbital error phase along the azimuth at the dotted line shown in (<b>a</b>), (<b>c</b>) is a schematic diagram of different blocks, and (<b>d</b>) is the overlap area between different blocks and the distribution of control points and connection points.</p>
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<p>Geographical location of the test sites: (<b>a</b>) Hunan and (<b>b</b>) Krycklan. The red boxes represent LuTan-1 InSAR data and E-SAR data, the blue box represents airborne LiDAR data, and the yellow dots represent the footprint of ICESat-2 ATL08 elevation.</p>
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<p>Orbital error phase analysis of LT-1 InSAR data from the Hunan test site: (<b>a</b>) interferometric coherence, (<b>b</b>) differential interferometric phase, (<b>c</b>,<b>d</b>) are the profiles of the orbital error phase in the azimuth and range directions, respectively.</p>
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<p>Time-varying orbit error phase estimation results: (<b>a</b>) original differential interferometric phase, (<b>b</b>) estimated orbital error phase, (<b>c</b>) differential interferometric phase after removing orbit error.</p>
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<p>DEM elevation validation. (<b>a</b>) InSAR DEM from interferogram inversion after removing orbit error, (<b>b</b>) difference between InSAR DEM and external DEM, (<b>c</b>) error histogram of InSAR DEM relative to ICESat−2 elevation.</p>
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<p>Analysis of airborne P−band time−varying orbit errors in the Krycklan test site. (<b>a</b>) Airborne SAR intensity map, (<b>b</b>) azimuth profiles of five differential interferograms, (<b>c</b>) azimuth profile of interferometric pair 0101–0103 and extracted peaks and troughs, (<b>d</b>) range profile of interferometric pair 0101–0103.</p>
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<p>Results of the proposed method for estimating airborne time–varying orbital error phase. (<b>a1</b>–<b>a5</b>) are original differential interferometric phases, (<b>b1</b>–<b>b5</b>) are the estimated orbital error phases, (<b>c1</b>–<b>c5</b>) are differential interferometric phases after removing orbit error phases. From left to right, the five interferometric pairs shown in <a href="#remotesensing-16-03578-t001" class="html-table">Table 1</a> are represented.</p>
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<p>(<b>a</b>) InSAR DEM estimated by the interferometric pair numbered 0103–0111, (<b>b</b>–<b>f</b>) differences between InSAR DEM and LiDAR DTM estimated after correcting orbit errors for the five interferometric pairs in <a href="#remotesensing-16-03578-t001" class="html-table">Table 1</a>.</p>
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<p>(<b>a</b>–<b>e</b>) Error statistical histograms of InSAR DEM and LiDAR DTM estimated before and after orbital error correction for the five interferometric pairs in <a href="#remotesensing-16-03578-t001" class="html-table">Table 1</a>.</p>
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24 pages, 10209 KiB  
Article
An Attitude Determination and Sliding Mode Control Method for Agile Whiskbroom Scanning Maneuvers of Microsatellites
by Xinyan Yang, Zhaoming Li, Lei Li and Yurong Liao
Aerospace 2024, 11(9), 778; https://doi.org/10.3390/aerospace11090778 - 20 Sep 2024
Viewed by 545
Abstract
Microsatellites have significantly impacted space missions by offering advanced technology at a low cost. This study introduces an attitude determination and control algorithm for agile whiskbroom scanning maneuvers in microsatellites to enable wide-swath target detection for low-Earth-orbit microsatellites. First, an angular velocity calculation [...] Read more.
Microsatellites have significantly impacted space missions by offering advanced technology at a low cost. This study introduces an attitude determination and control algorithm for agile whiskbroom scanning maneuvers in microsatellites to enable wide-swath target detection for low-Earth-orbit microsatellites. First, an angular velocity calculation model for agile whiskbroom scanning is established. A methodology has been developed to calculate the maximum available time for whiskbroom scanning from one side of the sub-satellite point to the other while ensuring the seamless joining of adjacent strips to avoid missing targets. Thereafter, a gyro- and magnetometer-based cubature Kalman filter is put forward for microsatellite attitude estimation. Furthermore, for attitude control, a hybrid manipulation law capable of preventing singularities and escaping singularity surfaces is designed to ensure high-precision torque output from the control moment gyroscopes (CMGs) used as actuators. The benefits of the linear sliding mode and fast terminal sliding mode are integrated, and a non-singular sliding surface is designed, yielding a non-singular fast terminal sliding mode attitude control algorithm for tracking the desired trajectory. This algorithm effectively suppresses chattering and enhances dynamic performance without using a switching term. A semi-physical simulation experiment system is also conducted on the ground to validate the proposed algorithm’s high-precision tracking of the planned whiskbroom scanning path. The experimental results demonstrate an attitude angle control accuracy of 4 × 10−2 degrees and angular velocity control accuracy of 0.01°/s and thus the effectiveness of the proposed algorithm. Full article
(This article belongs to the Section Aeronautics)
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<p>Schematic of agile whiskbroom scanning maneuvers of microsatellites.</p>
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<p>Overlap increase for image stitching.</p>
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<p>Geometric relationship of satellite imaging regions.</p>
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<p>Correspondence of satellite travel distance and frame swath in one whiskbroom scanning cycle.</p>
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<p>Pyramid configuration of the four-SGCMG system.</p>
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<p>Space magnetic field simulator.</p>
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<p>Internal structure of the microsatellite electrical model.</p>
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<p>Magnetometer.</p>
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<p>Gyroscope.</p>
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<p>Desired Euler angles.</p>
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<p>Desired angular velocity.</p>
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<p>Attitude tracking error of NFTSMC in Euler angle.</p>
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<p>Angular velocity tracking error of NFTSMC.</p>
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<p>Control torques of NFTSMC.</p>
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<p>Singularity measure.</p>
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<p>Angular momentum of CMGs.</p>
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<p>Angles of CMGs rotation.</p>
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