Closure to “Discussion of ‘Kinematics
of the Translational 3-URC
Mechanism’ ” (2006, ASME J. Mech.
Des., 128, pp. 812–813)
Raffaele Di Gregorio
Department of Engineering,
University of Ferrara,
Via Saragat, 1,
44100 Ferrara, Italy
e-mail: rdigregorio@ing.unife.it
DOI: 10.1115/1.2205876
Kong and Gosselin 1, with reference to what this author wrote
in 2, disagree on the following two points: i the inverse posi-
tion analysis IPA of the translational 3-URC has only one solu-
tion, and ii the translational 3-URC belongs to the class of trans-
lational parallel mechanisms TPMs with linear input-output
equations presented in 3.
Point (i): Figure 1 of 1 shows the ith leg of type URC of a
translational 3-URC. With reference to the notations shown in that
figure, the inverse position analysis consists of calculating the
values of the angle
1i
i =1,2,3 compatible with an assigned
position of the axis of the cylindrical pair note that the cylindrical
pair axis passes through the platform point B
i0
and is parallel to
the unit vector, w
1i
, of the axis of the revolute pair that is embed-
ded in the base, whereas the unit vectors w
2i
and w
3i
of the axes
of the two intermediate revolute pairs are parallel to each other
and perpendicular to w
1i
. It can be shown through a simple geo-
metric reasoning that the ith translational URC leg can be as-
sembled in four different configurations assembly modes once
the position of the cylindrical pair axis is assigned. Such configu-
rations can be divided into two groups each of which is composed
of two configurations that are symmetric with respect to the plane,
the cylindrical pair axis and point A
i
belong to plane
i
of Fig. 1,
and correspond to only two values of the angle
1i
the values
1
1i
and
2
1i
shown in Fig. 1. Even though the axis of the
cylindrical pair keeps itself parallel to w
1i
during the platform
translation, suitable platform translations that bring the ith leg into
any configuration out of the four assembly modes, without dis-
mounting and reassembling the leg, exist. Therefore, both the val-
ues of
1i
that correspond to the four assembly modes are a solu-
tion of the IPA i.e., Kong and Gosselin are right, and the
formulas 16 and 17a, and 17breported in 2 must be
changed as follows:
w
2i
= ± sin
i
w
1i
B
i0
- A
i
w
1i
B
i0
- A
i
+ cos
i
w
1i
w
1i
B
i0
- A
i
w
1i
B
i0
- A
i
16
cos
1i
= v
i
· w
2i
17a
sin
1i
= w
1i
v
i
· w
2i
17b
where the angle
i
is shown in Fig. 1 it can be easily computed
through the relationships: cos
i
= d
i
/ g
i
and sin
i
= 1
- d
i
/ g
i
2
1/2
with g
i
= B
i0
- A
i
- B
i0
- A
i
· w
1i
w
1i
.
Point (ii): Since the IPA of the translational 3-URC has two
solutions per leg i.e., 8 solutions, the translational 3-URC pro-
posed in 2 does not belong to the class of TPMs presented in 3,
and Ref. 2 must be correct on this point as Kong and Gosselin
observed.
This author wishes to thank X. Kong and C. M. Gosselin for
having given him the opportunity to discuss and correct his work.
References
1 Kong, X., and Gosselin, C. M., 2006, “Discussion: ‘Kinematics of the Trans-
lational 3-URC Mechanism’,” ASME J. Mech. Des., 128, pp. 812–813.
2 Di Gregorio, R., 2004, “Kinematics of the Translational 3-URC Mechanism,”
ASME J. Mech. Des., 1266, pp. 1113–1117.
3 Kong, X., and Gosselin, C. M., 2002, “A Class of 3-DOF Translational Parallel
Manipulators With Linear Input-Output Equations,” Proceedings of the Work-
shop on Fundamental Issues and Future Research Directions for Parallel
Mechanisms and Manipulators, Québec City, Québec, Canada, October 3–4,
pp. 25–32.
Contributed by the Mechanisms and Robotics Committee of ASME for publica-
tion in the JOURNAL OF MECHANICAL DESIGN. Manuscript received May 15, 2005; final
manuscript received September 10, 2005. Review conducted by Q. Jeffrey Ge.
Fig. 1 Assembly modes of the ith translational URC leg pro-
jected onto a plane perpendicular to w
1i
; each of the projection
„„1… or „2…… corresponds to two leg configurations with the same
value of
1i
and different values of s
i
814 / Vol. 128, JULY 2006 Copyright © 2006 by ASME Transactions of the ASME
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