Regularizing a Time-stepping Method for Rigid
Multibody Dynamics
T. Preclik? , C. Popa† , U. Rüde?
? Chair of Computer Science 10 (System Simulation)
University of Erlangen-Nürnberg, Cauerstr. 6, 91058 Erlangen, Germany
e-mails: [email protected],
[email protected]
web page: http://www10.informatik.uni-erlangen.de/
† Faculty of Mathematics and Computer Science
Ovidius University of Constanta, 124 Mamaia Blvd., 900527 Constanta, Romania
e-mail: [email protected]
web page: http://math.univ-ovidius.ro/
July 3, 2011
Introduction
The Chair of Computer Science 10 (System Simulation) amongst
other topics deals with:
I
non-smooth rigid multibody dynamics with contact and
friction (e.g. granular flow)
I
particulate flows (e.g. fluidization and sedimentation)
Motivation
I
rigidity is a convenient idealization reducing the degrees of
freedom
I
in the stiff limit ambiguities appear
Hard constraints:
I
contact reactions statically
indeterminate
Soft constraints:
I
contact reactions uniformly
distributed
Motivation (cont.)
hard spherical objects deform mainly in
the contact region [1]
I in contrast e.g. ladders typically need
flexion to resolve static indeterminacy [2]
⇒ rigid bodies might need to be enriched by
additional degrees of freedom (e.g.
torsion springs)
I
[1] R. Cross. Differences between bouncing balls, springs, and rods. American Journal of
Physics, 76(10):908–915, 2008.
[2] A.G. Gonzalez and J. Gratton. Reaction forces on a ladder leaning on a rough wall.
American Journal of Physics, 64(8):1001–1005, 1996.
Outline
Motivation
Modeling
Differential Equations
Ball and Socket Joint
Unilateral Contact
Frictional Contact
Conclusion
Modeling: Differential Equations
Ordinary Differential Equation:
q̇
v
=
ϕ̇
Qω
v̇
f
M
=
+ JT λ
ω̇
τ − ω × Iω
Modeling: Differential Equations
Ordinary Differential Equation:
q̇
v
=
ϕ̇
Qω
v̇
f
M
=
+ JT λ
ω̇
τ − ω × Iω
Semi-implicit Time Discretization:
0
0 ṽ
q̃
q̃
= δt
+
ϕ̃0
ϕ̃
Q̃ω̃ 0
0
ṽ
f̃
ṽ
−1
−1 T
= δt M̃
+ δt M̃ J̃ λ̃ +
ω̃ 0
ω̃
τ̃ − ω̃ × Ĩω̃
Modeling: Ball and Socket Joint
I
I
I
hard constraint requires gap gi (q(t), ϕ(t)) = 0
implicit discretization and linearization
0
ṽ
0
0
gi (q̃ , ϕ̃ ) = gi (q̃, ϕ̃) + δt J̃i∗
ω̃ 0
inserting discretized differential equation
1
g(q̃, ϕ̃) + J̃M̃−1 J̃T δt λ̃ +
δt
„„ «
„
««
ṽ
f̃
J̃
+ δt M̃−1
=0
ω̃
τ̃ − ω̃ × Ĩω̃
Modeling: Ball and Socket Joint
I
I
I
hard constraint requires gap gi (q(t), ϕ(t)) = 0
implicit discretization and linearization
0
ṽ
0
0
gi (q̃ , ϕ̃ ) = gi (q̃, ϕ̃) + δt J̃i∗
ω̃ 0
inserting discretized differential equation
1
g(q̃, ϕ̃) + J̃M̃−1 J̃T δt λ̃ +
δt
„„ «
„
««
ṽ
f̃
J̃
+ δt M̃−1
=0
ω̃
τ̃ − ω̃ × Ĩω̃
Modeling: Ball and Socket Joint
I
I
I
hard constraint requires gap gi (q(t), ϕ(t)) = 0
implicit discretization and linearization
0
ṽ
0
0
gi (q̃ , ϕ̃ ) = gi (q̃, ϕ̃) + δt J̃i∗
ω̃ 0
inserting discretized differential equation
AT A x − AT b = 0
Modeling: Ball and Socket Joint
I
I
I
hard constraint requires gap gi (q(t), ϕ(t)) = 0
implicit discretization and linearization
0
ṽ
0
0
gi (q̃ , ϕ̃ ) = gi (q̃, ϕ̃) + δt J̃i∗
ω̃ 0
inserting discretized differential equation
AT A x − AT b = 0
I
possibly underdetermined solution
Modeling: Regularized Ball and Socket Joint
I
I
constraint equation is replaced by a Hookian
spring λi = − diag(ki )gi (q(t), ϕ(t))
implicit discretization
λ̃i = − diag(ki )gi (q̃0 , ϕ̃0 )
I
inserting discretized differential equation
“
”
1
g(q̃, ϕ̃) + J̃M̃−1 J̃T + δt12 diag(k)−1 δt λ̃ +
δt
„„ «
„
««
ṽ
f̃
J̃
+ δt M̃−1
=0
ω̃
τ̃ − ω̃ × Ĩω̃
Modeling: Regularized Ball and Socket Joint
I
I
constraint equation is replaced by a Hookian
spring λi = − diag(ki )gi (q(t), ϕ(t))
implicit discretization
λ̃i = − diag(ki )gi (q̃0 , ϕ̃0 )
I
inserting discretized differential equation
“
”
1
g(q̃, ϕ̃) + J̃M̃−1 J̃T + δt12 diag(k)−1 δt λ̃ +
δt
„„ «
„
««
ṽ
f̃
J̃
+ δt M̃−1
=0
ω̃
τ̃ − ω̃ × Ĩω̃
Modeling: Regularized Ball and Socket Joint
I
I
constraint equation is replaced by a Hookian
spring λi = − diag(ki )gi (q(t), ϕ(t))
implicit discretization
λ̃i = − diag(ki )gi (q̃0 , ϕ̃0 )
I
inserting discretized differential equation
AT A + D x − AT b = 0
Modeling: Regularized Ball and Socket Joint
I
I
constraint equation is replaced by a Hookian
spring λi = − diag(ki )gi (q(t), ϕ(t))
implicit discretization
λ̃i = − diag(ki )gi (q̃0 , ϕ̃0 )
I
inserting discretized differential equation
AT A + D x − AT b = 0
Modeling: Regularized Ball and Socket Joint (cont.)
I
AT A + D x − AT b = 0
drive spring constants homogeneously to infinity to approach
stiff limit ⇒ introduce regularization parameter s
Modeling: Regularized Ball and Socket Joint (cont.)
AT A + sD x − AT b = 0
I
drive spring constants homogeneously to infinity to approach
stiff limit ⇒ introduce regularization parameter s
I
solution converges to the (unique) weighted minimum norm
solution x∗ of the unregularized system
s→0
x −→ x∗
1
kD 2 x∗ k22 =
min
AT Ax=AT b
xT Dx
Modeling: Unilateral Contact
I
Signorini contact condition:
gin (q(t), ϕ(t)) ≥ 0 ⊥ λin ≥ 0
I
discretize implicitly, insert
discretized differential equation,
simplify notation
AT Ax − AT b ≥ 0 ⊥ x ≥ 0
I
possibly underdetermined solution
Modeling: Regularized Unilateral Contact
I
Regularized Signorini contact condition:
gin (q(t), ϕ(t)) + ki−1
λin ≥ 0 ⊥ λin ≥ 0
n
I
discretize implicitly, insert
discretized differential equation,
simplify notation
(AT A+ sD )x−AT b ≥ 0
I
⊥ x≥0
uniquely determined solution
Modeling: Regularized Unilateral Contact (cont.)
(AT A + sD)x − AT b ≥ 0 ⊥ x ≥ 0
I
drive spring constants homogeneously to infinity to approach
stiff limit
I
solution converges to the (unique) weighted minimum norm
solution x∗ of the unregularized system [3]
s→0
x −→ x∗
1
kD 2 x∗ k22 =
xT Dx
min
AT Ax−AT b≥0
⊥ x≥0
[3] T. Preclik, U. Rüde, and C. Popa. Resolving Ill-posedness of Rigid Multibody
Dynamics. Technical report, Friedrich-Alexander University Erlangen-Nürnberg.
Modeling: Frictional Contact
I
closed frictional contacts can be in the static or dynamic state
kλit,o k2 ≤ µi λ̄in
0 = ġit,o (q(t), ϕ(t)),
λit,o = −
I
ġit,o (q(t), ϕ(t))
kġit,o (q(t), ϕ(t))k2
µi λin ,
kλit,o k2 = µi λ̄in
conditions can be combined in a variation inequality (VI)
λit,o ∈ S(µi λ̄in ),
hġit,o (q(t), ϕ(t)), yit,o − λit,o i ≥ 0,
∀yit,o ∈ S(µi λ̄in ),
where S(r ) is the set of vectors in R2 within a disc of radius r
around the origin
Modeling: Regularized Frictional Contact
I
closed frictional contacts can be in the static or dynamic state
λit,o = −γi ġit,o (q(t), ϕ(t)),
λit,o = −
I
ġit,o (q(t), ϕ(t))
kġit,o (q(t), ϕ(t))k2
µi λin ,
kλit,o k2 ≤ µi λ̄in
kλit,o k2 = µi λ̄in
conditions can be combined in a variational inequality (VI)
λit,o ∈ S(µi λ̄in ),
hġit,o (q(t), ϕ(t)) + γi−1 λit,o , yit,o − λit,o i ≥ 0,
∀yit,o ∈ S(µi λ̄in ),
where S(r ) is the set of vectors in R2 within a disc of radius r
around the origin
Modeling: Regularized Frictional Contact (cont.)
I
Signorini contact condition can be reformulated into a VI and
combined with the frictional part (under some simplifying
assumptions)
I
discretization leads to an affine VI of the type
x ∈ δt F̃,
I
h(AT A + sD)x − AT b, y − xi ≥ 0,
∀y ∈ δt F̃.
solution converges to the (unique) weighted minimum norm
solution x∗ of the unregularized system [4]
s→0
x −→ x∗
1
kD 2 x∗ k22 =
xT Dx
min
x∈δt F̃ ,
hAT Ax−AT b,y−xi≥0,
∀y∈δt F̃
[4] Xiubin Xu and Hong-Kun Xu. Regularization and Iterative Methods for Monotone
Variational Inequalities. Fixed Point Theory and Applications, 2010:11, 2010.
Conclusion
I
physically motivated regularization of non-smooth rigid
multibody contact dynamics
I
regularization of ball and socket joints, frictionless unilateral
contacts and certain types of frictional contacts
I
the stiff limit of the smooth system corresponds to the
(unique) weighted minimum norm solution of the non-smooth
system
I
bridges the gap between smooth and non-smooth rigid
multibody dynamics
Thank you for your attention!
Questions? Comments?
References
[1] R. Cross. Differences between bouncing balls, springs, and
rods. American Journal of Physics, 76(10):908–915, 2008.
[2] A.G. Gonzalez and J. Gratton. Reaction forces on a ladder
leaning on a rough wall. American Journal of Physics, 64(8):
1001–1005, 1996.
[3] T. Preclik, U. Rüde, and C. Popa. Resolving Ill-posedness of
Rigid Multibody Dynamics. Technical report,
Friedrich-Alexander University Erlangen-Nürnberg, 2010.
[4] Xiubin Xu and Hong-Kun Xu. Regularization and Iterative
Methods for Monotone Variational Inequalities. Fixed Point
Theory and Applications, 2010:11, 2010.