Eventos Anais de eventos
COBEM 2017
24th ABCM International Congress of Mechanical Engineering
A COMPARISON OF TWO NUMERICAL ALGORITHMS FOR COMPUTING OPTIMAL LOW-THRUST TRAJECTORIES
Submission Author:
Francisco das Chagas Carvalho , SP , Brazil
Co-Authors:
Sandro da Silva Fernandes, João Victor Bateli Romão, Francisco das Chagas Carvalho
Presenter: Francisco das Chagas Carvalho
doi://10.26678/ABCM.COBEM2017.COB17-2598
Abstract
In this paper, two different numerical algorithms for computing optimal low-thrust limited power trajectories in an inverse square force field are discussed. Both algorithms are based on indirect approach of solving optimal control problems. In this approach the two-point boundary value problem resulting from the necessary conditions expressed by the Pontryagin Maximum Principle is solved by means of a neighboring extremals algorithm based on state transition matrix. The transfer problem to be analyzed is concerned to the transfers between coplanar circular orbits with large radius ratio and moderate time of flight. In the first algorithm, the optimization problem is formulated as a Mayer problem of optimal control with the radial distance and the components, radial and circumferential, of the velocity vector as state variables. In the second approach, the optimization problem is formulated as a Mayer problem of optimal control, with a set of nonsingular elements as state variables. Second order terms in eccentricity are then considered in the development of the state equations. In both cases, the fuel consumption is described by an auxiliary state variable that is a monotonic decreasing function of the mass of the space vehicle. The minimization of the final value of this consumption variable is equivalent to the maximization of the final mass of the vehicle or the minimization of the fuel quantity spent in the maneuver. In the first algorithm, the solution of the two-point boundary value problem of going from an initial circular orbit at time t0 to a final circular orbit at a prescribed final time tf is obtained by applying straightforwardly the neighboring extremals algorithm. But, in the second algorithm, the solution of such boundary value problem is obtained in two stages. In the first one, is solved a two-point boundary value problem defined by an average canonical system describing the secular behavior of the optimal trajectories. The maximum Hamiltonian governing the average canonical system is derived by applying Hori method. So, an infinitesimal canonical transformation is built, and, the short periodic terms can be included in the solution by computing the Poisson brackets of the generating function. This technique provides an approximation, at a first order in a small parameter closely related to the magnitude of the optimal thrust acceleration, to the time behavior of the state variables considering the complete maximum Hamiltonian. In other words, it provides an approximation to the numerical integration of the complete canonical system. However, when the short periodic terms are included in the solution of the two-point boundary value problem obtained in the first stage, small deviations from the prescribed final conditions arise. Accordingly, the initial values of the adjoint variables computed in the first stage must be adjusted. This last step is performed by a simple Newton-Raphson algorithm with the partial derivatives of terminal constraints computed by a method of differences in which the state variables are obtained by the approximate solution using the infinitesimal canonical transformation. The two algorithms previously described are applied in a preliminary analysis of some interplanetary missions. Numerical results show the great agreement between the algorithms, but they also show that the second algorithm converges faster than the first one. Although the second algorithm has been developed including second order terms in eccentricity in the state equations, time behavior of the eccentricity along the optimal trajectory shows excellent agreement to the exact results provided by the first algorithm. Using the continuation technique both algorithms can be applied to study transfers with large radius ratio and large time of flight.
Keywords
Numerical Algorithms, Optimal Low-Thrust, Pontryagin Maximum Principle, Low-thrust transfers, Infinitesimal canonical transformation
