Differential flatness in optimization-based motion planning and control

Abstract

Differential flatness has demonstrated numerous advantages and diverse applications in motion planning and control and while traditionally associated with robotic and mechanical systems, its applicability extends beyond these domains. By employing a change in the coordinate system, differential flatness transforms complex, nonlinear system dynamics into a linear representation. In essence, it provides a model inversion strategy that proves particularly useful for solving problems related to optimal trajectory generation and tracking under various constraints.

This work offers an overview of past and current advancements in optimization-based motion planning problems that employ differential flatness. To the best of our knowledge, no prior review has been conducted from this perspective. We begin with an in-depth analysis of the foundational principles of flatness, examining its key properties and its intrinsic connection to feedback linearization. Particular emphasis is placed on the available choices for flat parameterization, which facilitate the characterization of costs and constraints in optimization problems. Furthermore, we explore tools for curve parameterization, which are essential for addressing trajectory generation problems involving constraints such as saturation limits, collision avoidance or obstacle navigation. Our review is structured around the general framework of motion planning, which comprises three main components: guidance, control, and adaptation. For each of these components, we analyze and synthesize the use of differential flatness.

Rather than aiming for an exhaustive catalog or focusing narrowly on specific applications, this review seeks to highlight the benefits of differential flatness and aims to inspire new research directions in motion planning and control.