Efficient Nonlinear Spacecraft Trajectory Optimization via Overparameterized Monomial Coordinates and Fundamental Solution Expansions

Core Concepts

This paper introduces a framework to pose the nonlinear trajectory optimization problem as a path-planning problem in a space liberated of dynamics, by leveraging nonlinear expansion in terms of fundamental solutions and a minimal nonlinear basis of mixed monomials in problem initial conditions.

Abstract

The paper introduces a framework to efficiently solve nonlinear trajectory optimization problems by reformulating them as path-planning problems in a space of overparameterized monomial coordinates.
Key highlights:

The nonlinear trajectory optimization problem is posed as a path-planning problem in a space liberated of dynamics, by leveraging nonlinear expansion in terms of fundamental solutions and a minimal nonlinear basis of mixed monomials in problem initial conditions.
This framework enables a stable and highly rapid nonlinear guidance implementation without the need for collocation or real-time integration.
The overparameterized monomial coordinates are stationary in the absence of control, allowing the dynamics and kinematics to be encoded as linear constraints on the monomial states.
A successive convex programming scheme is proposed for delta-V minimizing trajectory optimization, leveraging the linear relationship between the monomial states and the control inputs.
The method is demonstrated on a nonlinear spacecraft rendezvous problem, showing a two-stage guidance approach (linear prediction, nonlinear correction) and a successive convex programming implementation.

Stats

The paper does not contain any explicit numerical data or statistics. The focus is on the theoretical framework and algorithmic developments.

Quotes

"This work enables a stable and highly rapid nonlinear guidance implementation without the need for collocation or real-time integration."
"The overparameterized monomial coordinates are stationary in the absence of control, allowing the dynamics and kinematics to be encoded as linear constraints on the monomial states."
"A successive convex programming scheme is proposed for delta-V minimizing trajectory optimization, leveraging the linear relationship between the monomial states and the control inputs."

Key Insights Distilled From

Rapid nonlinear convex guidance via overparameterized monomial coordinates and fundamental solution expansions

by Ethan R. Bur... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2403.19324.pdf

Deeper Inquiries

How can this framework be extended to handle more complex nonlinear dynamics, such as those involving aerodynamic forces or other environmental perturbations

To extend this framework to handle more complex nonlinear dynamics, such as those involving aerodynamic forces or other environmental perturbations, we can incorporate additional terms in the fundamental solution expansions. By including terms that account for these perturbations, we can capture the effects of aerodynamic forces or other environmental factors on the spacecraft's trajectory. This would involve computing the nonlinear fundamental solutions with these additional perturbations considered, allowing for a more comprehensive representation of the dynamics. Additionally, the monomial coordinates can be expanded to include variables that capture the influence of these perturbations, enabling a more accurate and detailed description of the system's behavior in the presence of these complexities.

What are the potential limitations or drawbacks of the overparameterized monomial coordinate representation, and how can they be addressed

One potential limitation of the overparameterized monomial coordinate representation is the challenge of ensuring the accuracy and convergence of the nonlinear expansions, especially as the order of expansion increases. As the complexity of the dynamics grows, higher-order expansions may be required, leading to increased computational complexity and potential numerical instability. To address this, careful consideration must be given to the choice of expansion order and the accuracy of the fundamental solutions. Additionally, techniques such as regularization methods or adaptive refinement of the expansion order can help mitigate these challenges and improve the robustness of the framework. Furthermore, validation and verification procedures should be implemented to ensure the reliability of the results obtained using the overparameterized monomial coordinates.

What other applications beyond spacecraft trajectory optimization could benefit from this type of nonlinear expansion and path-planning approach

Beyond spacecraft trajectory optimization, this type of nonlinear expansion and path-planning approach could benefit a wide range of applications in various fields. For example, in robotics, this framework could be utilized for motion planning of robotic systems operating in complex environments with nonlinear dynamics. By encoding the system's state constraints as linear constraints on overparameterized variables, the framework can enable rapid and efficient trajectory optimization for robotic motion planning tasks. Additionally, applications in autonomous vehicles, control systems, and optimization problems in physics and engineering could also benefit from the computational efficiency and stability offered by this approach. The ability to pre-compute useful information and leverage nonlinear expansions for guidance and optimization tasks opens up opportunities for implementing robust and rapid decision-making algorithms in a variety of domains.