A Bi-quintic Latitude/Longitude Spline and Lunar Surface Modeling for Spacecraft Navigation

A new bi-quintic latitude/longitude spline is developed and applied to the problem of Lunar surface modeling. The new spline provides a singularity-free parameterized function over an oblate spheroid. The bi-quintic spline uses 9 parameters at each of its regular latitude/longitude nodes. At the two poles there are only 6 parameters. The resulting function is continuous and has continuous first and second partial derivatives. Latitude and longitude partial derivatives go to zero at the poles in ways that guarantee continuous first and second partial derivatives of the function when evaluated with respect to Cartesian coordinates as its underlying inputs. The new spline model has been applied to the problem of navigating a Lunar lander. The lander measures slant-ranges to points on the Lunar surface. Its navigation filter uses a model of how the slant-ranges vary with its location and orientation. This model is based on a bi-quintic spline of the Lunar surface, and it returns slant-ranges and their first partial derivatives with respect to spacecraft position and sensor line-of-sight direction. The new sensor model works well in truth-model simulation tests of the navigation filter.