[ODE] Distance, or "stick" constraints

[i-photon@comcast.net]

I have need of a distance constraint for a robot simulator. Instead of using joints and bodies for actuators, I'd prefer to use distance constraints as this greatly reduces the number of bodies in the simulation.

My current forulation looks like this (a modified ball and socket):

(1):  (p2+R2*a2)+(1/2*L) = (p1+R1*a1)-(1/2*L)

(2):  d(1)/dt = (v2+SSM(w2)*a2') + (1/2*dL/dt) = (v1+SSM(w1)*a1')-(1/2*dL/dt)

where L = r*D/|D|, D = (p2+R2*a2)-(p1+R1*a1), r is the desired distance, and SSM(v) means to build a Skew-Symmetric Matrix out of "v".

I have found the time derivative of L, however I don't know how to incorporate it into the constraint Jacobians:

dL/dt = s*dD/dt + ds/dt*D

s = r/|D|

ds/dt = r*(dot(D,dD/dt))/sqrt(dot(D,D)^3)

dD/dt = ((v2+SSM(w2)*a2') - (v1+SSM(w1)*a1'))
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