Just to clarify some of these relationships..

Back in the APollo era, NASA used guidance systems developed by the MIT Instrumentation Laboratory, run by Charlie Draper. At that time, the lab was a part of MIT. The lab also developed quite a few guidance systems for ICBMs and SLBMs (I forget the exact list of which missiles).

After the Vietnam War, MIT spun off military-linked labs like that, and so Draper Laboratories became its own independent private corporation, though located only a block from MIT. What they do is guidance. I don't think they were the prime contractor for Shuttle guidance, though wouldn't be surprised if they supplied something (e.g., gyros). They did continue to develop missile guidance systems, most notably the super accurate system for the MX "Peacekeeper" missile. Most of their work today is still military, though increasinly they do other work, such as developing micro-accelerometers for things like automobile airbag sensors. I didn't get the impression they do much NASA work. On the other hand, the outfit that runs Chandra is renting office space in Draper's building, so there is *some* NASA money at that mailing address.

Lincoln Labs is the descendent of the MIT Radiation Lab, the group that developed radar. Today they are located about 10 miles from MIT near Hanscom AFB in the Boston suburbs. Despite (or maybe because of) the distance, they weren't spun off as thoroughly as Draper; they are still connected to MIT. They do communications and sensors, again, mostly on DoD contracts. I'm not sure if they do any NASA work, but it's possible -- I could imagine them having things to discuss with the DSN people, or for radio telescopes, or radar, microwave, mm-wave, and other such instruments on probes.

The early Apollo era plan had been for a full scale NASA center in Boston that would do electronics, computing, and guidance, but I think this had been dropped well before Apollo 11, even. How much it being dropped had to do with Kennedy leaving the scene I'm not sure; it never was to the stage of being a done deal, more like part of a longer-range plan. What Cambridge got instead for that physical (an area of crumbling old factories near MIT that was cleared around 1970) was a Department of Transportation research center. Some of what they do is military logistics planning, but they also do development for FAA systems. When you see real-time flight tracking graphics -- any of the various commercial providers -- the raw data is always from building a block from MIT. Much of that parcel actually sat vacant until about 1990, but in the last ten years it's been filled in mostly with the Whitehead Institute and private biotech firms, and more importantly, the Cambridge Brewing Company.