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Research Papers

Embedding Desired Eigenstates into Active and Passive Dynamics of a Linear, Underactuated Feedback System

[+] Author and Article Information
Frank Saunders

Mechanical Engineering Department,
Tufts University,
Medford, MA 02155

Jason Rife

Mechanical Engineering Department,
Tufts University,
Medford, MA 02155

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received January 8, 2013; final manuscript received August 21, 2013; published online April 28, 2014. Assoc. Editor: Michael Kokkolaras.

J. Mech. Des 136(7), 071005 (Apr 28, 2014) (6 pages) Paper No: MD-13-1009; doi: 10.1115/1.4025295 History: Received January 08, 2013; Revised August 21, 2013

This paper introduces a novel methodology to embed desired reference trajectories into the modal dynamics of an underactuated system through eigenstructure assignment. A unique characteristic of the method is that it decomposes the control input into two parts: an open loop, periodic excitation signal, and a closed loop feedback signal. The periodic excitation causes the system’s natural modes to resonate in a fashion that matches the desired trajectory; modal dynamics, determined by the system’s eigenstates (eigenvectors and their corresponding eigenvalues), are shaped by tuning physical and control parameters concurrently. The method requires the solution of a dual-domain eigenstate factorization problem, in which it is necessary to compute certain unknown elements of a matrix and of its eigenvectors at the same time.

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Figures

Grahic Jump Location
Fig. 1

Mass spring damper configuration with tunable passive dynamical parameters (mass mi, linear damper ci, and linear spring ki) and feedback-controlled forces acting on the center of mass of mass one and three (ui). For this model, it is assumed that contact between blocks does not occur.

Grahic Jump Location
Fig. 2

Relative displacements of the masses with respect to time (normalized by 0.1 m); given the specified eigenstates for one period

Grahic Jump Location
Fig. 3

DDEF response over many periods, converging toward periodic steady state

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