Invariant-Based Inverse Engineering of Crane Control Parameters

S. González-Resines, D. Guéry-Odelin, A. Tobalina, I. Lizuain, E. Torrontegui, and J. G. Muga

Phys. Rev. Applied 8, 054008 – Published 6 November 2017

Abstract

By applying invariant-based inverse engineering in the small-oscillation regime, we design the time dependence of the control parameters of an overhead crane (trolley displacement and rope length) to transport a load between two positions at different heights with minimal final-energy excitation for a microcanonical ensemble of initial conditions. The analogy between ion transport in multisegmented traps or neutral-atom transport in moving optical lattices and load manipulation by cranes opens a route for a useful transfer of techniques among very different fields.

Received 19 May 2017

DOI:https://doi.org/10.1103/PhysRevApplied.8.054008

Physics Subject Headings (PhySH)

Classical mechanics Control & applications of chaos

Atom & ion trapping & guiding

General PhysicsAtomic, Molecular & Optical

Authors & Affiliations

S. González-Resines¹, D. Guéry-Odelin², A. Tobalina¹, I. Lizuain³, E. Torrontegui⁴, and J. G. Muga^1,*

¹Departamento de Química Física, UPV/EHU, Apartado 644, 48080 Bilbao, Spain
²Laboratoire de Collisions Agrégats Réactivité, CNRS UMR 5589, IRSAMC, Université de Toulouse (UPS), 118 Route de Narbonne, 31062 Toulouse CEDEX 4, France
³Department of Applied Mathematics, University of the Basque Country UPV/EHU, Plaza Europa 1, 20018 Donostia-San Sebastian, Spain
⁴Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain

^*Corresponding author. jg.muga@ehu.eus

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Issue

Vol. 8, Iss. 5 — November 2017

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Images

Figure 1
Length of the rope with respect to time for a hoisting from $l_{0} = 10 m$ to $l_{f} = 5 m$ . $t_{f} = 15 s$ , red solid line; $t_{f} = 10 s$ , black dashed-dotted line; and $t_{f} = 5 s$ , blue dashed line.
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Figure 2
Position of the trolley $x (t)$ with respect to time for a dual operation with $d = 15$ , $l_{0} = 10 m$ , and $l_{f} = 5 m$ . $t_{f} = 15 s$ , red solid line; $t_{f} = 10 s$ , black dashed-dotted line; and $t_{f} = 5 s$ , blue dashed line.
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Figure 3
Angle of the payload with respect to time for a dual operation with a translation of 15 m and a hoisting from $l_{0} = 10 m$ to $l_{f} = 5 m$ ). The exact dynamics are $t_{f} = 15 s$ , red solid line; $t_{f} = 10 s$ , black dashed-dotted line; and $t_{f} = 5 s$ , blue dotted line. The approximate dynamics (harmonic approximation) are $t_{f} = 5 s$ , green dashed line; for the two other times, the exact and approximate curves are undistinguishable in the scale of the figure. The payload is initially set at equilibrium.
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Figure 4
Maximal angle of the payload in the final configuration versus the initial angle for a payload initially at rest, $\dot{θ} (0) = 0$ . For the same line type, the (red) curve with a larger slope corresponds to transport with hoisting ( $l_{0} = 10 m$ , $l_{f} = 5 m$ , and $d = 15 m$ ) and the other (blue) curve to transport with lowering ( $l_{0} = 5 m$ , $l_{f} = 10 m$ , and $d = 15 m$ ). The solid lines are the (exact) adiabatic lines. $t_{f} = 10 s$ , dotted lines; $t_{f} = 7 s$ , dashed-dotted lines; and $t_{f} = 5 s$ , dashed lines.
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Figure 5
Final-to-initial-energy ratio for pure transport (no hoisting involved) versus the initial angle for a load initially at rest. Here, the initial energy is equal to the final adiabatic energy, as the cable length does not change. Protocols are as follows: (i) the three-step protocol in Ref. [65] in the harmonic approximation (the black dotted-dashed line) and for the exact dynamics (the green dashed line), (ii) an invariant-based protocol (the red dotted line) with a polynomial Ansatz (15) for $α$ (the blue solid line in the harmonic approximation, the red dotted line for the exact dynamics), and (iii) an invariant-based protocol with three more parameters in the polynomial adjusted to flatten the excitation curve as in Ref. [64] (the blue solid line, indistinguishable from 1 in the scale of the figure throughout the entire angle interval). The parameters are taken from Ref. [65]: $l = 1.2 m$ and $x_{f} = 4 m$ . The upper bounds to set the protocol (i) for trolley acceleration, trolley velocity, and payload angle are (the notation is the one in [65]): $a_{ub} = 0.5 m / s^{2}$ , $v_{ub} = 1 m / s$ , $θ_{ub} = 5 °$ . These parameters imply a total time $t_{f} = 6.45 s$ with $a_{\max} = 0.4276 m / s^{2}$ , $T = 2.1987 s$ , and $t_{c} = 2.0567 s$ . All protocols are adjusted for the same final total time.
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