Two-scale command shaping for arresting motion in nonlinear systems
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ORIGINAL PAPER
Two-scale command shaping for arresting motion in nonlinear systems Alexander Alyukov . Michael J. Leamy
Received: 27 March 2020 / Accepted: 27 August 2020 Ó Springer Nature B.V. 2020
Abstract This paper presents a feedforward technique for arresting motion in nonlinear systems based on two-scale command shaping (TSCS). The advantages of the proposed technique arise from its feedforward nature and ease of implementation in linear and nonlinear systems. Using the TSCS strategy, the control input required to arrest motion is decomposed into two scales—the first arrests dynamics associated with the linear subproblem, while the second eliminates response from the nonlinearities. Using direct numerical integration, the method is assessed using a traditional Duffing system and multi-degree-of-freedom nonlinear systems. Experiments are conducted on a compound pendulum attached to a servomotor, documenting effective arrest of the system in close agreement with theoretical predictions.
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11071-020-05923-w) contains supplementary material, which is available to authorized users. A. Alyukov M. J. Leamy (&) George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA e-mail: [email protected] A. Alyukov Automotive Engineering Department, South Ural State University, Chelyabinsk, Russia 454080
Keywords Nonzero initial conditions Motion arrest Nonlinear systems Perturbation methods Input shaping Two-scale command shaping
1 Introduction Compared to feedback controllers, feedforward controllers typically have decreased cost, minimal sensor requirements, and faster control response as they act directly on the input, not on the error signal [1, 2]. They are particularly attractive in control applications in which systems must be smoothly brought to motion from rest. Smith [3, 4] made the first attempts at reducing residual vibrations arising in linear systems following a control command. The proposed method, termed ‘posicast control,’ divides the reference command signal into two or three separate excitations, each spaced by one-half or one-eighth, respectively, of the natural period of the system. Singer, Seering, and Singhose [5–7] generalized posicast control, terming their approach ‘input shaping.’ Zero vibration (ZV) input shaping ensures that the residual vibration amplitude of the second-order system reaches zero in the time-optimal sense, and reaches the same set point as the reference command. As the ZV input shaper’s efficiency may suffer due to parametric uncertainty, various modifications have been proposed, including zero vibration and derivative (ZVD), and extra-
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insensitive (EI), input shapers. However, these approaches increase the impulse sequence duration to the full damped period in comparison with the half period for the ZV input shaper. Since its inception
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