# Development of a Coordinate Measuring Machine--Based Inspection Planning System for Industry 4.0
## 4 Inspection Planning Model
### Initial Measuring Path Generating  
The initial MP generating starts from the mathematical model presented in [27,28]. The model is adapted to the needs of the I4.0 concept that is basically related to the aspect of automatic generation of the MP. Primary benefit of the mathematical model is to create connections between CSs (Figure 3a) such as probe (O\({}_{\text{W}}\)X\({}_{\text{W}}\)Y\({}_{\text{W}}\)Z\({}_{\text{W}}\)), features (O\({}_{\text{F}}\)X\({}_{\text{F}}\)Y\({}_{\text{F}}\)Z\({}_{\text{F}}\)), and CMM (O\({}_{\text{M}}\)X\({}_{\text{M}}\)Y\({}_{\text{M}}\)Z\({}_{\text{M}}\)), as well as to generate an initial MP for each IF. According to Figure 3a, the basic Equation (1) of the model is:  
\[{}^{\text{M}}\mathbf{{}_{P_{i}}}={}^{\text{M}}\mathbf{{}_{W}}+{}^{\text{W}} \mathbf{{}_{F_{i}}}+{}^{\text{F}}\mathbf{{}_{P_{i}}}={}^{\text{M}}\mathbf{{}_ {F_{i}}}+{}^{\text{F}}\mathbf{{}_{P_{i}}} \tag{1}\]  
Figure 3: Mathematical model: (**a**) a PW and its tolerances, (**b**) initial MP for truncated sphere, (**c**) collision free path, (**d**) initial MP for plane. where: \({}^{\rm W}\)\({}_{\rm F}\)--GF system's position vector in the PW CS, \({}^{\rm F}\)\({}_{\rm F}\)\({}_{\rm I}\)--probe point's position vector in the GF CS, \({}^{\rm M}\)\({}_{\rm F}\)\({}_{\rm W}\)--PW system's position vector in the CMM CS, \({}^{\rm M}\)\({}_{\rm F}\)--GF (plane) system's position vector in the CMM CS. As aforementioned, the role of a mathematical model is automatic distribution of measuring points per IF and generation of initial MP. Two sets of points generating of the initial path: (i) the measuring points and (ii) nodal points. Sampling strategy of those points for different GFs that make up a single IF is obtained by modifying Hamersley sequences presented in [28]. An example of equations of measuring points coordinates \({\rm P}_{\rm i}({\rm s}_{\rm i},{\rm t}_{\rm i},{\rm w}_{\rm i})\) in _Cartesian_ CS for a plane and truncated hemisphere, according to [27; 28], is given as follows:  
Plane:  
\[{\rm s}_{\rm i}=\frac{{\rm i}}{{\rm N}}\cdot{\rm a} \tag{2}\]  
\[{\rm t}_{\rm i}=\left(\sum_{j=0}^{\rm k-1}\left(\left[\frac{{\rm i}}{2}\right] {\rm Mod2}\right)\cdot 2^{-(j+1)}\right)\cdot{\rm b} \tag{3}\]  
\[{\rm w}_{\rm i}=0 \tag{4}\]  
where: \({\rm a}[{\rm mm}]\)--distance per axis \(x\); \({\rm b}[{\rm mm}]\)--distance per axis \(y\)  
Truncated hemisphere:  
\[{\rm s}_{\rm i}=\sqrt{{\rm R}^{2}-\left(\left(\sum_{j=0}^{\rm k-1}\left(\left[ \frac{{\rm i}}{2}\right]{\rm Mod2}\right)\cdot 2^{-(j+1)}\right)\cdot{\rm R }\right)^{2}}\cdot\cos\left(\frac{{\rm i}}{{\rm N}}\cdot 360^{0}\right) \tag{5}\]  
\[{\rm t}_{\rm i}=\sqrt{{\rm R}^{2}-\left(\left(\sum_{j=0}^{\rm k-1}\left(\left[ \frac{{\rm i}}{2}\right]{\rm Mod2}\right)\cdot 2^{-(j+1)}\right)\cdot{\rm R }\right)^{2}}\cdot\sin\left(\frac{{\rm i}}{{\rm N}}\cdot 360^{0}\right) \tag{6}\]  
\[{\rm w}_{\rm i}=\left(\sum_{j=0}^{\rm k-1}\left(\left[\frac{{\rm i}}{2}\right] {\rm Mod2}\right)\cdot 2^{-(j+1)}\right)\cdot({\rm R}-{\rm h}_{1}) \tag{7}\]  
where \({\rm R}[{\rm mm}]\) is the radius and \({\rm h}_{1}[{\rm mm}]\) is the height of a truncated hemisphere. The set of nodal points implies two sub-sets \({\rm P}_{\rm i1}({\rm s}_{\rm i1},{\rm t}_{\rm i1},{\rm w}_{\rm i1})\) and \({\rm P}_{\rm i2}({\rm s}_{\rm i2},{\rm t}_{\rm i2},{\rm w}_{\rm i2})\), where is \({\rm i}=0,1,2,\ldots,({\rm N}-1)\) and \({\rm N}\)--number of sampling points. The points \({\rm P}_{\rm i1}({\rm x}_{\rm i1},{\rm y}_{\rm i1},{\rm z}_{\rm i1})\) presents points for the transition from fast to slow feed. The distance between points \({\rm P}_{\rm i1}({\rm x}_{\rm i1},{\rm y}_{\rm i1},{\rm z}_{\rm i1})\) and \({\rm P}_{\rm i}({\rm x}_{\rm i},{\rm y}_{\rm i},{\rm z}_{\rm i})\) is presented (Figure 3d) by \({\rm d}_{1}\)--slow feed probe path, and the distance between points \({\rm P}_{\rm i2}({\rm x}_{\rm i2},{\rm y}_{\rm i2},{\rm z}_{\rm i2})\) and \({\rm P}_{\rm i1}({\rm x}_{\rm i1},{\rm y}_{\rm i1},{\rm z}_{\rm i1})\) is \({\rm d}_{2}\)--rapid feed probe path.