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 (2,3,4,5,6) 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 have 
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.

Solution for θ1i

Taking the inner product of Eq. 1 with w2i, we obtain 
w2i(AiBi0)di=0
2
i.e., 
(Bi0Ai)(w1i×vi)sinθ1i+(Bi0Ai)vicosθ1i+di=0
3
Define an angle αi by
4
 
cosαi=[(Bi0Ai)(w1i×vi)]ai
 
sinαi=[(Bi0Ai)vi]ai
where ai=[(Bi0Ai)vi]2+[(Bi0Ai)(w1i×vi)]2.
Equation 3 can be rewritten as 
sin(θ1i+αi)=diai
5
Solving sin2(θ1i+αi)+cos2(θ1i+αi)=1, we obtain two solutions for cos(θ1i+αi) as 
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
7
 
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.

Solution for θ2i

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 obtain 
sinθ2i=aicos(θ1i+αi)hi
8
Solving sin2θ2i+cos2θ2i=1, we obtain two solutions for cosθ2i as 
cosθ2i=±(1sin2θ2i)12
9

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

Solution for si

Once θ1i and θ2i have been determined, si can be obtained by taking the inner product of Eq. 1 with w1i as 
si=[(Bi0Ai)w1i]hicosθ2i
10

Number of Solutions to the Inverse Displacement Analysis

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)2 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 (2,3,4,5,6). 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).

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.
Di Gregorio
,
R.
, 2004, “
Kinematics of the Translational 3-URC Mechanism
,”
ASME J. Mech. Des.
1050-0472,
126
(
6
), pp.
1113
1117
.
2.
Kong
,
X.
, and
Gosselin
,
C. M.
, 2002, “
A Class of 3-DOF Translational Parallel Manipulators With Linear Input-Output Equations
,”
Proceedings of the Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators
,
Québec, Canada
, October 3–4, pp.
25
32
.
3.
Gosselin
,
C. M.
, and
Kong
,
X.
, 2002, “
Cartesian Parallel Manipulators
,” International Patent No. WO 02/096605 A1.
4.
Gosselin
,
C. M.
, and
Kong
,
X.
, 2004, “
Cartesian Parallel Manipulators
,” U.S. Patent No. 6729202.
5.
Kong
,
X.
, and
Gosselin
,
C. M.
, 2002, “
Type Synthesis of Linear Translational Parallel Manipulators
,”
Advances in Robot Kinematics—Theory and Applications
,
J.
Lenarčič
and
F.
Thomas
, eds.,
Kluwer Academic
,
Dordrecht
, pp.
411
420
.
6.
Kong
,
X.
, and
Gosselin
,
C. M.
, 2004, “
Type Synthesis of 3-DOF Translational Parallel Manipulators Based on Screw Theory
,”
ASME J. Mech. Des.
1050-0472,
126
(
1
), pp.
83
92
.