Application of Taylor Series Integration to Reentry Problems

The determination of optimal reentry trajectories and the analysis of the sensitivity of these trajectories to various disturbances often requires large numbers of simulations, making a fast and reliable reentry propagation tool a valuable asset to reduce computational times. Taylor Series Integration (TSI) is a numerical integration technique, which generates the Taylor series expansions of the state variables to propagate them in time. The generation of the Taylor series is done through automatic differentiation, which rewrites the equations of motion into recurrence relations, with which the Taylor coefficients, i.e., the coefficients that make up a Taylor series expansion, can be obtained up to an arbitrary order. This allows TSI to adapt its order to the accuracy requirements, allowing it to propagate with high efficiency. For TSI, the step size can be computed after the Taylor coefficients are obtained, which means that no time steps have to be rejected during the step size selection. These advantages allowed TSI to be on average 15.8 times faster for celestial mechanics than the Runge-Kutta-Fehlberg integrators that are typically used for this type of problems. The purpose of this MSc. Thesis is to determine whether TSI is also faster for reentry trajectory propagation. TSI requires complete knowledge of the equations of motion, and all the equations that make up the environment and aerodynamic models. Furthermore, this method can only handle purely mathematical expressions, so the discrete data of these models are fitted with regression lines to obtain mathematical expression that approximate the data. Since the tables of the atmosphere model and the tabular form of the aerodynamic coefficients of the reentry vehicle, HORUS2B, were known before integration, regression could be applied to them before integration. Another important feature added to TSI is the step-size reduction in case it integrated over a discontinuity. This way, the integrator does not unnecessarily accumulate errors by integrating an equation set that is no longer valid after the discontinuity. Discontinuities for reentry include the layer boundaries of the atmosphere model, the limits in terms of Mach number of the aerodynamics regression model, the energy levels of the nodes that define the control profiles, bank reversals, and the Terminal Area Energy Management interface. The step-size reductions are done using root-finding methods that were specially selected for each type of discontinuity, most of which make use of the Taylor coefficients of the variables of which the root is to be found. Furthermore, a number of root-finding methods were added to determine when constraints are violated. In previous applications, it was found that the Runge-Kutta-Fehlberg 5(6) (RKF5(6)) method has the best performance for reentry of the traditional integration methods. The state-variable set yielding the lowest computational times for TSI is the spherical set for problems both with and without wind, whereas for RKF5(6), it is the Cartesian set in FR without wind and the spherical set with wind. When comparing the computational times of TSI and this integrator, TSI is faster for all cases, ranging from 1.24 to 4.48 times for trajectories with wind and 3.28 to 11.61 times for trajectories without wind. During optimization with high error tolerance, it was found that TSI also has better accuracy than RKF5(6) and is able to detect all constraint violations. During sensitivity analysis with low error tolerance, it was found that TSI is again more accurate and that the cause for the better accuracy is the fact that TSI adapts its step sizes to avoid integrating over discontinuities.