## 3 Results and Discussions
Measuring arms are characterized as portable devices, which is why they are equipped with additional battery power built into the base. The counterweight system used in the construction of the device allows for easy maneuvering with the device, including performing measurements with one hand [13; 24]. Unlike measuring machines equipped with impulse measuring heads, the arm has a rigid head. Performing a measurement in this case requires the operator to exercise carefulness and skill during the testing process [12]. Setting up the measuring arm for operation requires mounting it on a measuring plate and calibrating the measuring probe. Due to the rigid measuring probe, it is necessary to securely mount the element onto the measuring plate. The next step involves defining the coordinate system on the element being tested. In the measurements, an auxiliary grid was also used to determine the position of the point being examined. The results, in the form of XYZ coordinates in Figure 5c, were exported as text files. Characteristic parameters:  
* Assembly and disassembly of the measuring arm: 6 min;
* Time required to prepare the measurement setup: 6 min;
* Measurement time of the plate: 7 min;
* Total research time amounted to 19 min. The compiled results in Figure 5c show significant differences in the \(Z\)-axis between adjacent points. There are clear shape errors of the investigated plate visible. Values for the \(Z\)-axis range from \(0.00\)\(\upmu\)m to \(60\)\(\upmu\)m. The characteristic parameters for the three measurement methods are shown in Table 1.  
Measurements of manual and automatic machines largely overlap both in shape and in error values. The chart obtained from the results of the measuring arm shows smaller values for the \(Z\)-axis as well as significant variation for adjacent points. To determine the flatness deviation in coordinate measuring machines (CMMs), we use the least squares method of deviations. In this method, we determine a reference plane from the collected points, and then we define the distances between it and the individual measurement points. This distance is the deviation for the partial area of the tested surface. Points located above the surface are called positive locations, and those below are called negative locations. The surface flatness error is taken as the sum of the absolute values of the positive and negative locations [25]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Activities**} & **Automatic Measuring** & **Manual Measuring** & **Measuring Arm** \\ \cline{2-4}  & **Machine** & **Machine** & **Measuring Arm** \\ \cline{2-4}  & & **Time t, [min]** & \\ \hline Assembly and disassembly the device & - & - & 6 \\ Time needed to prepare the device & 4 & 6 & 6 \\ Time to write the measurement program & 16 & - & - \\ Measurement time of the plate & 12 & 13.5 & 7 \\ \hline Total research time amounted & 32 & 19.5 & 19 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The characteristic parameters for the three measurement methods [own elaboration]. To determine the average plane from an n-element set of real points, it is necessary to establish the formula defining this plane. Here, we can use the formula for the plane given below (1):  
\[z=ax+by+c \tag{1}\]  
In the least squares method, defining the plane requires calculating the parameters \(a\), \(b\), and \(c\). To obtain them, it is necessary to perform calculations of three equations, which in matrix form are expressed by Formula (2) given below:  
\[ \begin{bmatrix}\sum_{i=1}^{n}x_{i}^{2}&\sum_{i=1}^{n}x_{i}y_{i}&\sum_{i=1}^{n }x_{i}\\ \sum_{i=1}^{n}x_{i}y_{i}&\sum_{i=1}^{n}y_{i}^{2}&\sum_{i=1}^{n}y_{i}\\ \sum_{i=1}^{n}x_{i}&\sum_{i=1}^{n}y_{i}&n\end{bmatrix}\begin{bmatrix}a\\ b\\ c\end{bmatrix} = \begin{bmatrix}\sum_{i=1}^{n}x_{i}z_{i}\\ \sum_{i=1}^{n}y_{i}z_{i}\\ \sum_{i=1}^{n}z_{i}\end{bmatrix}  \tag{2}\]  
Through this, we obtain an equation with which we can describe the plane.