Optimal trajectory determination for increased relative navigation observability of air vehicles

This paper first derives the full equations of motion for relative navigation of air vehicles. An extended Kalman filter is used for estimating the relative position and attitude of two air vehicles, designated leader and follower. All leader states are assumed known, while the relative states are estimated using line-of-sight measurements between the vehicles along with acceleration and angular rate measurements of the follower. Noise is present on all measurements, while biases are present only on the latter two. The global attitude is parameterized using a quaternion, while the local attitude error is given by a three-dimensional attitude representation. An application of the new theoretical developments is also given, which involves determining an optimal trajectory to improve the estimation accuracy of the system. A cost function is derived based upon the relative position elements of the estimator covariance. State constraints are appended to the cost function using an exponential term. Results show that minimization of this cost function yields a trajectory that improves accuracy of both the position and attitude state estimates.