Safe and Efficient Trajectory Planning Under Control Barrier Function Constraints with Bernstein Polynomials

Abstract

Autonomous mobile robots (AMRs) are increasingly deployed in dynamic, cluttered environments, where ensuring safety without sacrificing real-time computational efficiency remains a critical challenge. Nonlinear Model Predictive Control (NMPC) combined with Discrete-Time Control Barrier Functions (DCBFs) provides a rigorous framework for safety-critical navigation. However, standard approaches impose safety constraints at every prediction step, causing the number of constraints to grow linearly with the horizon length. This often leads to excessive computational overhead and feasibility issues, particularly in complex scenarios that require long-horizon planning. To address these limitations, this thesis proposes a Bernstein polynomial-based trajectory planning framework termed Bernstein-Polynomial NMPC with Discrete-Time CBFs (BP-NMPC-DCBF). Rather than enforcing safety constraints at every prediction step, the proposed method parameterizes the reference trajectory using Bernstein polynomials. By exploiting the convex hull property of Bernstein polynomials, the framework enforces safety constraints only on a finite set of control points within an online-constructed safe corridor. This strategy ensures forward invariance of the entire trajectory while decoupling the complexity of safety verification from the prediction horizon length, reducing the constraint complexity from O(mN) to O(m(d+1)). The proposed algorithm is validated through extensive simulations in narrow passage and complex field scenarios. Comparative experiments show that BP-NMPC-DCBF achieves up to 11 times faster optimization than interactive MPC (iMPC) baselines while maintaining real-time control frequencies. Moreover, in dynamic environments with limited prediction horizons, the proposed method attains a 92.4% success rate compared to 69.6% for a standard NMPC-DCBF baseline. Sensitivity analyses confirm that the Bernstein-based parameterization maintains high feasibility even under conservative safety settings. These results demonstrate that the proposed framework effectively balances theoretical safety guarantees with the high-frequency real-time performance required for modern autonomous navigation.