Research Papers

Two Natural Dexterity Indices for Parallel Manipulators: Angularity and Axiality

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

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

J. M. Rico-Martínez

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

V. H. Pérez-Muñoz

Department of Mechanical Engineering,
Universidad de Guanajuato,
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

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.

Copyright © 2014 by ASME
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Gosselin, C. M., 1990, “Dexterity Indices for Planar and Spatial Robotic Manipulators,” IEEE International Conference on Robotics and Automation, Cincinnati, OH, May 13–18, pp. 650–655. [CrossRef]
Pittens, K. H., and Podhorodeski, R. P., 1993, “A Family of Stewart Platforms With Optimal Dexterity,” J. Rob. Syst., 10(4), pp. 463–479. [CrossRef]
Bhattacharya, S., Hatwal, H., and Gosh, A., 1995, “On the Optimum Design of Stewart Platform Type Parallel Manipulators,” Robotica, 13(1), pp. 133–140. [CrossRef]
Zanganeh, K. E., and Angeles, J., 1997, “Kinematic Isotropy and the Optimum Design of Parallel Manipulators,” Int. J. Robot. Res., 16(2), pp. 185–197. [CrossRef]
Tsai, K. Y., and Huang, K. D., 1998, “The Manipulability and Transmissivity of Manipulators,” Int. J. Rob. Autom., 13(4), pp. 132–136.
Merlet, J. P., 2006, “Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots,” ASME J. Mech. Des., 128(1), pp. 199–206. [CrossRef]
Pond, G., and Carretero, J. A., 2007, “Quantitative Dexterous Workspace Comparison of Parallel Manipulators,” Mech. Mach. Theory, 42(10), pp. 1388–1400. [CrossRef]
Kim, S. G., and Ryu, J., 2003, “New Dimensionally Nonhomogeneous Jacobian Matrix Formulation by Three End-Effector Points for Optimal Design of Parallel Manipulators,” IEEE Trans. Rob. Autom., 19(4), pp. 731–737. [CrossRef]
Kong, M., Zhang, Y., Du, Z., and Sun, L., 2007, “A Novel Approach to Deriving the Unit-Homogeneous Jacobian Matrices of Mechanisms”, IEEE International Conference on Mechatronics and Automation, Harbin, China, August 5–8, pp. 3051–3055. [CrossRef]
Cardou, P., Bouchard, S., and Gosselin, C., 2010, “Kinematic-Sensitivity Indices for Dimensionally Nonhomogeneous Jacobian Matrices,” IEEE Trans. Rob., 26(1), pp. 166–173. [CrossRef]
Liu, H., Huang, T., and Chetwynd, D. G., 2011, “A Method to Formulate a Dimensionally Homogeneous Jacobian of Parallel Manipulators,” IEEE Trans. Rob., 27(1), pp. 150–156. [CrossRef]
Klein, C. A., and Miklos, T. A., 1991, “Spatial Robotic Isotropy,” Int. J. Robot. Res., 10(4), pp. 426–437. [CrossRef]
Mansouri, I., and Ouali, M., 2010, “The Power Manipulability—A New Homogeneous Performance Index of Robot Manipulators,” Rob. Comput. Integr. Manuf., 27(2), pp. 434–449. [CrossRef]
Fenton, R. G., and Willgoss, R. A., 1990, “Comparison of Methods for Determining Screw Parameters of Infinitesimal Rigid Body Motion From Position and Velocity Data,” ASME J. Dyn. Syst., Meas., Control, 112(4), pp. 711–716. [CrossRef]
Angeles, J., 2007, Fundamentals of Robotic Mechanical Systems: Theory, Methods and Algorithms, Springer, New York, p. 324.
Angeles, J., 1982, Spatial Kinematic Chains: Analysis, Synthesis, Optimization, Springer-Verlag, New York, pp. 130, 134–135.
Brand, L., 1947, Vector and Tensor Analysis, John Wiley & Sons, New York, p. 34.
Schwartz, M., Green, S., and Rutledge, W. A., 1960, Vector Analysis With Applications to Geometry and Physics, Harper & Row, New York, p. 17.
Phillips, J., 1984, Freedom in Machinery. Vol. 1: Introducing Screw Theory, Cambridge University Press, Cambridge, UK, pp. 69–72.
Angeles, J., 1989, Rational Kinematics, Springer-Verlag, New York, pp. 49–50.
Ginsberg, J., 2008, Engineering Dynamics, Cambridge University Press, New York, pp. 175–176.
WordReference.com, 2013, Online Language Dictionaries, http://www.wordreference.com/definition/angularity
Huston, R. L., 1990, Multibody Dynamics, Butterworth-Heinemann, Boston, MA, pp. 17–18.
Davidson, J. K., and Hunt, K. H., 2004, Robots and Screw Theory: Applications of Kinematics and Statics to Robotics, Oxford University Press, New York, pp. 28–29.
Hunt, K. H., 1990, Kinematic Geometry of Mechanisms, Oxford University Press, New York, pp. 268–269.
Kane, T. R., Likins, P. W., and Levinson, D. A., 1983, Spacecraft Dynamics, McGraw-Hill, New York, pp. 422–431.
Carretero, J. A., Podhorodeski, R. P., Nahon, M. A., and Gosselin, C. M., 2000, “Kinematic Analysis and Optimization of a New Three Degree-of-Freedom Spatial Parallel Manipulator,” ASME J. Mech. Des., 122(1), pp. 17–24. [CrossRef]
Ghosal, A., and Ravani, B., 2001, “A Differential-Geometric Analysis of Singularities of Point Trajectories of Serial and Parallel Manipulators,” ASME J. Mech. Des., 123(1), pp. 80–89. [CrossRef]
Nakamura, Y., 1991, Advanced Robotics: Redundancy and Optimization, Addison-Wesley Publishing Company, pp. 17.
Strang, G., 1988, Linear Algebra and Its Applications, Harcourt Brace & Company, Fort Worth, TX, pp. 50–331.
Gibbs, J. W., 1960, Vector Analysis, Dover Publications, Inc., New York, p. 76.


Grahic Jump Location
Fig. 1

Three noncollinear points on a moving rigid body

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Fig. 2

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

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Fig. 3

The angular velocity vector and its kinematic generators

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Fig. 4

The helicoidal velocity field

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Fig. 5

Solid model of the 3-PRS parallel manipulator

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Fig. 6

Auxiliary kinematic diagram of the 3-PRS parallel manipulator

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Fig. 7

Geometry associated with the 3-PRS parallel manipulator

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Fig. 8

Additional details of the manipulator's legs

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Fig. 9

Workspace of the 3-PRS parallel manipulator

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Fig. 10

Angularity index for the maximum velocities of the attachment points

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Fig. 11

Dexterous rotational workspace for 0.80 < η < 1.0

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Fig. 12

Axiality index over the global workspace

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Fig. 13

Dexterous translational workspace for 0.80 < σ < 1.0

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Fig. 15

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

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Fig. 16

Geometry associated with distance δP



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