Efficient Generation of Smooth Paths with Curvature Guarantees by Mollification

Most path following and trajectory tracking algorithms in mobile robotics require the desired path or trajectory to be defined by at least twice continuously differentiable functions to guarantee key properties such as global convergence, especially for nonholonomic robots like unicycles with speed constraints. Consequently, these algorithms typically exclude continuous but non-differentiable paths, such as piecewise functions. Despite this exclusion, such paths provide convenient high-level inputs for describing robot missions or behavior. While techniques such as spline interpolation or optimization-based methods are commonly used to smooth non-differentiable paths or create feasible ones from sequences of waypoints, they either can produce unnecessarily complex trajectories or are computationally expensive. In this work, we present a method to regularize non-differentiable functions and generate feasible paths through mollification. Specifically, we approximate an arbitrary path with a differentiable function that can converge to it with arbitrary precision. Additionally, we provide a systematic method for bounding the curvature of generated paths, which we demonstrate by applying it to paths resulting from linking a sequence of waypoints with segments. The proposed approach is computationally efficient, enabling real-time implementation on microcontrollers and compatibility with standard trajectory tracking and path following algorithms.