INSIA-ETSII-UPM 1/46 Multibody Systems Made Simple and Efficient J. García de Jalón IDETC/CIE 2011 MSNDC 23-1 Washington, USA, 28-31 Aug, 2011 Universidad

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Multibody Systems Made Simple and Efficient

J. García de Jalón

IDETC/CIE 2011MSNDC 23-1Washington, USA, 28-31 Aug, 2011

Universidad Politécnica de MadridDept. de Mat. Aplicada and INSIA

José Gutiérrez Abascal 228006 Madrid, Spain

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Deo omnis gloria!

Outline of this presentation

History of natural coordinates Acknowledgements Early developments and applications of natural coordinates

- Back to the basics- Multibody simulation plus realistic 3-D graphics- Global formulations: Improving the efficiency- Biomechanics- Flexible multibody systems

Improving the efficiency throughout topological formulations- Rigid multibody systems- Flexible multibody systems

Live demos Conclusive remarks

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Aknowledgements

Former PhD students- Rafael Avilés - José Antonio Tárrago- Jorge Unda- Alejandro García-Alonso- Alejo Avello- José Manuel Jiménez- Alejandro Gutiérrez- Jesús Cardenal- Javier Cuadrado- Gorka Álvarez- Antonio Avello- Nicolás Serrano- José Ignacio Rodríguez- Jesús Vidal

Current PhD Students - F. Javier Funes- Andrés F. Hidalgo- Alfonso Callejo- Mª Dolores Gutiérrez

Colleagues- Joaquín Casellas ()- Francisco Tejerizo ()- José María Bastero- Joaquín Aguinaga ()- Miguel Ángel Serna- Juan Tomás Celigüeta- Carlos Vera ()- Francisco Aparicio

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Part I: History of natural coordinates





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1.1 How the idea arose

F. Beaufait, Basic Concepts of Structural Analysis, Prentice Hall, 1977

G. Strang and G. Fix, An Analysis of the Finite Element Method, Wellesley-Cambridge Press (1973)

G. Strang, Linear Algebra and Its Applications, Academic Press, First Edition (1976)

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The beginning of natural coordinates 1/2

1971-77- Lecturer on Structural Analysis and Numerical Methods- Doctoral Thesis on “Finite Element Thermal Stresses in Railways Wheels”

1978-79- Strong pressure to complete in four weeks an original research work on

mechanisms - An idea urgently needed: A mechanism as an unstable structure!

- The stiffness matrix has the same null space as the Jacobian matrix of the constant distance constraint equations

Statically determinateStatically indeterminate Unstable

Kx 0

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The beginning of natural coordinates 2/2

Mathematical evolution- From minimum elastic potential to minimum constraint violation, and then to

nonlinear constraint equations: constant distances, angles, areas,… Extension to three-dimensional systems (1981)

- Straightforward when only points are considered- Each rigid body has at least three non-aligned points- Revolute joints introduced by sharing two points- Cylindrical joints introduced with four points aligned- High number of points needed

Unit vectors introduced (1984)- Points and unit vectors contribute to define 3-D orientation- A single unit vector may define the direction of several parallel axes- Revolute joints introduced by sharing a point and a unit vector- Cylindrical joints introduced with two points aligned with a shared unit vector- Important size reductions achieved

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Example 1: Constraint equations of robot ERA

u 3u 2

u1

u 1

1

2

3

45 u1

u 0

21 coordinates and 15 constraints:r2, r3, r4, r5, u1, u2 y u3

2 2 2 22 1 2 1 2 1 12

2 1 1 2 1 1 2 1 1

2 2 21 1 1

1

2 2 2 23 2 3 2 3 2 23

3 2 1 3 2 1 3 2 1

2 2 2 24 3 4 3 4 3 34

3 4 1

0

0Body 2

1 0

0

0Body 3

0

0

Body 4

x y z

x y z

z

x y z

x x y y z z d

x x u y y u z z u

u u u

u

x x y y z z d

x x u y y u z z u

x x y y z z d

x x u

3 4 1 4 3 1

3 4 2 3 4 2 4 3 2

2 1 2 1 2 1

2 2 22 2 2

2 2 2 25 4 5 4 5 4 45

5 4 2 5 4 2 5 4 2

5 4 3 5 4 3 5 4 3

0

0

0

1 0

0Body 5

0

0Body 6

x y z

x y z

x x y y z z

x y z

x y z

x y z

y y u z z u

x x u y y u z z u

u u u u u u

u u u

x x y y z z d

x x u y y u z z u

x x u y y u z z u

2 2 23 3 3 1 0x y zu u u

2

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Example 2: Robot ERA with mixed coordinates

Natural coordinates allow an easy addition of relative coordinates- Angles and distances in

Revolute and Prismatic joints, respectively, allow a direct introduction of actuator forces.

In the ERA robot, angles have been defined in the following way:- Angle y1 goes from u2 to u5 and it is measured on u1.- Angle y2 goes from u1 to (r2–r1) and it is measured on u2.- Angle y3 goes from (r1–r2) to (r3–r2) and it is measured on u2.- Angle y4 goes from (r2–r3) to (r4–r3) and it is measured on u2.- Angle y5 goes from (r3–r4) to (r5–r4) and it is measured on u3.- Angle y6 goes from u3 to u4 and it is measured on (r5–r4). - Each angle is defined with the dot product of vectors or with the largest

component of the cross product of vectors, according to its value.

u4 u 3

1

2

3

45

u1

u2

u2

u2

Y1

Y2

Y3

Y4

Y5

Y6

u5

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Part II: Early developments and applications of natural coordinates





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2.1 Back to the basics: singular mass matrices and redundant constraints

J. García de Jalón and E. Bayo, Kinematic and Dynamic Simulation of Multi-Body Systems: The Real-Time Challenge, Springer-Verlag, New-York (1993).

Udwadia, F. E. and Phohomsiri, P., Explicit Equations of Motion for Constrained Mechanical Systems with Singular Mass Matrices and Applications to Multi-body Dynamics, Proceedings of the Royal Society of London, Series A, 462, pp. 2097-2117, (2006).

J. García de Jalón, Multibody dynamics with redundant constraints and singular mass matrix: Theory and practice, paper in preparation, (2011).

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Aspects of formulations with natural coordinates

Singular mass matrices arise very often- A constant mass matrix (with no

velocity dependent inertia forces) can be defined with two points and two non coplanar unit vectors.

- If there are more points and vectors their inertia coefficients are null and the mass matrix singular.

Dependent constraint equations are also frequent- for implementation convenience

for instance:- in over-constrained systems

These difficulties need always to bedealt with to find an adequate solution, but, When does this solution exists? Is it unique?

ir

mujr

nu

lu

kr

, j k i j k u u 0 u r r 0

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Dynamic equations with Lagrange multipliers 1/3

Motion differential equations- Putting together the dynamic equations and all the kinematic constraints:

that is a system of 3N+M differential-algebraic equations of index 3.- It is also possible to set only the dynamic and the acceleration equations:

- Assume matrix M is positive definite and matrix has full rank

- Then,

- These conditions on M and are too restrictive and shall be relaxed

some stabilization necessaryT

q

q

M Φ q F

Φ 0 λ c

T qMq Φ λ F

, = , , t tt q q qΦ q 0 Φ q Φ b Φ q Φ Φ q c

qΦ

1 T qq M F Φ λ

11 1 1 T T q q q q q qΦ q c Φ M F Φ λ c λ Φ M Φ Φ M F c

, , rank , rankT n n m n n m q qM M Φ M Φ

qΦ

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Motion differential equations: the rank r of made visible- Index-1 ordinary differential equations may be transformed:

- Matrix M may be singular or semi-definite- Matrix may have redundant constraints:

- By taking into account that matrix is invertible,

- The existence and uniqueness of solutions are studied next.

T

q

q

M Φ q F

Φ 0 λ c

qΦ

r n

LU m mm r n

q

UP Φ L

0

T T Tr n m mm r n

r nLU LU

m m m mm r n

M U 0 Lq F

U P λ P cL 00

1, ,

T Tr n m r n

Tm m LU m m LUr n

m mm r n

M U 0q F

λ L P λ c L P cU λ c00

qΦ

m mL

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Dynamic equations: existence and uniqueness of solutions- As has seen, these equations are compatible if is compatible,

- Values for compatibility. Values are arbitrary.- Moreover, if matrix has full rank n, these equations are

determined in the acceleration vector and undetermined in the transformed Lagrange Multipliers In this case, the following matrix is invertible

even if matrix M is not invertible. The QR factorization and the SVD can be applied in a similar way

1, ,

T Tr n m r n

Tm m LU m m LUr n

m mm r n

M U 0q F

λ L P λ c L P cUλ c0

0

q.λ

Tr n M U

qΦ q c

Tr n

r n

M U

U 0

1 ... 0r mc c 1 2, ,...,r r m

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Three forms to set and solve the dynamic equations 1/4

Three equivalent formulations- Index 1 Lagrange equations

- Null space method

- Maggi’s method

- The conditions for the existence of a solution are:

T

q

q

M Φ q F

Φ 0 λ c

T T

q

R M R Fq

Φ c

T T T R MRz R F R MSc

qΦ R 0

, q Rz q Rz Sc

rank ker ker def. pos.Tn

q

q

MM Φ 0 R MR

Φ

q

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The existence of solutions- By applying the Rouché-Capelli theorem, in order to be able to

express any external force vector as a linear combination of inertia and constraint forces,

- This condition can be transformed into an equivalent one,

which has a more clear physical significance:

“Any physically possible velocity is associated with a non-zero (positive) kinetic energy “

1 rankT n T n q q

qM Φ F M Φ

λ

rank ker kern

q

q q

M M 0x x 0 M Φ 0

Φ Φ 0

ker qq Φ , 0 .T Mq 0 q Mq

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The existence of solutions (cont.)- For the null space equations, taking into account that

But matrix M is positive-semidefinite, so

Assume vector x belongs to the null spaces of and

and the previous condition is obtained.

,qΦ R 0

, rank ker kerT TT

Tn

q

q q

R M R MR Fq R M Φ 0

Φ Φc

, rank rank ker kerT M P P M P P M

1ker ,

0ker

f

T T T

T T

qx Φ x Rz zR MRz 0 z R MRz

x R M R Mx 0

2

2 ker kerT T T T T z R MRz z R P PRz PRz 0 Rz P x M

qΦ ,TR M

ker ker qM Φ 0

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The existence of solutions(cont.)- For the Maggi’s equations

The accelerations and are determined if matrix is positive-definite.

This condition can be written as,

Any vector cannot belong to Thus,

The positive-definiteness of has a physical meaning:

“Any physically possible velocity is associated with a positive kinetic energy “

, T T T R MRz R F R MSc q Rz Sc

z q TR MR

2

2 ker kerT T T T T z R MRz z R P PRz PRz 0 x Rz P M

ker , , qx Rz Φ x 0 ker .M

ker ker qM Φ 0

, T T r z R MRz 0 z z 0

ker , qq Φ q Rz

TR MR

2 0, .T T TT q Mq z R MRz z 0

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Deo omnis gloria!

The uniqueness of solutions

Assume the conditions for the existence are fulfilled,

- The uniqueness of the acceleration is better studied with the null space formulation

- As the columns are independent, the solution is unique, even if there are redundant constraints

- Also is unique if matrix is definite-positive- In the Lagrange equations of the first kind and are unique,

but the multipliers are not if there are redundant constraints

- The resultant constraint forces are unique, but the particular are not

rank rank ker ker def. pos.T

Tn

q

q q

M R MM Φ 0 R MR

Φ Φ

, rank rankT TT

n

qq q

MR M R MR Fq

ΦΦ Φc

q

q

z TR MR

, , rankT

T T n m T r m

qq q q

q

M Φ q FΦ λ F Mq Φ Φ

Φ 0 λ c

q TqΦ λ

λ

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Deo omnis gloria!

Redundant constraints and constraint forces

Assume the MBS is over-constrained and constraint forces are to be determined- Remember is determined

but is not. Then

- The individual constraint forces are

that are not determined. General and minimum norm solution of

- The general solution is where is the minimum norm solution and is a vector in

- The important point is that is a set of self-equilibrating constraint forces (see examples)

- This minimum norm solution can be computed from the system of equations:

TqΦ λ

λ

,T qΦ λ F q q Mq

1:,1 :,T T Tmm q q qΦ λ Φ Φ

T qΦ λ f

0 , ,m r λ λ Nα α 0λNα ker T

qΦTqΦ Nα

0

T

T

qfΦ

λ0N

kerA11 1 12 2 0a x a x

Im TA

x

u0x

px

11 1 12 2 1a x a x b Ax b

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Example 1: 2-D double quad

Over-constrained system- Bar 5-6 can be removed

without changing the kinematics of the motion

- The system cannot be assembled if unless bars are stretched

- The minimum norm solution makes the self-equilibrating forces null

Set of self-equilibrating constraint forces

Constraint forces for the three constant distances

1 2

3 4

5 6

34 56 ,L L

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-80

-60

-40

-20

0

20

40

60

80

100Constraint force in rods 1 (red), 2 (green) and 3 (blue)

time

For

ce

12 34 56, ,L L L

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Example 2: 8-link spatial parallelogram

Frączek & Wojtyra, ECCOMAS Warsaw (2009)

Platform 1-2-3-4 and bars modeled with natural coordinates- 6 rigid body dofs, 6 constant distance

conditions and 1 final constrained dof- The figure shows the set of self-

equilibrating constraint forces - The force systems in red and in blue

must self-equilibrate separately- The forces in blue can not self-

equilibrate, and so they are null (proved algebraically by Frączek & Wojtyra)

K

M5

6

4

32A

C E

1

G

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Deo omnis gloria!

Example 3: 3-D four bar system

3-D four-bar with 3 self-equilibrating sets of constraint forces

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2.2 Multibody simulation withrealistic 3-D graphics

J. García de Jalón, A. García–Alonso and N. Serrano, "Interactive Simulation of Complex Mechanical Systems", Eurographics'93, Barcelona, 1993.

G. Álvarez, A. García-Alonso, P. Urban, N. Serrano and J. García de Jalón, "Biomechanics: Dynamics & Playback", video presented in SIGGRAPH’93, Annual Conference Series, ISBN 0-89791-602-6, Anaheim CA, USA, 1993.

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Animated CG essential to understand MBS

Computer plots used from 1979 Tektronix 4114 y 4115 for

monochrome vector animations since 1982

Workstation HP 320SRX with realistic graphics since 1987- Motorola 69020 processor plus

3-D graphics accelerator- Some examples presented in the VII

IFTóMM World Congress on the Theory of Machines and Mechanisms, Seville, September, 17-22, 1987

Silicon Graphics 4D/240GTX since 1989- Four MIPS 3000 processors

Since 1987 computer graphics produced an interest boom on interactive or even real-time multibody systems

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Some MBS animations from the late 80s

Some of the first examples - Kinematics of

steering and suspension systems

- Complete model of a truck

- Kinematic deployment of an antenna

- Space dynamic maneuver

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2.3 Global formulations: Improving the efficiency

R. von Schwerin, “Multibody System Simulation: Numerical Methods, Algorithms and Software”, Springer, 338 pages, 1999.

J. Cuadrado and D. Dopico, “Penalty, semi-recursive and Hybrid Methods for MBS Real-Time Dynamics in the Context of Structural Integrators”, MULTIBODY DYNAMICS 2003, Jorge A.C. Ambrósio (Ed.), IDMEC/IST, Lisbon, Portugal, July 1-4 2003

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von Schwerin’s Formulation 1/3

Main characteristics- Natural coordinates are used for the system of linear equations- Two methods are considered to solve this system: RSM and NSM- The problem is assumed non-stiff and the index 1 DAE system is integrated

with a special ABM integrator based on the solution of the inverse dynamics that does not need the derivative of the state vector.

- Instead of using a predictor-corrector formula, von Schwerin enforces the discretized dynamic equations by a modified Newton-Raphson iteration, with an approximate tangent matrix that can be kept for many time steps. At the end, it is possible to obtain a O(n) method based on the descriptor form.

- The index 1 DAE system is stabilized for positions and velocities by using two mass-orthogonal projection with the aforesaid tangent matrix

- The efficiency can be further improved by considering the system topology through a distinction between the closure of the loop constraints and all the remaining constraints.

This method seems to have been discontinued in the bibliography

T

q

q

M q F

0 c

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The Range Space method (RSM)

An LDLT factorization is assumed and then computed- Factorization step (M and assumed full rank matrices)

- Substitution step

- This method takes into account the matrix symmetry and doesn’t need complicated pivoting strategies

1 1 21 1 1 1 21

21 2 2 21 1 21 21 2 2

T T T T Tm

T T T Tf

q

q

I 0M L 0 L L L L L L0 I0 L L 0 L L L L L L L

11 21 1 2 21 21chol , , cholT T T qL M L L L L L

1 1 21 1 21

21 2 2 2

, T T T T

m m

T Tf f

I 0 I 0L 0 q F x qL L L L0 I 0 IL L λ c μ λ0 L 0 L

1 212 1 21

2

\ , \T T

T T T

T

q xL Lλ L μ q L x L λ

λ μ0 L

11 2 21

21 2

\ , \

L 0 x Fx L F μ L c L x

L L μ c

q

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The Null Space method (NSM)

A transformation based on a null space basis R of is applied- The constraint equations for velocities and accelerations allow a

transformation based on the null space basis R,

- The accelerations can be computed from the equations

, ,

, ,, , , ,

, d i d i d dd

n n n m

iiim n m mf m f f f m f f f

q q q qq

I 0Φ Φ Φ Φ S RcqΦ q b

0 I0 I 0 I S Iqq

1

1 1

iT i T

dT dd d dT di id d ii i dT d i dT dd ii d

d d i i di

q

q q q

R MRq R F MSc

R M R R M M R M q R F F R M M Φ c

q Φ Φ q Φ c

qΦ

1 1 1

, ,

1

, ,

d i d d id d di

i i ii i if m f f f

d i d ddi

iif m f f f f m

q q q q q

q q

Φ Φ b b bΦ Φ Φq S Rq Sb Rq

0 I q q qq S R 0 I

Φ Φ R Scqq q

0 I I 0qq

1

1, i

d d i

i

d d

q q

q 0

q

R Φ Φc Rq Sc Sc q

S Φ

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Deo omnis gloria!

The RSM and NSM complexities compared

von Schwerin presented a theoretical study of the complexity of solving the descriptor system of equations using the three methods, depending on the relationship between the number of constraints m and the number of dependent coordinates n.

His findings are very significant and interesting:- For m/n > 0.63 the NSM is more efficient than the RSM. It is also more efficient

for m/n > 0.69 when matrix M is constant and the RSM takes advantage of this circumstance.

- When the ratio m/n is close to 1 (strongly constrained systems with very few degrees of freedom), the NSM requires 7 times less arithmetic operations than the Gauss method, and 3.5 times less operations than the RSM (2.5 if matrix M is constant).

- In the worst circumstances (m/n small and matrix M constant), the NSM never requires more than 15% of additional operations with respect to the RSM.

These results justify the superiority of the methods based on independent coordinates, which are equivalent to the RSM.

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Augmented Lagrangian (Cuadrado and Dopico) 1/4

Description of the multibody formalism- Based on natural coordinates (dependent) plus angles and distances.- Index-3 augmented Lagrangian equations with generalized- method.

- Newmark equations to approximate velocities and accelerations.

- Nonlinear system of equations:

1 1

1 12 2

ˆ ˆ ; 1 12

1 1 1 1ˆ ˆ ; 12

n n n n n n n

n n n n n n n

hh h

hh h

g g g gb b b b

b bb b

+ +

+ +

ì üæ ö æ öï ï÷ ÷ï ïç ç= + = - + - + -÷ ÷í ýç ç÷ ÷ç ç÷ ÷ç çï ïè ø è øï ïî þæ öæ ö ÷ç ÷ç ÷= + = - + + - ÷ç ç ÷÷ç ç ÷ ÷çç è øè ø

q q q q q q q

q q q q q q q

& & & & &&

&& && && & &&

( )t h+ =f q 0

( ){ } ( ) ( ){ }( ){ }

*1

1*

1 1 Tm n m n f

nT

fn

d d d

d

++

- + + - + -

+ + - =

q

q

M q q Φ αΦ λ Q

Φ αΦ λ Q 0

&& &&

1 1i i i+ += +λ λ αΦ

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Description of the multibody formalism (cont.)- Solution of the nonlinear system by the Newton-Raphson method

where

( )( )1 1 1;

iii i i i

t h t h t h t ht ht h

+ + ++ + + ++

+

é ù¶ê ú é ùD = - = +Dê ú ê úë û¶ê úë û

f qq f q q q q

q

( ) ( ){ } ( ) ( ){ }( ){ }

2 *1

1

*

1 1 Tm n m n f

n

Tf

n

hb d d d

d

++

é= - + + - + -ê

êëù

+ + - =úúû

q

q

f q M q q Φ αΦ λ Q

Φ αΦ λ Q 0

&& &&

( ) ( ) ( ) ( ) ( )2 T1 1

1 1 1m f n f nh hd d g d b+ +

é ù¶ê ú@ - + - + - +ê ú¶ê úë ûq q

f qM C Φ αΦ K

q

,¶ ¶

= - = -¶ ¶Q Q

Κ Cq q&

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Description of the multibody formalism (cont.)- Projections of velocities and accelerations onto their constraint manifolds.

where

,¶ ¶

= - = -¶ ¶Q Q

Κ Cq q&

( ) ( )* 21 Tf thd b

é ù¶ê ú = - -ê ú¶ê úë ûq

f qq Pq Φ αΦ

q& &

( ) ( ) ( ) 21 11 1 1m f n f nh hd d g d b+ += - + - + -P M C K

( ) ( ) ( )* 21 Tf thd b

é ù¶ê ú = - - +ê ú¶ê úë ûq q

f qq Pq Φ αΦ q Φ

q& &&& && &

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Implementation details- Sparse matrix techniques:

- CS: Coordinate format (MA27, MA57)- CCS: Compressed column format (Cholmod, KLU)

- Constant mass matrices and gravity forces in rigid-body case- Analytical evaluation of Jacobians of forces easy to obtain (spring-damper,

contact, tire, brake, etc.)- Usually symmetric positive-definite tangent matrices- Integration step-sizes attainable: stable with very large step sizes, problems

with very small step sizes (bad condition number)- Able to modify the system in real-time: adding (removing) bodies (constraints)

Conclusions on the efficiency obtained- Real time for systems with sizes of more than 1000 coordinates

- Assembly of four–bar linkages with “n” loops, CCS, h=1e-3, AMD Athlon 64.- Excavator simulator (152 coordinates): about 5 times faster than RT

- CS, h=5e-3, ASUS U6V laptop computer Intel Core 2 Duo.

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2.4 Biomechanics

ABAT’92 project: Análisis Biomecánico de Alta Tecnología (1988-1992) G. Álvarez, A. García-Alonso, P. Urban, N. Serrano and J. García de

Jalón, "Biomechanics: Dynamics & Playback", video presented at SIGGRAPH’93, Anaheim CA, USA, 1993.

G. Alvarez, A. Gutiérrez, N. Serrano, P. Urban and J. García de Jalón, "Computer Data Acquisition, Analysis and Visualization of Elite Athletes Motion", VIth International Symposium on Computer Simulation in Biomechanics, Paris, France, 1993.

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Natural coordinates and biomechanics

Step 1: Motion capture: points model - Using several synchronized cameras- Manual or automatic digitalization- 3D coordinates determination with

optional filtering Step 2: Data conditioning: consistent

MBS model- Definition of rigid bodies and joints- Consistent MBS model: natural

coordinates- Evaluation of relative angles and filtering

Step 3: MBS simulation- Kinematic simulation- Direct dynamic simulation- Inverse dynamic simulation

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2nd step: MBS model of the human body 1/3 The HUMAN BODY model

- Figure shows a typical multi-rigid-body model of the human body, including 37 degrees of freedom.

- The hip body (consisting of center, right and left points, and three unit vectors) is taken as the base body, in order to show easily the rigid body motions.

Natural coordinates- This model has 30 points (27 movable) and

23 unit vectors (20 movable).- The relative coordinates are the angles

between the different bodies.- So, this human body model has 37 degrees

of freedom and 141 dependent coordinates.

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2nd step: MBS model of the human body 2/3 Example: definition of a leg:

- Points 9, 25 and vectors 15, 16 are a rigid body:



- Sometimes foot twist is restricted:

226 27 26 27 26,27

17 26 27

18 26 27

( )( ) 0

( ) 0

( ) 0

T T

T

T

d

r r r r

u r r

u r r

17 18 0 u u

29 25 9 25 9,25

15 9 25

16 9 25

( )( ) 0

( ) 0

( ) 0

T T

T

T

d

r r r r

u r r

u r r

225 26 25 26 25,26

16 25 26

17 25 26

( )( ) 0

( ) 0

( ) 0

T T

T

T

d

r r r r

u r r

u r r

9

25

26

u5

u16

10

5

27

u17

u15

u18

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2nd step: MBS model of the human body 3/3

Relative coordinates used to control the 6 degrees of freedom:- goes from to measured on- goes from to

measured on - goes from to measured on - goes from to

measured on - goes from to measured on

- goes from to measured on

Angle definition constraints:- Angle from to measured on

- Angle from to measured on

( ) ( ) sin( )j i l k m ij klL L y r r r r u 0

( )sin( )l k

i jklL

y

r ru u 0

9

25

26

u5

u16

10

5

27

u17

u15

u18

27y

28y

29y

30y

31y

26y 5u 15u

27y 10 9( )r r15u

15u 16u

25 9( )r r

25 9( )r r

26 25( )r r 9 25( )r r

27 26( )r r25 26( )r r

17u16u16u

28y29y

30y

31y

19 10( )r r

mu

26 25( )r r

17u

( )j ir r ( )l kr r

( )l kr riu ju

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Deo omnis gloria!

3rd step: Kinematic and dynamic simulation

Kinematic simulation- After conditioning, the consistent model with the recorded

motion is visualized with realistic graphics from several fixed or moving cameras

- If necessary, new frames are obtained by interpolation Inverse and direct dynamic simulation

- The inertia properties of the athlete’s members are obtained from anthropometric data tables

- Knowing the motion as a function of time, the external reaction and internal motor-constraint forces can be computed and visualized

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Deo omnis gloria!

Barcelona’ 1992 Olympic Games 1/4

ABAT’92 Project: Análisis Biomecánico de Alta Tecnología (1988-1992)- Developed with CAR (High Performance Sport Training Center)- Research project aimed at the multibody analysis of athletes, using the

latest technologies in data capturing, analysis of motion and computer graphics

Two sports were studied: athletics and gymnastics Working procedure

- The trials were recorded in the afternoon- The images were manually digitized during the night- The digitized data was conditioned, produced and recorded in the

morning- The resulting images were broadcast at midday news services (too

late…) Some broadcast images will be shown next

- Gymnastics, high jump and long jump

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Deo omnis gloria!


Gymnastics

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High jump: Javier Sotomayor

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Long jump: Lewis vs. Powell

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Biomechanics and natural coordinates today 1/3

STT Ingeniería y Sistemas Automatic data capturing

- Gait analysis- Bull fighter- Golf

Manual data capturing- Gymnastic 1- Gymnastic 2

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Deo omnis gloria!


Kinematic Motion Reconstruction (KMR) at CEIT Optimization process

- The quadratic error between the experimental and theoretical position of the markers, subjected to the kinematic constraint equations, is minimized

In the next video we will see:- Red spheres ( ) in the center of joints, - Blue spheres ( ) in the model points associated to markers - Green spheres ( ) to represent marker’s position experimentally measured

The error between blue and green spheres is minimized subjected to the constant distance condition between blue and red spheres

Over-determined and under-determined problems- There is over-determined information when more points or more cameras

than necessary are used for a body- On the other hand, there is for instance an under-determined problem when a

body has fewer than necessary points or they are hidden for some cameras

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Deo omnis gloria!


A CEIT’s REALMAN example: A driver enters and exits a vehicle

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2.5 Flexible multibody systems

J. García de Jalón and E. Bayo, Kinematic and Dynamic Simulation of Multi-Body Systems: The Real-Time Challenge, Springer-Verlag, New-York (1993).

J. Cuadrado, J. Cardenal and J. García de Jalón, Flexible Mechanisms Through Natural Coordinates and Component Synthesis: An Approach Fully Compatible with the Rigid Case, International Journal for Numerical Methods in Engineering, Vol. 39, pp. 3535-3551, (1996).

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Deo omnis gloria!

Moving frame approach with natural coordinates

The classical moving frame approach separates the large rigid body motion and the small elastic deformations respect the moving frame- The elastic deformations are expressed as a linear combination of

a reduced set of static and dynamic modes Natural coordinates provide a simple way to define

boundary conditions for the static modes- The elastic static modes can be considered as the relaxation of

one or more rigid body constraint equations- It is not necessary to consider all the possible static nodes. Those

considered as irrelevant may be kept as constraint equations- It is not necessary to consider the amplitude of the static modes

so defined: they can be computed from the natural coordinates

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Static modes coming from a point displacement

There are three static modes related to the displacement of point

The static modal amplitudes can be defined as

0r 1r

u

v

w

1v

2v

1 0 1T

r n η A r r r

1r

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Static modes coming from a point rotation

There are three static modes related to the infinitesimal rotations of vectors and

The modal amplitudes are obtained from

rx ry rz G I u v w

A v GvTn1 1

A v GvTn2 2

0r 1r

u

v

w

1v

2v

1v 2v

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Deo omnis gloria!

A possible selection of static modes

A revolute joint on the right hand side allows a free rotation respect to the joint axis

Only two rotational and one translational static modes are considered

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Part III: Improving the efficiency throughout topological formulations





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3.1 Semi-recursive formulation basedon a double velocity transformation

J.I. Rodríguez (2000), PhD Dissertation (in Spanish). J. I. Rodríguez et al., Recursive and Residual Algorithms

for the Efficient Numerical Integration of Multi-Body Systems, Multibody System Dynamics, 11, pp. 295-320, (2004).

J. Cuadrado et al., A combined Penalty and Recursive Real-Time Formulation for Multibody Dynamics, ASME Journal of Mechanical Design, pp. 602-608 (2004).

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Deo omnis gloria!

Open-chain topology: path matrix

Path matrix T- Rows correspond to rigid bodies.- Columns correspond to joints.- Tij = 1 if body i is upwards of joint j.

Otherwise Tij = 0. Column i of T:

- Defines the bodies that are upwards of joint i.

Row j of T:- Defines the joints that are

downwards of body j. Numbering of joints and bodies

- Each rigid body has the same number as its input joint.

- Bodies and joints are numbered from the leaves to the root.- Matrix T is upper triangular.

1z3z

2z 4z

5z

1

2

3

4

5

6 6 6 6 6

6 6 6 6 6

6 6 6 6 6

6 6 6 6 6

6 6 6 6 6

1 1 0 0 1

0 1 0 0 1

, 0 0 1 1 1

0 0 0 1 1

0 0 0 0 1

I I 0 0 I

0 I 0 0 I

T T 0 0 I I I

0 0 0 I I

0 0 0 0 I

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Deo omnis gloria!

Geometry and position

The bodies’ geometry is defined through natural coordinates.

The position of each body is defined by the Cartesian coordinates of a point and the rotation matrix.

All the joints are defined as a combination of revolute (R) and prismatic (P) joints.

ri

gi

X Y

Z

1iX

i–1

i

1iY

1iZ

iY iX

iZ

1ir

ir

1ip

iu

X Y

Z

1iX

i–1

i

1iY

1iZ iY

iX

iZ

1ir

ir

1ip

iu

1i id

R P

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Deo omnis gloria!

Cartesian velocities and accelerations

Cartesian velocities and accelerations:- According to the point of the solid chosen as reference:

Center of gravity Point at the origin of global axis

- Simple relationships between them:

- Recursive Cartesian velocities and accelerations (open-loop systems):

- Matrix and vectors and depend on the joint type between elements i and i−1. For instance, with Z velocities:

, i ii i

i i

s sZ Z

ω ω

3 3

3 3

,

i i i i i i i i ii i i i i i i

i i i i

g I g s g I g s ω ω gY D Z Y D Z e

ω 0 I ω ω 0 I ω 0

1

1

i i i i

i i i i i

z

z

Z Z b

Z Z b d

1

1

i i i i i

i i i i i i

z

z

Y B Y b

Y B Y b d

, i ii i

i i

g gY Y

ω ω

iB , ,i i ib d b

11

1

22, ; ,

i i i i i iR R i i i i i i P Pj j j j

i i i i

s

r u u ω uω u r ub d b d

u 0 0ω u

id

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Deo omnis gloria!

1st velocity transformation (open-chain)

Transformation from Cartesian to relative coordinates- Cartesian velocities Z

are dependent and can be obtained from the relativeindependent velocities:

- Column i of matrix R contains the Cartesian velocities produced by a unit input velocity in joint i andnull velocities in the rest of the joints.

- All the bodies upwards of joint i have the same Cartesian velocity bi (the same vector that appeared in ).

- For the whole open-chain system:

1 0z 3 0z

2 1z 4 0z

5 0z

1

2

3

4

5

1 1 2 2 ... n nz z z Z Rz R R R

1 2 5 6 6 1

2 5 6 6 6 2

6 13 4 5 6 6 3

4 5 6 6 6 4

5 6 6 6 6 5

0 0 0 0 0 0

0 0 0 0 0 0 0

, 00 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

d

6 6 6

6 6

6 6 6

6 6

6

b b b I I 0 0 I b

b b 0 I 0 0 I b

R TR 0b b b 0 0 I I I b

b b 0 0 0 I I b

b 0 0 0 0 I b

1i i i iz Z Z b

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Deo omnis gloria!

Open-chain motion differential equations 1/2

Newton-Euler equations for each solid- Cartesian velocities and accelerations Y and Z can be used:

- By applying the Virtual Power Principle to the complete system:

- Introducing now the 1st velocity transformation:

in the virtual power equation, the following ODE system is obtained:

where t are the forces/torques applied in the joints.- These equations are further developed in the next slide.

, , i ii i i i i

i i i i i

m

3I 0 FM Y Q M Q

0 J n ω J ω

1 2 1 2 1 2diag , ,..., , , ,..., , , ,..,T T T T T T T Tn n n M M M M Q Q Q Q Z Z Z Z

0T Z MZ Q

, d d d Z Rz TR z Z TR z TR z

T T T Td d d d R T MT R z τ R T Q MTR z

, , i i iT Ti i i i i i i i i i i i

i i i i i i

m m

m m

3I gM Z Q M = D M D = Q D Q M e

g J g g

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Open-chain motion differential equations 2/2

Motion differential equations with relative coordinates:- The motion equations are:- By substituting matrices Rd and T from the previous example:

- Notice that the LU factorization will not destroy the zeroes-pattern.- For external and velocity dependent inertia forces:

- In both cases the accumulation of inertias and forces becomes very explicit. The path matrix T plays a fundamental role.

1 1 1 1 1 2 1 1 5

2 1 1 2 2 2 2 2 5

3 3 3 3 3 4 3 3 5

4 3 3 4 4 4 4 4 5

5 1 1 5 2 2 5 3 3 5 4 4 5 5 5

T T T

T T T

T T TT T Td d d d

T T T

T T T T T

b M b b M b 0 0 b M b

b M b b M b 0 0 b M b

R T MTR R M R0 0 b M b b M b b M b

0 0 b M b b M b b M b

b M b b M b b M b b M b b M b

1 1

2 2 1

3 3

4 4 3

5 5 2 4

M M

M M M

M M

M M M

M M M M

1 1

2 2

3 3

4 4

5 5

, T Td

Q P

Q P

Q T Q P T MTR zQ P

Q P

Q P

1 1

2 2 1

3 3

4 4 3

5 5 2 4

Q Q

Q Q Q

Q Q

Q Q Q

Q Q Q Q

1 1 1

2 2 12

3 3 3

4 4 34

5 5 2 45

d

d

d

d

d

P M TR z

P M TR z P

P M TR z

P M TR z P

P M TR z P P

T T T Td d d d R T MT R z τ R T Q MTR z

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Closed-chain differential equations 1/2

Closed-chain equations are obtained in two steps:- First, the loops are opened by removing some joints or special bodies (rods).- Second, the constraints are introduced by an appropriate method: Lagrange

multipliers, penalties, augmented Lagrangian or a 2nd velocity transformation The constraint equations are formulated in the simplest

way through natural coordinates:

1z3z

2z4z

5z

1

2

3

4

4

ju ku

jr kr

1z3z

2z4z

5z

1

2

3

4

5

3 independent equationsj k r r 0

only 2 independent equationsj k u u 0 2 0

T

j k j k jkl r r r r

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Closed-chain differential equations 2/2

Closure of the loop constraint equations- The Jacobian matrix in relative coordinates can be computed as:

- By using the coordinate partitioning method:

- Introducing in the motion equations:

- The parenthesis of this expression can be written as:

; 2 , 2

j j T T T Tk kj k j kj k j k j kz r r r r r r

r rr rΦ Φ Φ Φ Φ Φ r r Φ r r

z z z z

& && &

1,

dd i d d i i

i

z z z z

zΦ Φ 0 z Φ Φ z

z

1 1

, , d d i d i

i i

i

z z z zz z

z Φ Φ Φ Φz R z z R

z I I

i i z zz R z R z

T T i T T T T T T id d d d d d

z z z z z zR R M R R z R τ R R Q R R T MT R z R R z

di i i i id d d d d d

d

dt z

z z z z z

R RR z R R z R R z R R z R R R R z z

0

, , i

d di i i id d d

d d

dt dt z z

z z z

R R R RZ Rz TR z TR R z Z TR R z T z T z Z

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Deo omnis gloria!

Closed-chain equations with penalties 1/2

The closed-chain constraints are applied for the open-chain relative coordinates as in Cuadrado and Dopico- By using the penalty method for positions the following system of differential

equations is obtained

- These equations are integrated by the HHT implicit method

- The Newmark’s expressions are assumed for velocities and accelerations

- By substituting in the HHT equations, the following nonlinear system of equations is obtained

, ,T zM z z = Q z z Φ z αΦ z Q z z

( ) ( ) ( )1 1 11 , ,n n f fn n

M z Q z z Q z z = 0d d+ + +- - -&& & &

1 1

1 12 2

ˆ ˆ; 1 12

1 1 1 1ˆ ˆ; 12

n n n n n n n

n n n n n n n

hh h

h h h

+ +

+ +

æ öæ ö æ ö ÷ç ÷ ÷ç ç ÷= + = - + - + -÷ ÷ç ç ç ÷÷ ÷ç ÷ ÷ç ç ÷ç è ø è øè ø

æ öæ ö ÷ç ÷ç ÷= + = - + + - ÷ç ç ÷÷ç ÷ç ÷ç è øè ø

z z z z z z z

z z z z z z z

& & & & &&

&& && && & &&

g g g gb b b b

b b b b

2 2 21 1 1 1 1

ˆ( ) (1 )n n n n n f n f nh h hb d b d b+ + + + += + - - - =f z M z M z Q Q 0&&

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Numerical examples3.2 Numerical examples

(semi-recursive formulation)

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Coach

15 dof 33 bodies (14 auxiliary) 34 joints 11 rods Closed loops opened by

removing:- 2 spherical joints- 11 rods

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IVECO van

17 dof 43 bodies (17 auxiliary) 47 joints 4 rods Closed loops opened by

removing:- 4 rods- 2 spherical joints- 3 revolute joints

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Semi-trailer truck

40 dof 5 axles 81 bodies (34 auxiliary) 89 joints Closed loops opened

by removing:- 1 spherical joints- 8 revolute joints

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Computation times (explicit RK4 integrator) [s]

Algorithm Step

Coach Semi-trailer truck

Time step: 1x10-3 s Time step: 2.5x10-5 s

C++ BLAS SPARSE C++ BLAS SPARSE

0.276 0.279 0.276 26.410 26.361 26.436

0.120 0.215 0.408 18.071 32.926 28.763

0.355 0.183 0.089 223.253 29.733 28.228

0.122 0.133 0.200 13.360 14.647 7.671

0.193 0.192 0.182 13.861 13.849 14.126

0.151 0.143 0.100 12.972 9.925 9.371

0.087 0.086 0.637 8.638 8.543 9.080

0.649 0.606 0.637 151.284 73.521 73.715

0.073 0.074 0.083 13.696 13.765 13.927

Elapsed time [s] 2.026 1.912 2.461 481.543 223.27 211.316

, , , ,

j i i i i i ij i

b A J M M M M

, [ , ] lud i d z z zΦ Φ L U Φ

1d iz z

z

Φ ΦR

I

1, recVel: , ,d d i i

i i i

z zz Φ Φ z Z d d

, ,i i iQ Q τ

, \d z z z zc Φ z R z Φ Φ z

recAccel: , , iZ Q Q

T Td d

T T T id d d d

z z

z z

M R R M R Rb R R Q R M R z R R z

, , , ,\ , d d d d g g g g d d g g gz M M z b M z M z

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Computation times (Implicit method) [s]

Algorithm Step

Iveco van Semi-trailer truck

Time step: 5x10-3 s Time step: 5x10-3 s

Serial TBB Serial TBB

0.001 0.001 0.001 0.001

0.153 0.194 0.193 0.236

0.333 0.115 1.748 0.243

4.437 1.581 9.199 2.802

0.028 0.033 0.081 0.096

0.179 0.164 0.674 0.706

Convergence evaluation 0.001 0.001 0.002 0.002

Velocities projections 0.019 0.020 0.072 0.073

Accelerations projections 0.013 0.014 0.030 0.031

Elapsed time [s] 5.162 2.122 11.999 4.191

0z

,K Q z C Q z

( 1) 1 1\

k kkn n n

z f(z) z f(z)

Tz zΦ Φ

1

k

nf(z) z

, , , , ,zM(z) Φ(z) Φ (z) Q(z z) f(z)

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Computation times

Slalom manoeuvre (15 s) Without graphics MATLAB graphics 3D graphics

Coach 16.31 1440 19.17

FSAE racing car 31.12 1419 43.86

15 s simulation time Intel Core 2 Duo 1.8 GHz, GeForce 8

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3.2 Semi-recursive formulation for flexible bodies: method of the 24 constant matrices

F.J. Funes and J. García de Jalón, Simulation of Flexible Multibody Systems Based on Recursive Topological Methods and Modal Synthesis, ECCOMAS Multibody Dynamics 2009, Warsaw, Poland.

F.J. Funes and J. García de Jalón, Efficient Simulation of Complex Flexible Multibody Systems Based on Recursive Methods and Implicit Integrators, ECCOMAS Multibody Dynamics 2011, Brussels, Belgium.

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Objectives

Motivation and objectives- To present a work in progress to analyze large size, real-life flexible multibody

systems - This formulation uses two efficient methods for implicit integration of the

dynamic equations of systems with flexible bodies (not described here) Outline

- Semi-recursive formulation for flexible bodies: a review- Kinematics and dynamics of the flexible body- Method of the “24 constant matrices” (ECCOMAS 2009, Warsaw)- Dynamic equations of the multibody system

- Definition of the reduced model of a flexible body- Recursive computation of the dynamic equations

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Kinematics of a flexible body 1/2

Combines large rigid body motions with small deformations

- Based on the classical Moving Frame Approach- Deformed position of a point

- Relative position and velocity between two contiguous bodies

1 1

, s dn n

n i i j ji j

r g A r g A r q q φ ψ Φη Ψε

1 1 1 , 1 1, 1 , , 21 2

f f f fi i i i t i i k j j i i t i i k

k j k

z

g g A r φ u A r φ

11, 1 , 2

1 21 1

i i t ti k j j i k

k j ki i r ri i

z

s s φ φb

ω ω φ φ

r0

r y

x

z

nrq

gi-1 gi

ri-1 ri

body i-1

body i

zi-1,1

zi-1,2

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Kinematics of a flexible body 2/2

Transformation between Cartesian/FEM and Cartesian/modal coordinates- Cartesian linear and angular velocity of a FEM node

- For the n nodes of a flexible body:

, ,1 1

, ,1 1

s d

s d

n n

i i t j j t k kj k

n n

i r j j r k kj k

r s ω r A φ ψ

ω ω A φ ψ

1 1 1 1,1 , ,1 ,

11 1 1 1

,1 , ,1 ,1

2 2 2 22 ,1 , ,1 ,2

2 2 2 22 ,1 , ,1 ,

,1

f i

f i

f i

f i

t t n t t n

r r n r r n

t t n t t n

f r r n r r n

nn

n t

n

3 1

3 3

3

3 3

3

I r Aφ Aφ Aψ Aψr

0 I Aφ Aφ Aψ AψθI r Aφ Aφ Aψ Aψr

v θ 0 I Aφ Aφ Aψ Aψ

r I r Aφ Aφθ

1 1 1

1 1 1

, ,1 ,

,1 , ,1 ,

s s s

f i

f i d d d

RBn n n

n n nt n t t n

n n n nr r n r r n n n n

3 3

s s s

ω ω ω

D D D D

Aψ Aψ

0 I Aφ Aφ Aψ Aψ

f

Dq

f f f v Dq Dq

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Dynamics of a flexible body

By using the Principle of Virtual Power and a non-consistent interpolation for velocities, the dynamic equations can be obtained- Motion equations of a flexible body in Cartesian coordinates

- The mass matrix can be expressed in partitioned form:

The inertia matrix coming from FEM has been computed in the global reference frame and in the initial position:

(constant matrices/vectors are typed in blue color) The evaluation of terms and can be very

expensive.

T FEM T T FEMext ext vel elastic qD M Dq Φ λ Q D M Dq Kq Q Q Q

1 1s d

T T T T T Tn n q s ω Z η ε

, diag , , ,FEM FT FEM T T Tn n n

EM D M D D A A D DM D A A A AM

T FEMD M D T FEMD M Dq

, ,

, RB RB RB RB

Text

vel elastsym

q

M M M Z Q 0

M M η Φ λ Q Q Q

M ε Q Q

T FEMD M D

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Efficient computation of the term 1/3

Computation of matrix

- By taking into account that

1 1 1 1,1 , ,1 ,

1 1 1 1,1 , ,1 ,

2 2 2 22 ,1 , ,1 ,

2 2 2 2,1 , ,1 ,

,1

f i

f i

f i

f i

T T T T T Tt t n t t n

T T T T Tr r n r r n

T T T T T Tt t n t t n

T T T T T Tn r r n r r n

T T T nn t

1

3

3

A A r A Aφ A Aφ A Aψ A Aψ

0 A A Aφ A Aφ A Aψ A Aψ

A A r A Aφ A Aφ A Aψ A Aψ

D A D 0 A A Aφ A Aφ A Aψ A Aψ

A A r A Aφ

, ,1 ,

,1 , ,1 ,

f i

f i

RB

T n T n T nt n t t n

T T n T n T n T nr r n r r n

3

D

A Aφ A Aψ A Aψ

0 A A Aφ A Aφ A Aψ

D

ψ

D

A A

constant

3 3

3 3

3 3

3 3 3

3

1

2

3

2

3

T T T

T

T T TT

TRB T

T T Tn n

T

1

3 3

33 3

3

3 3

3 3

3 3

3 3

3 3 3

3

3

A A r A r

0 A 0

A A r A rA 0

D 0 A 00 A

A A r A r

0 A 0

I 0 I 0

0 I 0 I

I 0 I 0

0 I 0 I 0 I

I 0 I

0 I

1

21 2

3

TT

R RBT

n

B

3

3

3

3

3

3

33

r

0

rA 0

0 A D D0 A

0

0 r

0 I 0

Ti iA rA r

T FEMD M D

D

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The global position of a node, expressed in the moving frame is

Substituting this equation in the expression of

3 01 11

2

1 1 11 1

3 02 21 2 21 2

3 0

2

1 1

t

T

t t tnf ni

t t t tnf ni

t t t tn n nnf n n i

RB

nn

3 3 3 3 3 3

3 3 3 3 3 3

3 3 3 3

3

33

3 3 3

I r φ φ ψ ψ

0 0 0 0 0 0

I r φ φ ψ ψ

0 0 0 0 0 0

I r φ φ ψ ψ

0 0 0 0 0 0

r

0

r

D 0 A0

r

0

0

3

1 3

1

3

1 3

3

23,RB m

T

nf

ni

3

r

I

I

0 A ΓI

I

I

D 0

0 01 1

s dn n

T T T t ti i i i i ij j ij j

j j

A r A r r A r A r r φ ψ

2RBD

T FEMD M D

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The matrix to compute is- It depends on 10 precomputed, constant matrices- Sub-matrix MRB

- Sub-matrix MRB

- Sub-matrix M

1 2 1 2

3 33,3 1

3 33

3, 3

41 23

52 33

3 3,3 364 5

3,

,3,3 1

1 13

FEM

T

T TRB RB RB RB RB

T

mT

Tm m

mT T

mTT TT

m

M D D D D

A 0 A 0A 00

M

0 MM M

0 MΓ

0 A 0 A0 A

0 0A 0

M M

0 00 Γ

Γ Γ0

0

0 MA

0

M M

3 3,7

9

8

1 2

3 1

mT TRB R

FEB RB B

MRB R T

A 0 0M M M D D

0 A Γ

MM D D M

M

10 11

11 12

T T FEM T FEMFEM

T T FEM T FEM T

M M D D M D D M D M MM M D D

M M D D M D D M D M M

RB RBT FEMTRB

M MD M D

M M

T FEMD M D

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Dq

Velocity dependent inertia forces

The term can be computed as follows:

with

For each node there are two kind of terms: and

T FEMD M Dq

T FEM T FEM Tvel n n Q D M Dq D A M A Dq

1 1 1 1,1 , ,1 ,

1 1 1 1,1 , ,1 ,

2 2 2 22 ,1 , ,1 ,

2 2 2 2,1 , ,1 ,

,1 ,

f i

f i

f i

f i

f

t t n t t n

r r n r r n

t t n t t n

r r n r r n

n nn t t n

3 1

3 3

3

3 3

3

0 r s Aφ Aφ Aψ Aψ

0 0 Aφ Aφ Aψ Aψ

0 r s Aφ Aφ Aψ Aψ

Dq 0 0 Aφ Aφ Aψ Aψ

0 r s Aφ Aφ A

1 1

1

1

2 2

2

1

1,1 ,

,1 , ,1 ,

2

2

2

s

i

df i

r

r

r

r

n

rn nnt t n

rn n n nnnr r n r r n

3 3

s ω ω r ω vω

ω ω

ω ω r ω v

ω ω

ω ω r ω vψ Aψ

ω ω0 0 Aφ Aφ Aψ Aψ

, , , ,1 1 1 1

, s s s sn n n n

r rt i i t j j r i i r j j

i j i j

v φ ψ ω φ ψ

j ω ω r 2 r

j

rj

ω v

ω ω

Dq

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Computation of the centrifugal term at each node:

For each one of the three adding terms

with:

2 2 2 2 2 2 2 21 2

1,3 1,3

3 1 2 31,3 1,3

1,3 1,3

Ti i i

iT

ii

T

Ti

ω ω r A ω A r A ω r A ω ω ω r A

r 0 0

0 r 0

0 0 r

ω ω ω

1,3 123 0 ,1 , ,1 ,

1,3 1,3 1,3 1,3 1,3 1,3 1,3 1,3

1,3 1,3 1,3 1,3 1,3 1,

3

0 3

3

1 32 21 2 1 2

3

1 3

3 1,3 1,3

3

, s d

T T i T i T i T i TTi t

nf

ni

t n t t ni

0 e r φ φ ψ ψr

0 0 0 0 0 0 0 0

0 0 0 0 0 0

0

r I

I

I

A ω A Γ

0 0

ω Γ

I

I

I

0

0 0 123

0

1 1 1, x

Ty

z

r 0 0

r I 0 r 0 e

0 0 r

T FEMD M Dq

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Computation of the Coriolis acceleration term

Finally, the velocity dependent inertia forces are:

There 12 constant matrices that can be pre-computed

,1 , ,1 ,

,1 , ,1 ,

1

3

3 1

2 2 2 22 2 2s d

s

s

d

d

T

Tr r r i i i it t n t t n

i i i ir r n r r

Tni i i

r r T Tri ii

T

n

n

φ φ ψ ψ

φ φ ψ

A

Aω v A 0v A v Aω ω

0 Aω A ω A Aω ψω

A

ω

1

3 2 2 2, 2 1 2 2 2 3 3

3

3

13, 2 2 22 1

13 1

2 2 2 3 31

4 15 16

17 18 19 20

s

d

T

vel RB Tn

TmT

Tn

A

A 0Q Γ ω Γ ω Γ ω Γ ω

A0 AΓ

A0Γ ω Γ ω

M M M M

M M M Γ ω Γ ωMΓ

A

2 2 2

, 221 22 23 24

1 2 2 2 3 3vel M MQ Γ ω Γ ω ΓM Mω Γ ω

,

,

vel RB

velvel

QQ

Q

T FEMD M Dq

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1st velocity transformation 1/4

First, the closed loops are open. The Cartesian velocities are expressed in terms of the relative velocities and the static and dynamic modal amplitudes (computations carried out recursively):

The transformation matrix R can be partitioned in the form:

1 11

2 21 2 2,1

3 31 2 3 2,1 2,3

4 41 4 4,1

5 1 4 5 4,1 5,6

6 1 4 6 4,1 5,6

2

3

4

5

6

2

4

Z zb

Z zb b φ

Z zb b b φ φ

Z zb b φ

Z b b b φ φ

Z b b b φ φ

η I

η I

η I

η I

η I

ε I

ε I

5

6

2

3

4

5

6

2

4

z

z

η

η

η

η

η

ε

ε

q4

5

6

q5

q64

q3

0

q1

3

2

q2

1

, RB

Z q R R 0

η R η R 0 I 0

ε ε 0 0 I

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The transformation matrix R relate Cartesian and relative coordinates for velocities and accelerations:

By assembling the dynamic equations of each body and setting them in terms of the relative and modal coordinates, the final expression for the open-chain dynamic equations is:

All the terms in this equation can be evaluated efficiently, using a recursive algorithm.

,

Z q Z q q

η R η η R η R η

ε ε ε ε ε

T T T

q q

R MR η R MR η R Q 0

ε ε

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By substituting matrix R partitioned in the expression :

The first term of this expression is the mass accumulation in the relative coordinates and in the amplitudes of the boundary nodes.- A single recursive calculation can evaluate all expressions.

The second and third terms have analogous meanings.

TR MR

,

,

, ,

, , , ,

, ,

T TRB RB RB RB RB

T T TRB RB RB

T TRB RB RB RB

T TRB RB RB RB RB

RB RB RB

R M R R M R 0 0 0 0

R MR R M R R M R 0 0 M M

0 0 0 0 M M

0 R M R M

M R R M M R R M

M R M R 0

, , ,

, , n n n

T T k T T k T T kRB RB RB i RB j RB i RB j RB RB i RB ji j i j i j

k i k i k i

R M R b M b R M R φ M φ R M R φ M b

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The projection of the forces can be evaluated similarly:

The evaluation is similar to that corresponding to the mass matrix:- All the terms are evaluated recursively

following the open-chain tree structure.- The forces applied to the relative

coordinates and modal amplitudes are added.

The remaining expressions also can be evaluated recursively.

TR Q

T TRB RB RB RB

T T TRB

T TRB

R 0 0 Q R Q

R Q R I 0 Q R Q Q

R 0 I Q R Q Q

i ii

i i ii

i ii

nT TRB RB z i k zz

k i

nT

RB kk i

R Q Q b Q Q

R Q Q φ Q Q

Q Q

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2nd velocity transformation

Closed-chain systems:- First the closed-loops are open by removing some joints- A subset of independent relative coordinates is chosen

- The constraint equations are enforced by a 2nd velocity transformation:

- Substituting in the open-chain differential equations:

i z B z

, , and

i

z ηz η

z η z

z

Φ Φ 0Φ Φ

Φ z η ε 0 Φ Φ 0 I 0 R 0B 0

0 0 I

z R z

T T i T T id

dt

zz z z

RRR R MRR z R R Q M z

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Numerical examples 4. Live demos and concluding remarks

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Mbs3d: Implementation details

Program mbs3d- Computer program that implements the semi-recursive

formulation previously described- Written in MATLAB- Available at http://mec21.etsii.upm.es/mbs/

mbs3d/MEX- The core functions for computations and graphics written in C++

and called from MATLAB - Speed up of two orders of magnitude: real-time attainable!- Use of BLAS and sparse matrix libraries

Some advantages of the MATLAB/C++ combination- MATLAB is faster for the first implementation and testing- Afterwards, C++ functions may be implemented faster

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Concluding remarks

Natural coordinates provide a very simple way to define multibody systems.

Some of their characteristics (constant mass matrices and not velocity dependent inertia forces, quadratic constraint equations and linear Jacobians, …) allow fast global formulations.

For faster simulations topological methods with relative coordinates are preferable. Even in this case, natural coordinates may help in the definition of the system geometry and in the closure of the loop constraint equations, keeping the formulation simpler.

Natural coordinates and MATLAB may be useful for education in multibody systems under severe time limitations.

Algorithm implementation and testing are very easy with MATLAB, but the numerical performance is very poor. The combined implementation MATLAB/C++ is fast, reliable and efficient.

Keeping things as simple as possible always has an added value.

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Numerical examples Thank you very much!(Slides available in mec21.etsii.upm.es/mbs

in a few days)

Documents

INSIA-ETSII-UPM 1/46 Multibody Systems Made Simple and Efficient J. García de Jalón IDETC/CIE 2011 MSNDC 23-1 Washington, USA, 28-31 Aug, 2011 Universidad