3D Impact Time and Angle Control Based on Apollo Guidance and Bézier Curves

This paper proposes the development of a three-stage guidance law for controlling both impact time and impact angle for a missile. While the impact angle law ensures that the missile is directed to the target at the right approach angle, impact time control ensures that the missile hits the target at the desired time. The approach here utilizes Apollo guidance in the first phase, Bézier curve-based impact time control in the second phase, and proportional navigation as a terminal stage. The method initially adjusts the impact angle by aligning the missile’s velocity vector with the impact plane, while impact time control is implemented in the second stage. Previous methods have often focused on two-dimensional systems for impact time control, whereas this work extends the system to a three-dimensional framework. The core idea involves defining an impact plane relative to the target, and by shaping the missile’s path according to this plane, the missile is guided to settle on the predefined impact plane, achieving the desired impact angle through an Apollo descent-inspired guidance approach. To manage impact time control for the 3D engagement, a polynomial-based guidance approach using Bézier curves is employed. In this setup, an acceleration command derived for a two-dimensional engagement is adapted to a three-dimensional engagement through a simple rotational transformation, aligning it with the specified impact plane. Since the guidance law in the second stage involves time-to-go and look-angle terms in the denominator, a proportional navigation guidance phase is added to prevent singularity issues, thereby ensuring an effective collision with the target. The contribution of the proposed method lies in its ability to achieve 3D impact time control, along with the capability to reach the target at the desired approach angle and time. The kinematic engagement equations were used under ideal conditions assuming a stationary target. The performance of the proposed method is given with different impact time and impact plane scenarios.