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6.3. VELOCITY-LEVEL CONSTRAINTS 65
large magnitudes exerted during the period of the collision change the body velocity 7 . Since
this would be hard to simulate (the corresponding ODE would have to be integrated with an
extremely small time step) we will postulate a concept of impulsive force and torque (impulses)
that will allow to change body velocities instantly without producing stiff ODEs. The collision
in our example could then be treated as if its time duration was zero.
Impulsive Force and Torque
Let us assume that we have a rigid body with the center of mass at the world-space position
x(t) and P (t) and L(t) are the body linear and angular momentum. Impulsive force (impulse)
JF is postulated as a force with “units of momentum”. When JF is applied to the body at the
world-space position r, then the linear momentum P (t) and the angular momentum L(t) change
by ∆ P (t) and ∆ L(t), where
∆ P (t) = JF
∆ L(t) = Jτ
Jτ = (r − x(t)) × JF
(6.21)
and Jτ is the impulsive torque due to impulsive force JF exerted on the rigid body at the worldspace
position r. More precisely if P − (t) and L − (t) is the linear and angular momentum before
the application of the impulse, then the linear and angular momentum P + (t), L + (t) after the
application of the impulse equals P + (t) = P − (t) + JF and L + (t) = L − (t) + (r − x(t)) × JF .
Similarly to forces, JF can be imagined as if it acted at the rigid body’s center of mass ( JF
affects the linear momentum only) and Jτ as capturing the rotational effects due to JF actually
exerted at r ( Jτ affects the angular momentum only).
Impulsive forces and torques can be seen as “ordinary” forces and torques that change the
body’s linear and angular momentums directly, instead of affecting their time derivatives. Remember
that if Ftotal(t) and τtotal(t) are the total external force and torque exerted on the body,
then ∂
∂t P (t) = Ftotal(t) and ∂
∂t L(t) = τtotal(t). In the case of impulses, however, the impulsive force
JF and impulsive torque Jτ change the linear and angular momentums directly, ∆ P (t) = JF and
∆ L(t) = Jτ . Note the correspondence of the impulsive torque Jτ = (r −x(t)) × JF due to the impulsive
force JF exerted on the body at the world-space position r to the torque τ = (r−x(t))× F
due to the force F exerted on the body at the world-space position r.
Notation
We will extend our notation of generalized quantities by
• generalized momentum F total
imp
A block vector F total
of a body
consisting of two blocks storing the rigid body’s linear momentum
imp
P ∈ R3 and angular momentum L ∈ R3 , F total
imp = ( P , L) ∈ R6 . Generalized momentum
serves the same purpose like the linear momentum P of particles.
The reason for naming the generalized momentum in this way will become clear soon.
• generalized impulsive force (impulse) Fimp exerted on a body
A block vector Fimp consisting of two blocks storing the impulsive force JF ∈ R 3 (imagined
as acting at the body’s center of mass) and impulsive torque Jτ ∈ R 3 due to JF , Fimp =
( JF , Jτ ) ∈ R 6 .
7 The restoration forces can be imagined as constraint forces due to ¨ Cp(r(t)) ≥ 0 constraints with position
stabilization terms like in (4.8).