GUARANTEED CONVERGENCE IN TRAJECTORY PLANNING USING UNIVARIATE OPTIMIZATION

Trajectory planning is a critical challenge in motion-based systems, requiring efficient and reliable guidance of agents—such as robots or drones—from an initial to a target position. Traditional methods often involve complex multivariate optimization or heuristic approaches, which may lack simplicity or convergence guarantees. This work proposes a univariate optimization framework for trajectory planning in two-dimensional space, leveraging iterative refinement of a quadratic error function based on Euclidean distance to ensure guaranteed convergence. The method is validated through simulations, demonstrating its effectiveness in guiding systems to desired endpoints while maintaining computational simplicity. Key findings reveal that fixed-step implementations are robust but may require adjustments for non-multiplicative distances, and non-identity weight matrices offer negligible benefits in such cases. The framework serves as both a practical tool for trajectory planning and an accessible pedagogical resource for introducing optimization principles. Limitations include suboptimal trajectories and fixed-step inefficiencies, suggesting future work in adaptive step sizes and hybrid methodologies. The study underscores the potential of univariate optimization as a foundational approach for low-dimensional, deterministic systems.