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.

翻译：路径生成——即从离散规划输出（如航点序列）生成平滑、可执行路径的问题，是自主机器人、工业机器人及数控机床控制中的基础步骤，因为路径跟踪与轨迹跟踪控制器对其参考输入施加了严格的微分性要求以保障稳定性与收敛性，尤其对于非完整系统而言。平滑化方法最近被提出作为路径生成的一种计算高效且解析可处理工具，可提供形式化的平滑性与曲率保证，相较于样条插值和基于优化的方法具有优势。然而，这种平滑化受限于一个基本几何约束：平滑后的路径被限制在原路径的凸包内，因此无法实现精确的航点插值——即使任务规范或上游规划器明确要求此功能。我们提出方向性平滑化这一新型算子，在保留经典平滑化解析可处理性的同时解决了这一局限。该算子可生成无限可微路径，严格插值预设航点，以任意精度收敛至原始不可微输入，并通过闭式表达式满足显式曲率界限，从而满足受控自主系统路径生成的核心要求。我们进一步建立了路径生成算子的参数化族，其中经典平滑化与方向性平滑化均为特例，为从不可微规划输出中系统生成平滑、可行路径提供了统一的理论框架。