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We have developed a new method for solving low-thrust fuel-optimal orbit transfer problems in the vicinity of a large body (planet or asteroid), considering a high-fidelity spherical harmonic gravity model. The algorithm is formulated via the indirect optimization method, leading to a two-point boundary value problem (TPBVP). We make use of a hyperbolic tangent smoothing law for performing continuation on the thrust magnitude to reduce the sharpness of the control switches in early iterations and thus promote convergence. The TPBVP is solved using the method of particular solutions (MPS) shooting method and Picard-Chebyshev numerical integration. Application of Picard-Chebyshev integration affords an avenue for increased efficiency that is not available with step-by-step integrators. We demonstrate that computing the particular solutions with only a low-fidelity force model greatly increases the efficiency of the algorithm while ultimately achieving near machine precision accuracy. A salient feature of the MPS is that it is parallelizable, and thus further speedups are available. It is also shown that, for near-Earth orbits and over a small number of en-route revolutions around the Earth, only the zonal perturbation terms are required in the costate equations to obtain a solution that is accurate to machine precision and optimal to engineering precision. The proposed framework can be used for trajectory design around small asteroids and also for orbit debris rendezvous and removal tasks. [ABSTRACT FROM AUTHOR]

A modified Chebyshev Picard iteration method is proposed for solving orbit propagation initial/boundary value problems. Cosine sampling techniques, known as Chebyshev-Gauss-Lobatto (CGL) nodes, are used to reduce Runge’s phenomenon that plagues many series approximations. The key benefit of using the CGL data sampling is that the nodal points are distributed nonuniformly, with dense sampling at the beginning and ending times. This problem can be addressed by a nonlinear time transformation and/or by utilizing multiple time segments over an orbit. This paper suggests a method, called a multisegment method, to obtain accurate solutions overall regardless of initial states and albeit eccentricity by dividing the given orbit into two or more segments based on the true anomaly. [ABSTRACT FROM AUTHOR]

Modified Chebyshev-Picard iteration methods are presented for solving boundary value problems. Chebyshev polynomials are used to approximate the state trajectory in Picard iterations, while the boundary conditions are maintained by constraining the coefficients of the Chebyshev polynomials. Using Picard iteration and Clenshaw--Curtis quadrature, the presented methods iteratively refine an orthogonal function approximation of the entire state trajectory, in contrast to step-wise, forward integration approaches, which render the methods well-suited for parallel computation because computation of force functions along each path iteration can be rigorously distributed over many parallel cores with negligible cross communication needed. The presented methods solve optimal control problems through Pontryagin's principle without requiring shooting methods or gradient information. The methods are demonstrated to be computationally efficient and strikingly accurate when compared with Battin's method for a classical Lambert's problem and with a Chebyshev pseudospectral method for an optimal trajectory design problem. The reported simulation results obtained on a serial machine suggest a strong basis for optimism of using the presented methods for solving more challenging boundary value problems, especially when highly parallel architectures are fully exploited. [ABSTRACT FROM AUTHOR]

The article discusses a computational methodology for solving the nonlinear stochastic optimal control based on the policy iteration algorithm and a finite dimensional approximation of the controlled Fokker-Planck-Kolomogorov/diffusion operator. It stresses the lack of stochastic equivalent to the two-point boundary value problems resulting from the Pontryagin minimum principle applied to the optimal control problem. The methodology was found to demonstrate satisfactory performance in several test cases.

The implicit function theorem due to Lagrange is generalized to enable high order implicit rate calculations of general implicit functions about pre-computed solutions of interest. The sensitivities thus calculated are subsequently used in determining neighboring solutions about an existing root (for algebraic systems) or trajectory (in case of dynamical systems). The generalization to dynamical systems, as a special case, enables the calculation of high order time varying sensitivities of the solutions of boundary value problems with respect to the parameters of the system model and/or functions describing the boundary condition. The generalizations thus realized are applied to various problems arising in trajectory optimization. It was found that useful information relating the neighboring extremal paths can be deduced from these implicit rates characterizing the behavior in the neighborhood of the existing solutions. The accuracy of solutions obtained is subsequently enhanced using an averaging scheme based on the Global Local Orthogonal Polynomial (GLO-MAP) weight functions developed by the first author to blend many local approximations in a continuous fashion. Example problems illustrate the wide applicability of the presented generalizations of Lagrange's classical results to static and dynamic optimization problems. [ABSTRACT FROM AUTHOR]

Indirect optimization methods convert optimal control problems (OCPs) into two- or multi-point boundary-value problems. A highly desirable feature of indirect methods, specifically for space applications, is that high-resolution trajectories can be generated, which satisfy the first-order necessary conditions of optimality. A recently developed Composite Smoothing Control (CSC) framework is utilized to formulate and solve the problem of simultaneous trajectory optimization and propulsion sub-system design of spacecraft. A reasonable parameterized breakdown of the spacecraft mass is adopted, which captures the impact of power produced by the solar arrays and its contribution to the total spacecraft mass. Thus, the implicit trade-offs can be considered in the indirect optimization approach. The function space co-optimization problem of spacecraft power subsystem parameters along with the main trajectory is solved with the objective to maximize the payload delivered. The proposed framework amounts to an invariant embedding that reduces the original, difficult-to-solve, multi-point boundary-value problem into a two-point boundary-value problem with continuous, differentiable control inputs. Utility of the proposed construct is demonstrated through a low-thrust, multi-revolution, multi-year rendezvous maneuver to asteroid Dionysus with a variable-specific-impulse, variable-thrust modeled engine. This is the first time that indirect optimization methods have tackled such a complex co-optimization problem using the CSC framework. • Co-optimization of trajectory and solar electric propulsion system parameters. • Comparison of constant and variable- l s p thruster for low-thrust trajectory design. • Solve multi-point boundary value problem using Composite Smoothing Control (CSC). • Optimization over bounds of the l s p , variable thruster efficiency, and power is achieved. • Linear relation between thruster efficiency and l s p is near-accurate for early studies. [ABSTRACT FROM AUTHOR]