Variable–Time–Domain Neighboring Optimal Guidance, Part 1: Algorithm Structure

Abstract

This paper presents a general purpose neighboring optimal guidance algorithm that is capable of driving a dynamical system along a specified nominal, optimal path. This goal is achieved by minimizing the second differential of the objective function along the perturbed trajectory. This minimization principle leads to deriving all the corrective maneuvers, in the context of a closed-loop guidance scheme. Several time-varying gain matrices, referring to the nominal trajectory, are defined, computed offline, and stored in the onboard computer. Original analytical developments, based on optimal control theory, in conjunction with the use of a normalized time scale, constitute the theoretical foundation for three relevant features: (i) a new, efficient law for the real-time update of the time of flight (the so called time-to-go), (ii) a new termination criterion, and (iii) a new analytical formulation of the sweep method. This new guidance, termed variable-time-domain neighboring optimal guidance, is rather general, avoids the usual numerical difficulties related to the occurrence of singularities for the gain matrices, and is exempt from the main disadvantages of similar algorithms proposed in the past. For these reasons, the variable-time-domain neighboring optimal guidance has all the ingredients for being successfully applied to problems of practical interest.Communicated by Anil Rao.M. Pontani (B) · G. Cecchetti · P. Teofilatto University "La Sapienza",

Variable-Time-Domain Neighboring Optimal Guidance, Part 2: Application to Lunar Descent and Soft Landing

“…6, to lunar descent and soft landing, for the purpose of proving its effectiveness, accuracy, and robustness. The new algorithm is quite general and can be adopted for dynamical systems with a generic number of state and control components and an arbitrarily large parameter set.…”

“…This means that the Pontryagin minimum principle and the EulerLagrange equations, together with the related boundary conditions, are to be derived and used, in order to translate the original optimal control problem into a two-point boundary value problem (cf. [6]). Once the optimal solution has been found, the remaining steps (c) and (d) lead to determining all the offline quantities.…”

“…With the intent of obtaining the necessary conditions for an optimal solution, the Hamiltonian H and the function of the boundary conditions for the problem at hand are introduced [6,8],…”

“…υ 10 T ) represent, respectively, the adjoint variable conjugate to the dynamics equations (4)- (9) and to the boundary conditions (10)- (11). The first-order necessary conditions for (local) optimality [6,8] include the adjoint equations…”

“…(16), λ 2 = 0 ∀τ . The optimal control u * can be expressed as a function of the costates through the Pontryagin minimum principle [6,8],…”