The Method - the theory behind the computer program
There are four fundamental aspects to the operation of the "Mars Path Finder" computer program that allow it to determine an optimal trajectory quickly and accurately.
The first is that the process is divided into two parts; the integration is started at conjuction and integrated backward in time to Earth (which is ok since all the forces are central forces) and forward in time to Mars. This divides the two point boundary value problem with variable endpoints in space and time, into two end point boundary value problems in which only the final conditions are unknown. This is far easier to solve, analytically as well as numerically.
What makes it possible to segregate this problem in this way is that the Hohmann Transfer in two body theory is used to determine the first guess of the initial position of the spacecraft at conjunction. This is a relative point, however; because of how the two integrations are done to Earth and Mars, so it is only necessary to have a good first guess. The theory behind this guess is developed in detail in a formal technical paper available elsewhere on this web site.
The second feature of the program that makes it work efficiently and without the need for the usual nonlinear optimization algorithms is a special mapping of the space in the vicinity of Earth and Mars. Consider the approach geometry to Earth. The code is set up to drive the trajectory into a specific quadrant. This is done by assigning penalty values to trajectories as a function of their distance from the best trajectory; ie one that passes near Earth. This coarse selection process is in the form of a one dimensional search for the lowest total thrust to reach a final parking orbit around Earth. The last few iterations optimize the trajectory at each step of the integrator, and when the spacecraft is far past Earth, the lowest total thrust of all the integrator steps for this particular pass is saved.
The program is very sensitive, in that it is capable of using the Earth's gravity to best advantage. That is, the optimum trajectory is not to start the interplanetary trajectory from the initial 200 Km parking orbit (which is a fixed parameter of the problem) but to do a small two thrust Hohmann transfer to a slightly higher orbit, then do the major thrust to escape Earth's gravity from this "phasing" orbit. The geometry of this flight path is reproduced schematically. This allows a little extra free velocity to be gained in a gravity assisted Earth flyby. The same process is used on the Mars end of the algorithm.
The third feature of the program is that the Mars capture trajectory is configured to seek a special type of incomming flight path called a free return trajectory. That is, the Apollo missions to the moon (detailed here in a page from Dr. Schutz' notes) were designed to approach the moon along a flight path that, if something went wrong, would bring the spacecraft right back toward Earth - no thrusts or maneuvers were needed for the spacecraft to return. It was a safety precaution, and it is also a very efficient trajectory.
To get an idea for just how helpful it is to put a spacecraft on this kind of "free return" incoming trajectory, or at least to seek a flight path that draws the spacecraft into this kind of a semi stable Mars orbit; consider the following graph. The usual Mars mission is the higher values. As the trajectory gets closer and closer to a "free return" type of approach, the total thrust to reach the final 100 Km parking orbit (again, a fixed parameter of the problem) drops off dramatically. The total savings is on the order of 5 Km/sec, a savings of almost 20% of the typical total flight allowance. A schematic of this approach trajectory is shown here. The free return loop is dashed in, where instead of the Earth-moon system, the program seeks an approach so that it finds a free return in the Mars-Sun system. That is, upon approach to Mars the central body is the sun instead of Earth as in the Apollo mission analogy; but the principle is the same.
The fourth feature of the program helps with all of the above three strategies. That is, the program always considers the forces from at least three major bodies. On approach to the final Mars parking orbit, for example, the gravity of the sun - small though it may be - is still considered in the optimization process, and it is used to help pull the spacecaft into the free return approach. A similar process happens at Earth. Both effects are quite subtle and they are not explicitly designed in the code, other than conveniently adding the third body forces at each step of the process.
These force calculations are external to the actual optimization routines, so it would be easy to add additional bodies such as Jupiter or Saturn, to make the analysis even more realistic. In fact, Jupiter is already included in the source code as a fifth perturbing body, but is zeroed out in the distribution copy of the program and code. It would also be easy to set up the program to be further optimized by a nonlinear optimizing computer routine. Since the existing algorithm is fast and accurate, the nonlinear routines will converge faster to a better solution than is possible with existing trajectory optimization methods.
