Optimal-Command-Direction Guidance with Fixed-Time-Convergent Relative Error Dynamics

Addressing the challenge of intercepting high-speed and highly maneuvering endoatmospheric targets, this paper dissects the issue into two components: the command direction and the magnitude design of the interception guidance law. For the former, through geometric and algebraic analysis, an optimal command direction is determined for the generalized differential geometric guidance law. This direction not only satisfies the constraint that the command acceleration vector is perpendicular to the velocity, but also meets the requirement of countering target maneuvers by maximizing the projection of the command acceleration along the line-of-sight normal. For the latter, tackling the matter of nonconvergence in zero-effort-miss (ZEM) due to high-speed engagements and significant target maneuvers, a fixed-time-convergent relative error dynamics (FxT-RED) method is proposed. This method introduces the relative error quantity and refines the parameters form, in turn, offering benefits, i.e., fewer parameters, a more accurate upper bound on the settling time, and stable convergent behavior of the relative error. Incorporating both, an optimal-command-direction guidance law with FxT-RED is designed, which ensures the ZEM convergence before terminal time, despite various target maneuver disturbances, validated by the theoretical proof and numerical simulations.