Martin W. Lo

Connecting orbits and invariant manifolds in the spatial restricted three-body problem

The invariant manifold structures of the collinear libration points for the restricted three-body problem provide the framework for understanding transport phenomena from a geometrical point of view. In particular, the stable and unstable invariant manifold tubes associated with libration point orbits are the phase space conduits transporting material between primary bodies for separate three-body systems. These tubes can be used to construct new spacecraft trajectories, such as a ‘Petit Grand Tour’ of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. This work extends the results to the three-dimensional case. Besides providing a full description of different kinds of libration motions in a large vicinity of these points, this paper numerically demonstrates the existence of heteroclinic connections between pairs of libration orbits, one around the libration point L_1 and the other around L_2. Since these connections are asymptotic orbits, no manoeuvre is needed to perform the transfer from one libration point orbit to the other. A knowledge of these orbits can be very useful in the design of missions such as the Genesis Discovery Mission, and may provide the backbone for other interesting orbits in the future.

The Lunar L1 Gateway : portal to the stars and beyond

Our Solar System is interconnected by a vast system of winding tunnels generated by the Lagrange Points of all the planets and their moons. These passageways are identified by portals around L1 and L2, the halo orbits. By passing through a halo orbit portal, one enters the ancient and colossal labyrinth of the Sun. This natural Interplanetary Supher highway System (IPS) provides ultra-low energy transport throughout the Earths Neighborhood, the region between Earths L1 and L2. This is enabled by an accident: the current energy levels of the Earth L1 and L2 Lagrange points differ from that of the Earth-Moon by only about 50 rn/s (as measured by AV). The significance of this happy coincidence to the development of space cannot be overstated. For example, this implies that Lunar L1 halo orbits are connected to halo orbits around Earths L1 or L2 via low energy pathways...

Transport in Dynamical Astronomy and Multibody Problems

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.

Resonance and Capture of Jupiter Comets

A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L1and L2, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L1and L2is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L1and the other around L2) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold ‘tubes’ associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiters orbit.

Halo orbit mission correction maneuvers using optimal control

This paper addresses the computation of the required trajectory correction maneuvers for a halo orbit space mission to compensate for the launch velocity errors introduced by inaccuracies of the launch vehicle. By combining dynamical systems theory with optimal control techniques, we are able to provide a compelling portrait of the complex landscape of the trajectory design space. This approach enables automation of the analysis to perform parametric studies that simply were not available to mission designers a few years ago, such as how the magnitude of the errors and the timing of the first trajectory correction maneuver affects the correction @DV. The impetus for combining dynamical systems theory and optimal control in this problem arises from design issues for the Genesis Discovery Mission being developed for NASA by the Jet Propulsion Laboratory.

Geometric Mechanics and the Dynamics of Asteroid Pairs

Abstract: The purpose of this paper is to describe the general setting for the application of techniques from geometric mechanics and dynamical systems to the problem of asteroid pairs. The paper also gives some preliminary results on transport calculations and the associated problem of calculating binary asteroid escape rates. The dynamics of an asteroid pair, consisting of two irregularly shaped asteroids interacting through their gravitational potential is an example of a full‐body problem or FBP in which two or more extended bodies interact. One of the interesting features of the binary asteroid problem is that there is coupling between their translational and rotational degrees of freedom. General FBPs have a wide range of other interesting aspects as well, including the 6‐DOF guidance, control, and dynamics of vehicles, the dynamics of interacting or ionizing molecules, the evolution of small body, planetary, or stellar systems, and almost any other problem in which distributed bodies interact with each other or with an external field. This paper focuses on the specific case of asteroid pairs using techniques that are generally applicable to many other FBPs. This particular full two‐body problem (F2BP) concerns the dynamical evolution of two rigid bodies mutually interacting via a gravitational field. Motivation comes from planetary science, where these interactions play a key role in the evolution of asteroid rotation states and binary asteroid systems. The techniques that are applied to this problem fall into two main categories. The first is the use of geometric mechanics to obtain a description of the reduced phase space, which opens the door to a number of powerful techniques, such as the energy‐momentum method for determining the stability of equilibria and the use of variational integrators for greater accuracy in simulation. Second, techniques from computational dynamic systems are used to determine phase space structures that are important for transport phenomena and dynamic evolution.

Role of Invariant Manifolds in Low-Thrust Trajectory Design

This paper demonstrates the significant role that invariant manifolds play in the dynamics of low-thrust trajectories moving through unstable regions in the three-body problem. It shows that an optimization algorithm incorporating no knowledge of invariant manifolds converges on low-thrust trajectories that use the invariant manifolds of unstable resonant orbits to traverse resonances. It is determined that the algorithm could both change the energy through thrusting to a level where the invariant manifolds could more easily be used, as well as use thrusting to move the trajectory along the invariant manifolds. Knowledge of this relationship has the potential to be very useful in developing initial guesses and new control laws for these optimization algorithms. In particular, this approach can speed up the convergence of the optimization process, retain the essential geometric and topological characteristics of the initial design, and provide a more accurate estimate of the A V and fuel usage based on the initial trajectory.

The role of invariant manifolds in lowthrust trajectory design (part III)

This paper is the third in a series to explore the role of invariant manifolds in the design of low thrust trajectories. In previous papers, we analyzed an impulsive thrust resonant gravity assist flyby trajectory to capture into Europa orbit using the invariant manifolds of unstable resonant periodic orbits and libration orbits. The energy savings provided by the gravity assist may be interpreted dynamically as the result of a finite number of intersecting invariant manifolds. In this paper we demonstrate that the same dynamics is at work for low thrust trajectories with resonant flybys and low energy capture. However, in this case, the flybys and capture are effected by continuous families of intersecting invariant manifolds.

Lunar sample return via the interPlanetary superhighway

The Lunar Sample Return mission consists of two spacecraft, a communications module, and a lander/sample return module carried to the Moon by another ship. Knowledge of the InterPlanetary Superhighway tunnels and their dynamics provided a quick back-of-the-envelope estimation of the timing and costing of such libration missions which compared well with fully integrated solutions.

Unstable Resonant Orbits near Earth and Their Applications in Planetary Missions

This paper explores the uses of planar, simple-periodic symmetrical families of orbits in mission designs in the Earth-Moon system. This classification is defined as the planar periodic orbits that pierce the x-axis in the rotating frame exactly twice per orbit where each piercing is orthogonal to the x-axis. A continuation method has been used to explore several families of this class of orbit in the Earth-Moon restricted three-body system. The invariant manifolds of the unstable orbits in each of these families are then produced and several mission designs are discussed that take advantage of these manifolds. Focus is given to mission designs that implement resonant orbits that periodically fly by the moon.