Discrete adjoint method for variational integration of constrained ODEs and its application to optimal control of geometrically exact beam dynamics
Direct methods for the simulation of optimal control problems apply a specific discretization to the dynamics of the problem, and the discrete adjoint method is suitable to calculate corresponding conditions to approximate an optimal solution. While the benefits of structure preserving or geometric...
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| Published in | Multibody system dynamics Vol. 60; no. 3; pp. 447 - 474 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.03.2024
Springer Nature B.V |
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| Online Access | Get full text |
| ISSN | 1384-5640 1573-272X 1573-272X |
| DOI | 10.1007/s11044-023-09934-4 |
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| Abstract | Direct methods for the simulation of optimal control problems apply a specific discretization to the dynamics of the problem, and the discrete adjoint method is suitable to calculate corresponding conditions to approximate an optimal solution. While the benefits of structure preserving or geometric methods have been known for decades, their exploration in the context of optimal control problems is a relatively recent field of research. In this work, the discrete adjoint method is derived for variational integrators yielding structure preserving approximations of the dynamics firstly in the ODE case and secondly for the case in which the dynamics is subject to holonomic constraints. The convergence rates are illustrated by numerical examples. Thirdly, the discrete adjoint method is applied to geometrically exact beam dynamics, represented by a holonomically constrained PDE. |
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| AbstractList | Direct methods for the simulation of optimal control problems apply a specific discretization to the dynamics of the problem, and the discrete adjoint method is suitable to calculate corresponding conditions to approximate an optimal solution. While the benefits of structure preserving or geometric methods have been known for decades, their exploration in the context of optimal control problems is a relatively recent field of research. In this work, the discrete adjoint method is derived for variational integrators yielding structure preserving approximations of the dynamics firstly in the ODE case and secondly for the case in which the dynamics is subject to holonomic constraints. The convergence rates are illustrated by numerical examples. Thirdly, the discrete adjoint method is applied to geometrically exact beam dynamics, represented by a holonomically constrained PDE. |
| Author | Nachbagauer, Karin Leyendecker, Sigrid Ober-Blöbaum, Sina Sato Martín de Almagro, Rodrigo T. Schubert, Matthias |
| Author_xml | – sequence: 1 givenname: Matthias surname: Schubert fullname: Schubert, Matthias email: matthias.schubert@fau.de organization: Institute of Applied Dynamics, Friedrich-Alexander-Universität Erlangen-Nürnberg – sequence: 2 givenname: Rodrigo T. surname: Sato Martín de Almagro fullname: Sato Martín de Almagro, Rodrigo T. organization: Institute of Applied Dynamics, Friedrich-Alexander-Universität Erlangen-Nürnberg – sequence: 3 givenname: Karin surname: Nachbagauer fullname: Nachbagauer, Karin organization: Faculty of Engineering and Environmental Sciences, University of Applied Sciences Upper Austria, Institute for Advanced Study, Technical University of Munich – sequence: 4 givenname: Sina surname: Ober-Blöbaum fullname: Ober-Blöbaum, Sina organization: Universität Paderborn – sequence: 5 givenname: Sigrid surname: Leyendecker fullname: Leyendecker, Sigrid organization: Institute of Applied Dynamics, Friedrich-Alexander-Universität Erlangen-Nürnberg |
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| Cites_doi | 10.1002/oca.912 10.1016/j.ifacol.2021.06.123 10.1137/S0036144504446096 10.1002/nme.1639 10.1115/1.4025476 10.1088/1361-6544/aab630 10.1080/00207216408937740 10.1016/j.compstruc.2018.12.007 10.1007/s10107-004-0559-y 10.1016/j.cma.2020.113475 10.1002/nme.6951 10.5194/ms-4-79-2013 10.1051/cocv/2010012 10.1016/0045-7825(85)90050-7 10.1093/imanum/8.1.141 10.1007/s00211-005-0661-y 10.3934/dcds.2015.35.4193 10.1007/0-387-24255-4_10 10.1007/s11044-019-09695-z 10.1137/151002769 10.1002/nme.5548 10.1115/1.4035197 10.1017/S096249290100006X 10.1007/s11044-017-9600-9 10.1115/1.4041237 |
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| Keywords | Variational integrators Geometrically exact beam Optimal control Discrete adjoint method Holonomically constrained system |
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| SubjectTerms | Automotive Engineering Constraints Control Dynamical Systems Dynamics Electrical Engineering Engineering Mechanical Engineering Optimal control Optimization Vibration |
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| Title | Discrete adjoint method for variational integration of constrained ODEs and its application to optimal control of geometrically exact beam dynamics |
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