An automatic node placement strategy for optimal control problems discretized using third-degree Hermite polynomials

A new strategy for automating the node placement in an optimal control problem, solved using third-order Herrnite polynomials to model the state time histories, is discussed. This Hermite polynomial results in an approximate solution to the system differential equations by taking on the form of the Simpson’s quadrature rule. The automatic node placement strategy uses an error function based on an adaptive quadrature method for Simpson's rule. The method compares solutions generated using two successive nodal distributions to determine how to refine a nodal distribution. The algorithm is applied to a continuously thrusting spacecraft trajectory problem where the final energy of the trajectory is maximized for a fixed final time. The use of this automatic node placement algorithm yields a more accurate solution than that obtained with an equal number of evenly spaced nodes.