Optimal guidance around circular trajectories for impact angle interception

A new linear optimal guidance law for impact angle interception is presented. The linearization of the non-linear equations of motion is performed around a nominal circular trajectory, thus allowing the linearization to be valid far from the initial line-of-sight. The proposed law can be viewed as a generalization of the optimal rendezvous problem. Closed form expressions for the miss distance, the intercept angle error, and the control effort are derived for the linear model. The transformation from the relative position and the relative velocity, with respect to the circular trajectory, to the relative inscribed angle and the relative inscribed angle rate is presented. Using this transformation, the implementation of the optimal impact angle guidance law based on the inscribed angle is derived. The application in various scenarios enforcing different impact angles is studied via nonlinear simulations, and it is shown that for the investigated parameters the proposed law performs better when compared to biased proportional navigation guidance. The nonlinear simulations also show that for investigated parameters, although the optimality is guaranteed only around the nominal circular trajectory, the proposed guidance law enables to impose the desired impact angles even when the deviations from the nominal trajectory are not small.