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: $w1i\u2219vi=0$, $w1i\u2219w2i=0$, and $w2i\u2219hi=0$.

The inverse position analysis of the 3-URC TPM can be performed by solving Eq. 1 for $\theta 1i$, $\theta 2i$, and $si$ in sequence.

## Solution for θ1i

## Solution for θ2i

## Solution for si

## 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 $\theta 1i$ 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 $\theta 11$, $\theta 12$, and $\theta 13$ 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).