IN THE field of guidance of aerospace vehicles, attention is consistently drawn to solutions that are effective in constrained circumstances. Optimal control enables the generation of the most favorable control commands with respect to a cost function, while adhering to the system dynamics and constraints [1]. The implementation of optimal control methods in aerospace guidance design has opened new perspectives and possibilities with the purpose to provide effective capability in constrained scenarios. Optimal controlbased guidance approaches are emerging to accomplish missions of various vehicles, e.g., missiles [2--4], spacecraft [5--7], hypersonic glide vehicles [8--10], and aircraft [11--13]. Some methods are designed for specific classes of systems to achieve particular guidance purposes, which limits the scope of their applications. The recently articulated concept of computational guidance [14] has used advanced numerical approaches with high computational efficiency. Applied optimization techniques in aerospace guidance and control have been steadily evolving in recent years, from which convex optimization approaches stand out in respect ofefficiency and complexity. Fast convex-programming algorithms enable online computation of the guidance commands subject to various constraints [15]. In previous studies [9,16--20], a fixed final time has been considered. The convex-optimization-based guidance algorithm tailored to the extended flight envelope in [16] considers a fixed final time for generating trajectories for reusable launchers. Some optimal control methods, considering a fixed final time, would require modifications on the dynamics to be able to tackle free finaltime problems [9]. Otherwise, the effectiveness would be restricted by an appropriately given guess value ofthe final time. To achieve the modifications on the dynamics, a group of approaches relies on introducing new independent variables and their monotonicity. By application category, these design philosophies can be seen in missile guidance [21,22], interplanetary transfer [23], and entry flight [24,25]. However, this philosophy is problem specific, as it relies on analyzing the system dynamics to find a proper variable, which may not be suitable for other problems. Yet a more general approach is that the normalized time is used and the dynamics are rewritten with respect to the normalized time via the chain rule from a continuous time perspective [9,26,27]. A class of bilevel strategies in the literature that help to avoid manually specifying a final time solves the fixed final-time trajectory optimization problem in the inner loop, while determining the optimal final time in the outer loop. In [28], the outer loop searches for the overall optimal final time with respect to propellant usage, and a fixed final-time convex optimization algorithm in the inner loop generates propellant usage-minimum asteroidlanding trajectories for the given final times. In this Note, a sequential-quadratic-programming (SQP)--based optimization technique is developed for state and input constrained guidance ofaerospace vehicles with a free final time. The philosophy of the proposed method is to reformulate a class of nonlinear guidance problems with path and terminal constraints into a generic quadratic programming (QP) form and solve the subproblems in sequence starting from an initial guess solution, following the basic concept of SQP. The differences mainly exist in the objective of the constrained subproblem, which is to update the control history and time step by determining their increments, instead of all variables including the states. Moreover, the algorithmic design to address infeasibility is tailored to guidance problems with terminal constraints. Each QP subproblem includes a convex quadratic cost function that can be selected freely. Increments ofstates are expressed as a function of the optimization variables, namely, the increment vector of control commands and time step. Therefore, states can be considered in the constraints and the cost function of the QP problems, but are not explicitly present. The proposed approach introduces the increment of time step rather than the final time as a variable. In addition, a revised augmented Lagrangian algorithm for solving infeasible QP problems [29] is adapted and used to address the infeasibility that might occur during iterations due to approximations. It can find a solution to the infeasible subproblem with the smallest shift, and thereby enable the SQP algorithm to proceed, distinct from methods that require the feasibility of linearized subproblems [30]. The original revised augmented Lagrangian algorithm relaxes every constraint, whereas in the proposed method, terminal constraints are considered solely as equality constraints without being relaxed. This formulation ensures that the terminal constraints are satisfied over iterations, which corresponds to the requirement of guidance applications that are often terminally constrained. To show the effectiveness of the proposed method, a simulation study of a three-dimensional obstacle-avoidance constrained missile guidance problem is presented. The proposed method can provide the solutions that satisfy the path constraints, while achieving different terminal hard constraints, at corresponding final times. Furthermore, a comparison with off-the-shelf optimization software is presented to demonstrate the performance and efficiency of the proposed method. The rest of this Note is organized as follows. Section II introduces the formulation and the solution of the QP problem for guidance command generation. Section III presents a missile guidance task and its simulation results, illustrating the solving process of the proposed method and the effectiveness. Section IV draws the conclusions.     «

IN THE field of guidance of aerospace vehicles, attention is consistently drawn to solutions that are effective in constrained circumstances. Optimal control enables the generation of the most favorable control commands with respect to a cost function, while adhering to the system dynamics and constraints [1]. The implementation of optimal control methods in aerospace guidance design has opened new perspectives and possibilities with the purpose to provide effective capability in constrained sce...     »