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RESEARCH PAPERS

# Identification of the Workspace Boundary Of a General 3-R Manipulator

[+] Author and Article Information
Erika Ottaviano

LARM Laboratory of Robotics and Mechatronics, DiMSAT, University of Cassino, Via Di Biasio 43-03043 Cassino (Fr), Italyottaviano@unicas.it

Manfred Husty

Institute for Engineering Mathematics Geometry and Computer Science, University of Innsbruck, Technikerstr. 13 A-6020 Innsbruck, Austriamanfred.husty@uibk.ac.at

Marco Ceccarelli

LARM Laboratory of Robotics and Mechatronics, DiMSAT, University of Cassino, Via Di Biasio 43-03043 Cassino (Fr), Italyceccarelli@unicas.it

J. Mech. Des 128(1), 236-242 (Sep 05, 2005) (7 pages) doi:10.1115/1.2120807 History: Received December 30, 2004; Revised September 05, 2005

## Abstract

In this paper an algebraic formulation is presented for the boundary workspace of 3-R manipulators in Cartesian Space. It is shown that the cross-section boundary curve can be described by a 16th order polynomial as function of radial and axial reaches. The cross-section boundary curve and workspace boundary surface are fully cyclic. Geometric singularities of the curve are identified and characterized. Numerical examples are presented to show the usefulness of the proposed investigation and to classify the design characteristics.

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## Figures

Figure 1

Kinematic scheme of a 3R manipulator

Figure 2

A descriptive proof of the 16th degree of the cross-section boundary

Figure 3

The generating manifold for the workspace with characteristic points: (a) A general case; (b) a cuspidal manipulator; (c) manipulator with double points but no void

Figure 4

Manifolds for inner branches of the workspace boundary envelope of 3R manipulators

Figure 5

A numerical example for a 3R manipulator with a1=1u; a2=1u; a3=1u; d2=3u; d3=5u; α1=π∕4; α2=π∕4: (a) Cross-section boundary curve f and curves of the one-parameter family; (b) f, fr, fz, fw plots. (u is the unit length, and angles are expressed in radians)

Figure 9

A numerical example for a 3R manipulator with a1=1u; a2=0.5u; a3=0.2u; d2=0.5u; d3=0.5; α1=π∕3; α2=π∕6: (a) Cross-section boundary curve f and curves of the one-parameter family; (b) f, fr, fz, fw plots. (u is the unit length, angles are expressed in radians)

Figure 10

A numerical example for a 3R manipulator with a1=1u; a2=3u; a3=4u; d2=3u; d3=0; α1=−π∕2; α2=π∕2: (a) Cross-section boundary curve f and curves of the one-parameter family; (b) f, fr, fz, fw plots. (u is the unit length, angles are expressed in radians)

Figure 11

A numerical example for a 3R manipulator with a1=1u; a2=1u; a3=1u; d2=1u; d3=2; α1=π∕4; α2=π∕4: (a) Cross-section boundary curve f and curves of the one-parameter family; (b) f, fr, fz, fw plots. (u is the unit length, angles are expressed in radians)

Figure 12

A numerical example for a 3R manipulator with a1=1u; a2=2u; a3=6u; d2=1u; d3=0; α1=−π∕2; α2=π∕2: (a) Cross-section boundary curve f and curves of the one-parameter family; (b) f, fr, fz, fw plots. (u is the unit length, angles are expressed in radians)

Figure 13

A numerical example for Mitsubishi MRP-700A manipulator with a1=250; a2=1000; a3=1209; d2=0; d3=0; α1=π∕2; α2=0: (a) Cross-section boundary curve f and curves of the one-parameter family; (b) f, fr, fz, fw plots. (lengths are expressed in mm, angles are expressed in radians)

Figure 6

A numerical example for a 3R manipulator with a1=1u; a2=1u; a3=1u; d2=3u; d3=2u; α1=π∕4; α2=π∕4: (a) Cross-section boundary curve f and curves of the one-parameter family; (b) f, fr, fz, fw plots. (u is the unit length, and angles are expressed in radians)

Figure 7

A numerical example for a 3R manipulator with a1=1u; a2=0.5u; a3=0.8u; d2=0.2u; d3=0; α1=−π∕2; α2=π∕2: (a) Cross-section boundary curve f and curves of the one-parameter family; (b) f, fr, fz, fw plots. (u is the unit length, angles are expressed in radians)

Figure 8

A numerical example for a 3R manipulator with a1=3u; a2=1u; a3=3u; d2=1u; d3=1u; α1=−π∕6; α2=π∕3: (a) Cross-section boundary curve f and curves of the one-parameter family; (b) f, fr, fz, fw plots. (u is the unit length, angles are expressed in radians)

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