Constrained Fuel- and Time-Optimal Six-Degree-of-Freedom Powered Descent Guidance Using Indirect Optimization

Powered descent guidance (PDG) problems subject to six-degree-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 open-loop 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. Pure state-path inequality constraints (SICs) result in a multipoint boundary-value problem, which can be challenging to solve. To overcome this, the SICs are enforced with interior penalty functions, resulting in a multiparameter family of two-point boundary-value problems solved with multiple shooting and numerical continuation. An empirical relation is derived for the directly adjoined Lagrange multipliers associated with SICs. The fuel- and time-optimal trajectories are compared against solutions from a general pseudospectral-based collocation optimal control problem solver.