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A Review of Pseudospectral Optimal Control: From Theory to Flight

I. M. Ross, M. Karpenko

TL;DR

The paper surveys pseudospectral optimal control within Sobolev spaces ($W^{m,p}$) and the Covector Mapping Principle (CMP), connecting theory to flight practice and onboard implementation. It develops convergence and consistency results, and details design choices ($\\delta^N$, $\\pi^N$, $W$) and spectral algorithms (as in DIDO/OTIS) that enable reliable solutions to problems with endpoint and path constraints, such as those encountered in aerospace trajectories. Ground and flight demonstrations, including Bedrossian’s zero-propellant maneuver and NASA TRACE STM, illustrate singularity handling, momentum management, and telemetry validation, while embedded PS controllers demonstrate real-time feasibility and SWaP considerations. The work highlights significant practical impact by enabling high-speed, certified optimal trajectories on aircraft, spacecraft, and autonomous systems, and outlines future directions toward hybrid PS, stronger theory under weaker assumptions, and compact embedded hardware solutions.

Abstract

The home space for optimal control is a Sobolev space. The home space for pseudospectral theory is also a Sobolev space. It thus seems natural to combine pseudospectral theory with optimal control theory and construct ``pseudospectral optimal control theory,'' a term coined by Ross. In this paper, we review key theoretical results in pseudospectral optimal control that have proven to be critical for a successful flight. Implementation details of flight demonstrations onboard NASA spacecraft are discussed along with emerging trends and techniques in both theory and practice. The 2011 launch of pseudospectral optimal control in embedded platforms is changing the way in which we see solutions to challenging control problems in aerospace and autonomous systems.

A Review of Pseudospectral Optimal Control: From Theory to Flight

TL;DR

The paper surveys pseudospectral optimal control within Sobolev spaces () and the Covector Mapping Principle (CMP), connecting theory to flight practice and onboard implementation. It develops convergence and consistency results, and details design choices (, , ) and spectral algorithms (as in DIDO/OTIS) that enable reliable solutions to problems with endpoint and path constraints, such as those encountered in aerospace trajectories. Ground and flight demonstrations, including Bedrossian’s zero-propellant maneuver and NASA TRACE STM, illustrate singularity handling, momentum management, and telemetry validation, while embedded PS controllers demonstrate real-time feasibility and SWaP considerations. The work highlights significant practical impact by enabling high-speed, certified optimal trajectories on aircraft, spacecraft, and autonomous systems, and outlines future directions toward hybrid PS, stronger theory under weaker assumptions, and compact embedded hardware solutions.

Abstract

The home space for optimal control is a Sobolev space. The home space for pseudospectral theory is also a Sobolev space. It thus seems natural to combine pseudospectral theory with optimal control theory and construct ``pseudospectral optimal control theory,'' a term coined by Ross. In this paper, we review key theoretical results in pseudospectral optimal control that have proven to be critical for a successful flight. Implementation details of flight demonstrations onboard NASA spacecraft are discussed along with emerging trends and techniques in both theory and practice. The 2011 launch of pseudospectral optimal control in embedded platforms is changing the way in which we see solutions to challenging control problems in aerospace and autonomous systems.

Paper Structure

This paper contains 26 sections, 6 theorems, 55 equations, 25 figures, 1 table.

Table of Contents

  1. Introduction
  2. Practical Optimal Control: Challenges and Curses
  3. Search Space
  4. Problem Formulation
  5. Necessary and Sufficient Conditions
  6. Challenges and Curses
  7. Pseudospectral Foundations
  8. Domain Transformation
  9. Interpolation
  10. Differentiation
  11. Integration
  12. Overview of Pseudospectral Optimal Control
  13. Theoretical Overview
  14. Design of Pseudospectral Methods
  15. Choosing the consistency parameter, $\delta^N$
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\mathbb R \supset [t_0, t_f] \mapsto y(t) \in \mathbb R$ be a continuous function. Then, there exists an algebraic polynomial sequence, $\left\{t \mapsto y^N\right\}_{N=0}^\infty$ on $[t_0, t_f]$ such that,

Figures (25)

  • Figure 1: Flight results of TRACE's minimum-time maneuver (left) in projected "quaternion space" illustrating its analogy with the classical Brachistochrone problem (right).
  • Figure 2: "Snapshots" of the shortest-time (left) and industry-heritage (right) maneuvers reconstructed from TRACE's telemetry data of the 2010 flight demonstrations.
  • Figure 3:
  • Figure 4: An arbitrary grid, $\pi^N$, for $N = 7$, and the Lagrange polynomial $\phi_2(t)$ over $\pi^N$.
  • Figure 5: A weighted interpolating polynomial over the same arbitrary grid, $\pi^N$ of Fig. \ref{['fig:Arbitrary_grid']}.
  • ...and 20 more figures

Theorems & Definitions (9)

  • Theorem 1: Stone-Weierstrass
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2: Gong-Kang-Ross-Fahroo
  • Theorem 3: Weierstrass
  • Theorem 4: Arzelà-Ascoli
  • Theorem 5: Jackson
  • Theorem 6: Kang-Ross-Gong