Multiplicative Extended Kalman Filter Research Articles

A framework for developing an attitude determination and control system simulator for Cubesats: Processor-in-loop testing approach

The outbreak of miniaturized satellites is increasingly demanding well-performing attitude determination and control systems (ADCS). Testing an ADCS directly using hardware is extremely costly and risky, whereas testing it using only computer simulation does not yield realistic enough results. Therefore, a detailed framework for developing an ADCS simulator with processor-in-the-loop testing for low-earth orbit Cubesats is established in this paper. In particular, the study focuses on the implementation of the algorithms on a STM32 with ARM Cortex CPU for efficient testing and prototyping. This is done by formulating a quaternion-based satellite dynamic model equiped with a three-axis magnetorquers and a set of pyramidal cluster of reaction wheels for high pointing capabilities. Prominently used sensors are modeled so that the attitude can be estimated using a multiplicative extended Kalman filter. For attitude control, several modes are developed based on different mission stages, and a simplified less burdensome control switching strategy is proposed to maintain stability with minimal computation and energy resources. To test the applicability of the simulator, an ADCS is designed to meet the requirements of the recent low-atmosphere measurement missions. The results are analyzed in accordance with the ESA ECSS standards, and the assessment shows that the specifications can be met with a safe margin, leaving room for unmodeled errors to be taken into account in later stages of the design. Additionally, the proposed control strategy gives various possibilities for missions with different types of objectives, furthermore the framework's flexibility provides easy modification of any part for the user's specific needs.

A meticulous covariance adaptive Kalman filter for satellite attitude estimation

Aiming at the problems of model errors, non-Gaussian noise and measurement anomaly in the spacecraft attitude estimation system, this article proposes an improved adaptive filtering method based on covariance matching, which solves the problems of simultaneous dynamics model error and measurement model error in the attitude estimation system, and at the same time, effectively reduces the effects of non-Gaussian noise and large outlier situations occurring in the vector measurement sensor. Firstly, an adaptive filtering algorithm based on the innovation sequence estimation covariance is investigated under the framework of multiplicative extended Kalman filter (MEKF), which is used to correct process noise covariance, then the Sage–Husa adaptive Kalman filtering (SHAKF) method is combined to correct the measurement noise covariance, and finally the meticulous covariance adaptive multiplicative extended Kalman filter is designed. The proposed algorithm uses both innovation and SHAKF methods to correct the two covariance matrices simultaneously. Several attitude estimation simulation scenarios are set up to simulate the proposed algorithm in the presence of model errors, non-Gaussian noise, and large outlier. The simulation results demonstrate that the proposed algorithm outperforms the conventional algorithms in terms of estimation accuracy and robustness.

Autonomous relative navigation using stereo-vision in a dual-agent system for proximity operations in a lunar orbit

This paper presents a novel autonomous relative navigation architecture for inspector spacecraft in proximity operations to a known structure in a lunar orbit. Autonomous on-orbit servicing and assembly of large space structures are at the forefront of research for advancing space exploration beyond low-earth orbit. Communication delays with ground stations and limited computational power located on the inspector spacecraft make current relative navigation methods unfeasible and in turn, make it difficult to maintain an accurate estimate of the relative position and orientation of the inspector spacecraft with respect to the structure. The proposed method utilizes the camera measurements from two identical inspector spacecraft, each equipped with a single camera, to estimate the relative position and orientation of the structure. This shared measurement strategy overcomes the deficiencies in monocular vision-based relative navigation techniques, such as the range ambiguities, as is proven in the simulated results of this paper. The proposed stereo-vision system in this work allows for a deterministic solution of the relative position and orientation from the measurements of each inspector spacecraft to the structure. The stereo-vision system was tested with a Multiplicative Extended Kalman Filter (MEKF) for each spacecraft to estimate the relative states of the structure while maintaining a fixed relative position to the structure. The performance of the proposed stereo-vision system was evaluated in terms of accuracy and reliability of the estimated relative position and orientation of two inspector spacecraft to the structure over 100 Monte Carlo simulations for three case studies. The results showed that the stereo-vision system significantly outperforms the monocular vision system even when fewer points on the structure are visible, as is the case for proximity operations. Additionally, the stereo-vision system is consistently more accurate for the relative velocity, attitude, and angular velocity states regardless of how many points are visible. The proposed stereo-vision relative navigation system has the potential to be applied autonomously in proximity operations of spacecraft for various space missions, such as rendezvous and docking, inspection, and on-orbit servicing.

Phased Array Radio Navigation System on UAVs: In-Flight Calibration

Global Navigation Satellite Systems (GNSS) has been the primary positioning solution for Unmanned Aerial Vehicles (UAVs) due to their worldwide coverage, high precision and lightweight receivers. However, GNSS is prone to electromagnetic interference and malicious assaults, including jamming or spoofing because of its low signal-to-noise ratio (SNR). To ensure the continuity and protection of UAV operations, using redundant navigation systems is essential. In recent years, the phased array radio system (PARS) has established itself as a local navigation solution. PARS is robust towards malicious assaults because of an much higher SNR than GNSS regarding directed and encrypted transmission. An essential factor of PARS is that the orientation of the radio antenna at a ground station needs to be precisely determined to obtain the correct positioning of UAVs. This paper presents a method for extending a previously proposed calibration algorithm to estimate the ground antenna orientation with an inertial navigation system (INS) aided by redundant positioning sensors (GNSS, PARS or barometer) using a multiplicative extended Kalman filter (MEKF) so that the calibration can be activated during flights whenever GNSS is available. In other words, the proposed navigation system is essentially an aided-INS which switches between two modes depending on the availability of GNSS: calibration and GNSS aiding mode when GNSS is available (Mode 1) and PARS and barometer aiding mode when GNSS is unavailable (Mode 2). Considering that the navigation system needs to include the effect of Earth’s curvature for a long-distance flight, PARS horizontal measurement and the barometer measurement were treated independently, and the navigation equations were propagated in Earth Centred Earth Fixed (ECEF) frame. The independent treatment of barometer measurement, and the propagation in ECEF frame were also beneficial when using multiple ground antennas to have a common reference point and reference frame. The proposed method was validated using data (Inertial Measurement Unit (IMU), GNSS, PARS, Pixhawk autopilot (including barometer) measurements) collected during a field test. In the validation, GNSS was made available at the middle of the flight and the calibration mode was activated for 200s. The proposed navigation system successfully estimated the precise orientation of multiple ground antennas and the navigation solutions were verified using GNSS and Pixhawk autopilot solutions as ground truth.

Extended Kalman filters for close-range navigation to noncooperative targets

This work presents a set of dynamic filters for estimating the relative roto-translational state and the main parameters of a noncooperative target from an observing chaser satellite during close proximity operations. The proposed different options address a wide range of design possibilities for the architecture of the relative navigation system. All filters are derived from a common, general, core shaped as a dynamic multiplicative extended Kalman filter using dual quaternions. This allows exploiting the advantages of handling the pose (i.e., attitude and position) in a multiplicative fashion, while improving the accuracy in the estimation of the angular and linear relative velocities, as well as enabling the estimation of some meaningful parameters of the target spacecraft (e.g., the ratios of the moments of inertia, position and orientation of the principal axes frame). Moreover, by adopting relative kinematics and dynamics equations in dual quaternions, the inherent coupling of the six degrees-of-freedom motion is addressed with no approximations. All filters take as observations only the noisy pose measurements from an electro-optical device. For each proposed formulation, numerical simulations are carried out to show the behavior of the filter within a scenario representative of close-range target inspection at conclusion of the mid-range rendezvous.

A disturbance rejection adaptive filtering approach for human motion tracking *

This article studies the human motion tracking problem using the disturbance rejection adaptive filter with an inertial sensor. Due to the complexity of human motion, conventional inertial methods easily suffer from motion accelerations (MAs) and ferromagnetic disturbances (FDs). Here, a disturbance rejection adaptive filter is presented to segregate the exterior disturbances from the inertial sensor observations before attitude estimation. This method estimates the MA and FD by Kalman filters. According to disturbance intensity, Sage–Husa adaptive strategies based on fuzzy rules are designed to adjust noise covariance. Moreover, gravity and geomagnetic field estimation are applied as multiplicative extended Kalman filter observations to solve external disturbances problems. Finally, the effectiveness and superiority of the proposed method are verified by an example of human motion tracking.

Error-Covariance Reset in the Multiplicative Extended Kalman Filter for Attitude Estimation

This paper presents a study of the reset step in the multiplicative extended Kalman filter (MEKF). This filter is widely used for spacecraft attitude estimation, which typically involves estimating the attitude and gyro drift in real time using external sensors such as star trackers. The basic idea of the MEKF is to use the quaternion or direction-cosine matrix as the “global” attitude parameterization and a three-component state vector for the “local” parameterization of attitude errors. The true attitude is expressed as the product of the error attitude and the estimate rather than as the sum of the error and the estimate. The reset operation moves the local error to the global variable. This reset does not add new information, but it changes the reference frame for the attitude error covariance. This results in an error-covariance reset that is very different from the measurement update of the error covariance in the MEKF. The effects of using an error-covariance reset in the MEKF are analyzed in this work. The results from this work can be applied to any application involving attitude estimation as part of its process, such as inertial navigation.

Doppler Light Detection and Ranging-Aided Inertial Navigation and Trajectory Recovery

A sensor fusion approach to terrain relative navigation is proposed in this paper. Sensor data from Global Position System (GPS), Inertial Navigation System (INS), and Doppler light detection and ranging (LIDAR) provide a relative state estimate through a novel multiplicative extended Kalman filter. The measurement model of the Doppler LIDAR is derived from first principals and employed by the filter to update the estimates propagated by the GPS/INS data stream. The line-of-sight Doppler velocity measurement provides direct feedback to the velocity states and improves sensitivity to these state estimates. A novel Rauch–Tung–Striebel smoother framework is derived that is compatible with states exhibiting multiplicative error that enables seamless postprocessing of state estimates filtered with a multiplicative extended Kalman filter. This new filter and smoother framework is tested with suborbital flight data from Blue Origin’s New Sheppard ascent and landing missions.

Satellite Attitude Determination Using ADS-B Receiver and MEMS Gyro

Automatic dependent surveillance-broadcast (ADS-B) is a very important communication and surveillance technology in air traffic control (ATC). In the future, more and more satellites will carry out ADS-B technology to perform a global coverage. In order to make full use of the resources in the satellite, this paper proposes a solution for satellite three-axis attitude determination using the ADS-B receiver. The principle of ADS-B-based attitude determination is presented first. On this basis, ADS-B-based methods are employed to solve the problem. To achieve a higher attitude determination precision, gyro is combined with the ADS-B receiver using a multiplicative extended Kalman filter (MEKF). Finally, a simulation is carried out and the result is presented.

Velocimeter Light-Detection-and-Ranging–Informed Terrain Relative Navigation

Velocimeter light-detection-and-ranging (LIDAR)–informed batch and sequential estimation approaches for terrain relative navigation applications are developed in this paper. State-of-the-art velocimeter LIDAR sensors are capable of delivering simultaneous three-dimensional relative position, relative Doppler velocity, and reflectivity measurements for every point in the field of view. The proposed batch estimation algorithm yields accurate and statistically consistent vehicle velocity estimates in unknown (uncooperative) environments. This work presents novel pointwise relative position and Doppler velocity measurement models that can be utilized in the presence of known (cooperative) environments. Rate estimates obtained by the batch estimator are fused with the pointwise measurement models and classical inertial measurement unit formulations in the form of a multiplicative extended Kalman filter. This Doppler LIDAR-based sensor fusion approach yields informative terrain relative navigation solutions suitable for both cooperative and uncooperative environments. Results from extensive emulation robotics testing performed at NASA Johnson Space Center and Texas A&M’s Land, Air, and Space Robotics laboratory are presented to validate the proposed methodologies.

Incorporating delayed star tracker measurements in Gokturk-2 satellite attitude filter

Star trackers (STRs) are widely used for fine attitude estimation. Due to the processing time for tasks such as star identification, tracking, and attitude solution, a STR can provide the attitude quaternion measurements with a time delay, which may be as high as a few 100 ms. Different methods are available for incorporating such delayed measurements in the attitude filter, such as the most straightforward method of propagating the measurements to the actual filter time. This study evaluates different methods to incorporate the delayed STR measurements in the Gokturk-2 satellite attitude filter, which is structured as a Multiplicative Extended Kalman Filter (MEKF). Methods are compared in terms of accuracy, computational load, and impact on the filter's optimality. They are evaluated using the simulated data for an Earth observation spacecraft in Sun-synchronous orbit and the actual STR data for the operational Gokturk-2 satellite.

Spacecraft attitude estimation under attitude tracking maneuver during close-proximity operations

This paper presents a real-time relative three-axis attitude estimation using a camera and two gyros under momentum changes during close-proximity operations. The camera sensor provides quaternion measurements that relate the body frame of the deputy spacecraft with respect to the body frame of the chief spacecraft. The quaternion measurements are coupled with gyro measurements and attitude dynamics model in a multiplicative extended Kalman filter to determine relative attitude and gyro biases under momentum changes. The relative quaternion kinematics is augmented with spacecraft relative motion dynamics to represent the filter process dynamics. The quaternion measurements from a camera sensor and inertial measurement units (IMU) are utilized to the filter measurement model. The fault-tolerant attitude controller actuated by four reaction wheels, which has no unwinding problem is used for attitude tracking maneuvers during close-proximity operations in the presence of external disturbances, uncertain inertia parameter and actuator faults. This controller is combined with the extended Kalman filter to filter noisy measurements and to estimate gyro biases, which leads to a full-state feedback control system. Numerical simulations are performed to verify the effectiveness of the attitude estimation and control systems of the spacecraft with four reaction wheels during close-proximity operations of spacecraft in low-Earth orbit.

Multiplicative extended Kalman filter ignoring initial conditions for attitude estimation using vector observations

AbstractIn this paper, the well-known multiplicative extended Kalman filter (MEKF) is re-investigated for attitude estimation using vector observations. From the Lie group theory, it is shown that the attitude estimation model is group-affine and its error state model should be trajectory-independent. Moreover, with such a trajectory-independent error state model, the linear Kalman filter is still effective for large initialisation errors. However, the measurement model of the traditional MEKF is dependent on the attitude prediction, which is therefore trajectory-dependent. This is also the main reason why the performance of traditional MEKF is degraded for large initialisation errors. Through substitution of the attitude prediction related term with vector observations in the body frame, a trajectory-independent measurement model is derived for MEKF. Meanwhile, MEKFs with reference attitude error definition and with global state formulating on special Euclidean group have also been studied, with the main focus on derivation of the trajectory-independent measurement models. Extensive Monte Carlo simulations of spacecraft attitude estimation implementations demonstrate that the performance of MEKFs can be much improved with trajectory-independent measurement models.

Uniting Attitude Estimation With Global Asymptotic Stability

This paper approaches the problem of attitude estimation using biased gyro and body-referenced measurements of inertial vectors. First, a new observer for attitude estimation is proposed through introducing a family of synergistic potential functions with explicit expression of synergistic gap. Furthermore, it is convenient to adjust the design by adding additional state-dependent gains without adapting the switching rules. The proposed observer exhibits faster convergence for large attitude estimation errors than similar work in literature. Second, to achieve better steady-state performance, we unite this global observer with a traditional local observer, i.e., multiplicative extended Kalman filter. The new uniting observer inherits the desired transient and steady-state performance from the two sub-observers respectively. The performance of proposed methods has been evaluated via simulation and experiments.

Robust Attitude Estimation Using Magnetic and Inertial Sensors

Lightweight and low-cost magnetic and inertial sensors are commonly used for attitude estimation in a wide range of applications, from motion tracking to autonomous navigation. However, the inherent and external sensor errors and the accelerations due to the actual motion of the platform highly affect the estimation accuracy. This study proposes a robust attitude estimation algorithm that compensates the sensor errors and the external accelerations at the algorithm level. The attitude estimation filter, which is structured as a Multiplicative Extended Kalman Filter (MEKF), is modified to compensate for both the long-term and short-term measurement uncertainties. Modification is applied mainly by estimating and compensating the bias and external accelerations for long-term uncertainties and tuning the measurement noise covariance matrix for short-term uncertainties. Simulations test the proposed algorithm, and the results are compared with benchmark algorithms.

Phased Array Radio Navigation System on UAVs: Real-Time Implementation of In-flight Calibration⋆

Unmanned aerial vehicles (UAV) require an alternative positioning system to the global navigation satellite systems (GNSS) for safety-critical operations since GNSS is prone to unintentional or malicious electromagnetic interference. As a possible redundant positioning solution independent of GNSS, the phased array radio system (PARS) has proved its potential as a local navigation system for UAVs. Previous work suggested an automatic calibration algorithm to estimate the precise PARS ground antenna orientation to improve PARS positioning accuracy. This algorithm was then integrated with PARS- and barometer-aided inertial navigation system (INS) based on multiplicative extended Kalman filter (MEKF) to enable in-flight calibration when GNSS is available. This paper presents the implementation of the previously suggested aided-INS with in-flight calibration mode for real-time operation.

Relative Stereovision-Based Navigation for Noncooperative Spacecraft via Feature Extraction

Spacecraft proximity operations play an important role in guaranteeing the capture success of noncooperative targets. This article technically presents a relative stereovision-based navigation scheme that ensures accurate relative pose measurement as well as practical motion estimation for proximity operations. Specifically, features parameters of the line and circle on the noncooperative target are first obtained by the different image processing extraction algorithms. Then, the center position and normal of the circle are derived by orthogonal transformation from the circle feature parameters, and two features descriptors are defined by the redundant parameters in the stereovision to evaluate error levels of the extraction algorithms. In particular, the unavailable roll angle around the circle normal is recovered by incorporating the line and the descriptors by utilizing the least squares method, wherein the relative attitude is proved less affected by the features extraction errors, and thus, the measurement accuracy is theoretically improved. Furthermore, considering the Mahalanobis distances of the attitude and position, two outlier detectors are coupled with the 3 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> rule in a multiplicative extended Kalman filter to achieve outlier rejection and motion estimation, respectively. Finally, numerical simulation and physical experiment are given to validate the effectiveness of the proposed approach.

A model-free method for attitude estimation and inertial parameter identification of a noncooperative target

Determining the attitude and inertial parameters of a noncooperative target is essential in an on-orbit servicing mission. Various methods based on machine vision have been proposed, but most of them require the 3D model of the target. This paper proposes a model-free method through sequentially registering point clouds captured by a depth camera. Our main contributions are the avoidance of the ambiguity in registration, and the combination of the multiplicative extended Kalman filter and the pose graph optimization to reduce the effect of measurement noise and drift error. A hardware experiment was performed to generate the sequence of point clouds of a three-axis free-floating target and validate our method. The result shows that the proposed method outperforms existing methods and effectively identifies the inertial parameters, including the normalized principal moments of inertia and the orientation of principal axes.

HMM Based Adaptive Kalman Filter for Orientation Estimation

An adaptive orientation estimation method based on hidden Markov model (HMM) using multiple Magnetic Angular Rate Gravity (MARG) sensor is proposed to address the problem of linear acceleration and ferromagnetic disturbance. Two parallel Kalman filters containing first-order Gauss-Markov disturbance models are implemented to estimate the disturbances before estimating the orientation. It is important that the noise covariance of the disturbance model should match the magnitude of the disturbance to ensure the estimation performance; otherwise, the disturbance estimation accuracy will reduce significantly. Therefore, an adaptive Kalman filter based on HMM is designed to adjust the disturbance noise covariance. After the disturbances are separated, the estimated gravity acceleration and geomagnetic field as well as the gyroscope measurement are taken as the inputs, and the orientation estimation procedure is established by using the multiplicative extended Kalman filter (MEKF). Experiments show that the proposed method achieves a higher level of accuracy in comparison with other orientation estimation methods under various conditions.

High-precision attitude determination using spaceborne gravity gradiometer and gyroscope

This study investigates the application of a spaceborne gravity gradiometer to spacecraft attitude determination. Considering that the gradiometer raw measurements are influenced by angular rotation, a gyroscope is integrated to provide angular velocity information. To deal with unknown biases in both gradiometer and gyroscope measurements, a multiplicative extended Kalman filter (MEKF) is developed to estimate the attitude states and the instrument biases simultaneously. In addition, the cross-correlation between gyroscope measurements and gyroscope-compensated gradiometer measurements is analyzed. The MEKF algorithm is modified accordingly to cope with the correlation problem. Simulation scenarios with two different attitude modes are built to test the algorithm. With a gradiometer of 0.1 E (1 E = 10−9 s−2) accuracy and a navigation-grade gyroscope, attitude determination accuracy of several arcsecs is obtained. Real data from the GOCE (gravity field and steady-state ocean circulation explorer) spacecraft are also used to test the algorithm. Yaw, pitch, and roll angle errors of 2.8 arcsec, 20.9 arcsec, and 11.3 arcsec are achieved, respectively. The method can directly estimate a spacecraft's attitude relative to the Earth-centered Earth-fixed (ECEF) frame and is promising for high-precision spacecraft attitude determination in Earth-oriented missions.