Programação
Data: 11/12/2018 - Terça-feira
Horário: 9:30 - 11:00
This talk shows how to derive all possible analytical functions, f(x), subject to n constraints on the function and its derivatives defined at any specified values. These expressions, called "constrained expressions," can be then adopted to describe trajectories satisfying specific constraints (e.g., path planning). The talk first shows general explicit functions passing through a single point in three distinct ways: linear, rational, and additive. Then, functions with constraints on single, two, and multiple points are introduced. In particular, a generalization of the Waring's interpolation form is derived to obtain expressions passing through a set of points.
The capability of deriving constrained expressions allows to obtain least-squares solutions to initial, boundary, and multivalued problems of nonlinear differential equations of any order. The constrained expressions contain a freely to choose function, g(x), which is then expanded in terms of basis functions (e.g., Chebyshev and Legendre orthogonal polynomials). The procedure leads to a set of equations in terms of the unknown coefficients vector that is then computed by least-squares. Numerical comparisons are then provided quantifying speed and accuracy versus the-state-of-art differential equation solvers.
The Theory of Connections is then applied to obtain all possible surfaces connecting functions and all manifolds connecting surfaces. This extension is then used to solve partial differential equations such as Poisson, Wave, and Heat equations.
Data: 13/12/2018 - Quinta-feira
Horário: 9:30 - 11:00
Star trackers have been successfully used in space for spacecraft attitude estimation. Now, thanks to the k-vector range searching technique, the Star-ID problem can be solved in real time on onboard computers with limited performance. This talk shows how to use star trackers to estimate the spacecraft position in the two distinct cases of interplanetary and cislunar trajectory:
Interplanetary OpNav
A single-point position estimation technique for interplanetary missions by observing visible planets using star trackers is shown. Closed-form least-squares solution is obtained by minimizing the sum of the expected object-space squared distance errors. Then, a weighted least-squares solution is provided by an iterative procedure. The weights are evaluated using the distances to the planets estimated by the least-squares solution. It is shown that the weighted approach only requires one iteration to converge and results in significant accuracy gains compared to simple least-squares approach. The proposed method is numerically validated by a statistical scenario on a grid of test cases in the ecliptic plane and through time, from January 1, 2018 to January 1, 2043.
Cislunar OpNav
When close to a celestial body (Moon/Earth) a novel algorithm to accurately estimate the observer-to-body relative position is presented. The main motivation is to provide a backup autonomous navigation capability in a loss of communications scenario with Earth. The algorithm is derived for the general case of a triaxial ellipsoid. The image processing approach derives the centroid and the distance to the body by a nonlinear iterative least-squares approach where the illuminated edge of the body (limb) is radially modelled as Gaussian in the gradient image. Numerical examples using real Moon pictures and Synthetic Earth images are provided to clarify the image processing steps and to validate the proposed theory. The proposed OpNav approach has been adopted by the NASA Orion missions.
Data: 14/12/2018 - Sexta-feira
Horário: 9:30 - 11:00
This year the Flower Constellations theory celebrates its 16th birthday. Many years were needed to fully understand the implications and to develop the theory. This new satellite constellations design tool is now ready for applications.
The theory introduces a new class of space objects characterized by shape preserving configurations where the whole constellation behaves as a rigid object. By using minimal parameterization (Hermite normal form) the 2D Lattice Flower Constellations allows to include all spatial and temporal symmetric solutions, while the extension to 3D Lattice allows designers to use any inclination when selecting elliptical orbits under J2 perturbation. Recently, the Necklace theory applied to 2D and 3D Flower Constellations exponentially increases the space of potential solutions while keeping limited the number of satellites and launches (costs).
The evolution of the mathematical theory is presented, showing some potential configurations to improve existing applications as well as configurations proposing new applications! The number of applications are many, including, positioning, communication, radio occultation, interferometric, and surveillance systems. In particular, the Flower Constellations theory allows to design conjunction-free constellations with many thousands of satellites and a new class of orbits/constellations, called J2 propelled systems, where the Earth oblateness perturbation is used (rather than control) to cover spatial volumes around the Earth to measure or monitor physical quantities.
