PhD Seminar Series: “Guaranteeing Convergence in Trajectory Shaping Guidance for Arrival-Time-Constrained Unicycle Vehicles”

Unicycle vehicles are widely used to represent the planar motion of various systems, such as fixed-wing uncrewed aerial vehicles, mobile robots, and uncrewed ground vehicles. Guiding multiple unicycle vehicles to reach a target simultaneously is critically important for missions such as formation flight and coordinated control. The pioneering work by Jeon et al. in 2006 developed a linear optimal guidance law enabling individual vehicles to reach the target at the same time. The primary challenge in this guidance problem lies in the highly nonlinear nature of the vehicle kinematics. Since then, researchers in robotics and aerospace engineering have explored this problem using various approaches, including linear optimal control, nonlinear control, and machine learning. However, despite nearly 20 years of extensive research, no guidance law with convergence guarantees has been established, which poses a major concern for safety-critical applications.
To address this gap, this seminar presents a convergence-guaranteed guidance law developed by integrating trajectory shaping guidance with optimal control theory. The trajectory shaping approach parametrizes the state trajectory using a cubic polynomial. Consequently, the guidance problem reduces to finding an unknown guidance gain by solving a root-finding problem without requiring linearization. However, the multi-root nature of the root-finding problem leads to convergence difficulties. To overcome this limitation, key properties of the optimal trajectory are embedded into the trajectory shaping guidance design, leading to a suboptimal solution among multiple candidates. Furthermore, by exploiting solution uniqueness and existence conditions, bounds on the guidance gain are established, thereby providing convergence guarantees for the root-finding algorithm. Moreover, this enables verification of guidance performance a priori before implementation. In addition, an analytical initializer for the guidance gain is constructed to facilitate convergence. Finally, the effectiveness of the proposed framework is demonstrated through extensive numerical simulations, showing promising results for practical implementation on uncrewed vehicles.