Adaptive Picard-Chebyshev Methods for Bang-Bang Control Dynamics

Numerical integration of dynamical systems is vital to science and engineering applications as it allows solutions to be obtained numerically when analytical solutions do not exist. In this paper we present an improved methodology for propagating high-fidelity low-thrust trajectories, that exhibit a bang-bang dynamical behavior, using Picard-Chebyshev methods. The approach combines a true anomaly informed segmentation scheme as well as early anticipation and accommodation of on/off engine switches which are a challenge for implicit integrators. Adaptation of segment length and Chebyshev polynomial degree is carried out using the accuracy of the acceleration fit (force model), prior to integration with Picard iteration, as an error metric. The magnitude of the smallest Chebyshev coefficients of the acceleration is also monitored to determine if their contribution is significant to warrant inclusion. We demonstrate the utility of this novel method for solving a fuel-optimal low-thrust transfer problem from a geostationary transfer orbit to geosynchronous orbit. A radially adaptive, variable fidelity spherical harmonic gravity model is included in our simulations. We compare the performance of the algorithm to MATLAB’s ode89 and find that APC requires about 15% of the computation time, compared with ode89, when a solution with a 5 minute output timestep is required.