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

Nicholas P. Nurre, Ehsan Taheri

TL;DR

The authors address constrained 6DOF PDG by formulating fuel- and time-optimal problems with state-path inequality constraints (SOPICs) on angular velocity, glideslope, tilt, thrust, and gimbal angle. They develop a regularized indirect optimization framework that enforces SOPICs with interior-secant penalties, derives closed-form thrust and steering expressions under a gimbal constraint, and solves the resulting MPBVPs via an indirect multiple-shooting scheme with numerical continuation. The approach yields high-accuracy fuel- and time-optimal trajectories and shows close agreement with DIDO, while providing insights into the Lagrange multipliers associated with SOPICs through empirical validation and Breakwell-type comparisons. The work demonstrates that indirect methods can solve challenging, constrained 6DOF PDG problems and offers practical guidance for leveraging SOPICs in real-time-like settings, including robustness checks via multiplier behavior.

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.

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

TL;DR

The authors address constrained 6DOF PDG by formulating fuel- and time-optimal problems with state-path inequality constraints (SOPICs) on angular velocity, glideslope, tilt, thrust, and gimbal angle. They develop a regularized indirect optimization framework that enforces SOPICs with interior-secant penalties, derives closed-form thrust and steering expressions under a gimbal constraint, and solves the resulting MPBVPs via an indirect multiple-shooting scheme with numerical continuation. The approach yields high-accuracy fuel- and time-optimal trajectories and shows close agreement with DIDO, while providing insights into the Lagrange multipliers associated with SOPICs through empirical validation and Breakwell-type comparisons. The work demonstrates that indirect methods can solve challenging, constrained 6DOF PDG problems and offers practical guidance for leveraging SOPICs in real-time-like settings, including robustness checks via multiplier behavior.

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.
Paper Structure (13 sections, 43 equations, 31 figures, 2 tables)

This paper contains 13 sections, 43 equations, 31 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Dynamical Model
  3. Constrained Fuel- and Time-Optimal Trajectory Optimization Problems
  4. Derivation of Necessary Conditions
  5. Regularization of Control Expressions
  6. Fuel- and Time-Optimal Regularized Two-Point Boundary-Value Problems
  7. Results
  8. Fuel-Optimal Solution
  9. Time-Optimal Solution
  10. Analysis of the Lagrange Multipliers associated with State-Only Path Inequality Constraints
  11. 6DOF PDG State-Only Path Constraints Lagrange Multipliers
  12. Breakwell Problem
  13. Conclusion

Figures (31)

  • Figure 1: Definitions of the inertial and body reference frames and some of the variables and parameters.
  • Figure 2: Schematic of the gimbal constraint.
  • Figure 3: Schematic of the indirect multiple-shooting solution scheme.
  • Figure 4: Fuel-optimal: trajectory solution and its projection on different inertial planes.
  • Figure 5: Fuel-optimal: state time histories.
  • ...and 26 more figures

Theorems & Definitions (3)

  • Remark
  • Conjecture 1
  • Remark