Low-frequency Errors Research Articles

Estimating the frequency response of a system in the presence of an integrator

A system with an integrator is one which does not have a steady-state gain at zero frequency. An example is a battery; when subjected to a constant charge or discharge current the voltage will continuously increase or decrease until the cell reaches its maximum/minimum cut-off voltage and not reach a steady-state value. Frequency response estimation techniques that minimises leakage errors lead to significant errors at low frequencies of the response. This paper develops and presents a technique whereby the low frequency errors are eliminated. The technique is applied over the frequencies of interest, except DC frequency, and gives better results over windowing and a local polynomial frequency response estimation method. As such, an accurate low frequency response and noise power spectrum can now be obtained which in turn can be used for estimating accurate parametric models.

Attitude Measurement Low Frequency Error Identification and Compensation for Star Tracker with Gyros

The low-frequency error (LFE) of star tracker is the most critical problems in the high accuracy attitude determination for the satellite. In this paper a novel approach is proposed to identify and compensate the attitude measurement LFE of the star trackers. The main difficulty in the attitude LFE identification is to distinguish the attitude LFE from the attitude motion. To overcome this difficulty, the gyro data, which is sensitive to the attitude motion, was used to fit the measurement attitude data to obtain the reference attitude. It is shown that, the LFE can be identified by comparing the measurement attitude and the reference attitude. The attitude LFE repeated feature is characterized well by the proposed method. By utilizing the orbital repeated feature, the LFE of the star trackers can be estimated. Finally, a LFE compensation strategy is presented base on the LFE estimation results. The validity and efficiency of the proposed approach are demonstrated by the relative Euler angle results from two test star trackers on-board the STECE satellite.

Attitude aberration correction for space technology experiment and climate exploration (STECE) satellite star tracker

The space technology experiment and climate exploration satellite is a small satellite mission of China. A new star tracker and an ASTRO 10 star tracker were loaded on the space technology experiment and climate exploration satellite to test the new star tracker’s in-flight performance. But the aberration correction function of the test star tracker is off during its experiment mission. Thus, the aberration effect should be considered in the evaluation of the test star tracker’s performance. This study researches on the aberration correction for the star tracker’s attitude measurement and deduces the aberration correction method based on attitude quaternion data. The inter-boresight angle and the relative Euler angle results show that the aberration of the test star tracker’s attitude has been corrected successfully. The low-frequency errors of the test star tracker are also analyzed in this paper.

In-flight quality evaluation of attitude measurements from STECE APS-01 star tracker

The STECE satellite is a small satellite mission of China for space technology experiment and climate exploration. To test the in-flight performance of the CMOS Active Pixel Sensor (APS) based star trackers, the APS-01 star tracker has been loaded on the STECE satellite. The APS-01 star tracker payload delivers 1Hz real time telemetry attitude data during its in-orbit experiment mission. These attitude data have been analyzed and evaluated down to the specification requirements like random noise, systematic bias and low frequency error in this paper. The systematic bias of APS-01, including precession–nutation, aberration and relative installation error, has been calibrated. An approach is proposed to analyze the attitude low frequency errors of APS-01. The results of the evaluation can contribute to a better understanding as well as further refinements of the CMOS APS based star trackers.

Precession–nutation correction for star tracker attitude measurement of STECE satellite

Space Technology Experiment and Climate Exploration (STECE) is a small satellite mission of China for space technology experiment and climate exploration. A new test star tracker and one ASTRO 10 star tracker have been loaded on the STECE satellite to test the new star tracker’s measurement performance. However, there is no autonomous precession–nutation correction function for the test star tracker, which causes an apparent periodic deflection in the inter-boresight angle between the two star trackers with respect to each other of up to ±500 arcsec, so the precession and nutation effect needs to be considered while assessing the test star tracker. This paper researches on the precession–nutation correction for the test star tracker’s attitude measurement and presents a precession–nutation correction method based on attitude quaternion data. The periodic deflection of the inter-boresight angle between the two star trackers has been greatly eliminated after the precession and nutation of the test star tracker’s attitude data have been corrected by the proposed method and the validity of the proposed algorithm has been demonstrated. The in-flight accuracy of the test star tracker has been assessed like attitude noise and low-frequency errors after the precession–nutation correction.

基于经验模态分解-Wigner分布的光学元件中频误差识别

The mid-spatial frequency error of high precision optical surface is crucial to the performance of high-energy laser system. To assure the performance and stability of the system, the low frequency error and high frequency error of the optical lens surface must be strictly controlled, the mid-spatial frequency error must also be strictly limited. In this paper, the relationship between empirical mode decomposition-Wigner-Ville distribution (EMD-WVD) method and power spectrum density is analyzed, and EMD-WVD diagnosis method is applied to identifying and locating the mid-spatial frequency error of the precision optical surface. The experimental results show that, with EMD-WVD method, the mid-spatial frequency error distribution of optical surface can be located in 15-27 mm and the spatial frequency is 0.1 mm-1, the interference of cross-term caused by multi-component signal whose spatial frequency is about 1.0-1.5 mm-1 can also be reduced, which enhances the identification accuracy of mid-spatial frequency error.

Performance evaluation of star sensor low frequency error calibration

This paper studies the star sensor low frequency error (LFE) in-flight calibration problem. The star sensor low frequency error, which is caused primarily by the periodic thermal distortion, has a great impact on the satellite attitude determination accuracy. It is formulated as a Fourier series in this paper. The low frequency error calibration is fulfilled by estimating the Fourier coefficients, which are assumed time-constant. The performance of the calibration method is evaluated through the derivation of the estimation error covariance. It is specified that the attitude determination accuracy can be improved by using the calibration method in the case that the calibration error is less than the true low frequency error parameters. The low frequency error model for numerical simulation are established based on the telemetry data from real star sensors operating under in-orbit conditions. The simulation results illustrate the efficiency of the proposed methods.

Accurate integration of segmented x-ray optics using interfacing ribs

Future lightweight and long-focal-length x-ray telescopes must guarantee a good angular resolution (e.g., 5 arc sec HEW) and reach an unprecedented large effective area. This goal can be reached with the slumping of borosilicate glass sheets that allow the fabrication of lightweight and low-cost x-ray optical units (XOU). These XOUs, based on mirror segments, have to be assembled together to form complete multishell Wolter-I optics. The technology for the fabrication and the integration of these XOUs is under development in Europe, funded by European Space Agency, and led by the Brera Observatory (INAF-OAB). While the achievement of the required surface accuracy on the glass segments by means of a hot slumping technique is a challenging aspect, adequate attention must be given to the correct integration and coalignment of the mirror segments into the XOUs. To this aim, an innovative assembly concept has been investigated, based on glass reinforcing ribs. The ribs connect pairs of consecutive foils, stacked into a XOU, with both structural and functional roles, providing robust monolithic stacks of mirror plates. Moreover, this integration concept allows the correction of residual low-frequency errors still present on the mirror foil profile after slumping. We present the integration concept, the related error budget, and the results achieved so far with a semi-robotic integration machine especially designed and realized to assemble slumped glass foils into XOUs.

Analysis of corrective characteristics of various polishing methods for mid-frequency errors

The theoretical analysis of corrective characteristics of three kinds of polishing methods for mid-frequency errors was studied, which was aimed to confirm the possibility that computer control optical surfacing and computer control active-lap can be replaced by bonnet polishing in the machining process. The first step was to calculate the removal functions of three kinds of polishing technologies and use fast Fourier transform to figure out the frequency spectrum of each method. After that, according to the frequency spectra, curves of cut-off frequencies related to the working ranges of spatial frequencies errors were obtained. It revealed that the affected scope of spatial frequencies is determined by the polishing method, diameter size of polishing tool and shape of removal function. Moreover, only low-frequency errors could be modified and mid-frequency errors could not be corrected or created by computer control active-lap, and computer control optical surfacing can correct part of the mid-frequency errors and low-frequency errors in the polishing process, but at the same time can produce some new mid-frequency errors; as for bonnet polishing, it can be computer control active-lap-like in smoothing which only modified and created the low-frequency errors or computer control optical surfacing-like which corrected and created the mid-frequency errors in local polishing. Otherwise, the efficiency of bonnet polishing is higher than the other two methods. As a result, seen from the point of correction ability of mid-frequency or polishing efficiency, bonnet polishing could replace computer control active-lap and computer control optical surfacing for finishing two polishing stages by only one tool, which is significant to extending the application of bonnet polishing in optical manufacturing.

A Fast and High-Accuracy Compass Alignment Method to SINS with Azimuth Axis Rotation

Azimuth axis rotating modulation was introduced to improve the alignment accuracy of strapdown inertial navigation system (SINS) through compass algorithm, in which the limit accuracy was determined by equivalent sensor errors in the eastern and northern direction. In this modulation, horizontal sensor errors were modulated into zero mean periodic variables. Furthermore, two methods were introduced to ensure alignment accuracy and speed: (1) shortened rotating cycle and redesigned compass parameters were selected to eliminate or ease the amplification to low-frequency senor error inputs in compass loop caused by rotation and (2) a data repeated calculation method was designed to shorten prolonged alignment time caused by the above redesigned parameters. Based on a certain SINS, turntable test proves that alignment accuracy and time were significantly improved and slightly shortened in comparison with the classical compass alignment.

Improvement in Steady-State Accuracy of Time-Delayed Control Systems with Communication Disturbance Observer by Low-Frequency Model Error Feedback

This paper proposes a method to improve the steady-state accuracy of time-delayed control systems with a communication disturbance observer (CDOB) by low-frequency model error feedback. The advantageous feature of time delay compensation by the CDOB is that it can be utilized without a delay time model. However, the CDOB needs a model of controlled system, and the model error occurs as the steady-state error. To solve this problem, this paper proposes model error feedback only in the low-frequency area. The proposed method achieves both high robust stability and steady state accuracy.

Bimorph mirrors: The Good, the Bad, and the Ugly

Bimorph mirrors are widely used by the X-ray, Laser, Space, and Astronomy communities to focus or collimate photon beams. Applying voltages to the embedded piezo ceramics enables the user to globally bend the optical substrate to a range of figures (including cylindrical, parabolic, and elliptical), and finely correct low spatial frequency errors, thus improving optical performance. Bimorph mirrors are employed on numerous synchrotron X-ray beamlines, including several at Diamond Light Source. However, many such beamlines were not achieving the desired size and shape of the reflected X-ray beam. Metrology data from ex-situ, slope measuring profilometry (using the Diamond-NOM) and in-situ, synchrotron X-ray “pencil-beam” scans, revealed sharp defects on the optical substrate directly above the locations at which the piezo ceramics are bonded together. This so-called “junction effect” has been observed on a variety of bimorph mirrors with different numbers of piezos, substrate length, and thickness. To repair this damage, three pairs of bimorph mirrors were re-polished at Thales-SESO. We review the re-polishing process, and show that it successfully removed the junction effect, and significantly improved beamline performance. Since the internal structure of the bimorph mirrors was not modified during re-polishing, it is hoped that the mirrors will retain their surface quality, and remain operational for many years. We also highlight the combination of super-polishing techniques with bimorph technology to create the “Ultimate” mirror, and discuss a next generation, bimorph mirror which is predicted not to suffer from the junction effect.

Frequency Analysis and Filtering Processing of Gravity Gradient Data from GOCE

AbstractThe removal of low‐frequency errors is an important task in Gravity field and steady‐state Ocean Circulation Explorer (GOCE) data processing, which suffers from a large number of errors. This paper analyzes the frequency characteristics of gradient data in the time domain and then derives a formula that can represent the relationship between degree and frequency. Moreover, two primary filtering strategies, namely the remove‐restore method and the forward‐backward filtering technique, are presented. The former can solve low‐frequency signal loss during filtering, while the latter can resolve the phase‐drift problem. The final result shows that the proposed formula can accurately show the relationship between the maximum cut‐off frequency and degree of the spherical harmonic model. The proposed filtering method is proved to be effective in removing low‐frequency errors while saving signals in the measurement bandwidth.

Low-frequency Periodic Error Identification and Compensation for Star Tracker Attitude Measurement

The low-frequency periodic error of star tracker is one of the most critical problems for high-accuracy satellite attitude determination. In this paper an approach is proposed to identify and compensate the low-frequency periodic error for star tracker in attitude measurement. The analytical expression between the estimated gyro drift and the low-frequency periodic error of star tracker is derived firstly. And then the low-frequency periodic error, which can be expressed by Fourier series, is identified by the frequency spectrum of the estimated gyro drift according to the solution of the first step. Furthermore, the compensated model of the low-frequency periodic error is established based on the identified parameters to improve the attitude determination accuracy. Finally, promising simulated experimental results demonstrate the validity and effectiveness of the proposed method. The periodic error for attitude determination is eliminated basically and the estimation precision is improved greatly.

Nonlinear iterative learning control applied to an aerostatic X–Y planar motion stage

The planar motion stage with multiple degrees of freedom, positioning in large travel ranges, has been widely used in high-accuracy applications such as semiconductor fabrication equipment and advanced scientific instruments. To achieve precision motion control, iterative learning control has been regarded as an effective means. However, linear iterative learning control techniques attenuate recurring disturbances while amplifying the non-recurring, suffering a fixed trade-off between convergence rates and noise amplification. In the present paper, an amplitude-based nonlinear iterative learning control is proposed with learning gain continuously updated to improve the control performance of the planar motion stage. For error levels beyond a predefined threshold, additional learning gain will be effectively used to diminish the low-frequency tracking error. Below the threshold, the original low gain value is maintained to avoid high-frequency noise amplification. Performance assessment on the developed non-contact planar motion stage shows that the amplitude-based nonlinear iterative learning strategy can realize a remarkable performance which includes micrometer positioning over large travel ranges, and provides a more desirable means to deal with the convergence rate and noise amplification.

Empirical Cross-Calibration of Coherent SWOT Errors Using External References and the Altimetry Constellation

This paper gives an overview of an empirical cross-calibration technique developed for the Surface Water Ocean Topography mission (SWOT). The method is here used to detect and to mitigate two spatially coherent errors in SWOT topography data: the baseline roll error whose signature is linear across track, and the baseline length error whose signature is quadratic across track. Assuming that topography data are corrupted by coherent error signatures that we can model, we extract the signatures, and we empirically use the error estimates to correct SWOT data. The cross-calibration is tackled with a two-step scheme. The first step is to get local estimates over cross-calibration zones, and the second step is to perform a global interpolation of local error estimates and to mitigate the error everywhere. Three methods are used to get local error estimates: 1) we remove a static first guess reference such as a digital elevation model, 2) we exploit overlapping diamonds between SWOT swaths, and 3) we exploit overlapping segments with traditional pulse-limited altimetry sensors. Then, the along-track propagation is performed taking the local estimates as an input, and an optimal interpolator (1-D objective analysis) constrained with a priori statistical knowledge of the problem. The rationale of this paper is to assume that SWOT's scientific requirements are met on all errors but the ones being cross-calibrated. In other words, the algorithms presented in this paper are not needed at this stage of the mission definition, and they are able to deal with higher error levels (e.g., if hardware constraints are relaxed and replaced by additional ground processing). Even in our most pessimistic theoretical scenarios of baseline roll and baseline length errors (up to 70 cm RMS of uncorrected topography error), the cross-calibration algorithm reduces coherent errors to less than 2 cm (outer edges of the swath). Residual errors are subcentimetric for very low-frequency errors (e.g., orbital revolution). Sensitivity tests highlight the benefits of using additional pulse-limited altimeters and optimal inversion schemes when the problem is more difficult to solve (e.g., wavelengths of less than 1000 km), but also to provide a geographically homogeneous correction that cannot be obtained with SWOT's sampling alone.

Quantitative cone‐beam CT imaging in radiation therapy using planning CT as a prior: First patient studies

Quantitative cone-beam CT (CBCT) imaging is on increasing demand for high-performance image guided radiation therapy (IGRT). However, the current CBCT has poor image qualities mainly due to scatter contamination. Its current clinical application is therefore limited to patient setup based on only bony structures. To improve CBCT imaging for quantitative use, we recently proposed a correction method using planning CT (pCT) as the prior knowledge. Promising phantom results have been obtained on a tabletop CBCT system, using a correction scheme with rigid registration and without iterations. More challenges arise in clinical implementations of our method, especially because patients have large organ deformation in different scans. In this paper, we propose an improved framework to extend our method from bench to bedside by including several new components. The basic principle of our correction algorithm is to estimate the primary signals of CBCT projections via forward projection on the pCT image, and then to obtain the low-frequency errors in CBCT raw projections by subtracting the estimated primary signals and low-pass filtering. We improve the algorithm by using deformable registration to minimize the geometry difference between the pCT and the CBCT images. Since the registration performance relies on the accuracy of the CBCT image, we design an optional iterative scheme to update the CBCT image used in the registration. Large correction errors result from the mismatched objects in the pCT and the CBCT scans. Another optional step of gas pocket and couch matching is added into the framework to reduce these effects. The proposed method is evaluated on four prostate patients, of which two cases are presented in detail to investigate the method performance for a large variety of patient geometry in clinical practice. The first patient has small anatomical changes from the planning to the treatment room. Our algorithm works well even without the optional iterations and the gas pocket and couch matching. The image correction on the second patient is more challenging due to the effects of gas pockets and attenuating couch. The improved framework with all new components is used to fully evaluate the correction performance. The enhanced image quality has been evaluated using mean CT number and spatial nonuniformity (SNU) error as well as contrast improvement factor. If the pCT image is considered as the ground truth, on the four patients, the overall mean CT number error is reduced from over 300 HU to below 16 HU in the selected regions of interest (ROIs), and the SNU error is suppressed from over 18% to below 2%. The average soft-tissue contrast is improved by an average factor of 2.6. We further improve our pCT-based CBCT correction algorithm for clinical use. Superior correction performance has been demonstrated on four patient studies. By providing quantitative CBCT images, our approach significantly increases the accuracy of advanced CBCT-based clinical applications for IGRT.

Identification of star sensor low-frequency error parameters

A new method is proposed to identify the parameters of the low-frequency error (LFE) of the star sensor. The LFE is one of the most dominant error sources in the high-precision attitude determination system for the satellite. It is shown that the gyroscope bias estimate, which is obtained by processing the star sensor measurement in the stellar inertial attitude determination Kalman filter, represents the variation in the LFE. As the LFE is a periodic signal, the LFE parameters can be identified from the frequency spectrum of the gyroscope bias estimate. The identified parameters are used for the LFE compensation, such that the effect of the LFE to the attitude estimate is partly eliminated. The efficiency of the proposed method is illustrated through numerical simulations with the use of the real gyroscope data.

Study on evaluation of RB-SiC aspherical mirror high frequency surface quality fabricated with fixed abrasive technology

The issue of the letter is focused on the RB-SiC aspherical mirror high frequency surface quality which is fabricated with the different type fixed abrasive pellets. The two-dimensional (2D) surface roughness of the mirror is simulated and experimented. The errors between the simulation and the experiment are 5.97%, 3.19%, 3.59%, 37.37% using W1.5, W3.5, W5, W7 pellets respectively. The error emerging reason is analyzed in detail after the comparison. Also the fractal theory is applied on the analysis of the optical mirror surface fabricated with the fixed abrasive technology. The analysis result shows that the good surface quality of the RB-SiC mirror can be obtained quickly with being polished with the fixed abrasive technology. So the fixed abrasive technology is very suitable for the fast optical surfacing, especially can obtain good surface roughness fast in the stage of grinding mirror which is not suitable for being tested with interferometer. OCIS codes: 220.0220, 220.4000, 220.4610, 220.5450. doi: 10.3788/COL201210.S22201. Generally, the surface error, the ripple and the surface roughness should be tested quantificationally when a SiC mirror is finished. The mirror surface quality is evaluated through the parameters tested. The surface error which is also called low frequency error is studied for decades [1 3] . So the surface error is not in the letter’s extension, it is focused on the high frequency error of the SiC mirror fabricated with the fixed abrasive technology, which is studied through several different methods. The existence of the high frequency error on the mirror will increase the scattering energy, decreasing the effective energy entering the camera, then the image quality will be decreased. So we should understand exactly what factor and which stage leads to the increase of the high frequency error on the mirror in the fabrication. In the most situation, the study is only focused on the method of testing the high frequency error, but the theoretical study on the high frequency error is seldom. The starting point of the letter is simulating the microcosmic formation the high frequency error in the fabrication process. And also the mirror surface fabricated with pellets is analyzed and evaluated using the fractal theory for the first time. Because of the complexity of simulating the fabrication process of SiC mirror with diamond abrasive actually, the following important assumption is made: the final mirror surface is formed by the indentation into the SiC mirror with the still diamond particles. In the fabrication of the SiC material, the size and depth of the indentation formed by the diamond particles are totally different. So we are on the assumption that the position distribution of the indentations is uniform, the size of the indentations are in Gaussian distribution, the depth is � [4]

Multiscale Local Map-Drift-Driven Multilateration SAR Autofocus Using Fast Polar Format Image Synthesis

This new autofocus method is based on multilateration by ranging to small target areas at independent directions on the ground. Range-clipped Doppler low-filtered profiles around target points are used to compute local images using frequency-domain polar format algorithm. Images obtained from adjacent subapertures are registered, and the displacements yield elevation, trajectory, or clock drift (bistatic case) errors. To alleviate the insensitivity of map drift to error fluctuation faster than subaperture duration, the algorithm is reiterated with coarse-to-fine resolution, yielding high to low frequency errors. This allowed true bistatic synthetic aperture radar (SAR) autofocus (without monostatic image), autofocus in circular SAR on remote areas, and, as a side product, our first successful air-to-air inverse SAR high-resolution imaging.