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6.5. CONTACT 85
where v + ni is the value of vni (relative body velocity along ni at pi) after the collision is resolved,
v− ni is the value before the collision is resolved and ɛi is called the coefficient of restitution. Setting
ɛi = 1 would model a perfectly elastic collision while ɛi = 0 would make the bodies not to bounce
at all, [13].
Since ˙ Ci p is the relative body velocity at contact i, C˙ i
p = vni , the collision law can be implemented
by the velocity-level constraint
C˙ i p = vni = Ji,Ai (ni) · vAi + Ji,Bi (ni) · vBi ≥ −ɛi · v − ni . (6.44)
Note that we have to use a greater-or-equal formulation of the constraint, because there might
be other impulses, say due to other collisions, acting on Bi that should not be cancelled.
Collisions With Impulsive Friction
To model collisions with friction we would like to (in addition to the collision constraint (6.44))
constrain vtx to zero but allow the constraint to break if the magnitude of the impulsive constraint
i
force due to the constraint had to exceed µi · fni , where fni was the magnitude of the impulsive
normal force due to the collision constraint, and do the same for v y
ti .
If f• and f• denote the impulsive quantities analogous to the force quantities f• and f• from
the previous section, then the velocity-level constraints we would like to model are given by
fni ≥ 0 vni ≥ −ɛi · v − ni
fni · vni = 0
vti = 0 ⇒ fti ≤ µi · fni
vti = 0 ⇒ fti = µi · fni ∧ fti · vti ≤ 0. (6.45)
We would like to approximate these equations analogously to the static friction case so that
the static friction’s algorithm could be reused (this time we would relate constraint impulses to
velocities instead of constraint forces to accelerations, though).
By following the derivation of (6.43) we get the formulation of the constraints we are looking
for, in the form of a 3-DOF velocity-level bounded-equality constraint (6.26),
Ji,Ai =
⎛
−n
⎜
⎝
T i
− t
−(rAi × ni) T
x T
i −(rAi × t x i )T
− t y T
i −(rAi × t y
i )T
⎞
⎟
⎠ Ji,Bi =
⎛
n
⎜
⎝
T i
t
(rBi × ni) T
x T
i (rBi × t x i )T
t y T
i (rBi × t y
i )T
⎞
⎟
⎠
ci = (−ɛi · v − ni , 0, 0) (6.46)
lo
λi = (0, −µi · ˆ fni , −µi · ˆ fni )
hi
λi = (+∞, µi · ˆ fni , µi · ˆ fni ),
where ˆ fni is a guess of fni = (λi)1, the first constraint DOF is due to the collision (impact) and
the other two DOFs are due to the impulsive friction along t x i and t y
i directions and are special.
Impulsive friction with other velocity-level constraints is handled analogously to the accelerationlevel
case of static friction with other acceleration-level constraints. All constraints are again
reduced to a LCP, the constraint rows due to the non-special DOFs are solved first, the values
of ˆ fni are obtained and impulsive friction limits are set. Finally, all constraint rows due to the
non-special and special DOFs are solved.