## Space Situational Awareness

**Space Situational Awareness**

**Maneuvering Spacecraft Tracking via State-Dependent Adaptive Estimation**

In this study, an adaptive estimation algorithm is developed to estimate the state of a spacecraft that performs impulsive maneuvers. The accurate tracking of a maneuvering spacecraft with impulsive burns is a challenging problem since the magnitude and the time of occurrence of impulsive maneuvers are usually unknown a priori. To deal with this problem, an adaptive state estimation algorithm is developed in this study using a bank of extended Kalman filters along with interacting multiple models that account for spacecraft motion with and without impulsive maneuvers. Motivated by the fact that impulsive maneuvers usually occur when certain conditions on the spacecraft state are satisfied, the multiple extended Kalman filters are systematically blended using a state-dependent transition probability. Since the information about the conditions based on which impulsive maneuvers occur is explicitly used in the state-dependent transition probability, the proposed algorithm can predict the impulsive maneuvers more accurately and thus produce more accurate state estimates. The proposed algorithm is demonstrated with an illustrative example: tracking of a spacecraft performing orbital transfers.

*Impulsive maneuver at a node for a noncoplanar transfer (inclination change)*

*Comparison of state estimation accuracy*

Through numerical simulation, it is validated that the proposed algorithm outperforms all the other algorithms (EKF1, EKF2, IMM), providing the smallest estimation errors even when the orbital maneuver occurs.

**Analytical Uncertainty Propagation for Satellite Relative Motion**

For satellites flying in close proximity, monitoring the uncertainties of neighboring satellites’ states is a crucial task because the uncertainty information is used to compute the collision probability between satellites with the objective of collision avoidance. In this study, an analytical closed-form solution is developed for uncertainty propagation in the satellite relative motion near general elliptic orbits. The Tschauner–Hempel equations are used to describe the linearized relative motion of the deputy satellite where the chief orbit is eccentric. Under the assumption of the linearized relative motion and white Gaussian process noise, the uncertainty propagation problem is defined to compute the mean and covariance matrix of the relative states of the deputy satellite. The evolution of the mean and covariance matrix is governed by a linear time-varying differential equation, for which the solution requires the integration of the quadratic function of the inverse of the fundamental matrix associated to the Tschauner–Hempel equations. The difficulties in evaluating the integration are alleviated by the introduction of an adjoint system to the Tschauner–Hempel equations and the binomial series expansion. The accuracy of the developed analytical solution is demonstrated in illustrative numerical examples by comparison with a Monte Carlo analysis.

*Relative motion in the local-vertical-local-horizontal frame*

*Evolution of the probability ellipsoid: proposed analytical solution (solid line), analytical solution using the CW equations (dashed line), and Monte Carlo simulation using 500 samples (dots)*

It is clear in the above figure that the proposed analytical solution (solid line) can accurately predict the probability distribution of the states (dots) obtained from Monte Carlo simulation. On the other hand, the solutions using the CW equations fail to capture the distribution. This is because the CW equations assume a circular chief orbit; thus, when applied to the case where the chief orbit is elliptic, their accuracy significantly decreases.

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