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Research Papers

Natural Coordinates in the Optimal Control of Multibody Systems

[+] Author and Article Information
Peter Betsch1

Ralf Siebert

Nicolas Sänger

Chair of Computational Mechanicse-mail: nicolas.saenger@uni-siegen.de University of Siegen, 57068 Siegen, Germany

One option is to introduce “mixed coordinates” in the sense of García de Jalón [1].

In the following the summation convention applies to lower case roman indices occurring twice in a term. Occasionally the summation is highlighted by a summation symbol.

1

Corresponding author.

J. Comput. Nonlinear Dynam 7(1), 011009 (Sep 26, 2011) (8 pages) doi:10.1115/1.4004886 History: Received May 11, 2011; Revised August 08, 2011; Published September 26, 2011; Online September 26, 2011

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Second, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

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References

Figures

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Figure 1

Planar sketch of the reference configuration of the rigid body

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Figure 2

External force vector acting on the rigid body in the current configuration

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Figure 3

Top: Kinetic energy of the spacecraft for the rest-to-rest maneuver. Middle: Energy consistency. Bottom: Angular momentum consistency.

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Figure 4

Snapshots of the spacecraft motion

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Figure 5

Optimal reaction wheel motor torques

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Figure 6

Cost function versus number of segments used in the optimization

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Figure 7

Comparison of angular momentum

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Figure 8

Comparison of kinetic energy

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