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Prof. dr. Zdravko Terze Dept. of Aeronautical Engineering,

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. CONSTRAINT GRADIENT PROJECTIVE METHOD. Introduction

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Prof. dr. Zdravko Terze Dept. of Aeronautical Engineering,

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  1. 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

  2. CONSTRAINT GRADIENT PROJECTIVE 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

  3. CONSTRAINT 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) n ODE ,

  4. CONSTRAINT GRADIENT PROJECTIVE METHOD 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

  5. CONSTRAINT GRADIENT PROJECTIVE METHOD • MBS with holonomic constraints • unconstrained system: , - trajectory in the manifold of configuration • holonomic constraints: , restrict system configuration space (‘positions’): n-r dimconstraint manifold: at the velocity level: linear in

  6. CONSTRAINT GRADIENT PROJECTIVE METHOD • Geometric properties of constraints - constraint matrix: constraint subspace tangent subspace , basis vectors:  - constraint submanifold : described by minimal form formulation : .... :

  7. CONSTRAINT GRADIENT PROJECTIVE METHOD • 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

  8. CONSTRAINT GRADIENT PROJECTIVE METHOD • 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

  9. CONSTRAINT GRADIENT PROJECTIVE METHOD • 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 accuracyalong manifold constraints will be satisfied anyway !!

  10. CONSTRAINT GRADIENT PROJECTIVE METHOD • Constraint gradient projective method projective criterion to the coordinatepartitioning method (Blajer, Schiehlen 1994, 2003), (Terze et al 2000), (Terze, Naudet 2003)

  11. CONSTRAINT GRADIENT PROJECTIVE METHOD • 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 ?

  12. CONSTRAINT GRADIENT PROJECTIVE METHOD • Structure of partitioned subvectors • System tangent bundle: dim = 2n Riemannian manifold • Holonomic constraints - ‘position’ constraint manifold  x correction gradient: , , 2 1

  13. CONSTRAINT GRADIENT PROJECTIVE METHOD - velocity constraint manifold  correction gradient : Holonimic systems:optimal partitioning returns ‘the same dependentcoordinates’ at the position and velocity level 2 1

  14. CONSTRAINT GRADIENT PROJECTIVE METHOD • 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 !!

  15. CONSTRAINT GRADIENT PROJECTIVE METHOD • Coordinates relative projections vs time

  16. CONSTRAINT GRADIENT PROJECTIVE METHOD • Non-holonomic mechanical system - dynamic simulation of the satelite motion (INTELSAT V)

  17. CONSTRAINT GRADIENT PROJECTIVE METHOD • Reference trajectories

  18. CONSTRAINT GRADIENT PROJECTIVE METHOD • Relative length of projections on constraint subspace

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