Standard Extended Kalman Filter Research Articles

Combined Kalman and sliding innovation filtering: An adaptive estimation strategy

This paper proposes a new adaptive estimation strategy for a nonlinear system with modeling uncertainties. The extended Kalman filter (EKF) and unscented Kalman filter (UKF) are optimal estimators which have been used extensively for state estimation in literature and industry. While the EKF uses a first order Taylor series expansion to approximate nonlinearities, the UKF uses sigma points from the projected probability distribution of states. The sliding innovation filter (SIF) is a suboptimal, yet robust estimation strategy which has recently been proposed. For nonlinear systems, the extended SIF (ESIF) is formulated by using a first order Taylor series expansion like the EKF. This work proposes a novel adaptive estimation strategy which combines and balances the optimality of the EKF and UKF with the robustness of the ESIF. These new methods are referred to as the EKF-ESIF and UKF-ESIF, respectively. A time-varying sliding boundary layer is used as a means of detecting the presence of faults or uncertainties and as a criterion for switching between the EKF or UKF and the ESIF. In normal operating conditions the algorithm computes estimates using an optimal KF-based gain, and an SIF-based gain when a fault is detected. The system examined in this study consists of a magnetorheological (MR) damper with a constant current. Faults or uncertainties are introduced as unwanted behavior in the power supply in the form of undercurrent and overcurrent. The robustness of the EKF-ESIF and UKF-ESIF was validated for force estimation exerted by the MR damper and the results were compared with the standard EKF and UKF.

A Static Reduced-Order Multiple-Model Adaptive Estimator for Noise Identification

A reduced-order multiple-model (MM) estimator for noise identification in dynamic stochastic systems is developed. The unknown noise statistics are assumed to be static. While a standard static MM estimator does not grow exponentially over time, its computational complexity grows exponentially with the number of modes. The proposed algorithm reduces the computational complexity of MM estimation from exponential to polynomial by constructing a significantly smaller set of mode models which are updated every time step. It is assumed that the constructed mode models do not change significantly between time steps, which in turn holds if the smoothness of the mode probabilities is guaranteed. It is shown that in the case where there is “enough” statistical distinction between the noise modes, the proposed reduced-order MM estimator converges to the standard MM estimator. The proposed reduced-order MM estimator is evaluated using Monte Carlo simulations, showing that it performs nearly similar to the standard MM estimator with a fraction of its complexity. The numerical example shows less than a 2% increase in the root mean-squared errors (RMSEs) of the reduced-order MM estimator from the standard one, while the reduction in the number of filters in the reduced-order MM estimator is 300%. To further validate the proposed filter, experimental results are presented of an unmanned aerial vehicle (UAV) navigating with terrestrial signals of opportunity. Opportunistic navigation serves a relevant application for MM-based estimation, as system parameters, namely the statistics of the clock error dynamics of opportunistic sources, are unknown and must be adaptively estimated. The experimental results show a UAV navigating for more than 5 minutes over a trajectory of more than 3 km, achieving a final position error of 6.21 m obtained using the standard MM estimator versus a final position error of 6.25 m obtained using the proposed reduced-order MM estimator. A standard extended Kalman filter (EKF) was implemented for comparative analysis, showing a final error of 40.03 m. In the experiments, the reduced-order MM estimator was implemented with 16 filters, while the standard MM was implemented with 256 filters.

On Quaternion Modeling and Kalman Filtering from Vector Observations

This paper presents novel results on quaternion state-space modeling and estimation from rate gyro measurements and vector observations. The spectral properties of the Lyapunov differential equation for the quaternion second-order moment are investigated and expressions for its solution are developed. The properties of a four-dimensional rank-degenerate measurement model of the quaternion are analyzed. Four different approaches are suggested in order to reduce the dimension of the measurement equation, hinging on information optimization or on the spectral properties of the measurement matrix. While the measurement dimension’s reduction saves computations, it does not impair the exact statistical treatment of state-dependent process and measurement noises. As a result, the proposed filters are optimal and statistically consistent without tuning. The performance of the maximum information rate quaternion filter is verified via extensive Monte Carlo simulations. The test cases of spinning and rapidly tumbling rotations of a small satellite on a low Earth orbit are simulated. The filters process measurements of the sun vector, of the Earth magnetic field, and of rate gyros. The novel filters converge where a standard extended Kalman filter fails to do so under similar conditions.

A Descriptor System Approach for the Nonlinear State Estimation of Li-Ion Battery Series/Parallel Arrangements

The nonlinear state estimation for a series/parallel arrangement of lithium-ion (Li-ion) battery cells is addressed in this article assuming limited information. First, a unified model for series/parallel arrangements in terms of a nonlinear descriptor model is proposed in order to account for Kirchhoff’s laws characterizing the interconnection. It relies on a simplified electrochemical model of the individual cells, the so-called equivalent-hydraulic model. Then, assuming that the system and output equation nonlinearities locally satisfy Lipschitz-like constraints, a linear matrix inequality (LMI)-based technique is proposed for designing a nonlinear state observer, which estimates the state of each individual cell (such as the state-of-charge and inner temperature) as well as the algebraic variables (currents or voltages for respectively parallel or series configurations). The proposed design conditions also provide an estimate of the region of guaranteed convergence while ensuring a peak-to-peak performance with respect to exogenous inputs and model uncertainties. The effectiveness of the approach is demonstrated on a detailed battery electrochemical simulator (based on the Doyle–Fuller–Newman model) for a series arrangement of two cells. Comparisons with a standard extended Kalman filter demonstrate the superior performance of the proposed approach.

Interacting Multiple Model Filter with a Maximum Correntropy Criterion for GPS Navigation Processing

In order to deal with the uncertainty of measurement noise, particularly for outlier types of multipath interference and non-line of sight (NLOS) reception, this paper proposes a novel method for processing the navigation states of the Global Positioning System (GPS) that combines the maximum correntropy criterion (MCC) and the interacting multiple model (IMM), with an extended Kalman Filter (EKF). Multipath mitigation is essential for increased positioning accuracy since multipath interference is one of the primary sources of errors. Nonlinear filtering with IMM configuration uses filter structural adaptation. In processing time-varying satellite signal standards for GPS navigation, it offers an alternative for creating the adaptive filter. A collection of switching models built on a method of multiple model estimation can be used to characterize the uncertainty of the noise. Even though most noise in real life is non-Gaussian, time-varying, and of fluctuating strength, the standard EKF operates effectively when the noise is Gaussian. The performance of EKF will drastically decline if the signals appear non-Gaussian. The underlying system disrupted by heavy-tailed, non-Gaussian impulsive sounds could be better since the EKF employs second-order statistical information. The MCC is a method for comparing two random variables based on higher-order signal statistics. The maximum correntropy-extended Kalman filter (MCEKF), which uses the MCC rather than the minimal mean square error (MMSE) as the optimization criterion, is used to enhance performance in non-Gaussian situations. Finally, a performance evaluation will be conducted to compare the effectiveness of the suggested strategy in improving positioning to alternative system designs.

Advanced ensemble modeling method for space object state prediction accounting for uncertainty in atmospheric density

For objects in the low Earth orbit region, uncertainty in atmospheric density estimation is an important source of orbit prediction error, which is critical for space traffic management activities such as the satellite conjunction analysis. This paper investigates the evolution of orbit error distribution in the presence of atmospheric density uncertainties, which are modeled using probabilistic machine learning techniques. The recently proposed “HASDM-ML,” “CHAMP-ML,” and “MSIS-UQ” machine learning models for density estimation (Licata and Mehta, 2022b; Licata et al., 2022b) are used in this work. The investigation is convoluted because of the spatial and temporal correlation of the atmospheric density values. We develop several Monte Carlo methods, each capturing a different spatiotemporal density correlation, to study the effects of density uncertainty on orbit uncertainty propagation. However, Monte Carlo analysis is computationally expensive, so a faster method based on the Kalman filtering technique for orbit uncertainty propagation is also explored. It is difficult to translate the uncertainty in atmospheric density to the uncertainty in orbital states under a standard extended Kalman filter or unscented Kalman filter framework. This work uses the so-called “consider covariance sigma point (CCSP)” filter that can account for the density uncertainties during orbit propagation. As a test-bed for validation purposes, a comparison between CCSP and Monte Carlo methods of orbit uncertainty propagation is carried out. Finally, using the HASDM-ML, CHAMP-ML, and MSIS-UQ density models, we propose an ensemble approach for orbit uncertainty quantification for four different space weather conditions.

On Performance Evaluation of Inertial Navigation Systems: The Case of Stochastic Calibration

In this work we address the problem of rigorously evaluating the performances of an inertial navigation system during its design phase in presence of multiple alternative choices. We introduce a framework based on Monte-Carlo simulations in which a standard extended Kalman filter is coupled with realistic and user-configurable noise generation mechanisms to recover a reference trajectory from noisy measurements. The evaluation of several statistical metrics of the solution, aggregated over hundreds of simulated realizations, provides reasonable estimates of the expected performances of the system in real-world conditions. This framework allows the user to make a choice between alternative setups. To show the generality of our approach, we consider an example application to the problem of stochastic calibration. Two competing stochastic modeling techniques, namely, the widely popular Allan variance linear regression, and the emerging generalised method of wavelet moments, are rigorously compared in terms of the framework’s defined metrics and in multiple scenarios. We find that the latter provides substantial advantages for certain classes of inertial sensors. Our framework allows to consider a wide range of problems related to the quantification of navigation system performances, such as the robustness of integrated navigation systems (such as INS/GNSS) with respect to outliers or other modeling imperfections. While real world experiments are essential to assess to performance of new methods, they tend to be costly and are typically unable to lead to a sufficient number of replicates to provide suitable estimates of, for example, the correctness of the estimated uncertainty. Therefore, our method can contribute in bridging the gap between these experiments and pure statistical consideration as usually found in the stochastic calibration literature.

An Expectation-Maximization-Based Estimation Algorithm for AOA Target Tracking With Non-Gaussian Measurement Noises

This paper considers angle-of-arrival (AOA) target tracking in the two-dimensional plane with non-Gaussian noise. Practical situations arise where sensor measurement noise is non-Gaussian and the standard extended Kalman filter (EKF) based tracker may not be robust, resulting in large tracking error or even lack of convergence. This motivated development of a new tracker, designed for white non-Gaussian noise cases. The noise is decomposed into a sum of Gaussian components using expectation maximization (EM), and the target state is estimated by adapting an extended Kalman filter (EKF). Adaptations are necessary to combine the EM with the iterative EKF. From the analysis of the estimation error, a closed-form expression was derived revealing the presence of a bias in the estimate. On investigation, a sensor path optimization scheme is found that can eliminate bias. Furthermore, a compensation algorithm is presented, which reduces bias for the case of static sensors. The new tracker is suitable for real-time implementation and simulation results demonstrate significant improvements for the target tracking with non-Gaussian noise.

State of Charge Estimation of Lithium-Ion Batteries Based on an Adaptive Iterative Extended Kalman Filter for AUVs

As a power source for autonomous underwater vehicles (AUVs), lithium-ion batteries play an important role in ensuring AUVs' electric power propulsion performance. An accurate state of charge (SOC) estimation method is the key to achieving energy optimization for lithium-ion batteries. Due to the complicated ocean environments, traditional filtering methods cannot effectively estimate the SOC of lithium-ion batteries in an AUV. Based on the standard extended Kalman filter (EKF), an adaptive iterative extended Kalman filter (AIEKF) method for the SOC in an AUV is proposed to address the traditional filter's problems, such as low accuracy and large errors. In this method, the adaptive update is introduced to deal with the uncertain noise from the lithium-ion battery. The iteration is used to improve the convergence speed and to reduce the computational burden. Compared with the EKF, iterative extended Kalman filter (IEKF) and adaptive extended Kalman filter (AEKF), the proposed AIEKF has a higher estimation accuracy and anti-interference capability, which is suitable for the AUV's SOC estimation. In addition, based on the second-order equivalent circuit model of the lithium-ion battery, a forgetting factor recursive least squares (FFRLS) method is proposed to deal with the multi-variability problem. In the end, four different methods, including EKF, IEKF, AEKF, and the proposed AIEKF, are compared in computational time. The experiment results show that the proposed method has high accuracy and fast estimation speed, meaning that it has good application potential in AUVs.

ECG enhancement using a modified Bayesian framework and particle swarm optimization

Extended Kalman filter (EKF) is a well-known nonlinear Bayesian framework deployed in various electrocardiogram (ECG) processing applications. However, it is not very effective in removing non-stationary noises, such as muscle artifacts (MA) that are common in ECG recordings. As a model-based framework, this filter relies on its predefined state space model. As a result, EKF fails when ECG dynamics don’t comply its state-space model. In order to solve the above issues, in this paper, we propose to use particle swarm optimization (PSO) to find a better and more accurate state-space model. In addition, in order to improve EKF performance in nonstationary environments, we propose a new measurement model. This Model is modified to include non-Gaussian non-stationary additive and stationary noises. However, in order for EKF to use this model, its equations should be reformulated. Two different approaches are proposed in this paper to reformulate EKF equations; 1-state augmentation and 2- measurement differencing strategies. The proposed formulations for the EKF algorithm in this paper enable it to perform better than standard EKF in removing non-stationary contaminants. The proposed filters also preserve the clinical characteristics of ECG signals better than standard EKF. In order to show the effectiveness of the proposed EKF algorithms, their denoising performances were evaluated on the MIT-BIH Normal Sinus Rhythm Database (NSRDB) in the presence of two different types of non-stationary contaminants; 1- synthetic pink noise and 2- real muscle artifact noise extracted from Physionet noise stress database. The results showed that the proposed modified EKF frameworks significantly outperformed the standard EKF framework in non-stationary environments.

Reliability Assessment of an Unscented Kalman Filter by Using Ellipsoidal Enclosure Techniques

The Unscented Kalman Filter (UKF) is widely used for the state, disturbance, and parameter estimation of nonlinear dynamic systems, for which both process and measurement uncertainties are represented in a probabilistic form. Although the UKF can often be shown to be more reliable for nonlinear processes than the linearization-based Extended Kalman Filter (EKF) due to the enhanced approximation capabilities of its underlying probability distribution, it is not a priori obvious whether its strategy for selecting sigma points is sufficiently accurate to handle nonlinearities in the system dynamics and output equations. Such inaccuracies may arise for sufficiently strong nonlinearities in combination with large state, disturbance, and parameter covariances. Then, computationally more demanding approaches such as particle filters or the representation of (multi-modal) probability densities with the help of (Gaussian) mixture representations are possible ways to resolve this issue. To detect cases in a systematic manner that are not reliably handled by a standard EKF or UKF, this paper proposes the computation of outer bounds for state domains that are compatible with a certain percentage of confidence under the assumption of normally distributed states with the help of a set-based ellipsoidal calculus. The practical applicability of this approach is demonstrated for the estimation of state variables and parameters for the nonlinear dynamics of an unmanned surface vessel (USV).

Consistent Extended Kalman Filter-Based Cooperative Localization of Multiple Autonomous Underwater Vehicles.

In order to solve the problem of inconsistent state estimation when multiple autonomous underwater vehicles (AUVs) are co-located, this paper proposes a method of multi-AUV co-location based on the consistent extended Kalman filter (EKF). Firstly, the dynamic model of cooperative positioning system follower AUV under two leaders alternately transmitting navigation information is established. Secondly, the observability of the standard linearization estimator based on the lead-follower multi-AUV cooperative positioning system is analyzed by comparing the subspace of the observable matrix of state estimation with that of an ideal observable matrix, it can be concluded that the estimation of state by standard EKF is inconsistent. Finally, aiming at the problem of inconsistent state estimation, a consistent EKF multi-AUV cooperative localization algorithm is designed. The algorithm corrects the linearized measurement values in the Jacobian matrix for cooperative positioning, ensuring that the linearized estimator can obtain accurate measurement values. The positioning results of the follower AUV under dead reckoning, standard EKF, and consistent EKF algorithms are simulated, analyzed, and compared with the real trajectory of the following AUV. The simulation results show that the follower AUV with a consistent EKF algorithm can keep synchronization with the leader AUV more stably.

A New Triple Filtering Algorithm and Its Application for Aerial GNSS/INS-Integrated Direct Georeferencing System

A global navigation satellite system and inertial navigation system- (GNSS/INS-) integrated system is employed to provide direct georeferencing (DG) in aerial photogrammetry. However, GNSS/INS suffers from stochastic error, strong nonlinearity, and weak observability problems in high dynamic or less maneuver scenarios. In this paper, we proposed a new triple filtering algorithm for aerial GNSS/INS integration. The new algorithm implements filtering in the sequence of forward, backward, and forward directions. Each filter is initialized by a previous filter to get a quick convergence, and the final result is combination of the last two filtering to smooth error. The proposed triple filtering strategy avoids inaccuracy in the 1st forward filtering when the system has not reached convergence. Moreover, it facilitates engineering implementation because backward filtering can employ the same equations with forward filtering. To assess stochastic error of the inertial measurement unit, the Allan variance method is used and abbreviated stochastic model is built. A real aerial testing is conducted, and the result indicates that DG can achieve horizontal accuracy of 5 cm by the proposed algorithm, which has 63% improvement compared to standard extended Kalman filter.

Data Assimilation of Growing Stock Volume Using a Sequence of Remote Sensing Data from Different Sensors

Airborne Laser Scanning (ALS) has implied a disruptive transformation of how data are gathered for forest management planning in Nordic countries. We show in this study that the accuracy of ALS predictions of growing stock volume can be maintained and even improved over time if they are forecasted and assimilated with more frequent but less accurate remote sensing data sources like satellite images, digital photogrammetry, and InSAR. We obtained these results by introducing important methodological adaptations to data assimilation compared to previous forestry studies in Sweden. On a test site in the southwest of Sweden (58°27′N, 13°39′E), we evaluated the performance of the extended Kalman filter and a proposed modified filter that accounts for error correlations. We also applied classical calibration to the remote sensing predictions. We evaluated the developed methods using a dataset with nine different acquisitions of remotely sensed data from a mix of sensors over four years, starting and ending with ALS-based predictions of growing stock volume. The results showed that the modified filter and the calibrated predictions performed better than the standard extended Kalman filter and that at the endpoint the prediction based on data assimilation implied an improved accuracy (25.0% RMSE), compared to a new ALS-based prediction (27.5% RMSE).

Machine Learning Gyro Calibration Method Based on Attitude Estimation

This paper proposes a novel machine learning (ML) gyro calibration method that can achieve higher-accuracy gyro calibration and attitude estimation than the classical extended Kalman filter (EKF) approach when high-accuracy measurements are unavailable. The gyro calibration process is modeled as a time-series problem based on the standard EKF output. Then, a designed ML model is trained by the collected time-series data so that it can conduct gyro calibration by generating an ML correction to the gyro measurement. The proposed ML calibration method does not make assumptions in the form of gyro measurement errors, but directly learns it from data when high-accuracy information is available. Therefore, it is possible to outperform the EKF bias calibration when there is only low-accuracy information. To validate the method, a torque-free CubeSat is simulated using sun sensors and magnetometers to generate higher- and lower-accuracy attitude measurements, respectively. The simulation results show that the ML gyro calibration achieves smaller residual errors compared with the standard EKF. Meanwhile, the EKF attitude estimation accuracy is also improved, as the attitude integration is more accurate using the ML-calibrated gyro measurement. Four ML models based on different principles are examined, including multilayer perceptron, convolutional neural network, recurrent neural network, and Gaussian processes. It is found that, usually, a more sophisticated ML model can capture more gyro error information, but all four models can achieve similar performance with well-tuned parameters.

Model-based observers for vehicle dynamics and tyre force prediction

Advanced control and driving assistance systems play a major role in modern vehicles, ensuring higher standards of safety and performance. Their correct operation extensively depends on the knowledge of tyre forces and vehicle drift. However, these quantities are hard to measure directly, due to cost or technological reasons. One possible alternative that is attracting much attention in the last few years is represented by virtual sensing where the quantities of interest can be inferred using a physical model that maps the relationship between these quantities and other available direct measurements, like accelerations, velocities and rate-of-turns. In this research, model-based observation is adopted to predict tyre forces and slip angles. In contrast to existing systems, ours relies on direct causality equations without the need of any explicit tyre model. Different observers are developed that are grounded, respectively, in the Cubature Kalman and Particle filtering, and they are contrasted against the standard Extended Kalman filter (EKF). Results are presented to quantitatively assess the performance of the observers using a 14 Degrees Of Freedom (DOFs) full vehicle model that has been simulated in standard manoeuvres including constant radius cornering, increasing and swept-sine steering, and sine-dwell manoeuvring. Although all three embodiments allow model nonlinearities and measurement noise to be appropriately tackled, the two Kalman filters outperform the PF in terms of estimation accuracy, especially for tyre force prediction. In addition, the novel Cubature Kalman filter shows comparable accuracy and robustness, but higher stability when compared to the EKF.

SOC Estimation of Li-ION Battery Based on Improved EKF Algorithm

The state of charge (SOC) is one of the important performance indicators of battery, which provides an important basis for the management and control of Battery Management System (BMS). In view of the characteristics of lithium iron phosphate battery, considering the model accuracy and calculation amount, the equivalent circuit model of improved PNGV was selected. Based on that, an improved Extended Kalman Filter (EKF) algorithm was adopted to estimate the state of charge (SOC) of Li-ion battery, which covariance matrix was modified by the Levenberg-Marquardt method. At the end of this paper, the SOC estimation algorithm was verified by MATLAB simulations. The results show that compared with the standard EKF, the improved EKF has higher estimation accuracy and anti-interference ability, and has better convergence in the estimation process.

Transversely Stable Extended Kalman Filters for Systems on Manifolds in Euclidean Spaces

Abstract In this article, we devise a variant of the extended Kalman filter that can be generally applied to systems on manifolds with simplicity and low computational cost. We extend a given system on a manifold to an ambient open set in Euclidean space and modify the system such that the extended system is transversely stable on the manifold. Then, we apply the standard extended Kalman filter derived in Euclidean space to the modified dynamics. This method is efficient in terms of computation and accurate in comparison with the standard extended Kalman filter. It has the merit that we can apply various Kalman filters derived in Euclidean space including extended Kalman filters for state estimation for systems defined on manifolds. The proposed method is successfully applied to the rigid body attitude dynamics whose configuration space is the special orthogonal group in three dimensions.

Kernel Entropy Based Extended Kalman Filter for GPS Navigation Processing

This paper investigates the kernel entropy based extended Kalman filter (EKF) as the navigation processor for the Global Navigation Satellite Systems (GNSS), such as the Global Positioning System (GPS). The algorithm is effective for dealing with non-Gaussian errors or heavy-tailed (or impulsive) interference errors, such as the multipath. The kernel minimum error entropy (MEE) and maximum correntropy criterion (MCC) based filtering for satellite navigation system is involved for dealing with non-Gaussian errors or heavy-tailed interference errors or outliers of the GPS. The standard EKF method is derived based on minimization of mean square error (MSE) and is optimal only under Gaussian assumption in case the system models are precisely established. The GPS navigation algorithm based on kernel entropy related principles, including the MEE criterion and the MCC will be performed, which is utilized not only for the time-varying adaptation but the outlier type of interference errors. The kernel entropy based design is a new approach using information from higher-order signal statistics. In information theoretic learning (ITL), the entropy principle based measure uses information from higher-order signal statistics and captures more statistical information as compared to MSE. To improve the performance under non-Gaussian environments, the proposed filter which adopts the MEE/MCC as the optimization criterion instead of using the minimum mean square error (MMSE) is utilized for mitigation of the heavy-tailed type of multipath errors. Performance assessment will be carried out to show the effectiveness of the proposed approach for positioning improvement in GPS navigation processing.

Low-cost GPS/INS integration with accurate measurement modeling using an extended state observer

The extended Kalman filter (EKF) is widely used for the integration of the global positioning system (GPS) and inertial navigation system (INS). It is well known that the EKF performance degrades when the system nonlinearity increases or the measurement covariance is not accurate. For the loosely coupled GPS/INS integration, accurate determination of the GPS measurement and its covariance is not a simple task. An extended state observer (ESO) is proposed for the first time to improve the navigation performance of the loosely coupled GPS/INS integration with accurate measurement modeling. The performance of the proposed method is comprehensively evaluated and analyzed, and comparisons were made with respect to the standard EKF method. Simulations and a field vehicular test were conducted to evaluate the performance of the integrated system using the proposed algorithm. The results demonstrate that the navigation performance of the proposed method can be improved significantly in terms of position and velocity when compared with the EKF method. In the simulation test, the RMS values of the positioning errors are reduced by 52.57%, 48.56%, and 34.16% in the north, east, and vertical directions, respectively. The corresponding percentage for the velocity errors are 48.20%, 43.17%, and 22.65%, respectively. In the field test, the RMS values of the positioning errors are decreased by 40.91%, 47.63%, and 12.21%, respectively. The corresponding percentage for the velocity errors are 42.13%, 31.38%, and 33.86%, respectively. The improvement of attitude accuracy is not obvious as it mainly relies on the quality of the inertial sensors.