Quaternion Estimator Research Articles

Precision Quaternion Estimation with Partially Norm-Constrained Unscented Kalman Filtering

Abstract Accurate orientation estimation is a key challenge in dynamics and control, particularly for rigid body motion. Quaternions are commonly used to represent rotations due to their computational efficiency, but they must always maintain a unit norm to function correctly. If this constraint is not enforced properly, it can lead to significant errors in orientation estimation. This paper proposes an unscented Kalman filter that ensures the quaternion parameters, a subvector within the state vector, adhere to the unit norm constraint. The proposed method provides a closed-form solution without dividing the state vector into constrained and unconstrained vectors. In simulation studies under high noise conditions, the filter demonstrates significantly improved performance compared to standard unscented and pseudo-unscented Kalman filters, enhancing accuracy during both transient and steady-state phases. These results highlight the importance of enforcing quaternion constraints to reduce mean square error and improve convergence.

Adaptive prescribed performance control for spacecraft attitude synchronization subject to external disturbances with guaranteed steady‐state performance

AbstractThis paper proposes an adaptive prescribed performance control approach for multiple spacecraft attitude synchronization with the undirected topology. A novel distributed observer is proposed to estimate the states of the leader spacecraft while the estimation value of attitude is still a unit quaternion. Then we design the adaptive prescribed performance control algorithm restrain the external disturbances and parameter uncertainty. By the unit quaternion estimation value of attitude, we make the extension from single spacecraft to multiple spacecraft. Compared to some previous results, the bounds of attitude tracking error and angular velocity error in the steady state can be adjusted by users.

A GNSS-aided DVL calibration method based on quaternion estimation for underwater vehicles

Doppler velocity logger (DVL) error parameters can significantly influence the navigation accuracy of DVL/strapdown inertial navigation system (SINS) integration for unmanned underwater vehicles (UUV). To improve the navigation accuracy of UUV, this study proposes a two-stage DVL calibration method aided by global navigation satellite system (GNSS) measurements. First, utilizing the velocity of GNSS/SINS integrated navigation, the scale factor error of DVL is calculated by the moduli of velocities in UUV body frame and DVL instrument frame. Then, using the measurements throughout the calibration process, the calibration problem of the installation angle is converted to a nonlinear constraint optimization problem by describing the angle as a unit quaternion. Moreover, an easy-to-implement quaternion estimation algorithm is chosen to solve the problem and obtain the optimal quaternion. Simulation and sea trial indicate that the proposed method can rapidly and accurately estimate the DVL error parameters in different scenarios, and the position accuracy of the DVL/SINS system is improved.

An Observation Model From Linear Interpolation for Quaternion-Based Attitude Estimation

This article aims to solve multiple problems associated with attitude estimation; the complexity of Kalman filter (KF) calculations in the attitude and heading reference system (AHRS), the interference sensitivity of magnetic, angular rate, and gravity (MARG) sensors, and the low accuracy of the factored quaternion algorithm (FQA). It presents a two-layer linear KF using MARG sensors to obtain attitude estimation in quaternions. First, data from a triaxial accelerometer and magnetometer is processed by a novel algorithm, which fuses the quaternion estimator (QUEST) algorithm and FQA by the linear interpolation (LERP) to obtain an observation model. Second, the process model in the two-layer KF was obtained by using LERP to fuse an optimum quaternion obtained from a gyroscope and FQA. The LERP can eliminate gyro bias drift, and integral error, and compensate for unexpected conditions, such as fast rotation and temporary strong magnetic disturbances. The proposed algorithm presents higher accuracy and lower computational load than QUEST or FQA used with a KF alone. The performance of the proposed algorithm is verified through simulation and experiments.

Gaussian process regression-based quaternion unscented Kalman robust filter for integrated SINS/GNSS

High-precision filtering estimation is one of the key techniques for strapdown inertial navigation system/global navigation satellite system (SINS/GNSS) integrated navigation system, and its estimation plays an important role in the performance evaluation of the navigation system. Traditional filter estimation methods usually assume that the measurement noise conforms to the Gaussian distribution, without considering the influence of the pollution introduced by the GNSS signal, which is susceptible to external interference. To address this problem, a high-precision filter estimation method using Gaussian process regression (GPR) is proposed to enhance the prediction and estimation capability of the unscented quaternion estimator (USQUE) to improve the navigation accuracy. Based on the advantage of the GPR machine learning function, the estimation performance of the sliding window for model training is measured. This method estimates the output of the observation information source through the measurement window and realizes the robust measurement update of the filter. The combination of GPR and the USQUE algorithm establishes a robust mechanism framework, which enhances the robustness and stability of traditional methods. The results of the trajectory simulation experiment and SINS/GNSS car-mounted tests indicate that the strategy has strong robustness and high estimation accuracy, which demonstrates the effectiveness of the proposed method.

Research on Algorithms for Multi-Vector Attitude Determination

Various attitude estimation methods are based on an optimization problem posed in 1965 by Grace Wahba, which is called Wahba’s problem in the field of attitude estimation. As a key for attitude determination, many different algorithms for minimizing Wahba’s loss function have been proposed in the past 60 years. Among the most representative are Quaternion Estimator (QUEST), Singular Value Decomposition (SVD), and Fast Optimal Attitude Matrix (FOAM), which are briefly introduced in this research paper, and new algorithms proposed in recent years such as Fast Linear Quaternion Attitude Estimator (FLAE), are also included. The calculation principle and derivation process are given in the article, and simulation under high noise conditions for algorithms mentioned has been completed. Finally, several practical engineering applications related to Wahba’s problem are introduced and the future development trend of attitude estimation is discussed.

A Fast Robust In-Motion Alignment Method for Laser Doppler Velocimeter-Aided Strapdown Inertial Navigation System

With ultra-high velocity measurement accuracy, laser Doppler velocimeter (LDV) is promising to replace the odometer to provide vehicle velocity information in the process of land integrated navigation. This paper investigates the fast in-motion initial alignment for LDV-aided strapdown inertial navigation system (SINS). Aiming at the problem that the installation misalignment angles and scale factor error of LDV will reduce the alignment accuracy and the alignment speed, a Robust Double Gain Square-Root Unscented Quaternion Estimator (RDGSR-USQUE) method with good effect at large misalignment angle is proposed. In RDGSR-USQUE, the influence of unknown parameter error is considered in the construction of process model and measurement model and a double gain structure is proposed to improve the convergence speed based on the usually neglected posterior measurement residual information. RDGSR-USQUE method improves the defects of the traditional Unscented Quaternion Estimator (USQUE) method, such as poor noise resistance, slow convergence speed under large misalignment angle and easy to lead to non-positive definite covariance matrix. This will help to estimate and compensate unknown parameter errors while estimating attitude, so as to improve the accuracy of process model and measurement model, and finally improve the accuracy of attitude estimation. The performance of the proposed scheme is verified by vehicle field test. The results show that the proposed method has higher alignment accuracy, faster convergence speed and stronger robustness than three other compared methods, and the estimated installation misalignment angle and scale factor of LDV are satisfactory in general cases.

Novel Attitude Estimation Of Strapdown Inertial Navigation Systems With Singular Value Decomposition Technique

Davenport’s q method & the Singular Value Decomposition (SVD) method are the two vigorous estimators that reduces Wahba’s loss function. In these, the q method is slightly quicker due to its computation of optimum quaternion as an eigenvector of a symmetric 4x4 matrix through the prevalent eigenvalue. The ESOQ and ESOQ2 (EStimators of the Optimal Quaternion) and the QUEST (QUaternion ESTimator) algorithms are less determined as the extreme eigenvalue’s distinguishing polynomial equation is solved by them. These estimators are apt to track the undulations of the sea with equivalent precision and accurateness. The SVD method is chosen and shown to be the most robust of all the hostile methods for the orientation of SDINS (Strap-Down Inertial Navigation Systems) using rate matching observations at sea in this paper. SVD is known most robust decomposition of all the decompositions of a matrix. SVD based attitude estimation being a batch technique would suffer from much less computational issues.

In-motion initial alignment method for a laser Doppler velocimeter-aided strapdown inertial navigation system based on an adaptive unscented quaternion H-infinite filter

With its advantages of high velocity measurement accuracy and fast dynamic response, the laser Doppler velocimeter (LDVs) is expected to replace the odometer (OD) in combination with a strapdown inertial navigation system (SINS) to give a higher-precision integrated navigation system. Since a LDV has higher velocity measurement accuracy and data update frequency than an OD and Doppler velocity log, a LDV is used for the first time in this paper to aid a SINS in in-motion alignment. Considering that some approximation is used in the alignment model, the uncertainty noise of the sensors during the motion process and the unknown noise parameters during the filter process, an adaptive unscented quaternion H-infinite estimator (AUSQUHE) is proposed. The proposed AUSQUHE method has high robustness since it combines the advantages of an unscented quaternion estimator and H-infinite filter. The adaptive threshold of the H-infinite filter and the adaptive measurement noise covariance matrix are introduced to make the filter adapt to the changing environment and accelerate the convergence of errors. The performance of the proposed method is verified by a vehicle field test with a normal LDV signal and a vehicle test with the LDV signal disturbed by noise. The results show that the proposed method has higher alignment accuracy, faster convergence speed and stronger robustness than the four other compared methods.

A Discrete Quaternion Particle Filter Based on Deterministic Sampling for IMU Attitude Estimation

A discrete quaternion filtering algorithm following high-density uncertainty matching-based deterministic sampling scheme is proposed for IMU attitude estimation. Existing quaternion filters rely on the specific distributions directly or the on-tangent-plane Gaussian distribution indirectly to model the underlying uncertainty. The key idea of this paper is to apply a Dirac mixture at few deterministic weighted samples for non-parametric interpreting the high-density uncertainty of unit quaternion manifold. In contrast to standard Monte Carlo techniques based on random sampling, the proposed approach is deterministic. Furthermore, the deterministic quaternion particles are obtained by projecting on-sphere uniform points to the unit quaternion manifold via an exponential map. The on-manifold quaternions are then exploited in a discrete Bayesian filtering procedure. We compare the new filter with the genetic algorithm embedded quaternion particle filter (GA-QPF) and the unscented quaternion estimator (USQUE) for accuracy and computation cost by using simulated and practical IMU data. The results illustrate that the proposed estimator possesses lower computational cost than the GA-QPF while providing a comparable accuracy, higher accuracy than the USQUE, as well as more effective against large initialization error.

A Novel Positioning Module and Fusion Algorithm for Unmanned Aerial Vehicle Monitoring

Aiming to solve the monitoring problem of unmanned aerial vehicles (UAVs), a novel positioning module is designed and implemented. The hardware design meets the prerequisites of miniaturization and low-cost. Then, a calibration procedure based on six-position calibration and ellipsoid fitting is executed to eliminate the bad influence on the system’s accuracy and stability caused by deterministic errors. Also, a stochastic error modeling procedure based on Allan variance is performed to identify stochastic errors’ model type and model parameters. Finally, data from inertial measurement units (IMUs), global navigation satellite system (GNSS), magnetometers, and a barometer are fused by an extended Kalman filter (EKF) based sensor fusion algorithm. This algorithm is applied to a UAV flight test and comparison result using calibrated data and uncalibrated data respectively are shown. Besides, the proposed EKF algorithm is compared with unscented Kalman filter (UKF) and unscented quaternion estimator (USQUE) to show its practicability in the application. The main contribution of this research is the design of the positioning module and the validation of the sensor fusion algorithm in this hardware. It is found that, under the constraint of low-cost sensors, the accuracy of positioning can achieve around 7.5m horizontally, and 2m vertically. The frequency of positioning increases to 50Hz after sensor fusion, a large improvement over the previous 1Hz GNSS positioning frequency.

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).

Interlaced Attitude Estimation for Spacecraft Using Vector Observations

This paper proposes an interlaced attitude estimation method for spacecraft using vector observations, which can simultaneously estimate the constant attitude at the very start and the attitude of the body frame relative to its initial state. The arbitrary initial attitude, described by constant attitude at the very start, is determined using quaternion estimator which requires no prior information. The multiplicative extended Kalman filter (EKF) is competent for estimating the attitude of the body frame relative to its initial state since the initial value of this attitude is exactly known. The simulation results show that the proposed algorithms could achieve better performance compared with the state-of-the-art algorithms even with extreme large initial errors. Meanwhile, the computational burden is also much less than that of the advanced nonlinear attitude estimators.

A quaternion-based indirect Gaussian particle filter for nonlinear attitude estimation.

A recursively indirect quaternion estimator based on the Gaussian particle filter (GPF) is proposed for nonlinear attitude estimation. The key idea is to estimate an on-tangent-plane Gaussian distribution in the GPF scheme for interpreting the uncertainty of the unit quaternion manifold. The unit quaternion is provided with a global nonsingular attitude description in the prediction step, and the three-dimensional attitude error is estimated in the update step. Based on the framework of the GPF, the proposed filter does not need resampling and regularization compared with the PF. The performance of the proposed filter is verified theoretically and evaluated by experiments. The results show that the proposed filter has a faster convergence speed, lower complexity, and lower computational cost than the existing quaternion PF under a comparable accuracy.

Error Analysis of Accelerometer- and Magnetometer-Based Stationary Alignment

This paper revisits the stationary attitude initialization problem, i.e., the stationary alignment, of Attitude and Heading Reference Systems (AHRSs). A detailed and comprehensive error analysis is proposed for four of the most representative accelerometer- and magnetometer-based stationary attitude determination methods, namely, the Three-Axis Attitude Determination (TRIAD), the QUaternion ESTimator (QUEST), the Factored Quaternion Algorithm (FQA), and the Arc-TANgent (ATAN). For the purpose of the error analysis, constant biases in the accelerometer and magnetometer measurements are considered (encompassing, hence, the effect of hard-iron magnetism), in addition to systematic errors in the local gravity and Earth magnetic field models (flux density magnitude, declination angle, and inclination angle). The contributions of this paper are novel closed-form formulae for the residual errors (normality, orthogonality, and alignment errors) developed in the computed Direction Cosine Matrices (DCM). As a consequence, analytical insight is provided into the problem, allowing us to properly compare the performance of the investigated alignment formulations (in terms of ultimate accuracy), as well as to remove some misleading conclusions reported in previous works. The adequacy of the proposed error analysis is validated through simulation and experimental results.

Estimation of Quaternion Motion for GPS-based Attitude Determination Using the Extended Kalman Filter

In this paper, the Global Positioning System (GPS) interferometer provides the preliminarily computed quaternions, which are then employed as the measurement of the extended Kalman filter (EKF) for the attitude determination system. The estimated quaternion elements from the EKF output with noticeably improved precision can be converted to the Euler angles for navigation applications. The aim of the study is twofold. Firstly, the GPS-based computed quaternion vector is utilized to avoid the singularity problem. Secondly, the quaternion estimator based on the EKF is adopted to improve the estimation accuracy. Determination of the unknown baseline vector between the antennas sits at the heart of GPS-based attitude determination. Although utilization of the carrier phase observables enables the relative positioning to achieve centimeter level accuracy, however, the quaternion elements derived from the GPS interferometer are inherently noisy. This is due to the fact that the baseline vectors estimated by the least-squares method are based on the raw double-differenced measurements. Construction of the transformation matrix is accessible according to the estimate of baseline vectors and thereafter the computed quaternion elements can be derived. Using the Euler angle method, the process becomes meaningless when the angles are at 90° where the singularity problem occurs. A good alternative is the quaternion approach, which possesses advantages over the equivalent Euler angle based transformation since they apply to all attitudes. Simulation results on the attitude estimation performance based on the proposed method will be presented and compared to the conventional method. The results presented in this paper elucidate the superiority of proposed algorithm.

Full-Order Solution to the Attitude Reset Problem for Kalman Filtering of Attitudes

Kalman filtering of states including attitudes poses a challenge due to the constraints of the rotation manifold. One of the standard approaches is to consider a deviation attitude, in the form of a reduced three-vector parametrization, to some nominal reference attitude that the Kalman filter tracks. After an update to the statistics of this deviation, a reset step is performed that adjusts the reference attitude. This reset step adjusts the statistics of the deviation, which has commonly been ignored in the literature. This paper presents an algorithm for the case when the deviation is represented as a rotation vector. The adjustment to the mean and covariance after the reset operation is presented, assessed using Monte Carlo sampling, and compared to other approaches used in the literature. The final result may be easily implemented and is computationally inexpensive. Connections and comparisons to the multiplicative extended Kalman filter, the unscented quaternion estimator, and Lie-group Kalman filters are made, with simulations performed on a rigid-body example.

A Fast SINS Self-Alignment Method Under Geographic Latitude Uncertainty

To achieve high alignment accuracy of strapdown inertial navigation system (SINS) with the limited time and the unknown geographic latitude constraints, a novel fast self-alignment method with the real-time estimated latitude is proposed. In the stage of the coarse alignment, the latitude self-estimation and the coarse alignment are performed at the same time. The geometric relationship between the transition angle and the latitude is established, where the angle can be determined by the gravity acceleration vector at different times. Based on the reconstructed observation vector and the set dynamic window, a real-time latitude self-estimation algorithm is proposed for a swaying case, and the attitude determination procedure is accomplished by the improved filter quaternion estimator (filter-QUEST) algorithm, where the vectors are constructed based on a generalized integration regulation. With the deduced backtracking alignment error model, the forward fine alignment is performed with the statistical latitude estimate and the saved data during the process of the coarse alignment, thus will obviously speed up the overall alignment time. The results of the mathematical simulation and the turntable experiment are performed to validate the estimation and alignment performance of the proposed approach.

Observability analysis and design of two nested filters for the satellite attitude estimation with magnetometer‐only

This study aims to propose a scenario for determining the attitude of a nano-satellite with only a magnetometer in the case of temporal failure or faults in other sensors of attitude determination subsystem. For the sake of this matter, two nested filters are employed. The first filter estimates the magnetic field derivative by an extended Kalman filter (EKF). This information is applied along with the magnetometer outputs to obtain the attitude in the second one. A proposed algorithm based on the Square-Root UnScented QUaternion Estimator has been carried out to guarantee the positiveness of the covariance matrices considering the restriction about the unit norm of the quaternion. Furthermore, the non-linear observability of these filters is investigated analytically accompanied by condition number studies to show the functionality of these filters for magnetometer-only attitude determination. Finally, Monte Carlo simulation results indicate better performance of this approach in comparison with the traditional method of multiplicative EKF, particularly with large initialisation errors.

Design and implementation of a star-tracker for LEO satellite

An important task of the attitude determination and control subsystem (ADCS) is the determination of attitude of the satellite in reference to the Earth or any other planetary and correct any deviation from the expected attitude. In order to determine the attitude of the satellite, sensors are needed to sense the orientation of the satellite. The star-tracker is able to determine the attitude from the view field of the star set without any prior knowledge of sensor information. In this paper, the design, simulation and implementation of a star-tracker is presented for LEO satellites. For capturing image data, the MT9P031 CMOS image sensor from Aptina Imaging was used. The star tracker uses a Texas Instrument OMAP3530 using an ARM Cortex A8. The processor supports powerful graphic interfaces resulting in reduction of the size and consumption power of the star-tracker. For storage and processing, the star tracker used NAND flash memory. The communication interface to the satellite is done through CAN Bus. One of the key features of this sensor is processing various algorithms for image processing, positioning, and tracking according to mission requirements. Quaternion estimator (QUEST) algorithm was utilized for positioning. The tracking algorithm utilizes information from the previous frame of the star-tracking. This algorithm is a high-speed, low cost, and low memory implementation. In the field tests, the identification success rate of this system was 96.81 %.