Mobile robots have recently attracted the attention and applicability in field areas ubiquitously. Within the context of autonomous navigation, path planning is relevant for comfortability, safety, execution time and energy savings. In this paper, we propose an approach to suggest smooth paths from observed robot trajectories by optimizing fitting and smoothness criteria using Differential Evolution with distinct modes of initialization, selection pressure, exploration and exploitation. Our rigorous computational experiments using a relevant set of real-world robot trajectories from the Boe-Bot mobile robot architecture show the feasibility and efficiency of our approach in computing smooth curves, suggesting the superior performance of the greedy initialization scheme based on the triangular convex hull of the robot trajectory, and Differential Evolution with exploitative and parameter adaptation schemes such as Rank-Based Differential Evolution (RBDE), Adaptive Differential Evolution with External Archive (JADE) and Strategy Adaptation Differential Evolution (SADE). Our obtained results offer the building blocks to further advance towards developing data-driven curve fitting and path planning algorithms, which may find use in several real-world applications in Robotics and Operations Research.