Sequential Estimation of Spacecraft Relative Orbits: Improving Observability with Nonlinearities, Chief Eccentricity, and Consensus Feedback

Abstract

This dissertation addresses the problem of spacecraft relative orbit estimation using a various of estimation strategies. To study the effects of incorporating nonlinearities and the practicality of using different measurement types in relative orbit estimation, multiple extended Kalman filters are constructed with first order, second order, third order and full nonlinear dynamic models as well as angles-only and range-only measurements. The numerical results compare the performance of these estimation scenarios and illustrate the benefits of using higher order models. The observability is studied analytically using a nonlinear observability matrix obtained using Lie derivatives, as well as numerically with the observability index and condition number obtained from the observability Gramian. Later, in order to describe a manifold of ambiguous relative orbits that arise with linear relative dynamics, circular chief orbits, and range-only measurements, a formulation based on relative orbit elements is used to solve these ambiguous orbits. These ambiguous orbits are categorized into two cases: mirror orbits, which conserve the size and shape but transform the orientation of the true relative orbit, and deformed orbits, which both distort the shape and change the orientation of the true relative orbit. The multiplicity of mirror and deformed ambiguous orbits are shown to be three and four respectively. Numerical results using an extended Kalman filter are provided to confirm the existence of these ambiguous orbits. Since all natural chief orbits are not exactly circular, these ambiguous orbits are studied with elliptical chief orbits. Using a geometric solution of the Tschauner-Hempel equation, mirror ambiguous orbits are shown to exist. However, deformed ambiguous orbits are shown to vanish due to the non-zero chief orbit eccentricity although they persist in Kalman filtering simulations. This persistent phenomenon is explained through a perturbation analysis using variation of the geometric ambiguous states. Furthermore, it is shown that the inclusion of chief orbit eccentricity and higher order nonlinear terms in the filter model can both partially exclude these ambiguous orbits. As an alternate method of excluding these ambiguous orbits, the consensus estimation technique is employed by utilizing the communication network between different chief spacecraft. Basically, multiple extended Kalman filters mounted on different chief spacecraft cooperatively estimate the orbit of one deputy spacecraft by using information sharing in a communication network. The numerical results show that use of the consensus estimation strategy can improve the convergence performance of the filter and increase the system observability by excluding ambiguous orbits. The improvements on system observability are validated by the analytical observability criteria using Lie derivatives. Since the interruption and loss of communication is possible in the space environment, the effects of a time-varying communication topology are examined. Changes in the network topology of information sharing resulting from kinematic properties such as relative separation distances are shown to affect the observability gained from including the consensus feedback in the filter. The manner in which a time-varying communication topology diminishes the regions of attraction for the estimated relative orbits is illustrated through numerical examples. The virtual state method is proposed to reduce the divergence behavior in the consensus Kalman filter. The decentralized estimation strategy in which multiple spacecraft without known inertial orbits cooperatively estimating their relative orbits is also proposed. Through analytical discussion, the benefits of using the decentralized estimation strategy on system observability are illustrated. A series of numerical results verify that the decentralized estimation strategy can improve the filtering performance and resist the tendency to converge on ambiguous relative orbits. The diffusion of observability between different estimation loops in the decentralized estimation network is studied by numerical examples. Finally, the estimation of both translational and rotational motion of rigid body is considered. In the framework of geometric mechanics, the consensus extended Kalman filter is employed assuming position measurements from feature points on a rigid body. The position and attitude (pose) of the rigid body is represented by Lie algebra se(3). Numerical results demonstrate that the filter can converge on the true state when measurements of three feature points are processed by a single Kalman filter. When multiple filters are allowed to communicate their local estimates with each other within a connected communication network, the filtering performance (estimation error and covariance envelope) is greatly improved and the number of feature points required to guarantee the observability can be relaxed.