This paper considers solution strategies for “dynamical inverse problems,” where the main goal is to determine the excitation of a dynamical system, such that some output variables, which are derived from the system’s state variables, coincide with desired time functions. The paper demonstrates how such problems can be restated as optimal control problems and presents a numerical solution approach based on the method of steepest descent. First, a performance measure is introduced, which characterizes the deviation of the output variables from the desired values, and which is minimized by the solution of the inverse problem. Second, we show, how the gradient of this error functional can be computed efficiently by applying the theory of optimal control, in particular by following an idea of Kelley and Bryson. As the major contribution of this paper we present a modification of this method which allows the application to the case where the state equations are given by a set of differential algebraic equations. This situation has great practical importance since multibody systems are mostly described in this way. For comparison, we also discuss an approach which bases an a direct transcription of the optimal control problem. Moreover, other methods to solve dynamical inverse problems are summarized.

References

References
1.
Reichl
,
S.
,
Steiner
,
W.
, and
Steinbatz
,
M.
, 2009, “
Practical Approaches for Inverse Calculations of Drive Signals in a Virtual Test Rig with Regard to Agricultural Machines
,”
Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics
,
European Community on Computational Methods in Applied Sciences
,
Warsaw University of Technology
, Institute of Aeronautics and Applied Mechanics.
2.
Burger
,
M.
,
Dressler
,
K.
,
Marquardt
,
A.
, and
Speckert
,
M.
, 2009, “
Calculating Invariant Loads for System Simulation in Vehicle Engineering
,”
Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics
,
European Community on Computational Methods in Applied Sciences
,
Warsaw University of Technology
, Institute of Aeronautics and Applied Mechanics.
3.
Burger
,
M.
,
Speckert
,
M.
, and
Dressler
,
K.
, 2010, “
Optimal Control Methods for the Calculation of Invariant Excitation Signals for Multibody Systems
,”
Proceedings of the 1st Joint International Conference on Multibody System Dynamics
,
Lappeenranta
,
Finland.
4.
Blajer
,
W.
, and
Kolodziejczyk
,
K.
, 2004, “
A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE framework
,”
Multibody Syst. Dyn.
,
11
, pp.
343
364
.
5.
Betsch
,
P.
,
Uhlar
,
S.
, and
Quasem
,
M.
, 2009, “
Numerical Integration of Mechanical Systems With Mixed Holonomic and Control Constraints
,”
Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics
,
Warsaw
,
Poland.
6.
Reichl
,
S.
,
Steiner
,
W.
, and
Steinbatz
,
M.
, 2010, “
Optimal Control Methods for the Computation of Excitation Signals in Multibody Systems
,”
Proceedings of the 1st Joint International Conference on Multibody System Dynamics
,
Lappeenranta University of Technology.
7.
Seifried
,
R.
, 2010, “
Two Approaches for Designing Minimum Phase Underactuated Multibody Systems
,”
Proceedings of the 1st Joint International Conference on Multibody System Dynamics
,
Lappeenranta
,
Finland.
8.
Rothfuss
,
R.
, 1997, “
Anwendung der flachheitsbasierten Analyse und Regelung nichtlinearer Mehrgrössensysteme
,”
VDI Fortschrittsberichte
,
Reihe 8. Nr. 664
, p.
181
.
9.
Bastos
,
G. J.
, and
Brls
,
O.
, 2010, “
Trajectory Optimization of Flexible Robots Using an Optimal Control Approach
,”
Proceedings of the 1st Joint International Conference on Multibody System Dynamics
,
Lappeenranta
,
Finland.
10.
Bottasso
,
C. L.
, and
Croce
,
A.
, 2004, “
Optimal Control of Multibody Systems Using an Energy Preserving Direct Transcription Method
,”
Multibody Syst. Dyn.
,
12
(
1
), pp.
17
45
.
11.
Bottasso
,
C. L.
, and
Croce
,
A.
, 2004, “
On the Solution of Inverse Dynamics and Trajectory Optimization Problems for Multibody Systems
,”
Multibody Syst. Dyn.
,
11
, pp.
1
22
.
12.
Kirk
,
D. E.
, 1998,
Optimal Control Theory. An Introduction, Dover Publications
,
New York.
13.
Betts
,
J. T.
, 2010,
Practical Methods for Optimal Control and Estimation Using Nonlinear Programming (Advances in Design and Control)
,
Society for Industrial and Applied Mathematics
,
Philadelphia
.
14.
Kelley
,
H. J.
, 1962, “
Method of Gradients
,”
Optimization Techniques With Applications to Aerospace Systems
,
G.
Leitmann
, ed.,
Mathematics in Science an Engineering
,
Academic
,
New York
, Vol.
5
, Chap. 6, pp.
206
254
.
15.
Bryson
,
A. E.
, and
Denham
,
W. F.
, 1964, “
Optimal Programming Problems With Inequality Constraint II: Solution by Steepest Ascent
,”
AIAA J.
,
2
(
1
), pp.
23
34
.
16.
Blajer
,
W.
, and
Kolodziejczyk
,
K.
, 2007, “
Control of Underactuated Mechanical Systems With Servo-Constraints
,”
Nonlinear Dyn.
,
50
, pp.
781
791
.
17.
Bryson
,
A. E.
, and
Ho
,
Y. C.
, 1975,
Applied Optimal Control,
Hemisphere Publishing Corporation
,
Washington.
18.
Gerdts
,
M.
, 2003, “
Direct Shooting Method for the Numerical Solution of Higher-Index DAE Optimal Control Problems
,”
J. Optim. Theory Appl.
,
117
(
2
), pp.
267
294
.
19.
Bhatti
,
M. A.
, 2000,
Practical Optimization Methods
,
Springer
,
New York.
20.
Bottasso
,
C. L.
,
Maisano
,
G.
, and
Scorcelletti
,
F.
, 2010, “
Maneuvering Multibody Dynamics—New Developments for Models With Fast Solution Scales and Pilot-in-the-Loop Effects
,”
Multibody Dynamics – Computational Methods and Applications
, (Computational Methods in Applied Sciences),
Springer
,
Dordrech
, Vol.
23
, pp.
29
48
.
21.
Ober-Blöbaum
,
S.
, 2008, “
Discrete Mechanics and Optimal Control
,” Ph.D. thesis, Universitt Paderborn, Fakultät fr Elektrotechnik, Informatik und Mathematik.
22.
Shabana
,
A. A.
, 2005,
Dynamics of Multibody Systems
,
Cambridge University Press
,
New York.
23.
Gear
,
C. W.
, and
Gupta
,
B. L.
, 1985, “
Automatic Integration of the Euler-Lagrange Equations With Constraints
,”
J. Comput. Appl. Math.
,
12/13
, pp.
77
90
.
24.
Ascher
,
U. M.
, and
Petzold
,
L. R.
, 1998,
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
,
Society for Industrial and Applied Mathematics
,
Philadelphia
.
25.
Reichl
,
S.
,
Steiner
,
W.
,
Steinbatz
,
M.
, and
Hofer
,
M.
, 2009, “
Evaluation of Different Methods to Compute Excitation Signals Based on Measured Targets in Agricultural Machines
,”
Proceedings of the 80th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM)
,
Gdansk
,
Poland.
26.
Gattringer
,
O.
,
Riener
,
H.
, and
Dannbauer
,
H.
, 2007, “
Integration von Prfstandsmethoden in die Simulation
,”
34. Tagung des DVM-Arbeitskreises Betriebsfestigkeit. Lastannahmen und Betriebsfestigkeit
,
10./11. Oktober2007
,
Wolfsburg
.
27.
Blajer
,
W.
, and
Kolodziejczyk
,
K.
, 2009, “
Dependent Versus Independent Variable Formulation for the Dynamics and Control of Cranes
,”
Solid State Phenom.
,
147–149
, pp.
221
230
.
28.
Seifried
,
R.
, 2010, “
Optimization-Based Design of Minimum Phase Underactuated Multibody Systems
,”
Multibody Dynamics – Computational Methods and Applications
, (Computational Methods in Applied Sciences),
Springer
,
Dordrech
, Vol.
23
, pp.
261
282
.
29.
Quasem
,
M.
,
Uhlar
,
S.
, and
Betsch
,
P.
, 2009, “
Inverse Dynamics of Underactuated Multibody Systems
,”
Proceedings of the 80th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM)
,
Gdansk
,
Poland.
30.
Uhlar
,
S.
, and
Betsch
,
P.
, 2009, “
A Rotationless Formulation of Multibody Dynamics: Modeling of Screw Joints and Incorporation of Control Constraints
,”
Multibody Syst. Dyn.
,
22
, pp.
69
95
.
31.
Fliess
,
M.
,
Lévine
,
J.
, and
Rouchon
,
P.
, 1991, “
A Simplified Approach of Crane Control via a generalized State-Space Model
,”
Proceedings of the 30th Conference on Decision and Control
,
Brighton
,
England.
32.
Fliess
,
M.
,
Lévine
,
J.
, and
Rouchon
,
P.
, 1993, “
Generalized State Variable Representation for a Simplified Crane Description
,”
Int. J. Control
,
58
(
2
), pp.
277
283
.
33.
Isidori
,
A.
, 1995,
Nonlinear Control Systems
,
Springer
,
New York.
34.
Svaricek
,
F.
, 2006,
“Nulldynamik linearer und nichtlinearer Systeme: Definitionen, Eigenschaften und Anwendungen”
Automatisierungstechnik
,
54
, pp.
310
322
.
35.
Baumeister
,
J.
, 1987,
Stable Solution of Inverse Problems
,
Friedr. Vieweg & Sohn
,
Braunschweig/Wiesbaden.
36.
Louis
,
A. K.
, 1989,
Inverse und schlecht gestellte Probleme
,
B.G. Teubner
,
Stuttgart
.
37.
Lanczos
,
C.
, 1986,
The Variational Principles of Mechanics, Dover Publications
,
New York.
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