Directional Mollification for Controlled Smooth Path Generation

Abstract

Path generation, the problem of producing smooth, executable paths from discrete planning outputs, such as waypoint sequences, is a fundamental step in the control of autonomous robots, industrial robots, and CNC machines, as path following and trajectory tracking controllers impose strict differentiability requirements on their reference inputs to guarantee stability and convergence, particularly for nonholonomic systems. Mollification has been recently proposed as a computationally efficient and analytically tractable tool for path generation, offering formal smoothness and curvature guarantees with advantages over spline interpolation and optimization-based methods. However, this mollification is subject to a fundamental geometric constraint: the smoothed path is confined within the convex hull of the original path, precluding exact waypoint interpolation, even when explicitly required by mission specifications or upstream planners. We introduce directional mollification, a novel operator that resolves this limitation while retaining the analytical tractability of classical mollification. The proposed operator generates infinitely differentiable paths that strictly interpolate prescribed waypoints, converge to the original non-differentiable input with arbitrary precision, and satisfy explicit curvature bounds given by a closed-form expression, addressing the core requirements of path generation for controlled autonomous systems. We further establish a parametric family of path generation operators that contains both classical and directional mollification as special cases, providing a unifying theoretical framework for the systematic generation of smooth, feasible paths from non-differentiable planning outputs.

Figures (6)

• Figure 1: Example of mollification of the function $x \in \mathbb R \mapsto f(x)=|x|$ (black plot). The mollified function $F_{0.5}$ (green plot) is as in Theorem \ref{['thm:OurReferenceTheorem']} with the mollifier in Example \ref{['example:OurMollifier']}. We could consider the original (black) curve as an illustrative case of a linear interpolation that includes the origin $(0,0)$, $(-1,1)$ and $(1,1)$ as waypoints As it can be seen, the mollified (green) curve is contained in the convex hull of the original curve and is above it as Theorem \ref{['thm:OurReferenceTheorem']} states, but it does not include the origin. This is a trade-off between smoothness, convergence, and the approximation provided by the conventional mollification with \ref{['eq:OurMollifier']}. Nevertheless, it will be shown in \ref{['sec:ApplicationThreePointTwoSegments']} that for this kind of curve, our directional mollification method will generate a smooth curve that inherits the convergence and smoothness properties of the conventional mollification while intersecting all the (way)points that define the original curve. This is illustrated by the red plot, with the directional mollification $\widehat{F}_{0.5}$ introduced and studied in this paper.
• Figure 1: Representation of the conventional and directional mollification of the strictly increasing function $f(x)= x\mathop{\mathrm{ind}}\nolimits_{(-\infty,0)}(x) + 30x\mathop{\mathrm{ind}}\nolimits_{[0,\infty)}(x)$. The black plot represents the original function $f$, while the green one is its conventional mollification, and the red one its directional mollification, both with $\varepsilon = 0.5$, and mollifier of \ref{['example:OurMollifier']}.
• Figure 1: Representation of the conventional and directional mollification approaches for a curve defined by linearly interpolating three points in $\mathbb R^2$. The black plot represents the original function $f$, while the green and red ones represent, respectively, the conventional and directional mollification of $f$, with $\varepsilon = 0.5$, and the mollifier of \ref{['eq:OurMollifier']}
• Figure 1: Comparison and representation of a family of curves using \ref{['eq:Combination']}, where the mollifier used is the one presented in \ref{['example:OurMollifier']} with $\varepsilon = 0.4$. The left picture represents in black the path created by linearly interpolating points in $\mathbb R^2$ represented as black dots. It also shows in blue a cubic spline, in red a B-spline, and in green a Quintic Hermite polynomial spline. All of them are created such that they intersect the waypoints. The right picture also represents as a black plot the original function, while in red and blue the conventional and directional mollification of $f$, respectively. The dashed green lines represent $G_{\varepsilon}^{\gamma}$ as in \ref{['eq:Combination']}, for the values $\gamma \in \{-1,-0.5,0,0.5,1,1.5,2\}$.
• Figure 2: Representation of the directional mollification of the function $t \in [0,2\pi] \mapsto (2+\cos(2t))(\cos(t),\sin(t)) \in \mathbb R^2$, where the used mollifier $\varphi$ is as in \ref{['eq:OurMollifier']}. Black line represents the original function $f$, while the blue and red lines the directional mollified $\widehat{F}_{\varepsilon}$ for $\varepsilon = 1.25$ and $\varepsilon = 5$, respectively.

Theorems & Definitions (38)

• Definition 3.1: Mollifier
• Example 3.2
• Theorem 3.3: Evans2022-PDE
• Theorem 3.4: gonzalezcalvin2025efficientgenerationsmoothpaths
• Definition 3.5
• Theorem 3.6: jahn2007introduction
• Definition 4.2: Directional derivative term
• Remark 4.3
• Definition 4.4: Directional mollification
• Theorem 4.5