# Error Averaging Effect in Parallel Mechanism Coordinate Measuring Machine
## 2 Measuring Principle of 3-PUU PCMM  
The structure of the developed PCMM is shown in Figure 1. Three sliders are connected to the moving platform through universal joints and three pairs of link rods. The sliders share one horizontal linear guider and can move separately while driven by steel belts and stepper motors, respectively. The linear grating ruler is fastened to one side of the linear guide. Three reading heads are installed with respect to each slider. The moving platform is connected to the end of each link rod through universal joints, and the moving platform translates in the space parallel to the three sliders without rotation. The trigger probe assembled on the moving platform triggers three reading heads to collect the instantaneous positions synchronously, when the probe touches workpiece [5; 6]. The coordinate of the probe can then be calculated based on a measuring model [7; 8]. As shown in Figure 1, three sliders are coplanar, thus, the centers of six universal joints on three sliders also share one common plane, which is plane \(OCDB\) in Figure 2. The angle between the \(OCDB\) plane and the horizontal plane of \(OHPB\) is \(45^{\circ}\). \(B_{1}\), \(B_{2}\) are centers of two joints on the left slider; \(B_{3}\)  
Figure 1: 3-PPU (three pairs of prismatic-universal-universal joints) PCMM (parallel coordinate measuring machine). (1) Base, (2) table, (3) steel belt, (4) reducer, (5) motor, (6) linear grating, (7) reading head, (8) beam, (9) linear guider, (10) rod, (11) universal joints, (12) slider, (13) moving platform, (14) measuring head, and (15) limit switch. (**a**) front view; (**b**) upward view. \(B_{4}\) are centers of the middle slider; and \(B_{5}\) and \(B_{6}\) are centers of the right slider. A fixed coordinate system \(O\)-\(XYZ\) is established based on the plane \(OCDB\), and every center of universal joints \(B_{i}\) can be expressed as: \(B_{1}(x_{1},\,0,\,0)\), \(B_{2}(x_{1},\,c,\,0)\), \(B_{3}(x_{2},\,b+c,\,0)\), \(B_{4}(x_{2}+a,\,b+c,\,0)\), \(B_{5}(x_{3},\,c,\,0)\), \(B_{6}(x_{3},\,0,\,0)\). Here, \(x_{1}\), \(x_{2}\), \(x_{3}\) indicate the positions of three reading heads, respectively, and \(a\), \(b\), \(c\) are position parameters between joints on three sliders. The moving platform is in the form of an uneven six-sided polygon on which \(e_{i}\) is the center of universal joint \(i\) on the moving platform. Let point \(e_{1}\) be the unknown position in the coordinate system of \(O\)-\(XYZ\) and set it to \((x,y,\,z)\). Thus, \(e_{2}\)-\(e_{6}\) coordinates can be determined with position parameters \((e,f,g)\) between joints. According to the spatial two-point formula, a set of equations expressing all link lengths can be established as below [9]. \[\left\{ \begin{array}{l}(x-x_{1})^{2}+y^{2}+z^{2}={l_{1}}^{2}\\ (x-x_{1})^{2}+{(y+g-c)}^{2}+z^{2}={l_{2}}^{2}\\ (x+e-{x_{2}})^{2}+{(y+g+f-b-c)}^{2}+z^{2}={l_{3}}^{2}\\ (x+d+e-a-{x_{2}})^{2}+{(y+g+f-b-c)}^{2}+z^{2}={l_{4}}^{2}\\ (x+d+2e-{x_{3}})^{2}+{(y+g-c)}^{2}+z^{2}={l_{5}}^{2}\\ (x+d+2e-{x_{3}})^{2}+y^{2}+z^{2}={l_{6}}^{2}\end{array} \right. \tag{1}\]  
The group consists of six equations and only three unknowns. If two rods have the same length on each slider (i.e., \(l_{1}=l_{2}\), \(l_{3}=l_{4}\), \(l_{5}=l_{6}\)), the unknown position of point \(e_{1}\) can be solved as:  
\[\left\{ \begin{array}{l}x=\frac{x_{1}+x_{3}-2e-d}{2}+\frac{l_{2}^{2}-l_{1}^{ 2}}{2e+d-{x_{3}}+{x_{1}}}\\ y=\frac{l_{3}^{2}-l_{1}^{2}+(x-x_{1})^{2}-(x+e-{x_{2}})^{2}}{2(f-b)}+\frac{b-f} {2}\\ z=\sqrt{l_{1}^{2}-(x-x_{1})^{2}-y^{2}}\end{array} \right. \tag{2}\]  
The probing position in space can be obtained from point \(e_{1}\) with a transformation matrix containing sliders' positions \((x_{1},\,x_{2},\,x_{3})\) and structural parameters of link \(l_{i}\) (\(i=1\)-\(6\)), joints positions (\(a\), \(b\), \(c\), \(d\), \(e\),\(f\), \(g\)), and the stylus length, if the probe-mounting position on the moving platform is known.