A Universal Formulation
for
Path-Parametric Planning & Control

Jon Arrizabalaga1,3, Zbyněk ŠÍR2, Zachary Manchester3, Markus Ryll1
1Technical University of Munich, 2Charles University, 3Carnegie Mellon University
Under Review
Paper Code arXiv

Path-parametric ingredients


Path-parametric methods rely on a reference path parameterized by an auxiliary variable, σ, to design planning and control algorithms. Due to the wide range of formulations and applications, the literature on parametric methods remains fragmented, with existing approaches often presented in isolation. To unify these methods, we introduce a universal formulation that reveals the underlying connections, and thereby, brings together existing path-parametric techniques in literature. This formulation consists of two main components (shown in the yellow box): (i) a path-parameterization technique for computing moving frames, and (ii) a spatial projection of the Cartesian system dynamics onto the parametric path, formulated without imposing prior assumptions on the moving frame.

Abstract

We present a unified framework for path-parametric planning and control. This formulation is universal as it standardizes the entire spectrum of path-parametric techniques -- from traditional path following to more recent contouring or progress-maximizing Model Predictive Control and Reinforcement Learning -- under a single framework. The ingredients underlying this universality are twofold: First, we present a compact and efficient technique capable of computing singularity-free, smooth and differentiable moving frames. Second, we derive a spatial path parameterization of the Cartesian coordinates for any arbitrary curve without prior assumptions on its parametric speed or moving frame, and that perfectly interplays with the aforementioned path parameterization method. The combination of these two ingredients leads to a planning and control framework that unites existing path-parametric techniques in literature.

Bringing together all path-parametric formulations

This paper aims to demonstrate the unified nature of path-parametric planning and control methods. We show how these approaches are inherently linked by introducing a universal formulation. Central to this formulation are two fundamental components that underpin all existing works in the literature. These components are: (i) the method of path parameterization, which assigns a singularity-free and continuous moving frame to a give curve, and (ii) the representation of system dynamics relative to this path parameterization, crucial for describing how the system behaves along the specified path.

Why Path-Parametric? A robotic manipulator case-study

To highlight the appeal, practicality and universality of the presented path-parametric framework in designing planning and control algorithms, our analysis is structured into three parts. First, we explore the foundational reasons that led to the development of path-parametric methods by comparing temporal and spatial references. Second, we demonstrate how path parametric methods allow for intricate motions, such as convergence to a path while achieving a predefined velocity profile. Third, we show how these core ideas extend to broader motion planning scenarios, where the desired trajectories are intended to fully exploit the available free space.

Minimizing Time or Maximizing Progress ?

After exploring the fundamental advantages that have propelled the rise of the path-parametric paradigm, we shift our focus to one of its most notable applications: time-optimal navigation. Progress maximization has emerged as a leading approach for minimum-time motion control. Driven by advancements in numerical optimization and embedded solvers, progress maximization based prediction-based controllers have shown very promising results in real-world applications. These controllers are built on path-parametric methods, allowing systems to operate near their performance limits and achieve behavior that closely approximates time-optimality. Additionally, these formulations leverage the capacity to easily impose collision-free constraints, which would otherwise be non-convex or difficult to enforce without the path-parametric structure. The combination of these attributes has made progress maximization the preferred approach for achieving agile performance along a designated reference path. However, while progress maximization serves as an approximation to time minimization, the precise quantification of the gap between these two methods remains unresolved. In this study, we aim to shed some light on this question by performing an experimental comparison of both approaches.

Differentiable Parametric Corridors: A continuous spatial representation

As it is apparent in the previous examples, spatial coordinates derived from path-parametric methods inherently capture the concept of advancement along a path while enabling the imposition of spatial bounds as convex constraints in the orthogonal components of the spatial states. These features make path-parametric methods a compelling toolset for planning and control algorithms in navigation. For instance, in the motion planning example of the robotic manipulator, explicitly representing the admissible region allows the system to efficiently utilize the available space, guiding the end effector along the reference path. Similarly, in the racing scenario, where minimizing time and maximizing progress were compared, an explicit representation of the racetrack enabled controllers to fully exploit the road's width for optimal performance.These examples highlight the potential of parameterizing a system’s motion using spatial coordinates. However, they also raise a critical question: how can one formulate and compute a spatial representation that effectively describes the admissible region around an arbitrary reference path as a function of the orthogonal spatial component, η?

BibTex

@article{arrizabalaga2024universal,
      title={A Universal Formulation for Path-Parametric Planning and Control},
      author={Arrizabalaga, Jon and Ryll, Markus},
      journal={arXiv preprint arXiv:2410.04664},
      year={2024}
      }