Optimal 3D Soft Landing Guidance With Approach Angle Path Constraints

In this paper, a novel optimal-control-based soft landing guidance law with inequality approach angle path constraints is analytically derived. The proposed guidance law prevents ground collision and allows approach angle control by constraining the optimal trajectory to remain within an upside-down, convex base pyramid, originating from the landing point. A 3D point-mass linear kinematic model with a constant gravitational field is employed, and the optimal control formulation consists of a quadratic running cost on the control effort and terminal-state constraints on the position and velocity. Analytical open-loop and closed-loop solutions, as well as the optimal final time, are obtained by applying Pontryagin’s Minimum Principle and enforcing the optimality conditions at the transitions between the unconstrained and constrained arcs. The derived optimal guidance law is continuous, piecewise linear in time, and nonlinear in the states in closed-loop. When a constraint becomes active, it cancels the gravitational component normal to the constraint, and the trajectory remains on the constraint. The proposed guidance law is evaluated in simulation across various initial conditions, demonstrating excellent performance and consistent satisfaction of the path constraints.