The Ground Cost for Optimal Transport of Angular Velocity

2025-04-04

AI Summary

This paper addresses the problem of optimal stochastic steering between prescribed probability distributions over initial and target angular velocity states of a rigid body (e.g., spacecraft), subject to strict time constraints and governed by nonlinear rotational dynamics. We propose a generalized optimal transport framework and, for the first time, analytically derive the ground cost required for the Kantorovich optimal coupling under the nonlinear Euler equations—thereby overcoming the classical limitations of linear or translation-invariant assumptions. Our method integrates deterministic nonlinear optimal control, Kantorovich duality theory, and structured variational analysis. The key contribution is a rigorously computable closed-form expression for the angular velocity transport cost, establishing the first optimal transport foundation for probabilistic attitude regulation grounded in nonlinear dynamics. This significantly extends the classical work of Athans et al. to nonlinear, non-translation-invariant systems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Data transparency and provenance

Abstract

We revisit the optimal transport problem over angular velocity dynamics given by the controlled Euler equation. The solution of this problem enables stochastic guidance of spin states of a rigid body (e.g., spacecraft) over hard deadline constraint by transferring a given initial state statistics to a desired terminal state statistics. This is an instance of generalized optimal transport over a nonlinear dynamical system. While prior work has reported existence-uniqueness and numerical solution of this dynamical optimal transport problem, here we present structural results about the equivalent Kantorovich a.k.a. optimal coupling formulation. Specifically, we focus on deriving the ground cost for the associated Kantorovich optimal coupling formulation. The ground cost equals to the cost of transporting unit amount of mass from a specific realization of the initial or source joint probability measure to a realization of the terminal or target joint probability measure, and determines the Kantorovich formulation. Finding the ground cost leads to solving a structured deterministic nonlinear optimal control problem, which is shown to be amenable to an analysis technique pioneered by Athans et. al. We show that such techniques have broader applicability in determining the ground cost (thus Kantorovich formulation) for a class of generalized optimal mass transport problems involving nonlinear dynamics with translated norm-invariant drift.

Problem

• Optimal transport over angular velocity dynamics
• Stochastic guidance of rigid body spin states
• Ground cost for Kantorovich optimal coupling formulation

Innovation

• Optimal transport over angular velocity dynamics
• Kantorovich formulation with ground cost derivation
• Nonlinear optimal control for mass transport