The Kalman filter (KF) is the optimal estimator that minimizes the mean square error when the state and measurement dynamics are linear in nature, provided the process and measurement noise processes are modeled as white Gaussian. However, in the real world, one encounters a large number of scenarios where either the process or measurement model (or both) are nonlinear. In such cases a class of suboptimal Kalman filter implementations called extended Kalman filters (EKF) are used. EKFs operate by linearizing the nonlinear model around the current reference trajectory and then designing the Kalman filter gain for the linearized model. Recently, an alternative approach has emerged for a certain class of problems where the error in the states is estimated using a Kalman filter, rather than the state itself. This error state KF (ErKF) approach, by deriving the error state dynamics, via the perturbation of the nonlinear plant, lends itself to optimal updates in the error states and optimal prediction and updates in the error state covariance. This is because the error state dynamics are linear, thereby satisfying a condition for optimal Kalman filtering. This paper offers a comparison between the EKF and ErKF via simulations and shows that the ErKF performance is robust to a variety of aircraft maneuvers performed. Furthermore, this paper shows that the ErKF, unlike the EKF, need not be repeatedly tuned with respect to the noise covariances in order to obtain acceptable estimation performance.

This paper implements several methods for performing vision-based formation flight control of multiple aircraft in the presence of obstacles. No information is communicated between aircraft, and only passive 2-D vision information is available to maintain formation. The methods for formation control rely either on estimating the range from 2-D vision information by using Extended Kalman Filters or directly regulating the size of the image subtended by a leader aircraft on the image plane. When the image size is not a reliable measurement, especially at large ranges, we consider the use of bearing-only information. In this case, observability with respect to the relative distance between vehicles is accomplished by the design of a time-dependent formation geometry. To improve the robustness of the estimation process with respect to unknown leader aircraft acceleration, we augment the EKF with an adaptive neural network. 2-D and 3-D simulation results are presented that illustrate the various approaches.

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