The inverse dynamics of flexible multibody systems is formulated as a two-point boundary value problem for an index-3 differential-algebraic equation (DAE). This DAE represents the equation of motion with kinematic and trajectory constraints. For so-called nonminimum phase systems, the remaining dynamics of the inverse model is unstable. Therefore, boundary conditions are imposed not only at the initial time but also at the final time in order to obtain a bounded solution of the inverse model. The numerical solution strategy is based on a reformulation of the DAE in index-2 form and a multiple shooting algorithm, which is known for its robustness and its ability to solve unstable problems. The paper also describes the time integration and sensitivity analysis methods that are used in each shooting phase. The proposed approach does not require a reformulation of the problem in input-output normal form, which is known from nonlinear control theory. It can deal with serial and parallel kinematic topology, minimum phase and nonminimum phase systems, and rigid and flexible mechanisms.

References

References
1.
Cannon
,
H.
, and
Schmitz
,
E.
,
1984
, “
Initial Experiments on the End-Point Control of a Flexible One-Link Robot
,”
Int. J. Robot. Res.
,
3
(
3
), pp.
62
75
.10.1177/027836498400300303
2.
Book
,
W.
,
1993
, “
Controlled Motion in an Elastic World
,”
ASME J. Dyn. Syst., Measu., Control
,
115
(
2B
), pp.
252
261
.10.1115/1.2899065
3.
Da Silva
,
M.
,
Brüls
,
O.
,
Swevers
,
J.
,
Desmet
,
W.
, and
Van Brussel
,
H.
,
2009
, “
Computer-Aided Integrated Design for Machines With Varying Dynamics
,”
Mech. Mach. Theory
,
44
, pp.
1733
1745
.10.1016/j.mechmachtheory.2009.02.006
4.
Fliess
,
M.
,
Lévine
,
J.
,
Martin
,
P.
, and
Rouchon
,
P.
,
1995
, “
Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples
,”
Int. J. Control
,
61
, pp.
1327
1361
.10.1080/00207179508921959
5.
Isidori
,
A.
,
1995
,
Nonlinear Control Systems
,
3rd ed.
,
Springer
,
London
.
6.
Sastry
,
S.
,
1999
,
Nonlinear Systems: Analysis, Stability and Control
,
Springer
,
New York
.
7.
Van Nieuwstadt
,
M.
and
Murray
,
R.
,
1998
, “
Real-Time Trajectory Generation for Differentially Flat Systems
,”
Int. J. Robust Nonlinear Control
,
18
(
11
), pp.
995
1020
.10.1002/(SICI)1099-1239(199809)8:11<995::AID-RNC373>3.0.CO;2-W
8.
Asada
,
H.
, and
Slotine
,
J.-J.
,
1986
,
Robot Analysis and Control
,
Wiley-Interscience
,
New York
.
9.
Spong
,
M.
,
1987
, “
Modeling and Control of Elastic Joint Robots
,”
ASME J. Dyn. Syst., Meas., Control
,
109
(4), pp.
310
318
.10.1115/1.3143860
10.
Kwon
,
D.
, and
Book
,
W.
,
1994
, “
A Time-Domain Inverse Dynamic Tracking Control of a Single-Link Flexible Manipulator
,”
ASME J. Dyn. Syst., Meas., Control
,
116
(
2
), pp.
193
200
.10.1115/1.2899210
11.
Devasia
,
S.
, and
Bayo
,
E.
,
1994
, “
Inverse Dynamics of Articulated Flexible Structures: Simultaneous Trajectory Tracking and Vibration Reduction
,”
J. Dyn. Control
,
4
(
3
), pp.
299
309
.10.1007/BF01985076
12.
Seifried
,
R.
,
Held
,
A.
, and
Dietmann
,
F.
,
2011
, “
Analysis of Feed-Forward Control Design Approaches for Flexible Multibody Systems
,”
J. Syst. Des. Dyn.
,
5
(
3
), pp.
429
440
.
13.
Seifried
,
R.
,
Burkhardt
,
M.
, and
Held
,
A.
,
2013
, “
Trajectory Control of Serial and Parallel Flexible Manipulators Using Model Inversion
,”
Multibody Dynamics: Computational Methods and Applications, Computational Methods in Applied Sciences
, Vol.
28
,
J.
Samin
and
P.
Fisette
, eds.,
Springer
,
New York
.
14.
Devasia
,
S.
,
Chen
,
D.
, and
Paden
,
B.
,
1996
, “
Nonlinear Inversion-Based Output Tracking
,”
IEEE Trans. Autom. Control
,
41
(
7
), pp.
930
942
.10.1109/9.508898
15.
Taylor
,
D.
, and
Li
,
S.
,
2002
, “
Stable Inversion of Continuous-Time Nonlinear Systems by Finite-Difference Methods
,”
IEEE Trans. Autom. Control
,
47
(
3
), pp.
537
542
.10.1109/9.989157
16.
Seifried
,
R.
,
2012
, “
Two Approaches for Feedforward Control and Optimal Design of Underactuated Multibody Systems
,”
Multibody Syst. Dyn.
,
27
(
1
), pp.
75
-
93
.10.1007/s11044-011-9261-z
17.
Seifried
,
R.
,
2012
, “
Integrated Mechanical and Control Design of Underactuated Multibody Systems
,”
Nonlinear Dyn.
,
67
, pp.
1539
1557
.10.1007/s11071-011-0087-2
18.
Seifried
,
R.
, and
Eberhard
,
P.
,
2009
, “
Design of Feed-Forward Control for Underactuated Multibody Systems With Kinematic Redundancy
,”
Motion and Vibration Control: Selected Papers from MOVIC 2008
,
H.
Ulbrich
and
L.
Ginzinger
, eds.,
Springer
,
New York
.
19.
Blajer
,
W.
, and
Kolodziejczyk
,
K.
,
2004
, “
A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework Theory and a DAE Framework
,”
Multibody Syst. Dyn.
,
11
, pp.
343
364
.10.1023/B:MUBO.0000040800.40045.51
20.
Blajer
,
W.
, and
Kolodziejczyk
,
K.
,
2007
, “
Control of Underactuated Mechanical Systems With Servo-Constraints
,”
Nonlinear Dyn.
,
50
, pp.
781
791
.10.1007/s11071-007-9231-4
21.
Seifried
,
R.
, and
Blajer
,
W.
,
2013
, “
Analysis of Servo-Constraint Problems for Underactuated Multibody Systems
,”
Mech. Sci.
,
4
, pp.
113
129
.10.5194/ms-4-113-2013
22.
Bastos
,
G.
,
Seifried
,
R.
, and
Brüls
,
O.
,
2013
, “
Inverse Dynamics of Serial and Parallel Underactuated Multibody Systems Using a DAE Optimal Control Approach
,”
Multibody Syst. Dyn.
,
30
, pp. 359–376.10.1007/s11044-013-9361-z
23.
Morrison
,
D.
,
Riley
,
J.
, and
Zancarano
,
J.
,
1962
, “
Multiple Shooting Methods for Two-Point Boundary Value Problems
,”
Commun. ACM
,
5
, pp.
613
614
.10.1145/355580.369128
24.
Keller
,
H.
,
1968
,
Numerical Methods for Two-Point Boundary-Value Problems
,
Blaisdell
,
Waltham, MA
.
25.
Roberts
,
S.
, and
Shipman
,
J.
,
1972
,
Two-Point Boundary Value Problems: Shooting Methods
,
Elsevier
,
New York
.
26.
Chung
,
J.
, and
Hulbert
,
G.
,
1993
, “
A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method
,”
ASME J. Appl. Mech.
,
60
(2), pp.
371
375
.10.1115/1.2900803
27.
Newmark
,
N.
,
1959
, “
A Method of Computation for Structural Dynamics
,”
J. Eng. Mech. Div., Am. Soc. Civ. Eng.
,,
85
, pp.
67
94
.
28.
Arnold
,
M.
, and
Brüls
,
O.
,
2007
, “
Convergence of the Generalized-α Scheme for Constrained Mechanical Systems
,”
Multibody Syst. Dyn.
,
18
(
2
), pp.
185
202
.10.1007/s11044-007-9084-0
29.
Arnold
,
M.
,
Brüls
,
O.
, and
Cardona
,
A.
,
2011
, “
Convergence Analysis of Generalized-α Lie Group Integrators for Constrained Systems
,”
Proceedings of the Multibody Dynamics ECCOMAS Thematic Conference
.
30.
Gear
,
C.
,
Leimkuhler
,
B.
, and
Gupta
,
G.
,
1985
, “
Automatic Integration of Euler–Lagrange Equations With Constraints
,”
J. Comput. Appl. Math.
,
12–13
, pp.
77
90
.10.1016/0377-0427(85)90008-1
31.
Brüls
,
O. E. L.
,
Duysinx
,
P.
, and
Eberhard
,
P.
,
2011
, “
Optimization of Multibody Systems and Their Structural Components
,”
Multibody Dynamics: Computational Methods and Applications, Computational Methods in Applied Sciences
, Vol.
23
,
W.
Blajer
,
J.
Arczewski
,
K.
Fraczek
, and
M.
Wojtyra
, eds.,
Springer
,
New York
, pp.
49
68
.
32.
Wasfy
,
T.
, and
Noor
,
A.
,
2003
, “
Computational Strategies for Flexible Multibody Systems
,”
Appl. Mech. Rev.
,
56
(
6
), pp.
553
613
.10.1115/1.1590354
33.
Bayo
,
E.
, and
Ledesma
,
R.
,
1996
, “
Augmented Lagrangian and Mass-Orthogonal Projection Methods for Constrained Multibody Dynamics
,”
Nonlinear Dyn.
,
9
, pp.
113
130
.10.1007/BF01833296
34.
Jay
,
L.
, and
Negrut
,
D.
,
2007
, “
Extensions of the HHT-Method to Differential-Algebraic Equations in Mechanics
,”
Electron. Trans. Numer. Anal.
,
26
, pp.
190
208
.
35.
Géradin
,
M.
, and
Cardona
,
A.
,
2001
,
Flexible Multibody Dynamics: A Finite Element Approach
,
John Wiley and Sons
,
Chichester, UK
.
36.
Bastos
,
G.
,
Seifried
,
R.
, and
Brüls
,
O.
,
2011
, “
Inverse Dynamics of Underactuated Multibody Systems Using a DAE Optimal Control Approach
,”
Proceedings of the Multibody Dynamics ECCOMAS Conference
.
37.
Wenger
,
P.
, and
Chablat
,
D.
,
2009
, “
Kinematic Analysis of a Class of Analytic Planar 3-RPR Parallel Manipulators
,”
Computational Kinematics: Proceedings of the 5th International Workshop on Computational Kinematics
, pp.
43
50
.
You do not currently have access to this content.