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The general objective is the development of efficient techniques for preliminary design of trajectory arcs in nonlinear autonomous dynamic systems in which the desired solution is subject to algebraic interior and/or exterior constraints. For application to the n-body problem, trajectories must satisfy specific requirements, e.g., periodicity in terms of the states, interior or boundary constraints, and specified coverage. Thus, a strategy is formulated in a sequence of increasingly complex steps: 1) a trajectory is first modeled as a series of arcs (analytical or numerical) and general trajectory characteristics and timing requirements are established; 2) the specific constraints and associated partials are formulated; 3) a corrections process ensures position and velocity continuity while satisfying the constraints; and finally, 4) the solution is transitioned to a full model employing ephemerides. Though the examples pertain to spacecraft mission design, the methodology is generally applicable to autonomous systems subject to algebraic constraints. For spacecraft mission design applications, an immediate advantage of this approach, particularly for the identification of periodic orbits, is that the startup solution need not exhibit any symmetry to achieve the objectives.

Preliminary designs of low-thrust transfer trajectories are developed in the Earth-moon three-body problem with variable specific impulse engines and fixed engine power. The solution for a complete time history of the thrust magnitude and direction is initially approached as a calculus of variations problem to locally maximize the final spacecraft mass. The problem is then solved directly by sequential quadratic programming, using either single or multiple shooting. The coasting phase along the transfer exploits invariant manifolds and, when possible, considers locations along the entire manifold surface for insertion. Such an approach allows for a nearly propellant-free final coasting phase along an arc selected from a family of known trajectories that contract to the periodic libration point orbit. This investigation includes transfer trajectories from an Earth parking orbit to some sample libration point trajectories, including L 1 halo orbits, L 1 and L 2 vertical orbits, and L 2 butterfly orbits. Given the availability of variable specific impulse engines in the future, this study indicates that fuel-efficient transfer trajectories could be used in future lunar missions, such as south pole communications satellite architectures. Nomenclature c = constraint vector G = endpoint function H = Hamiltonian I sp = engine specific impulse i = instantaneous inclination angle m = total spacecraft mass P = engine power magnitude R = rotation matrix r, v = position and velocity vectors, Earth-moon rotating frame S = design variable vector T = engine thrust magnitude t = time U = pseudopotential function u c = control vector u T = thrust direction unit vector X = uncontrolled state vector, Earth-moon rotating framê = eigenvector at a fixed point on a periodic orbit = control constraint Lagrange multiplier vector = instantaneous argument of latitude M = anglelike manifold parameter = costate vector M = timelike manifold parameter = state transition matrix = kinematic boundary condition vector = instantaneous right ascension angle $, = kinematic boundary condition Lagrange multiplier vectors Subscripts c = control variable f = final condition FP = fixed point condition MS = multiple shooting phase p = powered phase PO = parking orbit condition s = stable manifold SS = single shooting phase u = unstable manifold 0 = initial condition

NASA's GRAIL mission employed twin spacecraft in polar orbits around the Moon to measure the lunar gravity field at unprecedentedly high accuracy and resolution. The low spacecraft altitude in the extended mission enables the detection of small‐scale surface or subsurface features. We analyzed these data for evidence of empty lava tubes beneath the lunar maria. We developed two methods, gradiometry and cross correlation, to isolate the target signal of long, narrow, sinuous mass deficits from a host of other features present in the GRAIL data. Here we report the discovery of several strong candidates that are either extensions of known lunar rilles, collocated with the recently discovered “skylight” caverns, or underlying otherwise unremarkable surfaces. Owing to the spacecraft polar orbits, our techniques are most sensitive to east‐west trending near‐surface structures and empty lava tubes with minimum widths of several kilometers, heights of hundreds of meters, and lengths of tens of kilometers.