Closure to “Discussion of ‘Kinematics of the Translational 3-URC Mechanism’ ” (2006, ASME J. Mech. Des., 128, pp. 812–813)

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 $θ1i$$(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 $Bi0$ and is parallel to the unit vector, $w1i$⁠, of the axis of the revolute pair that is embedded in the base, whereas the unit vectors $w2i$ and $w3i$ of the axes of the two intermediate revolute pairs are parallel to each other and perpendicular to $w1i$⁠). It can be shown through a simple geometric reasoning that the $ith$ translational URC leg can be assembled in four different configurations (assembly modes) once the position of the cylindrical pair axis is assigned. Such configurations 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 $Ai$ belong to (plane $σi$ of Fig. 1), and correspond to only two values of the angle $θ1i$ (the values $θ1i(1)$ and $θ1i(2)$ shown in Fig. 1). Even though the axis of the cylindrical pair keeps itself parallel to $w1i$ during the platform translation, suitable platform translations that bring the $ith$ leg into any configuration out of the four assembly modes, without dismounting and reassembling the leg, exist. Therefore, both the values of $θ1i$ that correspond to the four assembly modes are a solution of the IPA (i.e., Kong and Gosselin are right), and the formulas 16,17a, and 17breported in (2) must be changed as follows:where the angle $αi$ is shown in Fig. 1 (it can be easily computed through the relationships: $cosαi=di∕gi$ and $sinαi=[1−(di∕gi)2]1∕2$ with $gi=∥(Bi0−Ai)−[(Bi0−Ai)∙w1i]w1i∥$⁠).