X 0 1 2 3 4 5 6 7 8
Section Two
Mars Pathfinder
7. GRABET Orbits
X 0 1 2 3 4 5 6 7 8 |
|
Section Two |
| Mars Pathfinder |
| ___ |
| 7. GRABET Orbits |
| 8. Earth Escape |
| 9. Mars Capture |
| 10. Mars Pathfinder |
TAB 8
GRAVITY ASSISTED EARTH ESCAPE TRAJECOTORYABSTRACTThe computer algorithm is set up to optimize each half of the flight path from conjunction to Earth (integrating backwards in time) independently. A fixed thrust is applied at conjunction, then the minimum thrust to reach parking orbit around Earth is determined. The "terminal" thrust is assumed to have three components: a large thrust to circularize at the targeting orbit, and two small thrusts to reach the final parking orbit via the Hohmann Transfer. Typically, the large thrust is combined with one of the two small thrusts, into one instantaneous thrust. All forces applied are along the velocity vector.
The numerical implementation of the "Earth capture" strategy necessitates the creation of a unique mapping in the vicinity of the target. Since the objective is to find a global minimum, it is essential that every thrust value applied at conjunction generates a total thrust to reach parking orbit. This total thrust need not be accurate to 12 significant digits except near the actual minimum. However, it must have a constant slope downward to the minimum from all directions so that a one dimensional line search can find the minimum easily. A cross section of the mapping provided later.
The energy/thrust "well" simulated by the algorithm returns a terminal thrust in such a way that it drives the final endpoint to the target quadrant. This is a point on the far side of the planet from which the s/c approaches, located at or near the final parking orbit itself.
Conjunction to Earth
The technique for optimizing the conjunction to Earth flight path is simple. The initial state of the s/c is integrated back in time to Earth’s SOI. Then the coordinates are changed from heliocentric to geocentric and the path is integrated until the range from Earth is increasing.
The integration is done with a Runge-Kutta 7/8 variable step integrator. At each step the thrust needed to reach the 200 km final parking orbit is calculated (i.e. the orbit is circularized, then a small Hohmann Transfer is done to the final parking orbit).
It is assumed that the integrator stops for any significant changes in conditions on the spacecraft, thus no intermediate targeting orbits need to be considered. As the spacecraft nears Earth, the lowest total trust is saved in an array and when the range to Earth begins to increase, the loop is terminated and the minimum thrust scenario is retrieved from the array as the optimal trajectory.
After a single trajectory is optimized, the total thrust is compared to the total thrust of the previous run(s). If it is decreasing, the thrust at conjunction is decreased (and vice versa) and another trajectory is evaluated. The mapping is such that the global minimum is always in the direction of a negative gradient, so all the iterative loop must do is to conduct a one dimensional search until the minimum is reached within a specified tolerance.
Earth Gravity Well Target Mapping