I am mechanical engineer with very little knowledge of electrical engineering. I am working on a project where I need two have two sensors, one on each shoe to track the motion of the feet, and a third sensor on the wrist of a human. The goal is to measure the footwork and movement of arms. As certain times I want to know the relative position of the wrist to the two shoes (feet). What sensors and system should I use and how can I get the relative position of each sensor to the other two?
Relative position of three sensors
It's actually a fairly difficult problem, depending on your constraints and requirements. Determining local positioning is prone to high levels of interference and accumulation of error.
GPS will be way too inaccurate unless you setup a super expensive differential GPS system, which sounds like it would be too much for you.
You could use accelerometers/gyroscopes as you mentioned, but there is a reasonable amount of error depending on your funds and how much error you can tolerate.
I'm not exactly sure what the application of this project is, but a low cost solution could be to just mount some ultrasonic distance sensors to the shoes? Or you could use cameras and colored shoes to track the movement? Just some thoughts.
Personally I'd just tie a rope to the shoes and then I'd know how long every step would be :)
Trouble with ultrasound is that it doesn't stay in the box - you still have reflections from background. So for objects that are more than a few tens of cm from the sensor you need to do signal processing to find the target. In which case you might as well pick a solution using visible light to make it easier to debug. Computer vision tracking of moving objects against static background is a solved problem; see OpenCV for a free implementation. It works best against neutral backgrounds, but doesn't require that. Two cameras gives you stereo imaging and three positions.
If you just want ball speed along an axis you can get off-the-shelf radar guns for this purpose. Building your own radar system is an extremely difficult prospect.
Edit: 50m/s should have been in the original spec, that's quite fast. Ultrasonic usually works by sending out "chirps" and timing the response. Speed of sound is ~340m/s. So it takes 35ms for sound to traverse the 12m box. During that time a 50m/s target has moved 1.75m. You need to wait until your first pulse has gone out and back before sending the next, so pulse interval is at least 70ms. So you can only get one measurement every 3.5m.
You should also spec the latency: are you trying to interact with the moving object, or is it sufficient to work out where it was after is has passed? Do you care about the exact trajectory, or just speed of passing through the box (which can be found with doppler methods with ultrasound or radar).
This is not a trivial problem. Depending on what exactly your ultimate goal is and the resources at your disposal, there could be a number of different approaches.
IMUs (inertial measurement units) - sensor units with accelerators, gyroscopes, and sometimes magnetometers - will not be enough to accurately determine relative positions of wrist and shoes. They will give you a lot of information about motion but if you try to infer position from that information, the error will add up and you will get inevitable drift.
Normal GPS is not a good idea for something this small scale (as suggested in a question comment) since it just won't be accurate enough. Differential GPS is slightly better there but might be completely unsuitable for your application and you might find it isn't worth it. Similarly, most RF approaches will heavily fluctuate in accuracy simply based on how a person is standing or where there are walls. Those are all long range solutions.
Again, depending on your application (indoors? outdoors? large area?), your best bet is probably a computer vision approach. There's motion capture where you attach markers to the parts you want to track and put cameras at as many angles as possible such that as many as possible always have an a view that is not occluded of these markers. With those video streams and some maths, you can very accurately track the positions of the markers. This is done in the film industry for example to get motion data for CGI.
Otherwise you could always grab a Kinect and make do with the data that that gives you (which may still be better than attaching IMUs) and maybe even fuse that data with a different tracking method.