A Universal Formulation for Path-Parametric Planning and Control

TL;DR

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.

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.

Figures (26)

• Figure 1: Path-parametric methods rely on a reference path parameterized by an auxiliary variable, $\sigma$. These methods can be classified based on two key aspects: the system states and the navigation criterion (as indicated by the white boxes). However, the literature on parametric methods remains fragmented, with existing approaches often presented in isolation. To unify these methods, we introduce a universal formulation that highlights their underlying connections. 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.
• Figure 2: Comparison of Frenet Serret Frame (FSF, first column) and Parallel Transport Frame (PTF, second column) for a two-dimensional planar curve (first row) and three-dimensional spatial curve (second row). The first, second and third components of the moving frame are shown in red, green and blue, respectively. The third colum shows the angular velocity of the moving frames.
• Figure 3: Numerical validation of the continuity analysis conducted in eqs. \ref{['eq:angvel_cont']}, namely that a reference curve $\bm{\gamma}$ that is $C^n$ relates to an angular velocity $\bm{\omega}_\text{PTF}$ that is $C^{n-2}$. For this purpose we divide an exemplary curve into two sections and interpolate with various continuity degree ($C^0$ to $C^4$ from top to bottom). The left column shows the exemplary curve, while the remaining columns depict the angular velocity $\bm{\omega}^\Gamma$, acceleration $\bm{\omega}^\Gamma$ and jerk $\bm{\omega}^\Gamma$. The intersection is given by the red dot located in the middle of the curve at $\theta=0.5$. The evaluations associated to the first and second sections are depicted in blue and orange. The boxes in the upper right side of each plot provide a mode detailed look into the intersection, allowing us to differ the continuous and discontinuous cases. Additionally, the continuous cases have been labelled by a green tick, while the discontinuous ones are marked by a red cross.
• Figure 4: Spatial projection of the three-dimensional Cartesian coordinates $\bm{p}^W$, represented by the pink dot, onto a geometric path $\Gamma$ with an associated adapted-frame $\text{R}(\xi) = \{\bm{e_1}(\xi),\bm{e_2}(\xi),\bm{e_3}(\xi)\}$. The distance to the closest point on the path $\bm{\gamma}(\xi)$ is decomposed into the transverse coordinates $\bm{\eta} = \left[\eta_1,\eta_2\right]$.
• Figure 5: Comparison between a temporal and spatial reference-tracking for a two joint robotic manipulator when traversing a sinusoidal path. The right side depicts three successive sequences for motions of the robotic manipulator in a nominal scenario, without disturbances. In the right, we show the trajectories obtained with both methods in the presence of a disturbance. In the left, the temporal reference keeps on progressing while the disturbance is happening, forcing it to catch-up, resulting in a large deviation. In comparison, the spatial reference only depends on its location, and thus, is able to resume without generating a large error.