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Research Papers

# Two Natural Dexterity Indices for Parallel Manipulators: Angularity and Axiality

[+] Author and Article Information
J. Jesús Cervantes-Sánchez

Professor
Department of Mechanical Engineering,
Salamanca,
Guanajuato 36885, Mexico
e-mail: jecer@ugto.mx

J. M. Rico-Martínez

Professor
Department of Mechanical Engineering,
Salamanca,
Guanajuato 36885, Mexico
e-mail: jrico@ugto.mx

V. H. Pérez-Muñoz

Department of Mechanical Engineering,
Salamanca,
Guanajuato 36885, Mexico
e-mail: vperez@ugto.mx

It should be noted that for a proper definition of the instantaneous motion of any rigid body in a three-dimensional space, the specification of the position and velocity of any three noncollinear points pertaining to the body under study is sufficient [14,15].

While the angular velocity vector $ω$ is a first-order property related to the rotational motion of a rigid body, the axial sliding velocity [19], denoted by $v‖$, is a first-order property associated with the translational motion of the body. The axial sliding velocity is obtained by projecting the velocity vector of any point of the moving rigid body onto the corresponding ISA. Moreover, the ISA is the locus of all points of the body moving with one and the same velocity vector $v‖$, which is of minimum Euclidean norm [20].

An attachment point is usually located at the physical center of the joint that connects the terminal link of a leg with the mobile platform.

It should be noted that according to definition of vector product, the sign of sin α is always positive, since angle α ranges between 0 deg and 180 deg.

A typical point will be referred to as any point on the body, except for those lying on the ISA.

An interesting geometric construction of equations (44)(46) is illustrated in Ref. [19].

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received May 31, 2013; final manuscript received February 13, 2014; published online June 5, 2014. Assoc. Editor: Philippe Wenger.

J. Mechanisms Robotics 6(4), 041007 (Jun 05, 2014) (13 pages) Paper No: JMR-13-1104; doi: 10.1115/1.4027236 History: Received May 31, 2013; Revised February 13, 2014

## Abstract

This paper introduces two novel dexterity indices, namely, angularity and axiality, which are used to estimate the motion sensitivity of the mobile platform of a parallel manipulator undergoing a general motion involving translation and rotation. On the one hand, the angularity index can be used to measure the sensitivity of the mobile platform to change in rotation. On the other hand, the axiality index can be used to measure the sensitivity of the operation point (OP) of the mobile platform to change in translation. Since both indices were inspired by very fundamental concepts of classical kinematics (angular velocity vector and helicoidal velocity field), they offer a clear and simple physical insight, which is expected to be meaningful to the designer of parallel manipulators. Moreover, the proposed indices do not require obtaining a dimensionally homogeneous Jacobian matrix, nor do they depend on having similar types of actuators in each manipulator's leg. The details of the methodology are illustrated by considering a classical parallel manipulator.

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## References

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## Figures

Fig. 1

Three noncollinear points on a moving rigid body

Fig. 2

Location of kinematic generators v21 and v31: (a) in a velocity polygon and (b) on a moving rigid body

Fig. 3

The angular velocity vector and its kinematic generators

Fig. 4

The helicoidal velocity field

Fig. 5

Solid model of the 3-PRS parallel manipulator

Fig. 6

Auxiliary kinematic diagram of the 3-PRS parallel manipulator

Fig. 7

Geometry associated with the 3-PRS parallel manipulator

Fig. 8

Additional details of the manipulator's legs

Fig. 9

Workspace of the 3-PRS parallel manipulator

Fig. 10

Angularity index for the maximum velocities of the attachment points

Fig. 11

Dexterous rotational workspace for 0.80 < η < 1.0

Fig. 12

Axiality index over the global workspace

Fig. 13

Dexterous translational workspace for 0.80 < σ < 1.0

Fig. 16

Geometry associated with distance δP

Fig. 15

Perpendicular kinematic generators and velocity vectors of points 1, 2, and 3

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