Analytical impact-time guidance law under varying velocity and field-of-view constraint

Achieving precise impact-time control under proportional navigation guidance (PNG) is challenging due to the lack of closed-form time-to-go solutions, especially under varying velocity and field-of-view (FOV) constraint. This paper presents an analytically tractable impact-time control guidance (ITCG) law addressing these coupled challenges. First, a trajectory-increment factor is introduced to quantify how a guidance law elongates the flight path. Based on this metric, an explicit PNG impact-time expression is derived, and a varying-coefficient PNG law that retains conventional PNG’s smooth, low-effort behavior while enabling explicit impact-time computation is formulated. Through a trajectory-consistency theorem, the geometric-invariance condition needed to unify constant- and varying-velocity cases is established. By designing a bias term that both ensures FOV compliance and eliminates residual impact-time error, a unified ITCG law for both the two- and three-dimensional engagements is proposed. It provides a fully analytical guidance solution that accounts for varying velocity and FOV constraint, and eliminates the need for time-to-go estimation or numerical integration. Extensive numerical simulations demonstrate that the proposed method reduces the computation time and cumulative guidance effort relative to the optimal and varying-gain methods.