Frenkel & Shaferman's pyramid guidance law prevents lander crashes with optimal control

A new analytical guidance law for powered descent landings uses pyramid-shaped approach-angle constraints to ensure safe touchdown. The paper by Frenkel and Shaferman (Technion) models a 3D point-mass linear kinematic system under constant gravity with a quadratic control-effort cost and terminal position/velocity constraints. The key innovation is an inverted pyramid constraint originating at the landing point: the trajectory must remain inside this convex region, preventing ground collisions and allowing precise glideslope control. Using Pontryagin's Minimum Principle, the authors derive optimal open- and closed-loop solutions, including the optimal final time. They show that when path constraints become active, the controller cancels the gravitational component normal to the constraint surface, causing the vehicle to slide along it. The resulting guidance law is continuous, piecewise linear in time, and nonlinear in states in closed-loop. Notably, activating the constraints reduces the optimal final time, a useful property for time-critical landings.

Simulations under diverse initial conditions confirm that the guidance consistently satisfies the pyramid constraints while achieving accurate terminal conditions. Unlike prior work that used inequality constraints on altitude or cone angles independently, this method unifies approach-angle limits in a simple geometric form. The analytical nature of the solution makes it computationally lightweight and suitable for real-time onboard implementation. This work has implications for planetary landers (e.g., Moon or Mars missions), drone precision landing, and any vertical takeoff/landing vehicle that must avoid terrain while controlling approach angle. By providing a closed-form control law with provable constraint satisfaction, the research bridges the gap between optimal control theory and practical flight guidance systems.