Adaptive grids for trajectory optimization by pseudospectral methods

Recently, pseudospectral (PS) methods have been routinely used to solve complex trajectory optimization problems in a simple and efficient manner by way of the reusable software package, DIDO. While PS methods typically outperform other methods across a variety of performance metrics, the grid generated by a standard PS method is static over both time and an ensemble of problems. In this paper, we propose several new ideas for an adaptive grid. We propose how a grid may be adapted to a particular problem in both a static and dynamic manner. A generic domain transformation technique is developed to yield more control over the node distribution. It is shown that the domain transformation approach satisfies a transformed version of the covector mapping theorem thus maintaining the useful properties of the original Legendre PS method on costate determination. The Kosloff-Tal-Ezer Map and the Tangent Map are discussed in some detail as they pertain to the numerical solution of trajectory optimization problems. All of these ideas are demonstrated for the continuous thrust orbit transfer problem and a robotics problem.