Smooth path planning with safety margins using Piece-Wise Bezier curves

TL;DR

The paper tackles safe, smooth path planning for 2D mobile robots in obstacle-rich environments by introducing a quadratic programming framework that uses piece-wise quadratic Bezier curves (PWB) with explicit safety margins implemented as safe polytopes shifted inward by $\varepsilon$. It enforces $C^1$ continuity and start/end connectivity while balancing path length and curvature through a quadratic objective $J = \sum_{i=0}^M \left( \|P^i_1 - P^i_0\|^2 + \|P^i_2 - P^i_1\|^2 + \lambda \|P^i_2 - P^i_0\|^2 \right)$, enabling a tractable, real-time optimization suitable for embedded platforms. Key contributions include the safe polytope construction with dual variants for seamless connectivity, a linear-constraint, quadratic-cost formulation on Bezier control points, and demonstrated robustness and efficiency advantages over piecewise linear and nonlinear gradient methods in representative scenarios. The approach shows improved trajectory smoothness and safety margins with sub-100 ms computation, indicating strong potential for scalable, safe navigation in cluttered environments. This work thus provides a practical, scalable framework for real-time, safe robotic path planning using smooth curves compatible with embedded controllers and existing standard planners.

Abstract

In this paper, we propose a computationally efficient quadratic programming (QP) approach for generating smooth, $C^1$ continuous paths for mobile robots using piece-wise quadratic Bezier (PWB) curves. Our method explicitly incorporates safety margins within a structured optimization framework, balancing trajectory smoothness and robustness with manageable numerical complexity suitable for real-time and embedded applications. Comparative simulations demonstrate clear advantages over traditional piece-wise linear (PWL) path planning methods, showing reduced trajectory deviations, enhanced robustness, and improved overall path quality. These benefits are validated through simulations using a Pure-Pursuit controller in representative scenarios, highlighting the practical effectiveness and scalability of our approach for safe navigation.

Figures (3)

• Figure 1: Safe and smooth robotic motion example.
• Figure 2: Illustration of connecting safe polytopes, facilitating smooth continuity between adjacent path segments.
• Figure 3: Optimized trajectories for different environment scenarios.