PROJECTIVE CONSTRAINT VIOLATION STABILIZATION METHOD FOR MULTIBODY SYSTEMS ON MANIFOLDS Prof. dr. Zdravko Terze Dept. of Aeronautical Engineering, Faculty

PROJECTIVE CONSTRAINT VIOLATION STABILIZATION PROJECTIVE CONSTRAINT VIOLATION STABILIZATION METHOD FOR MULTIBODY SYSTEMS ON MANIFOLDSMETHOD FOR MULTIBODY SYSTEMS ON MANIFOLDS

Prof. dr. Zdravko Terze

Dept. of Aeronautical Engineering,

Faculty of Mechanical Eng. & Naval Arch.

University of Zagreb

Dr. Joris Naudet

Multibody Mechanics Group

Dept. of Mechanical Engineering

Vrije Universiteit Brussel

DEPARTMENT OF AERONAUTICAL ENGINEERING

CHAIR OF FLIGHT VEHICLE DYNAMICS

CONSTRAINT GRADIENT CONSTRAINT GRADIENT PROJECTIVEPROJECTIVE METHOD METHOD Introduction

Focus: constraint gradient projective method for numerical stabilization of mechanical systems holonomic and non-holonomic constraints

Numerical errors along constraint manifold optimal partitioning of the generalized coordinates to provide full constraint satisfaction while minimizing numerical errors along manifold optimal constraint stabilization effect

Numerical example



CONSTRAINT GRADIENT PROJECTIVE METHODCONSTRAINT GRADIENT PROJECTIVE METHOD

Unconstrained MBS on manifolds - autonomous Lagrangian system, n DOF ,

Differentiable-manifold approach:

- configuration space differentiable manifold

covered (locally) by coordinate system x (chart)

*

d

d

xx

LL

t xxQxx ,*M,

n ODE

nRnM



is not a vector space, at every point :

n-dimensional tangent space

+ union of all tangent spaces :

tangent bundle (‘velocity phase space’)

Riemannian metric (positive definite)

locally Euclidean vector space

,

dim = 2n , xM

nM Mx

MxT Mxx TnM MM

M

n

TTx

x:

nTM

MMM xxxxx TT ,:,

MxT





MBS with holonomic constraints unconstrained system: ,

- trajectory in the manifold of configuration

holonomic constraints: ,

restrict system configuration space (‘positions’):

n-r dim constraint manifold:

at the velocity level: linear in

t,,* xxQxx M

txx ii :T

0x t,

τxxx tt ,* x

Mx

Mxxxxxxx TEk ,

2

1

2

1 T2M

M

rnt RRR :,xΦ

txx ii :T

0xx ttrn ,,)( MS




Geometric properties of constraints - constraint matrix:

constraint subspace

tangent subspace

, basis vectors:

- constraint submanifold : described by minimal form formulation

**1

T* ,....,, rt xx

1ˆgrad 1Φ rr ̂grad Φ

rn rr ˆ,.....,1̂

2M 1S

2MxT1̂

1̂r

....

:

:

n-rRy

0rn-rT xx CS nrn-r TT MCS xxx

n-rT Sx

rxC

n-rT SxrnS

0xRxx ),(),(* tt




Mathematical model of CMS dynamics DAE of index 3:

DAE of index 1:

‘projected ODE’ : , ,

integral curve drifts away from submanifold

only if can be determined that describe

constraint stabilization procedure is not needed

λtt ,,,T** xxxQxx x M

0

Qx

0x

x*

*

T*

λ

M

0

QRx

R

x

*T

*

T

M

zRRQRzRR MM TTT zRx zRzRx

n-rRy rnS

rnS




MBS with non-holonomic constraints ‘r’ holonomic constraints:

additional ‘nh’ non-holonomic constraints :

do not restrict configuration space /‘positions’

impose additional constraints on /‘ velocities’

if linear in velocities (Pfaffian form) ,

- system constraints ,

DAE constraint stabilization procedure

0xx t,,

0x t,

0xxxB tt ,,*

xxB

xx t

t

,

,*

*

0xRx ttnh ,,*

rnS

n-rrnn-rnhrn TT SS xxxST




Stabilized CMS time integration Integration step (DAE or ‘projected’ ODE)

Stabilization step

generalized coordinates partitioning:

correction of constraint violation ,

Problem: inadequate coordinate partitioning

negative effect on integration accuracy along manifold

constraints will be satisfied anyway !!

ξ

Qx

0x

x*

*

T*

λ

M

rd Rx r-ni Rx

0x t,

,x ODE xx

rd Rx r-ni Rx

τxxx tt ,*




Constraint gradient projective method projective criterion to the coordinate partitioning method

(Blajer, Schiehlen 1994, 2003), (Terze et al 2000), (Terze, Naudet 2003)




Questions ?!

If optimal subvector for ‘positions’ is selected:

is the same subvector optimal choice for velocity stabilization level as well ?

is it valid in any case ?

Is the proposed algorithm applicable for stabilization of non-holonomic systems ?




Structure of partitioned subvectors System tangent bundle:

dim = 2n

Riemannian manifold

Holonomic constraints

- ‘position’ constraint manifold

x correction gradient:

xx MMM ,diagMT

1

2

rn -S x

x̂

),(),(grad * tt x0x x

,

,

MMM xxxxx TT ,:,

nTM

0xx trn ,,MS




- velocity constraint manifold

correction gradient :

Holonimic systems: optimal partitioning returns ‘the same dependent coordinates’ at the position

and velocity level

),(,grad ** tt xxx xx

rn -Vx

x̂

1

2

x

xxx0x xx ttt ,grad),(,grad **

τxxx xx tTrn ,, *MV




Non-holonomic constraints

linear (Pfaffian form):

H + NH constraints:

correction gradient:

x correction gradient:

Non-holonomic systems: correction gradients do not match any more. A separate partitioning procedure for stabilization at configuration and velocity level !!

0xxxB tt ,,*

xxxB

xx *

*

*

,

,nh

t

t

x ),(grad ** tnhnh xx

),(),(grad * tt x0x x




Coordinates relative projections vs time




Non-holonomic mechanical system

- dynamic simulation of the satelite motion (INTELSAT V)



CONSTRAINT GRADIENT CONSTRAINT GRADIENT PROJECTIVEPROJECTIVE METHOD METHOD

Reference trajectories



CONSTRAINT GRADIENT CONSTRAINT GRADIENT PROJECTIVEPROJECTIVE METHOD METHOD

Relative length of projections on constraint subspace

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PROJECTIVE CONSTRAINT VIOLATION STABILIZATION METHOD FOR MULTIBODY SYSTEMS ON MANIFOLDS Prof. dr. Zdravko Terze Dept. of Aeronautical Engineering, Faculty