On the Computation and Accuracy of Trajectory State Transition Matrices

Abstract

State transition matrices provide sensitivities or partial derivatives between states at different times along a trajectory and are used for a number of applications such as feedback controls, stability analysis, estimation, targeting, parameter optimization, and optimal control. The need for accurate state transition matrices is especially important for problems and applications that have highly sensitive and highly nonlinear dynamics. In this paper, examples are considered in the context of multiple-revolution and multiple-body space trajectories. Three techniques to compute both the first- and second-order state transition matrices are compared: 1) augmenting the state with the classic variational equations, 2) complex and bicomplex-step derivative approximation, and 3) multipoint stencils for traditional finite differences. Each of the methods are compared for accuracy and speed across a variety of problems and numerical integration techniques. The subtle differences between variable- and fixed-step integration for partial computation are revealed, common pitfalls are observed, and recommendations are made to enhance the quality of state transition matrices. A main result is the demonstration of small but potentially significant errors in the partials when they are computed with variational equations and a variable-step integrator.