Quasi-Interpolation Based Trajectory Optimization (QuITO v.2) with Uniform Error Guarantees under Path Constraints

Belangrijkste concepten

This article introduces a new direct multiple shooting algorithm, QuITO v.2, that provides uniform error guarantees for approximating optimal control trajectories under path constraints by employing a quasi-interpolation scheme on a piecewise uniform grid. The algorithm also includes a wavelet-based change point localization and adaptive mesh refinement mechanism to accurately capture irregularities in the optimal control profile.

Samenvatting

The key highlights and insights of this article are:

QuITO v.2 is a direct multiple shooting algorithm that parameterizes the control trajectory using a quasi-interpolation scheme on a piecewise uniform grid. This allows for tight uniform error guarantees between the approximate and the optimal control trajectories.

The algorithm employs a wavelet-based change point localization mechanism to identify regions of irregularity in the optimal control profile, such as discontinuities, kinks, or narrow peaks/valleys.

Based on the localized regions, the algorithm adaptively refines the mesh around those areas to better capture the control profile, while keeping the mesh coarse elsewhere. This saves computational effort compared to solving on a very fine uniform mesh.

Theoretical results are provided to establish the uniform approximation guarantees and the ability of the wavelet-based localization to identify all change points under mild assumptions.

Numerical examples are presented demonstrating the effectiveness of QuITO v.2 in accurately approximating optimal control trajectories with features like bang-bang or singular controls, outperforming state-of-the-art direct collocation methods.

The algorithms developed in this article have been implemented in a software package called QuITO v.2, which includes a graphical user interface for ease of use.

Statistieken

The article does not contain any explicit numerical data or statistics. The key results are theoretical guarantees on uniform approximation error and change point localization.

Citaten

"For every uniform error margin ε > 0, one can pick a triplet (h, D, ρ) ∈ (0,∞)^3 such that ‖u^*(·) - û_G,D(·)‖_u ≤ ε."
"The approximation error is measured in the uniform sense, and can be controlled via the parameters (h, D, ρ) ∈ (0,∞)^3."

Belangrijkste Inzichten Gedestilleerd Uit

$\mathsf{QuITO}$ $\textsf{v.2}$: Trajectory Optimization with Uniform Error Guarantees under Path Constraints

by Siddhartha G... om arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.13681.pdf

Diepere vragen

How can the quasi-interpolation scheme and the mesh refinement strategy in QuITO v.2 be extended to handle stochastic optimal control problems with path constraints

To extend the quasi-interpolation scheme and mesh refinement strategy in QuITO v.2 to handle stochastic optimal control problems with path constraints, we need to incorporate probabilistic elements into the algorithm. In stochastic optimal control, the dynamics of the system and the constraints are influenced by random variables.

Quasi-Interpolation Scheme:

The quasi-interpolation scheme can be adapted to handle stochasticity by incorporating random variables into the generation of control trajectories. This would involve modifying the interpolation process to account for the uncertainty in the system dynamics.
Instead of deterministic control inputs, the quasi-interpolation scheme would generate control trajectories that are probabilistic in nature, taking into consideration the stochastic elements of the problem.

Mesh Refinement Strategy:

In the context of stochastic optimal control, the mesh refinement strategy would need to dynamically adjust the time grid based on the evolving uncertainty in the system.
Regions of high uncertainty or variability in the state-action trajectories could be identified using probabilistic measures, and the mesh could be refined in these areas to ensure accurate representation of the stochastic optimal control solution.

By integrating probabilistic elements into the quasi-interpolation scheme and mesh refinement strategy, QuITO v.2 can be extended to effectively handle stochastic optimal control problems with path constraints.

Can the wavelet-based change point localization technique be adapted to identify regions of high curvature or other features in the state trajectory, in addition to the control profile

The wavelet-based change point localization technique in QuITO v.2 can be adapted to identify regions of high curvature or other features in the state trajectory by leveraging the properties of wavelet analysis. Wavelets are well-suited for detecting abrupt changes, sharp transitions, and high-frequency components in signals, making them versatile tools for analyzing complex data patterns.

High Curvature Detection:

Wavelets can be used to identify regions of high curvature in the state trajectory by focusing on wavelet coefficients that capture rapid changes in the signal.
Localized wavelet analysis can highlight areas where the state trajectory exhibits significant variations or sharp turns, indicating regions of high curvature.

Other Features Detection:

Wavelet transform can also be utilized to detect specific features in the state trajectory, such as spikes, oscillations, or irregular patterns.
By analyzing the wavelet coefficients at different scales, the change point localization technique can be adapted to identify various characteristics of the state trajectory beyond change points.

By customizing the wavelet-based approach in QuITO v.2 to target high curvature regions and other specific features in the state trajectory, the algorithm can provide valuable insights into the behavior of the system and improve the accuracy of the control optimization process.

What are the potential applications of the uniform approximation guarantees provided by QuITO v.2 in safety-critical domains like aerospace or robotics, where precise tracking of control inputs is crucial

The uniform approximation guarantees provided by QuITO v.2 have significant applications in safety-critical domains like aerospace or robotics, where precise tracking of control inputs is essential for ensuring system stability and performance.

Aerospace Applications:

In aerospace systems, precise control of aircraft, spacecraft, or drones is crucial for safe and efficient operation. The uniform approximation guarantees offered by QuITO v.2 can help in designing optimal control strategies that meet stringent performance requirements while ensuring stability under varying conditions.
Applications include trajectory optimization for space missions, flight path planning, and autonomous flight control systems where accurate control inputs are necessary for mission success.

Robotics Applications:

In robotics, especially in scenarios involving robotic manipulators or autonomous vehicles, precise control of motion trajectories is vital to avoid collisions, achieve desired tasks, and ensure safety.
QuITO v.2's uniform approximation guarantees can be utilized in path planning, obstacle avoidance, and motion control algorithms to enhance the accuracy and reliability of robotic systems in dynamic environments.

By leveraging the uniform approximation guarantees provided by QuITO v.2, engineers and researchers in aerospace and robotics can optimize control strategies with confidence, leading to improved performance, safety, and efficiency in critical applications.