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Research Papers

The Discrete Adjoint Gradient Computation for Optimization Problems in Multibody Dynamics

[+] Author and Article Information
Thomas Lauß

University of Applied Sciences Upper Austria,
Stelzhamerstrae 23,
Wels 4600, Austria;
Institute of Mechanics and Mechatronics,
Vienna University of Technology,
Getreidemarkt 9/E325,
Wien 1060, Austria
e-mail: thomas.lauss@fh-wels.at

Stefan Oberpeilsteiner

University of Applied Sciences Upper Austria,
Stelzhamerstrae 23,
Wels 4600, Austria;
Institute of Mechanics and Mechatronics,
Vienna University of Technology,
Getreidemarkt 9/E325,
Wien 1060, Austria
e-mail: stefan.oberpeilsteiner@fh-wels.at

Wolfgang Steiner

University of Applied Sciences Upper Austria,
Stelzhamerstrae 23,
Wels 4600, Austria
e-mail: wolfgang.steiner@fh-wels.at

Karin Nachbagauer

University of Applied Sciences Upper Austria,
Stelzhamerstrae 23,
Wels 4600, Austria
e-mail: karin.nachbagauer@fh-wels.at

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 15, 2016; final manuscript received October 24, 2016; published online December 5, 2016. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 12(3), 031016 (Dec 05, 2016) (10 pages) Paper No: CND-16-1336; doi: 10.1115/1.4035197 History: Received July 15, 2016; Revised October 24, 2016

The adjoint method is a very efficient way to compute the gradient of a cost functional associated to a dynamical system depending on a set of input signals. However, the numerical solution of the adjoint differential equations raises several questions with respect to stability and accuracy. An alternative and maybe more natural approach is the discrete adjoint method (DAM), which constructs a finite difference scheme for the adjoint system directly from the numerical solution procedure, which is used for the solution of the equations of motion. The method delivers the exact gradient of the discretized cost functional subjected to the discretized equations of motion. For the application of the discrete adjoint method to the forward solver, several matrices are necessary. In this contribution, the matrices are derived for the simple Euler explicit method and for the classical implicit Hilber–Hughes–Taylor (HHT) solver.

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References

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Figures

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Fig. 1

Interpolation of the Hessian

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Fig. 2

Nonlinear two mass oscillator

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Fig. 3

Solution of the nonlinear two mass oscillator (a) optimal input and (b) convergence of cost functional

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Fig. 4

Planar overhead crane (a) schematic of the crane and (b) optimal input

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Fig. 5

Convergence of the two optimization problems (a) convergence of version 1 and (b) convergence of version 2

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Fig. 6

PUMA industrial robot (a) the schematic of the PUMA robot and (b) the start and end configuration

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Fig. 7

Solution of the robot example (a) convergence of the cost functional of the PUMA and (b) optimal joint torques of the PUMA

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