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Technical Briefs

# Type Synthesis of Underactuated Wrists Generated From Fully-Parallel Wrists

[+] Author and Article Information
Raffaele Di Gregorio

Department of Engineering,
University of Ferrara,
Via Saragat,1,
44100 Ferrara, Italy
e-mail: rdigregorio@ing.unife.it

In a manipulator, the workspace dimensionality is the minimum number of geometric parameters necessary to locate the end effector in the operational space. If the manipulator is not redundant, such a number will coincide with the configuration (or finite) degrees-of-freedom (dof) [1] of the manipulator, which is the minimum number of geometric parameters necessary to uniquely identify the manipulator configuration [2]. It may be different from the instantaneous dof, also called velocity dof [1], of the same manipulator.

An nS pair could be obtained by simply introducing a roller-sphere contact into a passive S pair [5].

P and U stand for prismatic pair and universal joint, respectively; and the underscore denotes the actuated pair.

The nonactuated SPU limb has connectivity six. It is worth noting that a non-actuated limb with connectivity six does not constrain the relative motion between the platform and the base. It only affects the workspace borders [5]; hence, it can be eliminated without changing the configuration dof of the manipulator.

The instantaneous input-output relationships of wrists relate the actuated-joint rates to the rates of their platform’s orientation parameters.

In this case, the IPA simply consists of applying the Euclidean distance formula to each SPU limb to calculate the distance between the S-pair center, whose coordinates are geometric constants of the base, and the U-joint center, whose coordinates are known because the platform pose is assigned.

Hereafter, (P, a) will denote the oriented line passing through the point P and with the direction of the unit vector a. Moreover, the rotation angles are meant counterclockwise with respect to the oriented line given as rotation axis.

Here, for the sake of simplicity, the constructive scheme of the nS pair is assumed to be the one of Fig. 13. Simple formulas which replace formula (5) can be deduced for other nS constructive schemes.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received August 19, 2011; final manuscript received August 4, 2012; published online October 19, 2012. Assoc. Editor: James Schmiedeler.

J. Mech. Des 134(12), 124501 (Oct 19, 2012) (7 pages) doi:10.1115/1.4007399 History: Received August 19, 2011; Revised August 04, 2012

## Abstract

Design criteria and constructive schemes are presented for a nonholonomic spherical (nS) pair proposed previously, along with a reconfigurable version of the same pair, named nS pair. Type synthesis of the underactuated parallel wrists derived from the fully parallel wrist (FPW) topology is addressed by replacing passive spherical (S) pairs with nS or nS pairs. Ten novel topologies of underactuated spherical wrists with practically the same workspace as the original FPW are identified. Wrist architectures based on these novel underactuated-wrist topologies are proved to be globally controllable, and a general path planning algorithm is proposed for these wrists. Here, the substitution of holonomic constraints with nonholonomic ones reduces the number of actuators and, sometimes, the number of links, too.

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## References

Angeles, J., 2003, “Fundamentals of Robotic Mechanical Systems,” Springer-Verlag, New York, NY.
O’ReillyO. M., 2008, “Intermediate Dynamics for Engineers,” Cambridge University Press, New York, NY.
StammersC. W., PrestP. H., and MobleyC. G., 1991, “The Development of a Versatile Friction Drive Robot Wrist,” 8th World Congress on the Theory of Machines and Mechanisms, pp. 499–502, Prague, Czechoslovakia, Aug. 26–31.
StammersC. W., PrestP. H., and MobleyC. G., 1992, “A Friction Drive Robot Wrist: Electronic and Control Requirements,” Mechatronics, 2(4), pp. 391–401.
GroschP., Di GregorioR., and ThomasF., 2010, “Generation of Underactuated Manipulators With Nonholonomic Joints From Ordinary Manipulators,” ASME J. Mech. Rob., 2(1), p. 011005.
BiegL. F., and BenavidesG. A., 2001, “Double Slotted Socket Spherical Joint,” U. S. Patent No. US 6,234,703 B1, May 22.
BiegL. F., and BenavidesG. A., 2002, “Large Displacement Spherical Joint,” U.S. Patent No. US 6,409,413 B1, June 25.
InnocentiC., and Parenti-CastelliV., 1993, “Echelon Form Solution of Direct Kinematics for the General Fully-Parallel Spherical Wrist,” Mech. Mach. Theory, 28(4), pp. 553–561.
Choset, H., Lynch, K. M., Hutchinson, S., Kantor, G., Burgard, W., Kavraki, L. E., and Thrun, S., 2005, Principles of Robot Motion: Theory, Algorithms, and Implementations, MIT Press, Boston.
Murray, R. M., Li, Z., and Sastry, S. S., 1994, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL.
Di Gregorio, R., 2012, “Kinematic Analysis of the (nS)-2SPU Underactuated Parallel Wrist,” ASME J. Mech. Rob., 4(3), p. 031006.
Di Gregorio, R., 2012, “Position Analysis and Path Planning of the S-(nS)PU-SPU and S-(nS)PU-2SPU Underactuated Wrists,” ASME J. Mech. Rob., 4(2), p. 021006.
StammersC. W., 1993, “Operation of a Two-Motor Robot Wrist to Achieve Three-Dimensional Manoeuvres With Minimum Total Rotation,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci., 207(1), pp. 33–39.
Sancisi, N., Zannoli, D., Parenti-Castelli, V., Belvedere, C., Leardini, A., 2011, “A One-Degree-of-Freedom Spherical Mechanism for Human Knee Joint Modeling,” Proc. Inst. Mech. Eng., Part H: J. Eng. Med., 225(8), pp. 725–735.
Jackovitch, T., 2010, “Flexion Control Ankle Joint With Spherical Hinge,” U.S. Patent No. US 7,753,866, July 13.

## Figures

Fig. 1

Roller-sphere contact and nS pair. (S and R stand for spherical and revolute pairs, respectively).

Fig. 2

Symmetric nS pair: top view and cross section

Fig. 3

Asymmetric nS pair: cross section

Fig. 4

Geometric parameter γ to be considered in the path planning

Fig. 5

Fully-parallel wrist (S-3SPU architecture)

Fig. 13

nS: the actuated revolute pair, R, directly controls the orientation of the AP

Fig. 14

Notation: σ and C are the AP and the center of the nS pair, respectively; (C, n) and θ are the axis and the angle of the finite link rotation to carry out

Fig. 15

Isostatic structures: (a) of type (nS)-2(nS)PU and (b) of type S-3(nS)PU

Fig. 16

Hyperstatic structure of type (nS)-3(nS)PU

## Errata

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