Attitude Determination Problem Research Articles

Asymptotically stable optimal multi-rate rigid body attitude estimation based on lagrange-d'alembert principle

<abstract><p>The rigid body attitude estimation problem is treated using the discrete-time Lagrange-d'Alembert principle. Three different possibilities are considered for the multi-rate relation between angular velocity measurements and direction vector measurements for attitude: 1) integer relation between sampling rates, 2) time-varying sampling rates, 3) non-integer relation between sampling rates. In all cases, it is assumed that angular velocity measurements are sampled at a higher rate compared to the inertial vectors. The attitude determination problem from two or more vector measurements in the body-fixed frame is formulated as Wahba's problem. At instants when direction vector measurements are absent, a discrete-time model for attitude kinematics is used to propagate past measurements. A discrete-time Lagrangian is constructed as the difference between a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential energy-like term obtained from Wahba's cost function. An additional dissipation term is introduced and the discrete-time Lagrange-d'Alembert principle is applied to the Lagrangian with this dissipation to obtain an optimal filtering scheme. A discrete-time Lyapunov analysis is carried out to show that the optimal filtering scheme is asymptotically stable in the absence of measurement noise and the domain of convergence is almost global. For a realistic evaluation of the scheme, numerical experiments are conducted with inputs corrupted by bounded measurement noise. These numerical simulations exhibit convergence of the estimated states to a bounded neighborhood of the actual states.</p></abstract>

Constrained Wrapped Least Squares: A Tool for High-Accuracy GNSS Attitude Determination

Attitude determination is a popular application of Global Navigation Satellite Systems (GNSS). Many methods have been developed to solve the attitude determination problem with different performance offerings. We develop a constrained wrapped least-squares (C-WLS) method for high-accuracy attitude determination. This approach is built on an optimization model that leverages prior information related to the antenna array and the integer nature of the carrier-phase ambiguities in an innovative way. The proposed approach adopts an efficient search strategy to estimate the vehicle's attitude parameters using ambiguous carrier-phase observations directly, without requiring prior carrier-phase ambiguity fixing. The performance of the proposed method is evaluated via simulations and experimentally utilizing data collected using multiple GNSS receivers. The simulation and experimental results demonstrate excellent performance, with the proposed method outperforming the ambiguity function method, the constrained LAMBDA and multivariate constrained LAMBDA methods, three prominent attitude determination algorithms.

Image-based attitude determination of co-orbiting satellites using deep learning technologies

Active debris removal missions pose demanding guidance, navigation and control requirements. We present a novel approach which adopts deep learning technologies to the problem of attitude determination of an uncooperative debris satellite of an a-priori unknown geometry. A siamese convolutional neural network is developed, which detects and tracks inherently useful landmarks from sensor data, after training upon synthetic datasets of visual, LiDAR or RGB-D data. The method is capable of real-time performance while improving upon conventional computer vision-based approaches, and generalises well to previously unseen object geometries, enabling this approach to be a feasible solution for safely performing guidance and navigation in active debris removal, satellite servicing and other close proximity operations. The performance of the algorithm, its sensitivity to model parameters and its robustness to illumination and shadowing conditions, are analysed via numerical simulation.

In-Orbit Attitude Determination of the UVSQ-SAT CubeSat Using TRIAD and MEKF Methods.

Ultraviolet and infrared sensors at high quantum efficiency on-board a small satellite (UVSQ-SAT) is a CubeSat dedicated to the observation of the Earth and the Sun. This satellite has been in orbit since January 2021. It measures the Earth’s outgoing shortwave and longwave radiations. The satellite does not have an active pointing system. To improve the accuracy of the Earth’s radiative measurements and to resolve spatio-temporal fluctuations as much as possible, it is necessary to have a good knowledge of the attitude of the UVSQ-SAT CubeSat. The attitude determination of small satellites remains a challenge, and UVSQ-SAT represents a real and unique example to date for testing and validating different methods to improve the in-orbit attitude determination of a CubeSat. This paper presents the flight results of the UVSQ-SAT’s attitude determination. The Tri-Axial Attitude Determination (TRIAD) method was used, which represents one of the simplest solutions to the spacecraft attitude determination problem. Another method based on the Multiplicative Extended Kalman Filter (MEKF) was used to improve the results obtained with the TRIAD method. In sunlight, the CubeSat attitude is determined at an accuracy better than 3 (at one ) for both methods. During eclipses, the accuracy of the TRIAD method is 14, while it reaches 10 (at one ) for the recursive MEKF method. Many future satellites could benefit from these studies in order to validate methods and configurations before launch.

Static Attitude Determination Using Convolutional Neural Networks.

The need to estimate the orientation between frames of reference is crucial in spacecraft navigation. Robust algorithms for this type of problem have been built by following algebraic approaches, but data-driven solutions are becoming more appealing due to their stochastic nature. Hence, an approach based on convolutional neural networks in order to deal with measurement uncertainty in static attitude determination problems is proposed in this paper. PointNet models were trained with different datasets containing different numbers of observation vectors that were used to build attitude profile matrices, which were the inputs of the system. The uncertainty of measurements in the test scenarios was taken into consideration when choosing the best model. The proposed model, which used convolutional neural networks, proved to be less sensitive to higher noise than traditional algorithms, such as singular value decomposition (SVD), the q-method, the quaternion estimator (QUEST), and the second estimator of the optimal quaternion (ESOQ2).

Characterization of Degenerate Configurations in Attitude Determination of Three-Vehicle Heterogeneous Formations.

The existence of multiple solutions to an attitude determination problem impacts the design of estimation schemes, potentially increasing the errors by a significant value. It is therefore essential to identify such cases in any attitude problem. In this paper, the cases where multiple attitudes satisfy all constraints of a three-vehicle heterogeneous formation are identified. In the formation considered herein, the vehicles measure inertial references and relative line-of-sight vectors. Nonetheless, the line of sight between two elements of the formation is restricted, and these elements are denoted as deputies. The attitude determination problem is characterized relative to the number of solutions associated with each configuration of the formation. There are degenerate and ambiguous configurations that result in infinite or exactly two solutions, respectively. Otherwise, the problem has a unique solution. The degenerate configurations require some collinearity between independent measurements, whereas the ambiguous configurations result from symmetries in the formation measurements. The conditions which define all such configurations are determined in this work. Furthermore, the ambiguous subset of configurations is geometrically interpreted resorting to the planes defined by specific measurements. This subset is also shown to be a zero-measure subset of all possible configurations. Finally, a maneuver is simulated to illustrate and validate the conclusions. As a result of this analysis, it is concluded that, in general, the problem has one attitude solution. Nonetheless, there are configurations with two or infinite solutions, which are identified in this work.

Estimator for Deep-Space Position and Attitude Using X-ray Pulsars

In this article, we develop an algorithm for the estimation of a full six degree of freedom position, navigation, and timing solution in deep space by measuring the time and angle of arrival of x-rays photons from pulsars and other bright stars. We show that for pulsar navigation, the position and attitude determination problems are coupled. This is due in part to the small signal-to-noise ratio of pulsar signals and the fact that x-ray photons emanating from various pulsars and bright stars have no unique identifier that can be used to associate them with their source. To address this challenge, a joint probabilistic data association filter is developed. The filter fuses angular rate measurements from a three-axis rate gyro with time-of-arrival and angle-of-arrival measurements from an x-ray detector. The performance of the filter is validated in simulation, and the tradeoffs associated with detector size and initial conditions are evaluated. Additional validation of the algorithm is performed by playback of data from x-ray detectors on the Chandra spacecraft. The results show that positioning accuracy on the order of 1000 km and attitude accuracies on the order of 5 arcseconds can be achieved.

Application of artificial intelligence technology in the nanosatellite attitude determination problem

The paper proposes an approach of the nanosatellite attitude determination problem solution using elements of artificial intelligence technology. This approach consists in the reconfiguration of the algorithms of the nanosatellite attitude determination problem solution based on the knowledge database that can be enriched to ensure the robustness of the feedback loop (the nanosatellite attitude determination using different combinations of different types of measurement information by various algorithms). This research results are presented in relation to the nanosatellite of the SamSat family: values of measurement significance coefficients are chosen; then generalized algorithm significance coefficients are calculated for various combinations of operable measuring means using chosen coefficients.

Parameterization-switching GNSS attitude determination considering the success rate of ambiguity resolution

The two key elements of GNSS attitude determination are the correct integer ambiguity resolution (IAR) and the choice of appropriate method. The least-squares ambiguity decorrelation adjustment method is normally used to fix ambiguities using float estimates. However, it is difficult to get all ambiguities fixed in real time. Therefore, a method of partial ambiguity resolution which considers the success rate of IAR and makes full use of the ambiguity information at each epoch is proposed. The ambiguities after decorrelation can be divided into fixed ambiguities and unfixed ambiguities considering the fixing success rate. At the next epoch, the fixed ones are treated as constants, while the float estimates of the unfixed ones, along with the corresponding covariance matrix, are treated as prior information or pseudo-measurements. The parameters of the attitude determination problem can be baseline coordinates (BCs) or attitude angles (AAs). BC parameterization, without considering the baseline constraints, has the merit of resulting in a linear model; however, it involves parameter redundancy, which reduces the model strength. AA parameterization, although avoiding parameter redundancy, brings in nonlinearity and hence linearization errors in the least-squares solution. The linearization errors decrease as the number of fixed ambiguities increases. A switching-parameterization method is proposed; namely, at any epoch, as long as there are at least three fixed double-difference ambiguities, AA parameterization is adopted, otherwise BC is adopted. Even with AA, BC is firstly estimated and then transformed to attitude estimates at which the nonlinear measurement equations are linearized and solved. To verify the performance of the proposed method, a comparative study is implemented with BC and AA methods in the static experiment. The results show that the proposed method can fix ambiguity faster and have better stability and accuracy. In addition, the yaw angle is consistent with the actual running route of the vehicle in the kinematic experiment.

Precise 3-D GNSS Attitude Determination Based on Riemannian Manifold Optimization Algorithms

In the past few years, Global Navigation Satellite Systems (GNSS) based attitude determination has been widely used thanks to its high accuracy, low cost, and real-time performance. This paper presents a novel 3-D GNSS attitude determination method based on Riemannian optimization techniques. The paper first exploits the antenna geometry and baseline lengths to reformulate the 3-D GNSS attitude determination problem as an optimization over a non-convex set. Since the solution set is a manifold, in this manuscript we formulate the problem as an optimization over a Riemannian manifold. The study of the geometry of the manifold allows the design of efficient first and second order Riemannian algorithms to solve the 3-D GNSS attitude determination problem. Despite the non-convexity of the problem, the proposed algorithms are guaranteed to globally converge to a critical point of the optimization problem. To assess the performance of the proposed framework, numerical simulations are provided for the most challenging attitude determination cases: the unaided, single-epoch, and single-frequency scenarios. Numerical results reveal that the proposed algorithms largely outperform state-of-the-art methods for various system configurations with lower complexity than generic non-convex solvers, e.g., interior point methods.

Unified Attitude Determination Problem From Vector Observations and Hand–Eye Measurements

The hand-eye measurements have recently been proven to be very efficient for spacecraft attitude determination relative to an ellipsoidal asteroid. However, recent method does not guarantee full attitude observability for all conditions. This paper refines this problem by taking the vector observations into account so that the accuracy and robustness of the spacecraft attitude estimation can be improved. The vector observations come from many sources including visual perspective geometry, optical navigation and point clouds that frequently occur in aerospace electronic systems. Completely closed-form solutions along with their uncertainty descriptions are presented for the proposed problem. Experiments using our simulated dataset and real-world spacecraft measurements from NASA dawn spacecraft verify the effectiveness and superiority of the derived solution.

Dot Product Equality Constrained Attitude Determination from Two Vector Observations: Theory and Astronautical Applications

In this paper, the attitude determination problem from two vector observations is revisited, incorporating the redundant equality constraint obtained by the dot product of vector observations. Analytical solutions to this constrained attitude determination problem are derived. It is found out that the studied two-vector attitude determination problem by Davenport q-method under the dot product constraint has deterministic maximum eigenvalue, which leads to its advantage in error/perturbation analysis and covariance determination. The proposed dot product constrained two-vector attitude solution is applied then to solve several engineering problems. Detailed simulations on spacecrafts attitude determination indicate the efficiency of the proposed theory.

A robust and fast self-alignment method for strapdown inertial navigation system in rough sea conditions

In this paper, a robust and fast self-alignment method under an undesirable non-periodic unidirectional vessel motion is proposed. The proposed method is based on the attitude determination technique which transforms the alignment procedure into an attitude determination problem by creating infinite vector observations. The method implements the QUaternion ESTimator (QUEST) attitude determination technique which reduces the computational complexity in practical implementations and accomplishes the alignment task more robustly and accurately compared with the Kalman filter based alignment methods in a specific time interval. Simulation and also experimental results show that the proposed technique has more accurate and faster responses compared to the conventional methods based on the Kalman filter. • Introducing a vessel motion derived from experience during self-alignment of SINS. • Overcoming the challenge of self-alignment by creating vector observations. • Low computational complexity for attitude determination by using vector observations. • The robustness of the proposed method against the disturbing forces. • Obtaining the vector observations for compensation of the Lever Arm effect.

A solution for the attitude determination of three-vehicle heterogeneous formations

This paper proposes a solution to a new attitude determination problem for a three-vehicle formation, where there are restrictions on the detection of other vehicles. Vision-based sensors are considered, which measure line-of-sight (LOS) vectors between different vehicles and inertial vectors that can vary according to the vehicle. There is a constraint on the LOS vectors which are only measured in relation to a chief vehicle. Moreover, each vehicle can measure only one inertial vector. The solution for the different attitude relations is devised geometrically, making use of a multi-stage process. First, two candidates for the relative attitude of each branch are determined. Then, these relative attitude candidates are used to compute inertial attitude candidates for the chief vehicle. The comparison between these results disambiguates the problem, which in general has a unique solution. There are some degenerate solutions, which are also determined. Finally, simulations are carried out considering noise in the sensors. The results are coherent with state-of-the-art approaches to similar problems.

Reduction of angular velocities of AIST-2D spacecraft using a system of kinetic moment dumping

The efficiency of a magnetic system of kinetic moment dumping in solving the problem of stabilizing Aist-2D spacecraft without using information on angular velocity is investigated. We defined the conditions under which the magnetic moment produced by electromagnets constitutes a useful control mechanical moment. Boundary conditions for preventing emergence of the parasitic moment are also defined. The paper presents the solution of the problem of forming the vector of the kinetic moment of the system spacecraft + control moment gyro according to the information on the vector of geomagnetic induction of the Earth. We defined the dependence of the time taken by the algorithm of angular speed reduction on the value and direction of the projections of the vector of spacecraft angular speeds at the time of algorithm actuation. The results obtained in the course of mathematical modeling and natural experiment on the spacecraft, the process of dumping angular speeds in providing the solution of the problem of attitude determination with the use of star sensors are compared.

Analytical and Iterative Solutions to GNSS Attitude Determination Problem in Measurement Domain

Attitude determination using double-differenced GNSS carrier phase measurements is studied. A realistic stochastic model is employed to take the correlations among the double-differenced measurements into full consideration. Two important issues concerning iteratively solving the nonlinear least-squares attitude determination problem are treated, namely, the initial guess and the iteration scheme. An analytical and sub-optimal solution is employed to provide the initial guess. In this solution, the orthogonal and determinant constraints among the elements of the direction cosine matrix (DCM) of the attitude are firstly ignored, and hence a relaxed 3×3 matrix is estimated using the linear weighted least-squares method. Then a mathematically feasible DCM, i.e., orthogonal and with +1 determinant, is extracted from the relaxed matrix estimate, optimally in the sense of minimum Frobenius norm. This analytical initial guess estimation method can be used for all feasible cases, including some generated ones, e.g., the case with only 3 antennas and only 3 satellites, subject possibly to some necessary, yet minor modifications. In each iteration, an error attitude, whose DCM is parameterized using the Gibbs vector, is introduced to relate the previously estimated and the true DCM. By linearizing the measurement model at the zero Gibbs vector, the least-squares estimate of the Gibbs vector is obtained and then used to correct the previously estimated DCM. By repeating this process, the truly least-squares estimate of the attitude can be achieved progressively. These are in fact Gauss-Newton iterations. For the final estimate, the variance covariance matrix (VCM) of the attitude estimation error can be retained to evaluate or predict the estimation accuracy. The extraction of the widely used roll-pitch-yaw angles and the VCM of their additive estimation errors from the final solution is also presented. Numerical experiments are conducted to check the performance of the developed theory. For the case with 3 2-meter long and orthogonally mounted baselines, 5 visible satellites, and 5-millimeter standard deviations of the carrier phase measurements, the root mean squared errors (RMSE) of the roll-pitch-yaw angles in the analytical solution are well below 0.5 degrees, and the estimates converge after only one iteration, with all three RMSEs below 0.2 degrees.

Gyro-Free Satellite Attitude Determination Based on Special Orthogonal Group

The attitude determination problem of Earth Orientation gyro-less spacecraft is studied in this paper. Considering the sensitivity of the initial attitude error for EKF, a new non linear attitude determination filter is developed by using the Special Orthogonal Group SO (3) which avoids any algebraic reconstruction of the attitude from reference vectors. Lyapunov analysis results for the proposed the filter is derived that ensure global stability of the error. The simulation results show that the estimation of attitude and angular velocity converge to the real in the Earth Orientation mode.

Integrated attitude-orbit dynamics and control of spacecraft systems: State of the art and future trends

Spacecraft orbit and attitude determination and control (OADC) is described as the methodology of determining and controlling the motion and orientation (i.e., the state vector, ephemeris, or state) of an orbiting object such as a spacecraft relative to the Sun, the Earth, or the stars [1]. The motion and orientation of a space-craft are estimated by a set of equations with the state adjusted in response to a set of discrete sensor's data and subject to both random and systematic errors [2]. In the context of this article, the integrated orbit and attitude determination and control (IO-ADC) problem is generally described by introducing statistical estimation techniques of determining the state of a spacecraft as a function of time using the set of observations. The spacecraft is supposed to be influenced by a variety of external forces and torques, such as gravity, aerodynamic drag, solar radiation, third-body disturbances, and Earth tidal effects [3]. The complex description of these forces and torques results in a highly nonlinear set of dynamical equations. Since the OADC equations and observational data are inherently nonlinear, linearization technique is often performed in which linear estimation methods are used to resolve the OADC problem [4].

Attitude, Angle of Attack and Angle of Sideslip Determination Using Carrier Phase Integer Ambiguity Estimation

The Global Navigation Satellite System (GNSS) positioning solution may be computed to a cm level of error, if the signal carrier phase is properly considered during the calculations. With such a precision, even attitude angles can be precisely estimated if more than one antennae is available. However, using the carrier phase information depends on solving the carrier phase integer ambiguity problem which is intrinsic to this approach. Since only a fraction of the carrier phase is observed by the GNSS receivers, the solution for the integer ambiguity is usually obtained by means of search algorithms, based on least squares. This contribution investigates the performance of the integer ambiguity solution for the attitude determination problem and also proposes a novel application: angle of attack and angle of sideslip estimation for specific aeronautical application. This novel application benefits aeronautical industry by introducing a new alternative for measuring reliable airdata parameters, namely angle of attack and angle of sideslip, by using a nonsusceptible to icing conditions data. The results presented here are based on the constrained LAMBDA method. Dynamic real data are used for evaluation of the proposed novel application.

A semi-analytical solution to the global navigation satellite system attitude determination problem

Global navigation satellite system attitude determination based on carrier phase differencing technique is studied. A realistic stochastic model is followed to fully consider the correlations among different measurements in the least-squares problem formulation. A prior-free, computation-fixed, and averagely optimal solution is proposed suitable for real-time and high-dynamic situations. The prior-free property is achieved by developing an analytical sub-optimal solution. This solution follows first transforming the original problem into one with vector measurements and then further approximating it with a general Wahba’s problem. The fixed computation is guaranteed by performing only one or two rounds of correction of the analytical solution. Every single round follows a linearization-estimation-correction process. The process also provides an error or covariance analysis for the estimate. The average optimality in terms of the root mean squared errors is brought about by the relatively good quality of the analytical solution and the fast convergence of the correction processes. The numerical experiments, with three 4 m long baselines and 5 mm (standard deviation) carrier phase errors, show that the estimation errors (in magnitude) for all three channels are well below 0.4° for almost all epochs and within 0.2° for most epochs.