Description: We propose herein to augment current NASA spaceflight dynamics programs with algorithms and software from two domains. First, we propose to use numerical parameter continuation methods to assist in computation of trajectories in complicated dynamical situations. Numerical parameter continuation methods have been used extensively to compute a menagerie of structures in dynamical systems including fixed points, periodic orbits, simple bifurcations (where the structure of the dynamics changes), Hopf bifurcations (where periodic orbits are created), invariant manifolds, hetero/homoclinic connections between invariant manifolds, etc. Perhaps more importantly for the current work, such methods have already proven their worth in flight dynamics problems, especially those having to do with the complicated dynamics near libration points. Second, we propose to use advanced filtering techniques and representations of probability density functions to appropriately compute and manage the uncertainty in the trajectories. While advanced methods for understanding and leveraging the underlying dynamics are clearly necessary for effective mission design, planning, and analysis, we contend that they do not suffice. In particular, they do not, in and of themselves, address the issue of uncertainty. Herein we discuss methods that balance the accuracy of the uncertainty representation against computational tractability, including a discussion of the notorious ``curse of dimensionality'' for problems with large state vectors. We propose approachs that revolve around modifications of algorithms such as ``log homotopy'' particle filters and especially Gaussian sum filters. Finally, we propose to integrate all of the above algorithms into standard NASA software packages GEONS, GIPSY, and GMAT.

Benefits: The satellite market is large and growing. For example, analysis firm Forecast International is projecting worldwide deliveries of about 262 geostationary or medium-Earth orbit commercial communications satellites by 2019. This implies a strong market for the technology described in this proposal. In particular, as space becomes more crowded with commercial and government spacecraft, not to mention the large number of ``junk'' objects currently in the space catalogue, the robust calculation of trajectories along with accurate estimates of uncertainty can only become more important. The space and satellite market is expected to reach $158 Billion by 2010 and is a multi-billion dollar industry both in the US and around the world. This market involves numerous government agencies and permeates many parts of the U.S. Military as well as numerous commercial entities. In particular, large players in this market include Boeing, TerreStar, and Northrop Grumman to name but a few. The algorithms and software proposed herein will find applicability to many challenging problems for both the DoD and commercial entities where complicated dynamics and uncertainty play a role.

There are several current state-of-the-art software packages that are clear and direct transition paths for the proposed work. In particular, there are the GPS-Enhanced Onboard Navigation Software (GEONS), the GPS-Inferred Positioning System and Orbit Analysis Software (GIPSY), and the General Mission Analysis Tool (GMAT). Of the various packages, GMAT is the most directly applicable and will be the focus of the Phase I effort. Accordingly, these algorithms will find applicability in any pre-flight mission design, planning, and analysis activities that utilize these software. One domain of particular note is space craft missions in the neighborhood of libration points, where the underlying dynamics are rather complicated and the effect of the dynamics on the trajectory uncertainty is important. Another domain that we see of prime importance is that of spacecraft formation flying (such as the Terrestrial Planet Finder mission) where, again, the dynamics and the uncertainty play a key role.