I've been working on implementing a piston-type joint in the Chipmunk physics library. This is a joint that allows one object to move freely along a straight track, but stops it at the endpoints. The sliding object is allowed to rotate freely. Obviously this is to simulate wheels on suspension; it will be combined with a damped spring to do the suspension.
To do this, I need to solve for an impulse which, when applied to the two bodies at the joint, will eliminate relative velocity in the cross-track direction. Additionally, if the sliding object has reached a track extent, I must generate an impulse which will kill any relative velocity that would carry it past the end of the track. This latter velocity adjustment is essentially the same as a contact non-penetration constraint, in that we don't want to ever produce an impulse that would keep the sliding object from being able to move the other direction, away from the track's endpoint.
Here's some initial math going the opposite direction: given an impulse, what is the change in relative velocity at the joint? I've used the Greek letter Delta (the triangle) to indicate change, as in “change in velocity”. Here are the variable definitions:
And here is development of the equation for change in relative velocity at point r, given an impulse j at that same point:
It's built up from the changes in linear and angular velocities at the two bodies' centers of mass.
By inverting this, it's possible to solve for an impulse which will produce a desired change in local relative velocity. I'll also need to rotate the coordinate system so that velocities along and across the track are easily separated. As I said before, the along-the-track velocity changes are really like contact constraints so they'll need to be handled slightly differently from the cross-track velocity constraint.
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