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 two-point 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 & travel time, evaluation of the generalized k = 1 Legendre-Clebsch optimality condition, and comparison with competing routes, namely the constant yaw rate and constant bearing angle routes.
- Dynamic Systems and Control Division
Brachistochrone on a 2D Curved Surface Using Optimal Control and Spline-Based DEM Map Data
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Wright, NC, Hennessey, MP, & Shakiban, C. "Brachistochrone on a 2D Curved Surface Using Optimal Control and Spline-Based DEM Map Data." Proceedings of the ASME 2012 5th Annual Dynamic Systems and Control Conference joint with the JSME 2012 11th Motion and Vibration Conference. Volume 2: Legged Locomotion; Mechatronic Systems; Mechatronics; Mechatronics for Aquatic Environments; MEMS Control; Model Predictive Control; Modeling and Model-Based Control of Advanced IC Engines; Modeling and Simulation; Multi-Agent and Cooperative Systems; Musculoskeletal Dynamic Systems; Nano Systems; Nonlinear Systems; Nonlinear Systems and Control; Optimal Control; Pattern Recognition and Intelligent Systems; Power and Renewable Energy Systems; Powertrain Systems. Fort Lauderdale, Florida, USA. October 17–19, 2012. pp. 703-709. ASME. https://doi.org/10.1115/DSCC2012-MOVIC2012-8802
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