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The first is the use of Einstein's formulas. They allow for very precise trajectory and orbit calculations which would not have been possible with the old Newtonian physics.
The second and most important aspect are TCMs or trajectory correction maneuvers. All missions allow for a number of these maneuvers in the cruise and approach phases of the mission. These TCMs ensure that they hit the right orbit when they arrive.
How about some more details.
In order to make the TCMs the spacecraft needs to determine quite accurately both its position and velocity.
I can understand how the apparent position of the stars will allow for determination of the orientation of the craft relative to the relatively fixed coordinate system of the stars.
The apparent positions of the planets would allow for the position to be determined. Tracking the position over time allows for the velocity to be determined; that is, both the direction of travel and the speed.
Does the typical spacecraft use a telescope to determine the positions of objects in its field of view? The stars will appear as single points. Does the craft use the edges of the planets when its proximity to them permits it?
The craft would need to find many points to verify its position and velocity. Does the craft use a mechanical system to measure the angles for a particular position of the telescope relative to the body of the craft?
I actually attempted to determine this information from a person involved in these space missions but I found his answer far too minimal and didn't want to embarrass him by finding that perhaps he didn't know the details.