A Fast Semidefinite Convex Relaxation for Optimal Control Problems With Spatio-Temporal Constraints

TL;DR

This paper introduces a fast semidefinite convex relaxation method for solving optimal control problems with spatio-temporal constraints, improving accuracy and efficiency in autonomous agent navigation.

Contribution

It proposes a novel time-scaling multiple shooting scheme combined with a semidefinite relaxation exploiting sparsity, enhancing solution quality and computational speed.

Findings

Demonstrates near-optimal solutions in simulations

Achieves real-time performance in quadrotor experiments

Outperforms traditional nonconvex approaches

Abstract

Solving optimal control problems (OCPs) of autonomous agents operating under spatial and temporal constraints fast and accurately is essential in applications ranging from eco-driving of autonomous vehicles to quadrotor navigation. However, the nonlinear programs approximating the OCPs are inherently nonconvex due to the coupling between the dynamics and the event timing, and therefore, they are challenging to solve. Most approaches address this challenge by predefining waypoint times or just using nonconvex trajectory optimization, which simplifies the problem but often yields suboptimal solutions. To significantly improve the numerical properties, we propose a formulation with a time-scaling direct multiple shooting scheme that partitions the prediction horizon into segments aligned with characteristic time constraints. Moreover, we develop a fast semidefinite-programming-based convex relaxation that exploits the sparsity pattern of the lifted formulation. Comprehensive simulation studies demonstrate the solution optimality and computational efficiency. Furthermore, real-world experiments on a quadrotor waypoint flight task with constrained open time windows validate the practical applicability of the approach in complex environments.