New Lambert Algorithm Using the Hamilton-Jacobi-Bellman Equation

Abstract

T HE two-point boundary-value problem (TPBVP) for Keplerian motion,which is known asLambert’s problem, is a fundamental problem in space trajectory design. Lambert’s problemmay be stated as follows: given initial and final position vectors, determine the initial velocity which will allow a transfer in a specified flight time. The classical approach to Lambert’s problem is based on geometry of the radius vectors, which directly stems from Lambert’s theorem [1]. After the early works of Gauss [2], a number of numerical algorithms were proposedbyLancaster et al. [3],Battin [4], andother researchers [5–7]. However, these solution procedures cannot be applied to the perturbed problem, because the geometry of the dynamics changes. On theother hand, general TPBVPs are solvedbymeans of numerical methods such as the shooting method and the finite difference method. Lizia et al. [8] and Armellin and Topputo [9] developed an integration scheme for the solution of the TPBVP and astrodynamics applications concerning the computation of trajectories in thevicinity of the libration points in the restricted threeand four-body problem. Guibout and Scheeres [10] solved the TPBVP using the theory of canonical transformation. These approaches [8–10] rely on a power series expansion along a nominal trajectory. Hamilton’s principle [11] is an alternative formulation of the differential equations of motion of spacecraft, which states that the trajectory between two specified states at two specified times is an extremum of the action integral. Motivated by this observation, this paper attempts to show that a solution toLambert’sproblemisdirectly obtained by minimizing the action integral. This problem can be viewedasanoptimalcontrolproblembyreplacingkineticenergywith a quadratic performance index of the control input, so that the initial velocity is found as the optimal control. Then, the solution is given by the Hamilton–Jacobi–Bellman (HJB) equation. The approximation methods to the solutions to the HJB equation were studied by many authors based on series expansion techniques [12–14]. In [14], a closed-formsolutionof theHJBequation isobtainedbyexpandingthe value function as a power series in terms of the state and the constant Lagrange multipliers. Although higher-order approximations are possible to obtain by series expansion solutions, their computations are time-consuming and there is no guarantee that the resulting solution improve the performance. Another approach is through successive approximation, where the HJB equation is reduced to a sequence of the first-order linear partial differential equations [15– 17]. Mizuno and Fujimoto [18] showed that the HJB equation is effectively solved by the Galerkin spectral method with Chebyshev polynomials based on successive approximation. In this paper, the TPBVP of the Hamiltonian system is treated as an optimal control problemwhere the Lagrangian function plays a role as a performance index. Similar toMizuno and Fujimoto, our approach is based on the expansionof thevalue function in theChebyshevserieswithunknown coefficients, considering the computational advantages of the use of Chebyshev polynomials. The differential expressions that arise in the HJB equation are also expanded in Chebyshev series with the unknown coefficients. As a consequence, the algorithm is much simpler than the procedure based on series expansion, and higherorder approximations are possible to obtain for more complicated nonlinear dynamics. Our algorithm can provide a solution to the TPVBP using the spectral information about the gravitation potential function and is also applicable to the TPVBP under a higher-order perturbed potential function without any modification. The paper is organized as follows. Section II is the problem statement. Section III.A reviews the Hamilton principle and the motivation of our method, Sec. III.B introduces the HJB equation in optimal control theory, and Sec. III.C formulates the solution procedure. Section IV presents simulation results for the two-body problem and the circular restricted three-body problem.