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: $w1i∙vi=0$, $w1i∙w2i=0$, and $w2i∙hi=0$.

In the coordinate system $O−XYZ$, we have
$Ai+hihi−diw2i+siw1i=Bi0$
1
where
$w2i=cosθ1ivi+sinθ1iw1i×vi$

$w2i×w1i=−cosθ1iw1i×vi+sinθ1ivi$

$hi=cosθ2iw1i+sinθ2iw2i×w1i$

$Bi0=P+Rbp(Bi0−P)p$

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
$w2i∙(Ai−Bi0)−di=0$
2
i.e.,
$(Bi0−Ai)∙(w1i×vi)sinθ1i+(Bi0−Ai)∙vicosθ1i+di=0$
3
Define an angle $αi$ by
4

$cosαi=[(Bi0−Ai)∙(w1i×vi)]∕ai$

$sinαi=[(Bi0−Ai)∙vi]∕ai$
where $ai=[(Bi0−Ai)∙vi]2+[(Bi0−Ai)∙(w1i×vi)]2$.
Equation 3 can be rewritten as
$sin(θ1i+αi)=−di∕ai$
5
Solving $sin2(θ1i+αi)+cos2(θ1i+αi)=1$, we obtain two solutions for $cos(θ1i+αi)$ as
$cos(θ1i+αi)=±[1−sin2(θ1i+αi)]1∕2$
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

$sinθ1i=sin(θ1i+αi)cosαi−cos(θ1i+αi)sinαi$

$cosθ1i=cos(θ1i+αi)cosαi+sin(θ1i+αi)sinαi$

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
$sinθ2i={sinθ1i[(Bi0−Ai)∙vi]−cosθ1i[(Bi0−Ai)∙(w1i×vi)]}∕hi$
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
$cosθ2i=±(1−sin2θ2i)1∕2$
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
$si=[(Bi0−Ai)∙w1i]−hicosθ2i$
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.

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