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Mechanical Integrators for the Optimal Control in Multibody Dynamics

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Mechanical Integrators for the Optimal Control in Multibody Dynamics CORE Metadata, citation and similar papers at core.ac.uk Provided by Publikations- und Dokumentenserver der Universitätsbibliothek Siegen Ralf Siebert Mechanical integrators for the optimal control in multibody dynamics Herausgeber: Peter Betsch Schriftenreihe des Lehrstuhls für Numerische Mechanik Band V Mechanical integrators for the optimal control in multibody dynamics Dissertation zur Erlangung des akademischen Grades Doktor-Ingenieur vorgelegt von Dipl.-Math. Ralf Siebert aus Heinsberg eingereicht dem Department Maschinenbau der Universität Siegen Referent: Prof. Dr.-Ing. habil. Peter Betsch Korreferent: Prof. Dr. Moritz Diehl Vorsitzender: Prof. Dr.-Ing. Oliver Nelles Tag der mündlichen Prüfung: 31. 08. 2012 Impressum Prof. Dr.-Ing. habil. Peter Betsch Lehrstuhl für Numerische Mechanik Universität Siegen 57068 Siegen ISSN 1866-1203 URN urn:nbn:de:hbz:467-6520 Zugl.: Siegen, Univ., Diss., 2012 i Abstract The present thesis deals with inverse multibody dynamics problems. In particu- lar, optimal control problems will be treated, which are governed by differential- algebraic equations. The main goal therein is the minimization of the control effort, which is necessary for moving a multibody system from one configuration to another. A basic task therefore is the formulation of the underlying equations of motion. Main focus will be on the formulation of the equations of motion with natural coordinates, which facilitates the design of structure-preserving time- stepping schemes. It is well known that schemes preserving basic properties of the mechanical system algorithmically exhibit superior stability properties in comparison to standard integrators. The application of these schemes within op- timal control problems requires a consistent incorporation of the control torques. A convincing way for the incorporation of the control torques will be proposed in this contribution. In addition to the schemes based on the rotationless for- mulation, also an energy-momentum conserving time-stepping scheme based on quaternions will be presented. Although the quaternion-based scheme turns out to be competitive in the forward dynamics of rigid bodies, the extension to multibody problems is not as straightforward and easy to handle as with the rotationless formulation in terms of natural coordinates. Therefore quaternions will not be applied within optimal control problems in this thesis. Regarding optimal control of multibody systems, new energy-momentum consistent direct transcription methods in terms of natural coordinates will be presented. In a first step, the equations of motion obtained by a reduction process via the dis- crete null space method will be applied. The arising results will be compared with those achieved by a formulation of the equations of motion with the widely used minimal coordinates. In a second step, the original equations of motion in form of differential-algebraic equations will serve as basis for the formulation of the optimal control problem. In addition to the direct transcription methods mentioned before, a novel optimal control method based on indirect transcrip- tion will be presented. The newly proposed method conserves the Hamiltonian corresponding to the optimal control problem. It is worth mentioning that this method is directly related to previously developed energy consistent schemes in forward dynamics. Keywords: Multibody Systems, Differential-Algebraic Equations, Rotationless Formulation, Quaternions, Energy-Momentum Consistency, Optimal Control, Actuation, Direct Transcription, Indirect Transcription, Conservation of the Hamiltonian ii iii Zusammenfassung Die vorliegende Arbeit behandelt Problemstellungen der inversen Mehrkörper- dynamik. Insbesondere werden Optimalsteuerungsprobleme untersucht, die mit- tels differential-algebraischer Gleichungen beschrieben werden. Das Hauptziel besteht in der Minimierung des Steuerungsaufwands, der benötigt wird, um ein Mehrkörpersystem von einer Konfiguration in eine andere zu bewegen. Eine wichtige Aufgabe dafür ist die Formulierung der zu Grunde liegenden Bewe- gungsgleichungen. Hauptaugenmerk wird dabei auf die Formulierung der Be- wegungsgleichungen mit natürlichen Koordinaten gelegt, die die Konstruktion von Struktur erhaltenden Zeitintegratoren ermöglichen. Es ist bekannt, dass sich Verfahren, die grundlegende mechanische Eigenschaften algorithmisch er- halten, durch überlegene numerische Stabilitätseigenschaften im Vergleich zu Standardintegratoren auszeichnen. Die Anwendung von diesen Verfahren inner- halb von Optimalsteuerungsproblemen erfordert einen konsistenten Einbau von Steuerungsmomenten. Ein überzeugender Weg für den Einbau von Steuerungsmo- menten wird in dieser Arbeit vorgeschlagen. Zusätzlich zu den Integratoren auf- bauend auf der rotationsfreien Formulierung wird ergänzend ein Energie-Impuls- erhaltendes Zeitschrittverfahren basierend auf Quaternionen vorgestellt. Zwar stellt sich das Quaternionen gestützte Verfahren als konkurrenzfähig in der Vor- wärtsdynamik von Starrkörpern heraus, die Erweiterung auf Mehrkörperprob- leme ist allerdings nicht so direkt und einfach zu handhaben wie bei der rota- tionsfreien Formulierung basierend auf natürlichen Koordinaten. Daher werden die Quaternionen innerhalb von den Optimalsteuerungsproblemen in dieser Ar- beit nicht verwendet. In Bezug auf die optimale Steuerung von Mehrkörpersys- temen werden neuartige Energie-Impuls-konsistente direkte Transkriptionsver- fahren beruhend auf natürlichen Koordinaten vorgestellt. In einem ersten Schritt werden Bewegungsgleichungen verwendet, die über eine Reduktion mittels der diskreten Nullraum-Methode gewonnen werden. Die Resultate werden verglichen mit denen, die man durch eine Formulierung der Bewegungsgleichungen mit den weit verbreiteten minimalen Koordinaten erhält. In einem zweiten Schritt dienen die ursprünglichen Bewegungsgleichungen in Form von differential-algebraischen Gleichungen als Basis für die Formulierung von den Optimalsteuerungsproble- men. Zusätzlich zu den bereits erwähnten direkten Transkriptionsmethoden wird auch eine neuartige Optimalsteuerungsmethode basierend auf indirekter Transkription eingeführt. Die besagte Methode erhält die zu dem Optimals- teuerungsproblem zugehörige Hamiltonfunktion. Es ist erwähnenswert, das diese Methode direkten Bezug hat zu früher entwickelten Energie-konsistenten Ver- fahren in der Vorwärtsdynamik. iv Schlagwörter: Mehrkörpersysteme, Differential-Algebraische Gleichungen, Ro- tationsfreie Formulierung, Quaternionen, Energie-Impuls-Konsistenz, Optimals- teurung, Aktuierung, Direkte Transkription, Indirekte Transkription, Erhaltung der Hamilton-Funktion v Vorwort Die vorliegende Arbeit entstand in der Zeit von 2007 bis 2012 im Rahmen meiner Tätigkeit als wissenschaftlicher Mitarbeiter am Lehrstuhl für Numerische Mechanik der Universität Siegen. Besonders bedanken möchte ich mich bei Herrn Prof. Dr.-Ing. Peter Betsch für die fachliche Unterstützung und Motivation bei der Erarbeitung dieser Disserta- tion sowie die Übernahme des Hauptreferates. Ebenfalls danken möchte ich Herrn Prof. Dr. Moritz Diehl für die Übernahme des Korreferates, das damit gezeigte Interesse an dieser Arbeit und zahlreiche wertvolle Anmerkungen. Außerdem möchte ich allen Mitarbeitern des Lehrstuhls für Numerische Mechanik für die angenehme Arbeitsatmosphäre und Zusammenarbeit danken. Siegen, September 2012 Ralf Siebert vi vii Contents 1. Introduction 1 1.1. Literaturereview ........................... 2 1.2. Outline................................. 4 2. Basics of optimal control 9 2.1. Continuousoptimalcontrol . 10 2.2. Discreteoptimalcontrol . 11 2.3. Review of numerical optimal control methods . .. 14 3. Equations of motion 17 3.1. Dynamics of constrained mechanical systems with size reduction . 17 3.2. Basic and reduced energy-momentum scheme . 21 3.3. Connection with generalized coordinates . ... 24 4. Rigid body dynamics 27 4.1. Spatialrigidbody. .. .. .. 28 4.2. Kinematicpairs ............................ 37 4.3. Consistent incorporation of control torques . ..... 46 4.4. Consistent incorporation of linear viscous friction . ........ 49 5. Alternative rigid body formulation: Quaternions 51 5.1. Basicsofquaternions . 52 5.2. Rigidbodydynamics . 53 5.3. Hamiltonequationsofmotion . 55 5.4. Conservingdiscretization. 57 5.5. Steadyprecessionofagyrotop . 59 5.6. Concluding remarks on quaternions . 62 6. Optimal control with equations of motion in ODE-form 63 6.1. Optimal control formulation: Direct transcription . ....... 64 6.2. 3-linkmanipulator .. .. .. 66 6.3. Satellite ................................ 71 7. Optimal control with equations of motion in DAE-form 79 7.1. Optimal control formulation: Direct transcription . ....... 80 7.2. Overheadcrane ............................ 81 7.3. 3-linkmanipulator .. .. .. 84 8. Hamiltonian conserving indirect optimal control method 91 viii Contents 8.1. Systemswithonedegreeoffreedom . 92 8.2. Systems with several degrees of freedom . 97 9. Summary and outlook 107 A. Details of the implementation of the REM 111 B. Details of the implementation of the BEM 115 1 1. Introduction In the present thesis, numerical methods for treating optimal control of mechan- ical multibody systems will be developed. To preserve resources, energy-optimal movements become more and more important. Hence, the main focus will be on the minimization of the control effort which is necessary for moving a multibody system from a specific initial to a specific end position. Areas of application of such problems are the simulation
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