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Research Papers

Optimal Control Strategies for Robust Certification

[+] Author and Article Information
Sigrid Leyendecker2

Biocomputing Group, Berlin Mathematical School, Free University of Berlin, Berlin 14195, Germanysleye@zedat.fu-berlin.de

Leonard J. Lucas

Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125lenny.lucas@gmail.com

Houman Owhadi

Department of Applied and Computational Mathematics and Department of Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125owhadi@caltech.edu

Michael Ortiz

Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125ortiz@aero.caltech.edu

2

Corresponding author.

J. Comput. Nonlinear Dynam 5(3), 031008 (May 18, 2010) (10 pages) doi:10.1115/1.4001375 History: Received March 16, 2009; Revised October 26, 2009; Published May 18, 2010; Online May 18, 2010

We present an optimal control methodology, which we refer to as concentration-of-measure optimal control (COMOC), that seeks to minimize a concentration-of-measure upper bound on the probability of failure of a system. The systems under consideration are characterized by a single performance measure that depends on random inputs through a known response function. For these systems, concentration-of-measure upper bound on the probability of failure of a system can be formulated in terms of the mean performance measure and a system diameter that measures the uncertainty in the operation of the system. COMOC then seeks to determine the optimal controls that maximize the confidence in the safe operation of the system, defined as the ratio of the design margin, which is measured by the difference between the mean performance and the design threshold, to the system uncertainty, which is measured by the system diameter. This strategy has been assessed in the case of a robot-arm maneuver for which the performance measure of interest is assumed to be the placement accuracy of the arm tip. The ability of COMOC to significantly increase the design confidence in that particular example of application is demonstrated.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Relation of redundant forces fn−1+, fn− at tn to piecewise constant discrete generalized forces τn−1, τn

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Figure 2

Configuration of a rigid body (a) and initial configuration of the robot arm consisting of two rigid bodies combined into a spherical pair by the joint S1 and fixed in space by the spherical joint S2 (b)

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Figure 3

Final configuration of the robot arm showing the director triads {dIα}(α=1,2, I=1,2,3) (a) and the joint location vectors ρβα(β=1,2) (b)

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Figure 4

Evolution of the kinetic, potential, and total energies (a) and components of the angular momentum (b). The bottom plot on the right shows that the momentum maps are represented consistently.

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Figure 5

Evolution of the torques (a) and trajectory of the tip (b)

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Figure 6

Uncertain geometry: simulated-annealing iteration for the determination of the optimal controls; evolution of the (a) mean performance, (b) system diameter, and (c) concentration-of-measure probability of failure upper bound

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Figure 7

Uncertain wind forces and uncertain geometry: simulated-annealing iteration for the determination of the optimal controls; evolution of the (a) mean performance, (b) system diameter, and (c) concentration-of-measure probability of failure upper bound

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