Algorithms for spacecraft formation flying navigation based on wireless positioning system measurements

Spacecraft formation flying navigation continues to receive a great deal of interest. The research presented in this dissertation focuses on developing methods for estimating spacecraft absolute and relative positions, assuming measurements of only relative positions using wireless sensors. The implementation of the extended Kalman filter to the spacecraft formation navigation problem results in high estimation errors and instabilities in state estimation at times. This is due tp the high nonlinearities in the system dynamic model. Several approaches are attempted in this dissertation aiming at increasing the estimation stability and improving the estimation accuracy. A differential geometric filter is implemented for spacecraft positions estimation. The differential geometric filter avoids the linearization step (which is always carried out in the extended Kalman filter) through a mathematical transformation that converts the nonlinear system into a linear system. A linear estimator is designed in the linear domain, and then transformed back to the physical domain. This approach demonstrated better estimation stability for spacecraft formation positions estimation, as detailed in this dissertation. The constrained Kalman filter is also implemented for spacecraft formation flying absolute positions estimation. The orbital motion of a spacecraft is characterized by two range extrema (perigee and apogee). At the extremum, the rate of change of a spacecraft’s range vanishes. This motion constraint can be used to improve the position estimation accuracy. The application of the constrained Kalman filter at only two points in the orbit causes filter instability. Two variables are introduced into the constrained Kalman filter to maintain the stability and improve the estimation accuracy. An extended Kalman filter is implemented as a benchmark for comparison with the constrained Kalman filter. Simulation results show that the constrained Kalman filter provides better estimation accuracy as compared with the extended Kalman filter. A Weighted Measurement Fusion Kalman Filter (WMFKF) is proposed in this dissertation. In wireless localizing sensors, a measurement error is proportional to the distance of the signal travels and sensor noise. In this proposed Weighted Measurement Fusion Kalman Filter, the signal traveling time delay is not modeled; however, each measurement is weighted based on the measured signal travel distance. The obtained estimation performance is compared to the standard Kalman filter in two scenarios. The first scenario assumes using a wireless local positioning system in a GPS denied environment. The second scenario assumes the availability of both the wireless local positioning system and GPS measurements. The simulation results show that the WMFKF has similar accuracy performance as the standard Kalman Filter (KF) in the GPS denied environment. However, the WMFKF maintains the position estimation error within its expected error boundary when the WLPS detection range limit is above 30km. In addition, the WMFKF has a better accuracy and stability performance when GPS is available. Also, the computational cost analysis shows that the WMFKF has less computational cost than the standard KF, and the WMFKF has higher ellipsoid error probable percentage than the standard Measurement Fusion method. A method to determine the relative attitudes between three spacecraft is developed. The method requires four direction measurements between the three spacecraft. The simulation results and covariance analysis show that the method’s error falls within a three sigma boundary without exhibiting any singularity issues. A study of the accuracy of the proposed method with respect to the shape of the spacecraft formation is also presented.

A PETROPHYSICAL EVALUATION OF THE TRENTON-BLACK RIVER FORMATION OF THE MICHIGAN BASIN

This work is conducted to study the geological and petrophysical features of the Trenton- Black River limestone formation. Log curves, crossplots and mineral identification methods using well-log data are used to determine the components and analyze changes in lithology. Thirty-five wells from the Michigan Basin are used to define the mineralogy of Trenton-Black River limestone. Using the different responses of a few log curves, especially gamma-ray, resistivity and neutron porosity, the formation tops for the Utica shale, the Trenton limestone, the Black River limestone and the Prairie du Chien sandstone are identified to confirm earlier authors’ work and provide a basis for my further work. From these, an isopach map showing the thickness of Trenton-Black River formation is created, indicating that its maximum thickness lies in the eastern basin and decreases gradually to the west. In order to obtain more detailed lithological information about the limestone formations at the thirty-five wells, (a) neutron-density and neutron-sonic crossplots, (b) mineral identification methods, including the M-N plot, MID plot, ϱmaa vs. Umaa MID plot, and the PEF plot, and (c) a modified mineral identification technique are applied to these wells. From this, compositions of the Trenton-Black River formation can be divided into three different rock types: pure limestone, partially dolomitized limestone, and shaly limestone. Maps showing the fraction of dolomite and shale indicate their geographic distribution, with dolomite present more in the western and southwestern basin, and shale more common in the north-central basin. Mineral identification is an independent check on the distribution found from other authors, who found similar distributions based on core descriptions. The Thomas Stieber method of analysis is best suited to sand-shale sequences, interpreting hree different distributions of shale within sand, including dispersed, laminated and structural. Since this method is commonly applied in clastic rocks, my work using the Thomas Stieber method is new, as an attempt to apply this technique, developed for clastics, to carbonate rocks. Based on the original assumption and equations with a corresponding change to the Trenton-Black River formation, feasibility of using the Thomas Stieber method in carbonates is tested. A graphical display of gamma-ray versus density porosity, using the properties of clean carbonate and pure shale, suggests the presence of laminated shale in fourteen wells in this study. Combined with Wilson’s study (2001), it is safe to conclude that when shale occurs in the Trenton-Black River formation, it tends to be laminated shale.