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DISCUSSION

Disscussion: “Kinematics of the Translational 3-URC Mechanism” [, 2004, ASME J. Mech. Des., 126, pp. 1113–1117] OPEN ACCESS

[+] Author and Article Information
Xianwen Kong

Département de Génie Mécanique, Université Laval, Québec, Québec, Canada G1K 7P4

Clément M. Gosselin1

Département de Génie Mécanique, Université Laval, Québec, Québec, Canada G1K 7P4gosselin@gmc.ulaval.ca

From Eqs. 5,6,8,9,10 in this paper or Eq. (11b) in (1), it is learned that there are usually two solutions for si for a given position of the moving platform.

1

Corresponding author.

J. Mech. Des 128(4), 812-813 (Jan 19, 2006) (2 pages) doi:10.1115/1.2205875 History: Received April 29, 2005; Revised January 19, 2006
FIGURES IN THIS ARTICLE

The author of (1) proposed a 3-URC translational parallel mechanism (TPM) and presented a comprehensive study on the kinematics of the 3-URC TPM. He concluded that “only one solution exists both for the direct and for the inverse position analyses.” However, we do not agree with his result on the inverse position analysis and his statement that Ref. 2 “presented a class of TPMs with linear input-output equations that contain some translational 3-URC mechanisms.”

In this discussion, we will show that the 3-URC TPM does not belong to the class of TPMs with linear input-output equations (2-6) by investigating the inverse position analysis of the 3-URC TPM.

Leg i of a 3-URC TPM is shown in Fig. 1. In addition to the notations used in (1), hi is used to denote a unit vector directed from Ai to Ci. For leg i, the following holds: w1ivi=0, w1iw2i=0, and w2ihi=0.

In the coordinate system OXYZ, we haveDisplay Formula

Ai+hihidiw2i+siw1i=Bi0
(1)
where
w2i=cosθ1ivi+sinθ1iw1i×vi
w2i×w1i=cosθ1iw1i×vi+sinθ1ivi
hi=cosθ2iw1i+sinθ2iw2i×w1i
Bi0=P+Rbp(Bi0P)p

The inverse position analysis of the 3-URC TPM can be performed by solving Eq. 1 for θ1i, θ2i, and si in sequence.

Taking the inner product of Eq. 1 with w2i, we obtainDisplay Formula

w2i(AiBi0)di=0
(2)
i.e.,Display Formula
(Bi0Ai)(w1i×vi)sinθ1i+(Bi0Ai)vicosθ1i+di=0
(3)
Define an angle αi by
cosαi=[(Bi0Ai)(w1i×vi)]ai
sinαi=[(Bi0Ai)vi]ai
where ai=[(Bi0Ai)vi]2+[(Bi0Ai)(w1i×vi)]2.

Equation 3 can be rewritten asDisplay Formula

sin(θ1i+αi)=diai
(5)
Solving sin2(θ1i+αi)+cos2(θ1i+αi)=1, we obtain two solutions for cos(θ1i+αi) asDisplay Formula
cos(θ1i+αi)=±[1sin2(θ1i+αi)]12
(6)
Equations 5,6 show that there are two solutions for (θ1i+αi). For each (θ1i+αi), one solution for θ1i can be obtained as
sinθ1i=sin(θ1i+αi)cosαicos(θ1i+αi)sinαi
cosθ1i=cos(θ1i+αi)cosαi+sin(θ1i+αi)sinαi

From Eqs. 4,5,6,7, we learn that there are usually two solutions for θ1i.

For each θ1i obtained using Eq. 7, sinθ2i can be obtained by taking the inner product for Eq. 1 with w2i×w1i as

sinθ2i={sinθ1i[(Bi0Ai)vi]cosθ1i[(Bi0Ai)(w1i×vi)]}hi
Substituting Eq. 4 into the above equation, we obtainDisplay Formula
sinθ2i=aicos(θ1i+αi)hi
(8)
Solving sin2θ2i+cos2θ2i=1, we obtain two solutions for cosθ2i asDisplay Formula
cosθ2i=±(1sin2θ2i)12
(9)

Equations 8,9 show that there exist two solutions for θ2i for a given θ1i.

Once θ1i and θ2i have been determined, si can be obtained by taking the inner product of Eq. 1 with w1i asDisplay Formula

si=[(Bi0Ai)w1i]hicosθ2i
(10)

The above analysis shows that for a given position of the moving platform, there are usually two solutions (Eqs. 4,5,6,7) for the input θ1i for each leg i and four sets of solutions (Eqs. 4,5,6,7,8,9,10) for the joint variables in each leg i. Thus, for a given position of the moving platform, there are usually eight (=23) sets of solutions for the inputs θ11, θ12, and θ13 and 64 (=43) sets of solutions for all the joints variables in the 3-URC TPM.

In summary, it has been shown that for a given position of the moving platform, there are usually two solutions for each input and eight sets of solutions for all the inputs in the 3-URC TPM. Thus, we have proved that the 3-URC TPM does not belong to the class of TPMs with linear input-output equations (2-6). In fact, it belongs to the class of linear TPMs, whose forward displacement analysis can be performed by solving a set of linear equations, dealt with systematically in (5-6). The work reported in (5-6) is an extension of the work reported in (3-4).

Copyright © 2006 by American Society of Mechanical Engineers
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