Regularizing a Time-stepping Method for Rigid Multibody Dynamics T. Preclik? , C. Popa† , U. R¨ ude? ? Chair of Computer Science 10 (System Simulation) University of Erlangen-N¨ urnberg, 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)
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particulate flows (e.g. fluidization and sedimentation)
Motivation I
rigidity is a convenient idealization reducing the degrees of freedom
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in the stiff limit ambiguities appear
Hard constraints:
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contact reactions statically indeterminate
Soft constraints:
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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: q˙ v = ϕ ˙ Qω v˙ f M = + JT λ ω˙ τ − ω × Iω
Modeling: Differential Equations Ordinary Differential Equation: q˙ v = ϕ ˙ Qω v˙ f M = + JT λ ω˙ τ − ω × Iω Semi-implicit Time Discretization: 0 0 v˜ ˜ ˜ q q = δt ˜ 0 + ϕ ˜0 ϕ ˜ Qω ˜ 0 ˜f v˜ v˜ −1 −1 ˜T ˜ ˜ ˜ = δt M + δt M J λ + ω ˜0 ω ˜ τ˜ − ω ˜ × ˜Iω ˜
Modeling: Ball and Socket Joint I I
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hard constraint requires gap gi (q(t), ϕ(t)) = 0 implicit discretization and linearization 0 v˜ 0 0 ˜ gi (˜ q ,ϕ ˜ ) = gi (˜ q, ϕ) ˜ + δt Ji∗ ω ˜0 inserting discretized differential equation 1 ˜M ˜ −1 J ˜T δt λ ˜ + g(˜ q, ϕ) ˜ + J δt „„ « „ «« ˜f v˜ ˜ ˜ −1 J + δt M =0 ω ˜ τ˜ − ω ˜ × ˜Iω ˜
Modeling: Ball and Socket Joint I I
I
hard constraint requires gap gi (q(t), ϕ(t)) = 0 implicit discretization and linearization 0 v˜ 0 0 ˜ gi (˜ q ,ϕ ˜ ) = gi (˜ q, ϕ) ˜ + δt Ji∗ ω ˜0 inserting discretized differential equation 1 ˜M ˜ −1 J ˜T δt λ ˜ + g(˜ q, ϕ) ˜ + J δt „„ « „ «« ˜f v˜ ˜ ˜ −1 J + δt M =0 ω ˜ τ˜ − ω ˜ × ˜Iω ˜
Modeling: Ball and Socket Joint I I
I
hard constraint requires gap gi (q(t), ϕ(t)) = 0 implicit discretization and linearization 0 v˜ 0 0 ˜ gi (˜ q ,ϕ ˜ ) = gi (˜ q, ϕ) ˜ + δt Ji∗ ω ˜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 v˜ 0 0 ˜ gi (˜ q ,ϕ ˜ ) = gi (˜ q, ϕ) ˜ + δt Ji∗ ω ˜0 inserting discretized differential equation
AT A x − AT b = 0
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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 ˜ i = − diag(ki )gi (˜ λ q0 , ϕ ˜ 0)
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inserting discretized differential equation “ ” 1 ˜M ˜ −1 J ˜T + 12 diag(k)−1 δt λ ˜ + g(˜ q, ϕ) ˜ + J δt δt „„ « „ «« ˜f v˜ ˜ ˜ −1 J + δt M =0 ω ˜ τ˜ − ω ˜ × ˜Iω ˜
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 ˜ i = − diag(ki )gi (˜ λ q0 , ϕ ˜ 0)
I
inserting discretized differential equation “ ” 1 ˜M ˜ −1 J ˜T + 12 diag(k)−1 δt λ ˜ + g(˜ q, ϕ) ˜ + J δt δt „„ « „ «« ˜f v˜ ˜ ˜ −1 J + δt M =0 ω ˜ τ˜ − ω ˜ × ˜Iω ˜
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 ˜ i = − diag(ki )gi (˜ λ q0 , ϕ ˜ 0)
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 ˜ i = − diag(ki )gi (˜ λ q0 , ϕ ˜ 0)
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inserting discretized differential equation
AT A + D x − AT b = 0
Modeling: Regularized Ball and Socket Joint (cont.)
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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
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drive spring constants homogeneously to infinity to approach stiff limit ⇒ introduce regularization parameter s
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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
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discretize implicitly, insert discretized differential equation, simplify notation AT Ax − AT b ≥ 0 ⊥ x ≥ 0
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possibly underdetermined solution
Modeling: Regularized Unilateral Contact I
Regularized Signorini contact condition: gin (q(t), ϕ(t)) + ki−1 λin ≥ 0 ⊥ λin ≥ 0 n
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discretize implicitly, insert discretized differential equation, simplify notation (AT A+ sD )x−AT b ≥ 0
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⊥ x≥0
uniquely determined solution
Modeling: Regularized Unilateral Contact (cont.) (AT A + sD)x − AT b ≥ 0 ⊥ x ≥ 0
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drive spring constants homogeneously to infinity to approach stiff limit
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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. R¨ ude, and C. Popa. Resolving Ill-posedness of Rigid Multibody Dynamics. Technical report, Friedrich-Alexander University Erlangen-N¨ urnberg.
Modeling: Frictional Contact I
closed frictional contacts can be in the static or dynamic state ¯ in kλit,o k2 ≤ µi λ
0 = g˙ it,o (q(t), ϕ(t)), λit,o = −
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g˙ it,o (q(t), ϕ(t)) kg˙ it,o (q(t), ϕ(t))k2
µi λin ,
¯ in kλit,o k2 = µi λ
conditions can be combined in a variation inequality (VI) ¯ in ), λit,o ∈ S(µi λ hg˙ it,o (q(t), ϕ(t)), yit,o − λit,o i ≥ 0, ¯ in ), ∀yit,o ∈ S(µi λ 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 g˙ it,o (q(t), ϕ(t)), λit,o = −
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g˙ it,o (q(t), ϕ(t)) kg˙ it,o (q(t), ϕ(t))k2
µi λin ,
¯ in kλit,o k2 ≤ µi λ ¯ in kλit,o k2 = µi λ
conditions can be combined in a variational inequality (VI) ¯ in ), λit,o ∈ S(µi λ hg˙ it,o (q(t), ϕ(t)) + γi−1 λit,o , yit,o − λit,o i ≥ 0, ¯ in ), ∀yit,o ∈ S(µi λ 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)
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discretization leads to an affine VI of the type ˜ x ∈ δt F,
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h(AT A + sD)x − AT b, y − xi ≥ 0,
˜ ∀y ∈ δt F.
solution converges to the (unique) weighted minimum norm solution x∗ of the unregularized system [4] s→0
x −→ x∗ 1
kD 2 x∗ k22 =
x∈δt F˜ ,
min hAT Ax−AT b,y−xi≥0,
∀y∈δt F˜
xT Dx
[4] Xiubin Xu and Hong-Kun Xu. Regularization and Iterative Methods for Monotone Variational Inequalities. Fixed Point Theory and Applications, 2010:11, 2010.
Conclusion
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physically motivated regularization of non-smooth rigid multibody contact dynamics
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regularization of ball and socket joints, frictionless unilateral contacts and certain types of frictional contacts
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the stiff limit of the smooth system corresponds to the (unique) weighted minimum norm solution of the non-smooth system
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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. R¨ ude, and C. Popa. Resolving Ill-posedness of Rigid Multibody Dynamics. Technical report, Friedrich-Alexander University Erlangen-N¨ urnberg, 2010. [4] Xiubin Xu and Hong-Kun Xu. Regularization and Iterative Methods for Monotone Variational Inequalities. Fixed Point Theory and Applications, 2010:11, 2010.
