Optimal Control of Differentially Flat Systems is Surprisingly Easy

TL;DR

This paper introduces a simplified and efficient method for solving optimal control problems in differentially flat systems, reducing complexity and numerical issues, demonstrated on a planar manipulator in real-time.

Contribution

The paper presents a novel approach leveraging differential flatness to simplify optimal control equations and enable real-time trajectory generation for complex CPS.

Findings

Achieves optimal trajectory in 4.5 ms in simulation

Handles both constrained and unconstrained cases

Successfully switches between different joint configurations

Abstract

As we move to increasingly complex cyber-physical systems (CPS), new approaches are needed to plan efficient state trajectories in real-time. In this paper, we propose an approach to significantly reduce the complexity of solving optimal control problems for a class of CPS with nonlinear dynamics. We exploit the property of differential flatness to simplify the Euler-Lagrange equations that arise during optimization, and this simplification eliminates the numerical instabilities that plague optimal control in general. We also present an explicit differential equation that describes the evolution of the optimal state trajectory, and we extend our results to consider both the unconstrained and constrained cases. Furthermore, we demonstrate the performance of our approach by generating the optimal trajectory for a planar manipulator with two revolute joints. We show in simulation that our approach is able to generate the constrained optimal trajectory in $4.5$ ms while respecting workspace constraints and switching between a `left' and `right' bend in the elbow joint.