# Low-Thrust Trajectory Design Using Reachability Sets near Asteroid 4769 Castalia

## Abstract

We present a computational approach for the design of continuous low thrust transfers around an asteroid. These transfers are computed through the use of a reachability set generated on a lower dimensional Poincaré surface. Complex, long duration transfer trajectories are highly sensitive to the initial guess and generally have a small region of convergence. Computation of the reachable set alleviates the need to generate an accurate initial guess for optimization. From the reachable set, we chose a trajectory which minimizes a distance metric towards the desired target. Successive computation of the reachable set allows for the design of general transfer trajectories which iteratively approach the target. We demonstrate this method by determining a transfer trajectory about the asteroid 4769 Castalia.

## Main Idea

The motion of spacecraft around an asteroid is difficult to analyze but the dynamics are quite similar to the three-body problem. This paper demonstrate how we extend our previous work in the planar three-body problem to the more challenging situation around an asteroid. Instead of a planar problem we are now dealing with full three dimensional motion. In addition, the asteroid itself is non-spherical and possess a complex gravitational field.

We include in our spacecraft model an ideal thruster, which emulates many current electric propulsion systems. It is always difficult to determine optimal trajectories as the solutions are very sensitive to variations of the initial conditions. In addition, the control input is typically very small and therefore requires an accurate numerical integration method to ensure the results are correct.

By using our reachability method we alleviate the need to determine initial guess for optimization and instead offer a simple distance metric to choose trajectories which approach the target.

## Asteroid Model

We model the asteroid 4769 Castalia using a polyhedron shape model. This model is defined, using ground based radar measurements, by a series of body-fixed vectors. We then link any three vectors to define a single face, or facet, of the asteroid. By defining many faces, we can create a very accurate shape model of the exterior of the asteroid.

For example, here is a small portion of the shape model file:

v   0.000000e+00   0.000000e+00   2.893730e-01
v   7.342140e-01   0.000000e+00   0.000000e+00
v   0.000000e+00   4.104200e-01   0.000000e+00
v  -8.202460e-01   0.000000e+00   0.000000e+00
v   0.000000e+00  -4.391400e-01   0.000000e+00
v   0.000000e+00   0.000000e+00  -3.752060e-01

Many, many more vectors

f 1882  652   23
f   24 1135  641
f   70 1030  581
f 1966  249 1750
f  622 1763  365



Typically, there are a series of vectors, denoted by the v in the first column, followed by the mapping of the vectors into the triangular faces, denoted by the f in the first column. You need both sets of information in order to define the complete shape of the asteroid. Luckily, this process has been completed for many asteroids and there is good information that is easily accessible.

Using the work of Robert Werner:

• R. A. Werner, “The Gravitational Potential of a Homogeneous Polyhedron or Don’t Cut Corners,” Celestial Mechanics and Dynamical Astronomy, vol. 59, no. 3, pp. 253–278, 1994.
• R. A. Werner and D. J. Scheeres, “Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia,” Celestial Mechanics and Dynamical Astronomy, vol. 65, no. 3, pp. 313–344, 1996.

we have a simple closed form expression for the gravitaitonal potential about an asteroid. This is completely defined in terms of the shape model and assumes a constant density for the asteroid.

## Reachability Set on a Poincaré Section

The Poincaré section is a lower dimensional subspace used to investigate the properties of periodic orbits in dynamical systems. In addition, the reachable set is the set of states that are achievable over a fixed time and under the constraints of the system. For example, consider a vehicle that is on a road. It begins with some initial state, it’s position and velocity. We would like to know where it can possibly end up from that initial position over a fixed amount of time. We also know that the possible control inputs are constrained, i.e. it has a limited acceleration and steering ability. This type of information is frequently used in collision avoidance situations, e.g. air traffic control.

Here we compute the reachable set on a Poincaré section to determine transfers around asteroids. This alleviates the need to determine an accurate initial guess as we can simply choose states from our reachable set which move us toward the target. Repeating this process allows us to achieve large or complex transfers.

Body-fixed Frame Inertial Frame

@inproceedings{kulumani2016d,