^{1}

Using a solar sail, a spacecraft orbit can be offset from a central body such that the orbital plane is displaced from the gravitational center. Such a trajectory might be desirable for a single-spacecraft relay to support communications with an outpost at the lunar south pole. Although trajectory design within the context of the Earth-Moon restricted problem is advantageous for this problem, it is difficult to envision the design space for offset orbits. Numerical techniques to solve boundary value problems can be employed to understand this challenging dynamical regime. Numerical finite-difference schemes are simple to understand and implement. Two augmented finite-difference methods (FDMs) are developed and compared to a Hermite-Simpson collocation scheme. With 101 evenly spaced nodes, solutions from the FDM are locally accurate to within 1740 km. Other methods, such as collocation, offer more accurate solutions, but these gains are mitigated when solutions resulting from simple models are migrated to higher-fidelity models. The primary purpose of using a simple, lower-fidelity, augmented finite-difference method is to quickly and easily generate accurate trajectories.

When a permanent outpost on the Moon to support extended human expeditions is eventually established, the astronauts at the facility will require a continual communications link with the Earth. A leading candidate location for the lunar base is at the south pole, which is not always in view of either antennas on the Earth or space-based transceivers, such as the TDRSS satellites, in geosynchronous orbit. Therefore, at least two spacecraft in traditional polar orbits about the Moon are required for an Earth-Moon link [

In contrast to a constellation of multiple spacecraft, alternative communications strategies that rely on only one satellite do exist, using current or near-future technology. Advanced propulsion concepts, such as low-thrust ion engines, as well as solar sails, supply a force in addition to gravity and can actually offset an orbit from the Moon [

Trajectories can be represented as solutions to boundary value problems (BVPs) in terms of ordinary differential equations (ODEs). For a BVP, conditions are specified at the beginning and, also, at the end of the time domain. The specific values of the states at the extremes may or may not be fixed in a BVP with a periodicity constraint; periodicity only requires that the values at the extremes are equal. Differential-corrections schemes [

Shooting methods [

A more rudimentary method for solving BVPs is the finite-difference method (FDM), in which the derivatives that appear in the differential equations are replaced with their respective finite differences and evaluated at node points along the trajectory [

In the following sections, an FDM is described and applied to solar sail orbits in the Earth-Moon circular restricted three-body (CR3B) system. Also included is a modified FDM for the same application, where the velocity is incorporated as part of the solution at each node along the discretized trajectory. Neither method is strictly a straightforward, “textbook” FDM [

A description of the dynamical model in the Earth-Moon CR3B system is followed by an algorithm using the augmented FDMs for generating trajectories. The strategies are based on minimizing the difference between accelerations from the equations of motion (as well as the velocities as another alternative) and the corresponding values numerically derived from positions along a discretized trajectory. An error analysis is also included. A separate study uses these FDMs for surveying the solution space and assessing the required solar sail and spacecraft characteristics necessary for the lunar south pole (LSP) coverage problem [

The problem that represents this application is defined within the context of the CR3B system, that is, the problem is formulated in a frame,

Earth-Moon system model.

To generate the magnitude of the sail acceleration in dimensional units,

The sunlight direction is expressed relative to the rotating frame and is a function of time, that is,

The sail modeled here is a perfectly reflecting, flat solar sail. Billowing is not incorporated in this force model; however,

The derivation for an augmented finite-difference method begins with generic, nonlinear, second-order, two-point BVP that can be represented as

For the current problem, the FDM is augmented to incorporate path and system constraints, as well as a control history. Two variations of this FDM are developed. In the first, the position is the only explicitly discretized trajectory state (FDM-R). The second FDM is formulated to directly solve for position

The FDM-R formulation is based on approximating (

The FDM-RV process is similarly formulated, but there are two distinct sets of ODEs to solve, that is,

Both the FDM-R and the FDM-RV algorithms constrain the difference between the evaluated EOMs at node

Algebraically, these FDMs are equivalent. In practice, the two formulations yield slightly different results and, sometimes, dramatically different results. This problem is sensitive to the initial guess; because the initial condition may or may not explicitly include velocity, the vector of initial guesses is not the same for the two FDMs. Additionally, the least-norm update in (

At each epoch, the converged subvector

The augmented finite-difference methods sacrifice precision for simplicity. At each node, the error in the trajectory is proportional to

Prior to any analysis, the design space for this problem is not well known, so a trajectory that is precise to within

At the core of the augmented finite-difference methods is the algebraic constraint vector,

The first element set in (

The next constraints enforce periodicity and unit length of the control vector. To enforce periodicity, the goal is an originating and final state vector that are equal, which is represented as a set of constraints in the algebraic constraint vector,

Path constraints are included as inequality constraints, which are converted to equality constraints by use of slack variables, a successful numerical adaptation from nonlinear programming [

Path constraints for an orbit below the Moon (Moon image from nasa.gov).

Only the first

As mentioned, the Jacobian is a sparse matrix. A diagram of the sparsity pattern based on partial derivatives of

Sparsity pattern for the Jacobian

The advantage of a finite-difference approach is its ease of implementation and the speed improvements enabled by its simplicity. Partial derivatives are easily accessible via analytical derivation, especially for an idealized force model such as (

Computation times for various differentiation strategies.

Strategy | Time |
---|---|

Analytic | 0.003269 seconds |

PMAD | 0.013798 seconds |

NUMJAC | 0.014924 seconds |

TOMLAB/MAD | 0.051849 seconds |

In summary, a finite-difference method yields a solution for the path by replacing the path derivatives with their finite-difference approximations. The differences between the approximate derivatives and the derivatives determined by evaluating the equations of motion at a specific point along a path are minimized by iteratively solving a linearized system of equations. Inequality path constraints are added to this set of equality constraints by way of slack variables, and subsequently augmenting the constraint vector and linearized system of equations, resulting in a feasible path.

The objective when employing a finite-difference method is to quickly generate a feasible trajectory where system constraints and sail characteristics are given. Because model simplification into the CR3B system ignores lunar librations, the inclined orbits of the primaries, additional perturbations to the orbit, and lunar surface features, a conservative minimum elevation angle of

As mentioned, a recent sail design possesses a characteristic acceleration of ^{2} [^{2} and 10 m^{2}, resp.) and are designed to demonstrate in-space deployment of the sail. Designs for larger sailcraft exist, and future solar sail spacecraft are likely to be hybridized with other propulsion devices [^{2}, is used to generate example orbits.

To demonstrate the FDM-R and FDM-RV approaches, three sets of initial guesses for the path and control history are selected to initialize the process. The initial guesses corresponding to the slack variables are always established such that

Three example initial guesses.

3D view

Side view

When the estimates from Figure

Three converged orbits from the FDM-R process.

3D view

Side view

Three converged orbits from the FDM-RV process.

3D view

Side view

For all trials of the FDM-R and FDM-RV formulations (and in the collocation formulation described in the next section) it is observed that the corrections to the components of the initial guess of the control history are not smooth at the first and

Three converged orbits from the Hermite-Simpson collocation process.

3D view

Side view

Although the dark-blue and red orbits appear to be similar in both the FDM-R and FDM-RV results, they differ by up to 8000 km and 1800 km, respectively, at certain epochs along the respective paths. The difference in the light-blue orbits in Figures

As stated, prior to any analysis, the design space for this problem is unknown. Entry into the design space via analytic techniques is difficult because of coupling of the sail attitude and thrust vector. However, based on the FDM algorithms, solutions precise to

Using the same initial guesses as those represented in Figure

There is one noteworthy difference between the Hermite-Simpson collocation method employed by Ozimek et al. and the adaptation used for this analysis. Ozimek et al. employ a small step along the imaginary axis to calculate the partial derivatives related to the defects. For the current analysis, a simple central difference approximation (CDA) represents the same partial derivatives. Both are numerical schemes for computing difficult partial derivatives and subject to roundoff errors, but the CDA is also subject to truncation error, while the imaginary-step method is not. Thus, the error for the CDA employed in the Hermite-Simpson collocation method is minimized at a step size greater than zero (on the order of

The results from any procedure can be used to seed another, usually more precise, process. The solutions produced by the FDM-R and FDM-RV algorithms are used to initialize the Hermite-Simpson scheme. All orbits reconverge quadratically in three or four steps when the Hermite-Simpson strategy is incorporated. The differences in position and velocity, as well as the direction of the sail-face normal, along the 29.5 day trajectories appear in Figure

Differences between the solutions from the Hermite-Simpson method and their initial guesses supplied by the FDM-R (a) and FDM-RV (b) solutions.

FDM-R solutions as initial guesses

FDM-RV solutions as initial guesses

Augmented finite-difference methods are useful tools for generating trajectories in poorly understood dynamical regimes, such as the mechanics of flying a solar sail in a multibody system. These schemes are simple to understand and implement, notably in the presence of path constraints. While the theoretical errors of finite-difference methods may be larger than other similar methods that examine the trajectory as a whole, such as collocation, finite-difference methods can provide a reasonable entrance to the design space of a complicated nonlinear problem. They also quickly return results in a regime where little intuition concerning the trajectory and sail-angle history exist. Because of its speed and simplicity, this method may also serve as the basis for generating a large survey of orbit options. Once a viable trajectory is uncovered by the finite-difference method, other techniques may be employed to refine it further. The FDM approach developed for this analysis need not be applied to just solar sail problems, but should be applicable to a large variety of mission design problems.

The authors would like to thank Purdue Professor William Crossley for his explanation of optimization techniques in the literature used to solve similar problems and Professor Ahmed Sameh, also from Purdue University, for his insights on numerical methods used to solve boundary-value problems. Former Purdue graduate students Daniel Grebow and Zubin Olikara were invaluable to the initial development and understanding of the numerical methods examined in this paper. The authors also are grateful for the comments from the Editor and reviewers.