Kalman Filtering for Space Navigation

Each attendee receives extensive notes and reference materials.

This course is designed for spacecraft engineers, program managers, and other professionals who wish to enhance their knowledge of state estimation and Kalman filtering in order to better understand and appreciate the complexities of satellite navigation systems.  It is intended to familiarize the attendee with the fundamentals of estimation theory and the operational use of Kalman filtering for spacecraft navigation.

Definitions and fundamental concepts associated with dynamical systems, state estimation theory, stochastic systems, random processes, and Kalman filtering.  Historical information about the development of the Kalman filter and its implementation in spacecraft navigation systems.  Performance of the Kalman filter, including the interpretation of the covariance matrix.  Survey of variations of the Kalman filter, including the extended Kalman filter, the unscented Kalman filter, and the colored-noise Kalman filter.

1. Review of Fundamentals and Basic Definitions.
   Basic definitions. Review of dynamical systems, matrix-vector operations, and state-space notation.  State solution using the fundamental or state-transition matrix.  Numerical integration of the system dynamics.  Observability and the linear state estimator.  Perfect state estimation in the absence of sensor noise.
2. Estimation Theory.
   Least-squares methods.  Batch vs. sequential estimation methods.  Introduction to probability concepts including random variables, Gaussian distributions, white noise, and stochastic processes.  Autocorrelation functions and the covariance matrix.  Least-squares polynomials, filtering noisy measurements, and residuals.  Recursive least-squares filtering.
3. The Discrete-Time Kalman Filter.
   Converting a continuous-time system to a discrete-time system.    Discrete-time Kalman filter as a linear observer.  Covariance matrices for measurement noise, process noise, and estimation error.  The measurement update and the optimal Kalman gain matrix.  Optimal update to the error covariance matrix.  Presenting the discrete-time Kalman filter as a recursive algorithm.
4. Kalman Filter Examples for Low-Order Systems.
   Optimal estimation of noisy polynomial signals.  Introduction of process noise and filter performance.  Tracking a falling object using noisy measurements, tracking a noisy sinusoidal signal, and state estimation for a mass-spring system.
5. State Estimation for Spacecraft Systems.
   Coordinate systems and a review of the orbit and attitude determination problems.  Development of the basic equations of orbital mechanics and attitude dynamics.
6. Practical Issues with Implementing the Kalman Filter.
   History of the Kalman filter.  Initial implementations of the Kalman filter.  Computational and numerical issues.
7. Optimal Estimation for Spacecraft.
   Implementing the Kalman filter for orbit determination and spacecraft attitude determination.  Global positioning system navigation.
8. Advanced Topics in Kalman Filtering.
   Variations of the Kalman filter: the extended Kalman filter, unscented Kalman filter, ensemble Kalman filtering, and colored-noise Kalman filter.  The continuous-time Kalman filter.  Consider covariance analysis.