[ODE] ODE with non-penetration constraint

[Dirk Gregorius]

A hard constraint basically means that you have a "hard" algebraic 
constraint equation instead of plugging in a "soft" spring like in a 
penalty method. The ODE requires for each contact:

1) The penetration at the contact should vanish
C(p,q) = (p1 + r1 - p2 - r2) * n - d = 0

2) The relative velocity should vanish
dC/dt  = (v1 + cross( omega1, r1) - v2 - cross(omega2, r2) * n

-> J = ( n,  cross( r1, n ), -n, -cross( r2, n ) )

Since we apply only forces when we have an approaching relative velocity 
at the contact we basically end up with

w = A*x + b >= 0 and x >= 0 and xT * w = 0 which is a classic LCP or 
MLCP in presents of additonal holonome constraints.


Actually each position constraint imposes also a velocity- and 
acceleration constraint onto the system. Think of a ball socket joint. 
The position constraint requires that the anchors are at the same point 
in space. In order to maintain this constraint the relative velocity and 
acceleration must vanish at the anchor. Otherwise the anchors would get 
apart. So *all* constraints (position, velocity and acceleration) must 
always be satisfied. The ODE solves only at the velocity level. Note 
that even when you fully satisfied the velocity constraints, you still 
can end up with violated position constraints (e.g. because of high 
angular velocities). The ODE deals with this using Baumgarte 
stabilization. Erin Catto's has a nice presentaion about this at 
www.gphysics.com.

HTH,
-Dirk