# Calibration of a Hybrid Machine Tool from the Point of View of Positioning Accuracy
## 2 Materials and Methods  
### Kinematic Structure of the O-X Glide Mechanism  
The kinematic structure of the O-X glide mechanism is based on a planar parallel mechanism that moves translationally in the direction of one serial axis, which ensures the formation of a three-dimensional working space. The planar parallel mechanism is ingeniously constructed with a dual kinematic structure. This innovative design realizes two different arrangements of the supporting elements, forming \"O\" and \"X\" configurations with different shapes and dimensions of working space. The dual structure of the mechanism results in two kinematic configurations with different stiffness. The essential characteristics of mechanisms with parallel kinematics are represented by a mathematically relatively simple solution of inverse and complex solutions of direct kinematics. The planar parallel mechanism, which is the basis of the O-X glide mechanism due to its plane structure and simple conception, has analytically solvable kinematic solutions in both geometric configurations (extended \"O\" and crossed \"X\" configuration). Figure 1a shows the arrangement of vectors on the kinematic model of the parallel mechanism in extended configuration; Figure 1b shows this in crossed configuration. Figure 1 shows the crossed (Figure 1a) and extended (Figure 1b) configurations of the realized version of the planar O-X glide mechanism. The vectors that determine the position and orientation of the mechanism elements are shown in the X-Z plane, as performed on the machine prototype. The following symbols were used to denote vectors [21]:  
* P--position of the moving platform expressed in external coordinates of the mechanism (x\({}_{\text{P}}\), x\({}_{\text{P}}\));
* r\({}_{\text{P1}}\) and r\({}_{\text{P2}}\)--vectors of moving platform, which defines the orientation of platform and distance between the end of rods and position of point P (P\({}_{1}\) and P\({}_{2}\));
* r\({}_{\text{R1}}\), r\({}_{\text{R2}}\)--vectors of the positions of reference points of the particular sliders (R\({}_{1}\) and R\({}_{2}\));
* r\({}_{\text{u1}}\), r\({}_{\text{u2}}\)--vectors of the slide of the parallel mechanism in relation to the axis reference point, internal coordinates of the mechanism;
* l\({}_{1}\), l\({}_{2}\)--vectors of rods of constant length;
* d--distance between sliders. Based on Figure 1, the vector equations of the vector polygon can be set. These equations express the position vector of the moving platform as a function of the internal coordinates of the individual sliders. \[\overset{\rightarrow}{r_{p}}=\overset{\rightarrow}{r_{ref1}}+\overset{ \rightarrow}{r_{u1}}+\overset{\rightarrow}{r_{l1}}+\overset{\rightarrow}{r_{p1 }}=\overset{\rightarrow}{r_{ref1}}+\overset{\rightarrow}{r_{u1}}+\overset{ \rightarrow}{r_{l1}}+\overset{\rightarrow}{r_{p1}} \tag{1}\]  
Expressed in coordinates:  
\[\left\{\frac{x_{p}}{z_{p}}\right\}=\left\{\frac{x_{ref1}}{d}\right\}+\left\{ \frac{u_{1}}{0}\right\}+\left\{\frac{l_{1x}}{l_{1z}}\right\}+\left\{\frac{-p_{1 }}{0}\right\}=\left\{\frac{x_{ref2}}{d}\right\}+\left\{\frac{u_{2}}{0}\right\} +\left\{\frac{l_{2x}}{l_{2z}}\right\}+\left\{\frac{-p_{2}}{0}\right\} \tag{2}\]  
They obtain unique solutions by converting the components of vector l from the previous expressions and including them in the expressions that express the length of the rods through vector components d.  
\[{l_{1}}^{2}={l_{1}}^{2}+{l_{1}}^{2} \tag{3}\] \[{l_{2}}^{2}={l_{2}}^{2}+{l_{2}}^{2}\]  
Based on the previous expressions, the equations of inverse kinematics can be described in the mechanism's sprung (O) and crossed (X) kinematic configuration. For the sprung configuration of the mechanism \"O\":  
Figure 1: Kinematic model of the O-X glide parallel mechanism. \[ \begin{split} u_{1}=x_{p}-x_{R1}+p_{1}+\sqrt{l_{1}^{2}-\left(z_{p}-d \right)^{2}}\\ u_{2}=-x_{p}+x_{R2}+p_{2}+\sqrt{l_{2}^{2}-\left(z_{p}+d\right)^{2}}\\ s_{3}=y_{p}\end{split}  \tag{4}\]  
For cross mechanism configuration \"X\":  
\[