NASA Johnson Space Center
1 January 2009 – 31 May 2009
Co-P.I. Dr. John L. Junkins
Total award $51,913
The project supports current and future NASA goals in the area trajectory design and generation. It is based upon a novel extension and investigation of new analytical and computational methods for solutions of Lagrange’s Implicit Function Theorem for trajectory design and optimization. Compared to current methods, this new approach not only offers the promise of improved computation time, but more importantly insight into higher-order sensitivities. The focus of this work will be on investigating ways to make this method practical for designing complex lunar descent and ascent trajectories, with an emphasis on lunar descent and braking. These applications are of direct interest to our nation’s current and future vision for space exploration and national defense.
The Implicit function theorem originally due to Lagrange is an important result in analysis. Related methods for differentiation of implicit functions are even more important. The convergence of most versions of the Newton’s method to solve nonlinear equations and the theoretical foundations of virtually all algorithms for solving nonlinear differential equations (local existence and uniqueness of solutions) are closely related to these classical results. Other than serving as an important theoretical tool, the implicit function theorem and related concepts can be generalized to arrive at high order sensitivity equations for algebraic and differential equations about a given pre-computed solution. This perspective enables the conception of methods that construct families of neighboring solutions and blend them together in a fashion that simultaneously improves the accuracy of the local approximations and guarantees global piecewise continuity to a prescribed degree of partial differentiation.
In the light of these theoretical and computational advances, it is felt that it is important to develop general methods for trajectory optimization, investigate the high order sensitivities, and in particular, develop methods that enable computation of extremal field maps for neighboring optimal trajectories. Relatively simple benchmark problems we have run show that computing families of neighboring optimal solutions that previously took several hours to generate can now be generated in a small fraction of one hour. This is especially important as NASA returns to the moon with the “anytime, anywhere” mindset. That is, NASA is designing vehicles to leave Earth for the moon at anytime and land at any desired location on the lunar surface. The same holds for mission aborts back to Earth. A quick, efficient, and effective analytical and computational tool is a top priority for NASA mission planners and trajectory designers. Improved optimal solutions will increase efficiency and reduce flight time and fuel costs, which, in turn, facilitate a more flexible mission with increased payload capacity. Ultimately, we seek to address time varying constraints along the trajectory.
Working with me on this program is Graduate Research Assistant:
- Matthew Harris, M.S. student