Guaranteeing convergence in trajectory shaping guidance for impact time control

Trajectory shaping guidance provides autonomy for unmanned vehicles by parameterizing the state trajectory as polynomials or series of basis functions. By embedding boundary conditions, the guidance problem, without any linearization, reduces to a simple root-finding or optimization problem. However, the convergence of such a problem is often overlooked within the trajectory shaping framework. In this paper we present a method to guarantee the convergence of an impact time control guidance problem solved via trajectory shaping. First, by leveraging geometric properties, we derive explicit bounds on the guidance gain, thereby avoiding high-control-effort solutions. Next, by examining the multi-root nature of the resulting root-finding problem, we numerically establish tighter bounds on the guidance gain. Furthermore, the capture region of the proposed guidance law is analytically determined. This analysis ensures the existence and uniqueness of the guidance gain, allowing for a priori verification of the solution. This theoretical well-posedness ensures the convergence of the root-finding problem via the bisection method. On this basis, to achieve superior computational efficiency, an analytical initialization scheme is developed for Newton's method. Numerical simulations are finally presented to validate the proposed developments.