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Research Papers

# Isotropic Compliance in E(3): Feasibility and Workspace Mapping

[+] Author and Article Information
Matteo Verotti

Department of Mechanics
and Aerospace Engineering,
Sapienza University of Rome,
Rome 00149, Italy
e-mail: matteo.verotti@uniroma1.it

Nicola P. Belfiore

Department of Mechanics
and Aerospace Engineering,
Sapienza University of Rome,
Rome 00149, Italy
e-mail: belfiore@dima.uniroma1.it

1Corresponding author.

Manuscript received September 9, 2015; final manuscript received December 15, 2015; published online September 6, 2016. Assoc. Editor: Federico Thomas.

J. Mechanisms Robotics 8(6), 061005 (Sep 06, 2016) (9 pages) Paper No: JMR-15-1245; doi: 10.1115/1.4032408 History: Received September 09, 2015; Revised December 15, 2015

## Abstract

A manipulator control system, for which isotropic compliance holds in the Euclidean space $E(3)$, can be significantly simplified by means of diagonal decoupling. However, such simplification may introduce some limits to the region of the workspace where the sought property can be achieved. The present investigation reveals how to detect which peculiar subset, among four different classes, a given manipulator belongs to. The paper also introduces the concept of control gain ratio for each specific single-input/single-output joint control law in order to limit the maximum gain required to achieve the isotropic compliance condition.

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Topics: Manipulators

## Figures

Fig. 1

Isotropic compliance condition

Fig. 2

Nonisotropic compliance condition

Fig. 3

Three-link planar manipulator in three different configurations

Fig. 4

Flowchart representing the algorithm for the evaluation of the control gain ratios

Fig. 5

The workspace defined for θ1=0 and θ2,θ3∈[0,π] and the isotropic compliance subsets for the manipulator MA: in the dark zone there are no solutions, while, in the subsets in which solutions exist, the value of μ is proportional to the gradient bar

Fig. 6

The workspace defined for θ1=0 and θ2,θ3∈[0,π] and the isotropic compliance subsets for the manipulator MA: the dark zones are no-solution zones, while in the white zones the solutions do not verify condition (5). In the subsets verifying condition (5), the value of μc is proportional to the gradient bar.

Fig. 7

The workspace defined for θ1=0 and θ2,θ3∈[0,π] and the isotropic compliance subsets for the manipulator MB: in the dark zone there are no solutions, while, in the subsets in which solutions exist, the value of μ is proportional to the gradient bar

Fig. 8

The workspace defined for θ1=0 and θ2,θ3∈[0,π] and the isotropic compliance for the manipulator MB: the dark zones are no-solution zones, while in the white zones the solutions do not verify condition (5). In the subsets verifying condition (5), the value of μc is proportional to the gradient bar.

Fig. 9

Anthropomorphic RRR: semispherical surface subset in which isotropic compliance is verified, where the value of μ is proportional to the gradient bar

Fig. 10

Anthropomorphic RRR: in the white zones the solutions do not verify condition (5), while, in the subset verifying condition (5), the value of μc is proportional to the gradient bar

Fig. 11

RRP spatial manipulator: isotropic compliance subset, with a constant value of the control gain ratio μ

Fig. 12

RPP cylindrical manipulator: orthogonal cross section of the isotropic compliance subset, where the value of μ is proportional to the gradient bar

Fig. 13

RPP cylindrical manipulator: in the white zones the solutions do not verify condition (5), while, in the subset verifying condition (5), the value of μc is proportional to the gradient bar

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