Matthew Rhudy

Flight-Test Evaluation of Sensor Fusion Algorithms for Attitude Estimation

In this paper, several Global Positioning System/inertial navigation system (GPS/INS) algorithms are presented using both extended Kalman filter (EKF) and unscented Kalman filter (UKF), and evaluated with respect to performance and complexity. The contributions of this study are that attitude estimates are compared with independent measurements provided by a mechanical vertical gyroscope using 23 diverse sets of flight data, and that a fundamental difference between EKF and UKF with respect to linearization is evaluated.

Evaluation of Matrix Square Root Operations for UKF within a UAV GPS/INS Sensor Fusion Application

Using an Unscented Kalman Filter (UKF) as the nonlinear estimator within a Global Positioning System/Inertial Navigation System (GPS/INS) sensor fusion algorithm for attitude estimation, various methods of calculating the matrix square root were discussed and compared. Specifically, the diagonalization method, Schur method, Cholesky method, and five different iterative methods were compared. Additionally, a different method of handling the matrix square root requirement, the square-root UKF (SR-UKF), was evaluated. The different matrix square root calculations were compared based on computational requirements and the sensor fusion attitude estimation performance, which was evaluated using flight data from an Unmanned Aerial Vehicle (UAV). The roll and pitch angle estimates were compared with independently measured values from a high quality mechanical vertical gyroscope. This manuscript represents the first comprehensive analysis of the matrix square root calculations in the context of UKF. From this analysis, it was determined that the best overall matrix square root calculation for UKF applications in terms of performance and execution time is the Cholesky method.

Sensitivity Analysis of Extended and Unscented Kalman Filters for Attitude Estimation

The extended Kalman filter (EKF) and unscented Kalman filter (UKF) for nonlinear state estimation with both additive and nonadditive noise structures are presented and compared. Three different Global Positioning System (GPS)/inertial navigation system (INS) sensor fusion formulations for attitude estimation are used as case studies for the nonlinear state estimation problem. A diverse set of actual flight data collected from research unmanned aerial vehicles was used as empirical data for this study. Roll and pitch estimation results were comparedwith independent measurements from amechanical vertical gyroscope to evaluate the performance. The performance of the EKF and UKF is compared in terms of noise assumptions, covariance matrix tuning, sampling rate, initialization error, GPS outages, robustness to inertial measurement unit bias and scale factors, and linearization. Similar sensitivity for this GPS/INS attitude estimation problem was found between the EKF and UKF for most cases. Small differences were seen between EKF and UKF for initialization error and GPS outages: the UKF was found to be more robust to inertial measurement unit calibration errors, and the EKF was determined to be more computationally efficient.

UAV Attitude, Heading, and Wind Estimation Using GPS/INS and an Air Data System

A new attitude, heading, and wind estimation algorithm is proposed, which incorporates measurements from an air data system to properly relate predicted attitude information with aircraft velocity information. Experimental Unmanned Aerial Vehicle (UAV) flight data was used to validate the proposed approach. The experimental results demonstrated effective estimation of the roll, pitch, yaw, and heading angles, and provided a smoothed estimate of the angle of attack and sideslip angles. The wind estimation results were validated with respect to measurments provided by a local weather station. It was shown that this new method of attitude estimation is effective in distinguishing the yaw and heading angles of the aircraft, properly regulating the attitude estimates with air data system measurements, and provding a reasonable estimate of the local wind field.

On-line Modeling and Calibration of Low-Cost Navigation Sensors

In this paper, calibration modeling of a low-cost Inertial Measurement Unit (IMU) sensor for Small Unmanned Aerial Vehicle (SUAV) attitude estimation is considered. First, an Allan variance analysis method is used to determine stochastic noise model parameters for each sensor of a Micro-Electro-Mechanical-System (MEMS) IMU. Next, these models are included in a Global Positioning System/Inertial Navigation System (GPS/INS) sensor fusion algorithm for on-line calibration. In addition, an off-line magnetometer calibration is considered that uses a set of GPS/INS sensor fusion attitude estimates to derive a calibration model. This off-line magnetometer calibration model is then augmented on-line with sensor fusion estimates of the residual sensor biases. Finally, using the calibrated magnetometers, attitude estimation is considered that uses only a low-cost IMU with magnetometers. Each sensor fusion algorithm is formulated using an Unscented Kalman Filter (UKF). For performance validation, attitude estimates are calculated with data collected on-board a SUAV and are compared with high-quality vertical gyroscope measurements.

Online Stochastic Convergence Analysis of the Kalman Filter

This paper presents modifications to the stochastic stability lemma which is then used to estimate the convergence rate and persistent error of the linear Kalman filter online without using knowledge of the true state. Unlike previous uses of the stochastic stability lemma for stability proof, this new convergence analysis technique considers time-varying parameters, which can be calculated online in real-time to monitor the performance of the filter. Through simulation of an example problem, the new method was shown to be effective in determining a bound on the estimation error that closely follows the actual estimation error. Different cases of assumed process and measurement noise covariance matrices were considered in order to study their effects on the convergence and persistent error of the Kalman filter.

Microphone Array Analysis Methods Using Cross-Correlations

Due to civilian noise complaints and damage claims, there is a need to establish an accurate record of impulse noise generated at military installations. Current noise monitoring systems are susceptible to false positive detection of impulse events due to wind noise. In order to analyze the characteristics of noise events, multiple channel data methods were investigated. A microphone array was used to collect four channel data of military impulse noise and wind noise. These data were then analyzed using cross-correlation functions to characterize the input waveforms. Four different analyses of microphone array data are presented. A new value, the min peak correlation coefficient, is defined as a measure of the likelihood that a given waveform originated from a correlated noise source. Using a sound source localization technique, the angle of incidence of the noise source can be calculated. A method was also developed to combine the four individual microphone channels into one. This method aimed to preserve the correlated part of the overall signal, while minimizing the effects of uncorrelated noise, such as wind. Lastly, a statistical method called the acoustic likelihood test is presented as a method of determining if a signal is correlated or not.Copyright

Fusion of GPS and Redundant IMU Data for Attitude Estimation

Attitude estimation using Global Positioning System/Inertial Navigation System (GPS/INS) was used as an example application to study three different methods of fusing redundant multi-sensor data used in the prediction stage of a nonlinear recursive filter. Experimental flight data were collected with an Unmanned Aerial Vehicle (UAV) containing GPS position and velocity calculations and four redundant Inertial Measurement Unit (IMU) sensors. Additionally, the aircraft roll and pitch angles were measured directly with a high-quality mechanical vertical gyroscope to be used as a ‘truth’ reference for evaluating attitude estimation performance. A simple formulation of GPS/INS sensor fusion using an Extended Kalman Filter (EKF) was used to calculate the results for this study. Each of the three presented fusion methods was shown to be effective in reducing the roll and pitch errors as compared to corresponding results using single IMU GPS/INS sensor fusion. Additionally, the fusion methods were shown to be effective in estimating roll and pitch angles without the aid of GPS (dead reckoning).

Sensitivity Analysis of EKF and UKF in GPS/INS Sensor Fusion

This document presents a sensitivity analysis relative to different algorithm design parameters on the attitude performance of two different Global Positioning System/Inertial Navigation System (GPS/INS) sensor fusion algorithms for estimating aircraft attitude angles, namely the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF). The sensor fusion was performed using flight data acquired with three different WVU YF-22 research aircraft under a variety of flight conditions. The attitude estimates were compared with direct ‘truth’ measurements from an on-board mechanical vertical gyroscope. The sensitivity analysis was conducted on the following parameters: process and measurement noise covariance tuning, IMU and GPS sampling rates, GPS outages, time offset between GPS and IMU measurements, and acceleration due to gravity. Overall, the EKF and UKF performed very similarly in response to the different parameters for this study.

An Analytical Approach for Comparing Linearization Methods in EKF and UKF

The transformation of the mean and variance of a normally distributed random variable was considered through three different nonlinear functions: sin(x), cos(x), and xk, where k is a positive integer. The true mean and variance of the random variable after these transformations is theoretically derived within, and verified with respect to Monte Carlo experiments. These statistics are used as a reference in order to compare the accuracy of two different linearization techniques: analytical linearization used in the Extended Kalman Filter (EKF) and statistical linearization used in the Unscented Kalman Filter (UKF). This comparison demonstrated the advantage of using the unscented transformation in estimating the mean after transforming through each of the considered nonlinear functions. However, the variance estimation led to mixed results in terms of which linearization technique provided the best performance. As an additional analysis, the unscented transformation was evaluated with respect to its primary scaling parameter. A nonlinear filtering example is presented to demonstrate the usefulness of the theoretically derived results.