Brachistochrone on a 2D curved surface using optimal control and spline-based dem map data

Based on a bivariate spline representation of United States Geological Survey (USGS) digital elevation model (DEM) data, the brachistochrone on a 2D curved surface without friction was solved numerically using dynamic and control models in MATLAB^® in conjunction with the Spline Toolbox for surface modeling. This extends in a natural manner previous work by several of the authors (Hennessey and Shakiban) on both the 1D and 2D curved surface brachistochrone using optimal control and resulting in a twopoint boundary value problem. DEM data permits an accurate representation of the surface in question (30 m resolution data for Lone Mountain in MT) and the Spline Toolbox provides a sufficiently smooth version of the surface, including access to spatial partial derivatives needed in the minimum-time control law. Step-by-step results are reported, including the surface representation details, the minimum-time route and travel time, evaluation of the generalized = 1 Legendre-Clebsch optimality condition, and comparison with competing routes, namely the constant yaw rate and constant bearing angle routes.