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Research Papers

# Optimal-Regularity for Serial Redundant Robots

[+] Author and Article Information
Nir Shvalb

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

Tal Grinshpoun

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

Oded Medina

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

Manuscript received May 15, 2016; final manuscript received December 15, 2016; published online March 24, 2017. Assoc. Editor: Andreas Mueller.

J. Mechanisms Robotics 9(3), 031015 (Mar 24, 2017) (5 pages) Paper No: JMR-16-1144; doi: 10.1115/1.4035532 History: Received May 15, 2016; Revised December 15, 2016

## Abstract

A configuration of a mechanical linkage is defined as regular if there exists a subset of actuators with their corresponding Jacobian columns spans the gripper's velocity space. All other configurations are defined in the literature as singular configurations. Consider mechanisms with grippers' velocity space $ℝm$. We focus our attention on the case where m Jacobian columns of such mechanism span $ℝm$, while all the rest are linearly dependent. These are obviously an undesirable configuration, although formally they are defined as regular. We define an optimal-regular configuration as such that any subset of m actuators spans an m-dimensional velocity space. Since this densely constraints the work space, a more relaxed definition is needed. We therefore introduce the notion of k-singularity of a redundant mechanism which means that rigidifying k actuators will result in an optimal-regularity. We introduce an efficient algorithm to detect a k-singularity, give some examples for cases where m = 2, 3, and demonstrate our algorithm efficiency.

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## Figures

Fig. 1

Ignoring the obstacle, the depicted configuration is regular. Taking the obstacle into account will result with a singular configuration, since joints 2,3,…,7 are linear dependent.

Fig. 2

A three-singular configuration: rigidifying two joints of{3, 5, 8} and one of {1, 6} will result in an optimal-regular configuration

Fig. 3

An optimal-regular configuration

Fig. 4

A two-singular configuration: dropping joint 4 and one of the joints 3 or 5 will result in an optimal-regular configuration

Fig. 6

The calculation time needed when searching for a six vector linear dependencies of redundant serial robots with degrees-of-freedom ranging from 8 to 30. The dashed line presents the naïve approach. The vertical error bars indicate the 1.5 standard deviations of 20experiments.

Fig. 5

The bipartite graph where four vectors in J are perpendicular to two vectors in A

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