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Research Papers

# Kinematics for an Actuated Flexible n-Manifold

[+] Author and Article Information
Oded Medina

Department of Mechanical Engineering,
Ariel University,
Ariel 4076113, Israel
e-mail: odedmedina@gmail.com

Amir Shapiro

Department of Mechanical Engineering,
Ben-Gurion University,
Be'er Sheva 8410501, Israel
e-mail: ashapiro@bgu.ac.il

Nir Shvalb

Department of Industrial Engineering,
Ariel University,
Ariel 4076113, Israel
e-mail: nirsh@ariel.ac.il

Manuscript received February 8, 2015; final manuscript received July 27, 2015; published online November 24, 2015. Assoc. Editor: Jun Ueda.

J. Mechanisms Robotics 8(2), 021009 (Nov 24, 2015) (8 pages) Paper No: JMR-15-1028; doi: 10.1115/1.4031301 History: Received February 08, 2015; Revised July 27, 2015; Accepted August 08, 2015

## Abstract

Recent years show an increasing interest in flexible robots due to their adaptability merits. This paper introduces a novel set of hyper-redundant flexible robots which we call actuated flexible manifold (AFM). The AFM is a two-dimensional hyper-redundant grid surface embedded in $ℝ2$ or $ℝ3$. Theoretically, such a mechanism can be manipulated into any continuous smooth function. We introduce the mathematical framework for the kinematics of an AFM. We prove that for a nonsingular configuration, the number of degrees of freedom (DOF) of an AFM is simply the number of its grid segments. We also show that for a planar rectangular grid, every nonsingular configuration that is also energetically stable is isolated. We show how to calculate the forward and inverse kinematics for such a mechanism. Our analysis is also applicable for three-dimensional hyper-redundant structures as well. Finally, we demonstrate our solution on some actuated flexible grid-shaped surfaces.

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## Figures

Fig. 1

An 8×8×2 AFM grid in its initial configuration

Fig. 2

Crawling on the constraints. (A) Crawling on a cylindrical constraint defined by a single segment length by projecting the desired direction v on n's null space, k, where n=∇f. Motion proceeds in the general direction of v′. (B) represents a configuration in which two segments lengths are to be met.

Fig. 3

A portion of the AFM grid which constitutes a double four-bar mechanism

Fig. 4

Relaxation procedure: reducing edge 7–8 in a planar AFM with no relaxation (a) and with relaxation (b), and (c) simultaneously reducing two edges in a three-dimensional AFM

Fig. 5

Relaxation procedure: (a) the upper and lower grids are taken to be identical, sampling the desire function, and keeping the upper and the lower grids at a given distance and (b) the relaxed mechanism with its (encircled) angles slightly changed

Fig. 6

An 8 × 8 grid and its configuration change due to three external forces

Fig. 7

Design III of an AFM. The SMA actuator is initially curved and aligns when actuated. Such a design enhances the ability of the segment's ends to change their mutual distance.

Fig. 8

Mechanical experimentations with design III compared with the simulation results. The root mean square given in link length units (of nodes positions in both the simulation and the real experimentation) was found to be 0.24.

Fig. 9

Design II of an AFM. A 24DOF manifold attains three configurations.

Fig. 10

The SMA beam model with 7DOF actuating two corner actuators

Fig. 11

The 12DOF manifold using linear servo motors

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