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Cassini's Elaborate Orbital Mechanics
jamie found an article at the NY Times about the extreme orbital mechanics gyrations required to extend the Cassini mission at Saturn by seven more years. Here's a graphic of the mission extension, which NASA took two years to arrive at. "The plans are for Cassini to keep working for seven more years, but it currently has only 22 percent of the maneuvering propellant it had when it started. Figuring out how to more than double the duration of the mission with less than a quarter of the fuel is hard. Cassini's orbital mechanics present an astonishingly complex exercise in Keplerian physics and geometry. The enormous array of science objectives and targets — moons, rings, Saturn itself — makes it one of the most complex missions ever flown. ... 'Without Titan,' Mr. Seal [Cassini's mission planning supervisor] said, 'we would go into one orbit around Saturn and be stuck there.' Thus Titan, in the argot of orbital mechanics, is Cassini's 'tour engine.' [T]he final 'reference trajectory' ... now includes 56 passes over Titan, 155 orbits of Saturn in different inclinations, 12 flybys of Enceladus, 5 flybys of other large moons — and final destruction."
I have worked on this problem with a teacher at A&M that is working on this exact problem. Even in simple cases of move from here to here in more then 1 burn can not be numerically solved with current technology (damn computers too slow). So I worked on apply genetic algorithms and Lambert’s equation to solve for minimum delta V. These calculating become much more complex when you can enter a 3rd body (a moon) into this type of calculations.
I have also talked with people at Johnson Space Center about this and they use programs like Matlab to determine the orbit maneuvers and another program I can't recall offhand for visualizing it.
I had the good fortune to be working on the Galileo mission during its Mission Design phase. Many of the techniques used by the Cassini mission designers were developed for Galileo. Disclamer: I was not on the mission design team.
First of all, the Voyager encounters with Jupiter and Saturn were always when the spacecraft were moving away from the sun. However, during the Galileo satellite tour the mission designers realized that the Galileo spacecraft could encounter Callisto, Ganymede, and Europa when moving away and moving toward Jupiter. Furthermore, the closest approach ("encounter") could be targeted to be either in front of the moon (with respect its orbit around Jupiter) or behind it. These choices allowed the designers a great deal of freedom to use the moons' gravity to shape the spacecraft's orbit. As I understand it, they did not just plan the current encounter to obtain the next encounter, but also the encounter after that.
The ability to use a moon to shape a spacecraft orbit depends on the ratio of the mass of the planet to the mass of the moon (for all practical purposes the spacecraft is massless.) Only Io, Callisto, Ganymede, and Europa are able to provide gravity assists at Jupiter, and only Titan at Saturn.
I spoke to Bob Mitchell, Cassini Project Manager, a few years ago and asked him about this specifically. He told me that while it was true that having to go back to Titan every time to change the orbit was a constraint, it also provided the freedom to send the spacecraft out of the "plane" where the moons orbited. At Jupiter it was necessary to stay in the plane to make multiple visits to all the moons, but since at Saturn you must visit the same moon to change the spacecraft's orbit every time (Titan) there is fewer reasons to stay in the plane. And, as you can see from the orbit diagrams, Cassini has traveled outside of the plane many times.