Fractional differentiation-based observability analysis method for nonlinear X-ray pulsar navigation system

In this paper, a novel observability analysis method of a nonlinear navigation system is proposed, and its application to the optimization of beacon configuration for Mars entry navigation is demonstrated. A Lie algebra-based approach to nonlinear observability is used to compute the observability matrix. The quadratic approximation to the Lie derivatives is then used to recursively calculate the observability matrix efficiently. Next, the inverse of the condition number of the observability matrix is chosen as a metric to evaluate the observability quality. To verify the accuracy and effectiveness of the proposed approach, Mars navigation scenarios with ranging measurements from multiple ground-based beacons are considered in simulation examples, and the optimal beacon configurations are obtained using a genetic algorithm. The extended Kalman filter is finally used to demonstrate the navigation performance. It is concluded that the proposed observability analysis method is able to optimize the beacon configuration of the Mars entry navigation system accurately and efficiently. For Mars entry navigation using ground-based beacons, the geometric configuration of the entry vehicle and beacons is a main contributor to the observability. A diminishing return of observability is obtained solely by adding beacon numbers.

The high altitudes reached by HEO (highly eccentric orbits) spacecraft present a very unfavorable environment for the reception of GPS signals. This paper analyzed the GPS signal visibility viewed from running HEO spacecraft, and then investigated the observability of the autonomous navigation system with the pseudo-range measurements using a PWCS method , adopted the condition number of the observability matrix as a scalar measure of degree of observability, finally assessed the navigation performance. The results of simulation showed that, with measurements from only one GPS satellite, the observability of the navigation system can be observable; the error of position and velocity could be verified. Therefore, the HEO autonomous navigation using GPS is feasible.

X-ray pulsar navigation is a complete autonomous navigation system, which has broad application prospects. Because of the huge cost of the navigation system, the implementation of ground simulation system is essential to the application of X-ray pulsar navigation. At present, most of researches on the semi physical experiment system are static. The aim of this article is to develop the dynamic simulation experiment system as well as its performance test. Specifically, this system consists of the dynamic signal database, X-ray simulation source, vacuum system and detection system designed for different science purposes. The core component of the X-ray source is the gate controlled X-ray tube, which can simulate the pulse profile of arbitrary waveform. The detecting system is based on the silicon drift detector with high time response capability. It uses trapezoidal shape for signal processing, and the timing resolution of the detection system is better than 2 s. In addition, the dynamic signal generation method is given by analyzing the time transformation model while the SINC interpolation method is provided to generate the dynamic pulse profile. Finally, the spacecraft revolving around the earth for a circle and receiving a pulse signal of Crab is simulated. In the simulation, the orbital radius of satellite is 6578 km and the orbital period is 5400 s. The Crab pulsar is selected, and the pulse period is 33.4 ms, the number of photons received by the detector is 200 per second. As a contrast, a set of static experiments is also performed. The correlation coefficient between the cumulative pulse profile and the standard pulse profile is 0.9953. However, the correlation coefficient decreases gradually, from 0.9094 at 300 s to 0.4080 at 5400 s, in the dynamic experiment. Then, the pulse period is searched from the arrival time of photons. The periodicity of the pulse signal is sinusoidal when the search period is 60 s. The change rate of photon flux is less than 2\%, and the influence on the period search is negligible. The variation of pulse period is consistent with the motion law of spacecraft, which indicates that spacecraft motion is the dominant factor in time conversion. Finally, the arrival time of photons is transformed into the time at the solar system barycenter, indicating that the correlation coefficient between cumulative pulse profile and standard pulse profile is 0.9882. The result shows that the simulation system can simulate the X-ray pulse signal received by the spacecraft in orbit, which can provide the experimental basis for verifying the navigation algorithm and calibrating the detector performance.

X-Ray pulsar navigation is a new technology for astronavigation. It’s an effective supplement to the satellite navigation which is widely used at present. It can provide autonomous navigation services for spacecraft that cannot be covered by satellite navigation system. China has started researching on engineering application since 2005, and sent specific satellites to carry out space experiments in 2016. Compared to theoretical research and engineering application, the standardization of X-Ray pulsar navigation is left behind. There are no relevant standards and top-level planning for X-Ray pulsar navigation in the standard systems at national level. Standardization is an important guarantee condition to ensure development of an industry and progress of a major project. The general departments take standardization as an important work in the major national projects, such as CMS, BeiDou, CHEOS and so on. As an important part of navigation project, the standardization of X-Ray pulsar navigation is to meet the needs of navigation engineering construction, and is an effective way to implement the requirements of navigation standardization. This paper explains how to carry out standardization by combining the technical characteristics and development status of X-Ray pulsar navigation.

Implicit UKF and its observability analysis of satellite stellar refraction navigation system

The starlight Doppler navigation must be integrated with other navigation methods to operate normally since it is not completely observable. The star selection for the Doppler navigation system influences the whole integrated navigation performance greatly. The star magnitude-based star selection method is commonly used to select the star for the Doppler navigation. However, it can be found that the direction of star is more important than the measurement accuracy. To select the optimal navigation star, the star selection strategy for the starlight Doppler-based integrated navigation system, where both the direction and magnitude of star are considered, is proposed in this paper. Considering the impact of the star direction and measurement noise, the measurement coupling matrix is constructed to give the relationship between the star direction and the integrated navigation accuracy, and utilize this matrix as the star selection criterion. According to the perturbation matrix theory, the measurement coupling matrix is analyzed to obtain the theoretical maximum and minimum of its trace by formula derivation. Finally, the proposed star selection criterion is utilized to select the optimal navigation star from the first-magnitude stars. In this paper, the XNAV (X-ray pulsar navigation)/starlight Doppler integrated navigation system is taken as an instance to analyze and simulate. The simulation results demonstrate that using the star selected by our method as the navigation star, the XNAV/starlight Doppler integrated navigation has good performance. Compared with the X-ray sensor area and the spectrometer measurement accuracy, the direction of star is more important for the improvement of the integrated navigation performance.

Orbit propagation error effects are considered in precision analysis of X-ray pulsar navigation (XNAV) systems which integrating orbit dynamics model and pulse time of arrival observation. Precision analysis model is established, and the concept of optimal pulse observation accumulation time is proposed based on coupling analysis of orbit propagation error, pulsar observation process and pulse time of arrival observation precision. The estimation algorithm of optimal observation accumulation time is presented as an important tool for XNAV observation planning and scheme design.

This paper introduces the mechanism analysis of the large beacon calibration errors how to decrease the navigation performance for the AUV (Autonomous Underwater Vehicle). Generally, when the vehicle using inertial navigation system is performing specific tasks in the water, the navigation error accumulates as time goes on; sometimes the navigation algorithm may fail because of a lot of disturbances, such as outliers, noisy measurement, missed measurement. In order to obtain the better navigation performance for underwater navigation system in real time, the underwater integrated navigation system has been constructed with the underwater acoustic and inertial navigation system. But it is hard to obtain the accurate initial beacon position for the acoustic system in the complex and unknown underwater environment, which will seriously deteriorate the navigation performance. For the sake of obtaining the better navigation performance, we analyze the reasons how the large calibration errors decrease the navigation performance. At last we also provide three feasible methods to online correct the calibration errors. The lake trial data further explain the mechanism analysis.

Free vibration analysis of cylindrical shells using the Haar wavelet method

This paper presents a fast phased array calibration method which significantly reduces the number of antenna under test (AUT) amplitude and phase setting states. Multiple measuring probes are introduced to record transmission coefficients as AUT works at different states. A coefficient matrix with a small condition number is constructed to calculate the initial excitation coefficients of each antenna element. Simulation results show that the coefficient matrix condition number can be reduced to 1 by optimizing the geometry of the measurement system. When the signal-to-noise ratio is higher than 18dB, the amplitude error of the calibration result is less than 0.5dB and the phase error is less than 20 degrees in the optimal conditional number configuration. This method effectively reduces the measurement time for the phased array calibration and improves the stability of the calibration system.

This paper is concerned with the solution of ill-conditioned Systems of Linear Equations (SLE's) via the solution of equivalent SLE's which are well-conditioned. A matrix is first constructed from that of the given ill-conditioned system. Then, an adequate right-hand side is computed to make up the instance of an equivalent system. Formulae and algorithms for computing an instance of this equivalent SLE and solving it will be given and illustrated. © 2011 The Korean Mathematical Society.

This paper aims to provide an insight on the mechanism for the fast convergence of multigrid (MG) V-cycle algorithms. With this goal, the Fourier transform analysis is implemented to compute quantitatively the frequency spectrum of components of the residual resulted from the MG iterations. MG method has been extensively employed to solve large-scale systems of algebraic equations. Hence, one needs to understand its fast convergence property and scalability mechanism before applying MG method to a system equation, which affects the computational efficiency directly. In a first step, a number of novel features of the MG algorithm are studied, including the relationships among the iteration numbers needed for converged solutions, condition number of the system matrix and the grid-spacing in this paper. In a second step, the complexity of the MG method is researched, and various numerical experiments have been conducted in solving algebraic equations of models in finite element method for both heat transfer problems and solid mechanics problems. It is found that if the number of iterations in different grids is fixed, the optimal iteration mode is determined according to the attenuation of the high-frequency and low-frequency components of the residual. Our results also show that the complexity of the MG method is close to [Formula: see text], meaning that it is a close-to-linear scalable solver, which echoes the findings given by other literature. This study has revealed an important insight that the linear scalability is achieved by mitigating the negative effects (stagnation in convergence) of the large condition number of the system matrix created on the finest grid, using a sequence of coarser grids with smaller and smaller condition numbers.

BackgroundFractional-order models have become essential tools for representing nonlinear dynamical systems with memory and hereditary effects. While constant-order fractional operators have been widely studied, many real-world systems exhibit memory characteristics that evolve over time or system state. Variable-order fractional calculus addresses this limitation but introduces significant analytical and numerical challenges, particularly in nonlinear settings where stability and convergence are difficult to guarantee. Existing numerical approaches often suffer from accuracy loss and instability when fractional orders vary rapidly. MethodsThis study develops a hybrid fractional integral framework for nonlinear dynamical systems governed by variable-order operators. The methodology combines multiple variable-order fractional integrals within a unified formulation to balance short-term and long-term memory effects. A predictor–corrector numerical scheme is employed to handle nonlinearities and evolving fractional orders efficiently. Rigorous theoretical analysis is conducted to establish existence, uniqueness, convergence, and stability under standard smoothness and Lipschitz assumptions. The framework is evaluated through numerical experiments on benchmark nonlinear systems with different variable-order profiles. ResultsThe proposed hybrid framework demonstrates consistently improved numerical performance compared with constant-order and single-operator variable-order methods. Results show lower global error, higher observed convergence order, and enhanced stability under oscillatory and discontinuous fractional-order variations. The method also exhibits greater robustness in strongly nonlinear regimes, where conventional schemes experience significant error amplification. Computational efficiency is improved through faster convergence, enabling accurate solutions with fewer time steps and reduced memory overhead. ConclusionThe hybrid fractional integral framework provides a stable, accurate, and flexible approach for solving nonlinear variable-order dynamical systems. By structurally integrating complementary memory operators, the method overcomes key limitations of existing approaches and offers a reliable computational foundation for modeling complex systems with adaptive memory dynamics.

The study purpose was to evaluate the use of electromagnetic navigation in surgical treatment of patients with various brain lesions. The system of electromagnetic navigation was used for 102 operations in 98 patients (42 males and 56 females, including 18 children; median age, 34.8 years (min, 2.2 years; max, 69 years)) in the period from December 2012 to December 2016. In 36 patients, the system was used for endoscopic interventions. In 19 patients, electromagnetic navigation was used in combination with neurophysiological monitoring. In our series of cases, the frameless electromagnetic navigation system was used in 66 transcranial operations. The mean error of navigation was 1.9±0.5 mm. In 5 cases, we used the data of preoperative functional MRI (fMRI) and tractography for navigation. At the same time, in all 7 operations with simultaneous direct stimulation of the cortex, there was interference and significant high-frequency noise, which distorted the electrophysiological data. A navigation error of more than 3 mm was associated with the use of neuroimaging data with an increment of more than 3 mm, image artifacts from the head locks, high rate of patient registration, inconsequence of touching points on the patient's head, and unsatisfactory fixation to the skin or subsequent displacement of a non-invasive localizer of the patient. In none of the cases, there was a significant effect of standard metal surgical tools (clamps, tweezers, aspirators) located near the patient's head on the navigation system. In two cases, the use of massive retractors located near the patient's localizer caused noise in the localizer and navigation errors of more than 10 mm due to significant distortions of the electromagnetic field. Thirty-six transnasal endoscopic interventions were performed using the electromagnetic frameless navigation system. The mean navigation error was 2.5±0.8 mm. In general, electromagnetic navigation is an accurate, safe, and effective technique that can be used in surgical treatment of patients with various brain lesions. The mean navigation error in our series of cases was 1.9±0.5 mm for transcranial surgery and 2.5±0.8 mm for endoscopic surgery. Electromagnetic navigation can be used for different, both transcranial and endoscopic, neurosurgical interventions. Electromagnetic navigation is most convenient for interventions that do not require fixation of the patient's head, in particular for CSF shunting procedures, drainage of various space-occupying lesions (cysts, hematomas, and abscesses), and optimization of the size and selection of options for craniotomy. In repeated interventions, disruption of the normal anatomical relationships and landmarks necessitates application of neuronavigation systems in almost mandatory manner. The use of electromagnetic navigation does not limit application of the entire range of necessary intraoperative neurophysiological examinations at appropriate surgical stages. Succession in application of neuronavigation should be used to get adequate test results.

Since it is well-known that the Vandermonde matrix is ill-conditioned, this paper surveys the choices of other bases. These bases are data-dependent and are categorized into discretely $ell^2$-orthonormal and continuously $L^2$-orthonormal bases. The first one is defined via a decomposition of the Vandermonde matrix while the latter is given by a decomposition of the Gramian matrix corresponding to monomial bases. A discussion of various matrix decomposition (e.g. Cholesky, QR and SVD) provides a variety of different bases with different properties. Special attention is given to duality. Numerical results show that the matrices of values of the new bases have smaller condition numbers than the common monomial bases. It can also be pointed out that the new introduced bases are good candidates for interpolation.