1. 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

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 dim constraint 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 accuracy along manifold
 constraints will be satisfied anyway !!

10. CONSTRAINT GRADIENT PROJECTIVE METHOD
 projective criterion to the coordinate partitioning 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 dependent coordinates’ 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