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Research Papers

Path Following and Shape Morphing With a Continuous Slender Mechanism

[+] Author and Article Information
Javad S. Fattahi

Department of Mechanical Engineering,
University of Ottawa,
Ottawa, ON K1N 6N5, Canada
e-mail: sfatt032@uottawa.ca

Davide Spinello

Mem. ASME
Department of Mechanical Engineering,
University of Ottawa,
Ottawa, ON K1N 6N5, Canada
e-mail: dspinell@uottawa.ca

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 26, 2014; final manuscript received June 10, 2015; published online July 21, 2015. Assoc. Editor: Jwu-Sheng Hu.

J. Dyn. Sys., Meas., Control 137(10), 101007 (Jul 21, 2015) Paper No: DS-14-1188; doi: 10.1115/1.4030816 History: Received April 26, 2014

We present the continuous model of a mobile slender mechanism that is intended to be the structure of an autonomous hyper-redundant slender robotic system. Rigid body degrees-of-freedom (DOF) and deformability are coupled through a Lagrangian weak formulation that includes control inputs to achieve forward locomotion and shape tracking. The forward locomotion and the shape tracking are associated to the coupling with a substrate that models a generic environment with which the mechanism could interact. The assumption of small deformations around rigid body placements allows to adopt the floating reference kinematic description. By posing the distributed parameter control problem in weak form, we naturally introduce an approximate solution technique based on Galerkin projection on the linear mode shapes of the Timoshenko beam model that is adopted to describe the body of the system. Simulation results illustrate coupling among forward motion and shape tracking as described by the equations governing the system.

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Figures

Grahic Jump Location
Fig. 1

Kinematics of the system, with schematics of a rigid body placement and a small deformation (shape morphing). The time scales separation between deformation and rigid body placement translates into the floating frame assumption.

Grahic Jump Location
Fig. 2

Sketch of the coupling between the flexible mechanism and a rigid substrate described by the curve η. The coupling is exerted through a distributed system of compliant elements. The point η(s*(t)) is driven by the kinematics s*(t).

Grahic Jump Location
Fig. 3

Block diagram of the passivity based control system

Grahic Jump Location
Fig. 4

Sketch of the geometry of the path defining the rigid substrate

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Fig. 5

Head tracking errors |e1·η(s*(t))-d1(t)| (solid line) and |e2·η(s*(t))-d2(t)| (dashed line) for : (a) α3=20 and s*(t)=0.03t and (b) α3=20 and s*(t)=0.005t

Grahic Jump Location
Fig. 6

(a) Orientation tracking errors for (a) α3=20 and s*(t)=0.03t and (b) α3=20 and s*(t)=0.005t

Grahic Jump Location
Fig. 7

For α3=20 and s*(t)=0.03t, (a) four snapshots of the system and (b) snapshot around s = 10, with substrate represented by continuous line

Grahic Jump Location
Fig. 10

For α3=0.5, snapshot of the deformed shape (dashed line) with a near portion of the rigid substrate represented by the continuous line

Grahic Jump Location
Fig. 8

Head tracking errors |e1·η(s*(t))-d1(t)| (solid line) and |e2·η(s*(t))-d2(t)| (dashed line) for s*(t)=0.03t, (a) α3=20 and (b) α3=0.5. Results are computed for 0≤s≤15, but they are plotted only for 0≤s≤1.5 in order to zoom on the variation of the tracking error.

Grahic Jump Location
Fig. 9

Orientation tracking error for α3=0.5 and s*(t)=0.03t

Grahic Jump Location
Fig. 11

(a) L2 error norm, ||wn⋆-η·e2||L2 with number of basis functions n=1,3,5,7 and (b) L2 error norms ‖wn⋆-wn-2⋆‖L2 for n = 1, 3, 5

Grahic Jump Location
Fig. 12

Snapshot of the system for α3=20 and s*(t)=0.03t at s = 10: (a) deflection field approximated with one basis function and (b) deflection field approximated with seven basis functions

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