Constrained Fuel and Time Optimal 6DOF Powered Descent Guidance Using Indirect Optimization

Abstract: Powered descent guidance (PDG) problems subject to six-degrees-of-freedom (6DOF) dynamics allow for enforcement of practical attitude constraints. However, numerical solutions to 6DOF PDG problems are challenging due to fast rotational dynamics coupled with translational dynamics, and the presence of highly nonlinear state/control path inequality constraints. In this work, constrained fuel- and time-optimal 6DOF PDG problems are solved leveraging a regularized indirect method, subject to inequality constraints on the thrust magnitude, thruster gimbal angle, rocket tilt angle, glideslope angle, and angular velocity magnitude. To overcome the challenges associated with solving the resulting multipoint boundary-value problems (MPBVPs), the state-only path inequality constraints (SOPICs) are enforced through an interior penalty function method, which embeds the resulting MPBVPs into a multi-parameter smooth neighboring families of two-point BVPs. Extremal solutions are obtained using an indirect multiple-shooting solution method with numerical continuation. Moreover, an empirical relation is derived for the directly-adjoined Lagrange multipliers associated with SOPICs. The fuel- and time-optimal trajectories are compared against solutions of DIDO -- a capable pseudospectral-based software for solving practical constrained optimal control problems.