Statistical Process Control Charts Applied to Rock Disintegration Quality Improvement

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In the presented research, classical control charts were applied to the technological process of rock disintegration by rotary drilling. The significance of the submitted research is that so far, no significant results of control charts were applied and presented to such an extent. Despite the fact that this is required by industrial mining practice. Important parameters such as revolutions, pressure force, and vibration signal of process acceleration were investigated to determine statistical stability to improve drill core quality.

Abstract

At present, ever higher demands are placed on the quality of products. The success of organizations in the global market depends mainly on measuring and evaluating their products quality. A set of measurable criteria usually determines product quality. There are many technological processes in the structure of a production organization that is statistically unstable. The norms of ISO class 9000 emphasize statistical process control, known as SPC (Statistical Process Control). They represent a methodology for eliminating the causes of instability of production or technological process. The paper deals with the application of control charts for the technological process of rock disintegration by rotary drilling. The measured values of the dynamic system drilling tool-rock in working mode are processed. The control charts are applied to the input (control) variables of the pressure force-F (N), revolutions-n (rpm), and the output measured variable of the vibration signal of the acceleration. The article constructs and presents the resulting important control charts for the technological process of rock disintegration by rotary drilling. It is essential that for the technological process of rock drilling, the variables that enter and exit the dynamic system must be statistically manageable. The stable state of the input technical parameters (revolutions and pressure force) of the drilling tool is essential from the technological, performance, and economic point of view. The stable state of the output parameters is of significant importance in preventing the emergency state, excessive wear of the drilling equipment and optimizing the optimal operating mode. Industrial practice points out that the correct application of statistical regulation stabilizes the technological process, increasing the quality and productivity of work.

1. Introduction

The development of deep drilling technology rose sharply in recent decades. The emergence of new principally progressive technologies meant a significant increase in the quality of drilling work and a significant increase in the drilling process’s speed. With the development of new deep drilling technologies, new drilling rigs and their various modifications for deep drilling is being developed. They differ according to the technologies applied in practice. The basic principle of drilling with diamond bits into the rock is mainly in the drilling tool’s penetration into the rock and its subsequent disintegration by cutting. Practically all available technologies are used in the implementation of drilling works, which allow the necessary tests and measurements to be performed during drilling to obtain the required geological or mining information. The technological process of rock disintegration by rotary drilling is the subject of research from a qualitative point of view. The primary goal of rotary core drilling is to obtain a cylindrical drilling core, the compactness of which is determined by the drillability, physical-mechanical properties of the rock, and the drilling technology used. The quality of the work is evaluated by the core’s yield, which is the ratio of the drilled length of the borehole to the length of the obtained core. Rotary core drilling with direct flushing is the most common method of exploratory drilling. This method of rock disintegration is the subject of our scientific research from the point of view of statistical regulation in conjunction with industrial practice [1,2].

Important mode parameters for rotary drilling with diamond bits (tools) are the speed of revolutions of the bit n (rpm) and the pressure force F (N). Determining the speed of revolutions as the first input parameter in the working mode of drilling requires an empirical approach. Increasing or adjusting the speed of revolutions increases the progress of the drilling process [3,4,5,6,7].

It is recommended not to exceed the set optimal working speed of revolutions. Subsequently, it is necessary to select the optimal level of pressure force on the drilling tool as the second input parameter of the process. The pressure force has a great influence not only on the mechanical drilling speed but also on maintaining the borehole’s required (projected) direction. Improperly determined and the high-pressure force, despite relatively good drilling procedures, results in rapid wear of the drilling tool, damage to the drilling shaft, and deflection of the borehole in an undesired direction. When working with low-pressure force, undesirable conditions also arise, especially low drilling procedures and excessive wear of the drilling tool concerning the achieved drilling speed. Both cases of improperly selected pressure force do not allow the drilling tool to work in the optimal mode. Incorrect selection of revolutions speed and pressure force results in damage to the drilling rig. The drilling tool’s performance and its efficiency depend on many structural elements, such as the construction of the tool body, the geometry of the cutting part, the flushing system of the device, and the setting of the optimal working mode. One of the possible measurable output parameters of the drilling process is the vibration signal of the acceleration [8,9,10,11,12].

The vibrating effects of the drilling rig are undesirable. However, they represent an important technological parameter, which indicates the quality of the process and the drilling rig’s technical condition [13,14,15,16,17,18].

Therefore, it is useful to know the effect of vibration on the drilling process through statistical control. The research’s basic task was to apply control charts for the needs of statistical quality control of the drilling process. The main goal was to examine the variability of the process in order to achieve the required product quality. The control charts were first used by the American engineer W. A. Shewhart (1891–1961) in 1920 to control the technological process. They represent a graphical tool that ensures continuous monitoring of data on process variability and allows us to obtain information about the monitored process’s capability. The control chart aims to clarify whether the technological process is under statistical control and to signal the existence of definable causes, keep the state under statistical control, and demonstrate measures to improve the process capability. The control charts work with data obtained from the technological process at approximately regular intervals, which are determined in time. Usually, each subgroup consists of the same technological parameter, which has measurable units and the same range of the subgroup. From each such so-called logical subgroup, which represents a group of random measurements in which the action of only random causes can be assumed, one or more characteristics (e.g., sample arithmetic mean, sample standard deviation, number of discrepancies, and others) are obtained. A control chart is then a graphical tool showing the subgroup’s given selection characteristic to its sequence number. According to the international standard STN ISO 8258 and depending on the nature of the monitored quality feature, the classic control charts are divided into [19,20,21,22,23]:

• the control charts by measurement,
• the control charts by comparison (see Figure 1).

Control charts by measurement are about data that is continuous and measurable. Control charts, by comparison, are usually considered for data describing qualitative properties. Another division distinguishes between control charts (CC) in which:

• the basic values are specified,
• the basic values are not specified.

Basic values mean the specified requirements or the values stated in the technical documentation of the device. The aim is to reveal whether the monitored values plotted in the control chart fluctuate in a larger range than can be attributed to chance.

According to the STN ISO 8258 standard, several types of control charts are distinguished by measurement:

• the chart for the average $\overline{x}$ and chart for the pre range R (chart $\overline{x}-R$),
• the chart for the average $\overline{x}$ and chart for standard deviation s (chart $\overline{x}-s$),
• the chart for the average $\tilde{x}$ and chart for the range R (chart $\tilde{x}-R$),
• the chart for the individual values X and chart for moving range $R_{MR}$ (chart $X-R_{MR}$),
• the chart for the moving arithmetic means ${\overline{X}}_{RA}$.

Each typical control chart contains three baselines: a centerline CL (Center Line), an upper control limit UCL (Upper Control Limit), and a lower control limit LCL (Lower Control Limit). The centerline (CL) represents the average value of the regulated variable when the process is stable. The distance of the control limits from the central line is most often chosen as $3\sigma$ of the standard deviation of the given sample characteristic on both sides. This means that there will be approximately 99.73% of values within the regulatory limits (the so-called three-sigma rule). The control limits define the band in which the values of the sample characteristics of the individual subgroups lie with a preselected probability. It is assumed that only random causes of process variability affect the investigated process in a given time period. Obviously, the monitored process is under statistical control, i.e., the process is statistically mastered. If the technological process is under statistical control, the control chart provides a continuous verifying method that the technological process has not changed and is statistically mastered [24,25,26,27,28].

The technological process’s instability is indicated by points outside the regulatory limits or by points showing trends or non-random groupings. If such points exist in the control chart, it is necessary to perform process analysis, find and eliminate the identifiable causes that caused the signaled instability. In industrial practice, it often happens that even at a steady state of the technological process, the output variables still do not get the same results.

The causes of variability of output variables can be divided into two groups: random causes and definable causes of variability.

Random causes of variability are unidentifiable causes that do not change the shape and parameters of the distribution. These causes cannot be measured individually; they are part of every process and cannot be eliminated from the process [29,30,31,32].

Definable (identifiable, systematic) causes of variability are causes causing a fundamental change in the technological process (wear of the working tool, incorrect setting of the measuring instrument, failure of the human factor, etc.). Their impact can be limited by early intervention. The STN ISO 8258 standard requires that these causes be not only identified but that remedial action is taken, and effective measures are taken to prevent their recurrence and to prevent an emergency.

A stable and unstable technological process is distinguished. A stable process, i.e., a process in a statistically mastered state, is a process that is affected only by random causes. Natural variability is involved in the stable process, and the system of causes remains unchanged over time. An unstable process, i.e., a process in a statistically uncontrollable state, is a process that is affected by random and definable causes.

Analyzing a technological process is an examination of the factors affecting the process. Within the analysis, the factors were classified according to the significance of the impact on the process, while the aim was to manage them so that the required and steady-state of the technological process was achieved. Achieving a steady-state process is a prerequisite for quality improvement.

Classical control charts were applied in the present research. The significance of the present research is that so far, no control charts were applied to such an extent to the process of rock disintegration by rotary drilling despite the fact that it is required by industrial mining practice.

2. Theoretical Background and Calculation

A simple scheme of the dynamic indentor-rock system, which is affected by two crucial measurable input variables-pressure force F (N) and revolutions n (rpm), is shown in Figure 2 [33,34,35,36,37].

The measurable output of the system is a vibration signal of the acceleration of the drilling rig. All the mentioned variables were recorded as time series characterizing the process of rotary rock drilling [38,39,40].

The system also includes “faults” that are controllable (e.g., indenter, rock type) or uncontrollable inputs (e.g., geomechanical properties of rocks, flushing water).

At present, the drilling rig allows working in the range of the following parameters:

• pressure force, F              (0–15,690) N,
• revolutions, n                (0–2400) rpm,
• torque, $M_k$                 (0–196) Nm,
• drilled length, l             (0–0.3) m,
• volume flow of water flush, Q     (0–1) × 10⁻³ m³s⁻¹,
• speed (velocity) of drilling, v       (0–16.8) × 10⁻³ ms⁻¹.

A diamond drilling tool’s number of revolutions is a basic parameter that increases the drilling speed by good regulation. For small-diameter drilling, the speed reaches a maximum of 2400 (rpm). In surface drilling, it is significantly lower, and they move up to a maximum level of 800 (rpm). Due to the vibrations that occur at higher revolutions with increasing drilling diameter, the speed of revolutions must be reduced from the point of view of the drilling rig’s balance work. Therefore, drilling with diamond drilling tools is optimal at higher speeds of revolutions and lower pressure force. The optimal setting of the pressure force is strongly correlated with the revolutions’ speed. Obviously, the vibration response of the drilling rig is a random process. Thus, it is necessary to base the analysis and rely on control charts (CC) [41,42,43,44,45].

The horizontal drilling device consists of a support stand (12), on which a headstock (1) with a drilling spindle (2) is mounted. A drill bit (4) of standard size is screwed into the drilling spindle with the core barrel (3). The drilling spindle is driven by a DC motor (12.5 kW, 220 V) with external excitation (180 V) via V-belts (7). The source of direct current for driving the electric motor is a thyristor rectifier. The flushing fluid is fed from the pump via a rubber pressure hose to the drilling spindle. The rock sample (5) is clamped in a mechanical jig (6), which is mechanically connected to a strain gauge head of axial pressure force and torque. Parts of the device from the headstock (1) through the core barrel (3) to the clamping device (6) are located in a protective sheet metal housing (13). The strain gauge head (ten-so-metric head) is articulated to the piston rods (8) of the hydraulic cylinders (9). The handle of the sliding ram (10) is firmly attached to stand (12). The flushing water is drained through a high-pressure hose (12) (see Figure 3).

The following figures (see Figure 4 and Figure 5) describe and present essential parts of the drilling rig in more detail.

In addition to the measured process variables (revolutions, pressure force) and the vibration signal, the drilled rocks’ drilling cores are a practical result of drilling (see Figure 5).

Rotary disintegration is carried out on rocks andesite, granite, limestone. The operating modes for the revolutions parameter can be set to n = 800 (rpm), n = 1000 (rpm), n = 1200 (rpm), n = 1400 (rpm), and for the pressure force F = 5000 (N), F = 8000 (N), F = 10,000 (N), and F = 12,000 (N). Possible adjustable variants of operating modes are shown in Figure 6.

During the drilling process, several parameters that define this process are measured and recorded. However, they are not examined to determine statistical process control. A complete overview of the measured parameters and their values are shown in

Table 1. Drilling log.

Time (s)	Revolutions (rpm)	Pressure Force (N)	Torque (Nm)	Drilled Length (m)	Electric Current (A)	Electrical Voltage (V)	Acceleration (mms−2)
0.25	723	8137	6.2	0.177	15.0	217	−0.0791
0.50	778	8173	5.7	0.178	14.8	214	0.1489
0.75	810	8161	6.9	0.178	15.2	218	−0.1225
1.00	804	8185	5.5	0.178	15.7	213	−0.1076
1.25	804	7876	5.2	0.179	15.6	219	0.0425
1.50	811	7852	5.2	0.180	15.2	210	−0.0394
1.75	781	7792	5.0	0.180	14.9	220	−0.1079
2.00	749	7518	6.2	0.180	15.5	212	−0.0066
2.25	720	7388	5.4	0.181	15.0	213	−0.0140
2.50	695	7435	6.6	0.181	15.9	213	−0.0849
2.75	673	7328	5.7	0.182	15.1	221	−0.0447
3.00	672	7518	6.7	0.182	14.6	214	−0.0223
3.25	664	7316	5.8	0.182	15.9	211	0.0251
3.50	660	7328	6.0	0.183	15.4	228	−0.0984
3.75	660	7435	5.1	0.183	16.0	217	0.0517
4.00	665	7030	5.6	0.185	15.2	220	−0.0844
4.25	681	7447	5.6	0.186	15.9	219	−0.0614
4.50	705	7435	6.1	0.186	15.8	212	−0.0299
4.75	722	7316	5.0	0.187	15.5	224	0.0590
5.00	758	7328	5.4	0.188	15.0	212	0.0496
5.25	790	7459	6.2	0.188	15.0	224	−0.0200
5.50	811	7471	6.9	0.188	15.0	223	−0.0119
5.75	805	7697	5.8	0.189	15.3	213	−0.0644
6.00	792	7697	6.3	0.190	15.3	228	−0.0886
6.25	757	7780	5.9	0.190	15.0	222	−0.0529
6.50	683	7709	6.1	0.193	15.3	215	0.0037
6.75	654	7697	6.8	0.191	14.9	212	0.0838
7.00	648	8066	6.0	0.192	15.9	220	0.0132
7.25	670	8185	6.7	0.192	15.0	223	−0.0997
7.50	712	7780	5.3	0.193	15.1	225	0.0701
7.75	748	8114	4.9	0.193	14.8	226	−0.0600
8.00	773	8197	5.7	0.194	15.0	213	−0.0486
8.25	786	8518	6.4	0.194	15.6	220	0.0894
8.50	805	8566	6.2	0.195	15.7	215	0.0488
8.75	802	8613	5.1	0.195	15.0	215	−0.0968
9.00	777	8554	5.7	0.196	15.3	211	−0.0608
9.25	765	8530	5.8	0.196	15.2	224	0.0826
9.50	730	8161	5.5	0.197	15.9	220	−0.0333
9.75	683	8054	6.9	0.197	15.6	226	−0.0446
10.00	654	7792	5.4	0.198	15.0	229	−0.0340
10.25	650	7816	5.3	0.198	15.4	222	−0.0618
10.50	663	7709	6.9	0.199	15.6	224	−0.1032
10.75	677	7780	6.5	0.199	15.4	222	−0.0617
11.00	701	8244	6.8	0.201	15.4	211	0.1033
11.25	725	7792	6.3	0.201	15.4	219	−0.0389
11.50	765	7709	6.9	0.201	15.0	226	−0.0513
11.75	786	7673	6.2	0.201	15.6	223	−0.0888
12.00	801	7637	5.4	0.203	15.4	228	0.0806
...	...	...	...	...	...	...	...
45.50	731	7530	5.6	0.267	15.2	229	0.0496
45.75	750	7483	5.1	0.267	16.0	222	0.0741
46.00	802	7102	5.5	0.268	15.2	223	0.0989
46.25	810	7173	6.2	0.268	14.7	213	0.0138
46.50	823	7114	5.3	0.269	14.7	229	−0.0644
46.75	796	7114	6.0	0.269	15.7	223	−0.0662
47.00	738	7054	6.2	0.270	15.6	218	−0.0118
47.25	697	6864	5.5	0.270	14.9	218	−0.0529
47.50	668	7019	6.0	0.271	14.9	216	0.0812
47.75	666	7102	5.0	0.271	15.3	216	0.0336
48.00	688	7245	6.7	0.272	16.0	212	−0.0506
48.25	720	7066	5.1	0.272	14.7	211	0.0252
48.50	781	7102	6.5	0.273	15.8	213	0.1113
48.75	829	6995	6.7	0.273	15.2	218	0.0174
49.00	819	6864	5.7	0.274	14.5	227	−0.0562
49.25	777	7066	5.7	0.275	15.5	210	−0.1001
49.50	744	7042	6.2	0.274	15.9	227	−0.1106
49.75	715	7352	6.6	0.275	15.0	227	0.0396
50.00	671	7304	5.6	0.276	16.0	230	−0.0377
50.25	662	7614	5.3	0.276	15.8	211	0.0078
50.50	697	7768	5.6	0.277	14.8	218	0.0496
50.75	763	7649	5.9	0.277	14.9	212	0.0741
51.00	795	8018	6.5	0.278	15.8	214	0.0989
51.25	819	7780	5.6	0.278	14.9	214	0.0138
51.50	816	7983	6.1	0.278	15.5	214	−0.0644
51.75	785	8173	6.1	0.281	15.9	212	−0.0662
52.00	756	8161	6.0	0.281	14.8	216	−0.0118
52.25	707	8125	6.3	0.281	14.9	220	−0.0529
52.50	668	8256	5.5	0.282	14.5	229	0.0812
52.75	656	8233	6.2	0.283	15.1	211	0.0336
53.00	666	8471	5.9	0.283	15.0	219	−0.0506
53.25	660	8732	4.9	0.283	15.1	212	0.0252
53.50	694	8280	5.8	0.283	15.2	229	0.1113
53.75	818	7721	5.2	0.284	14.6	217	0.0174
54.00	853	7900	6.9	0.284	15.8	225	−0.0562
54.25	820	7888	5.0	0.284	15.6	210	−0.1001
54.50	790	7698	6.2	0.284	15.9	216	−0.1106
54.75	785	7412	7.0	0.284	15.0	228	0.0396
55.00	806	7352	6.6	0.284	15.3	216	−0.0377

.

Rock hardness is one of the longest-used properties for rock classification. Hardness is understood as the rock’s mechanical property, expressed by the resistance to deformation of its surface, caused by the action of a foreign body.

Rock hardness as a mechanical property is not a physical quantity because this value depends more than any other mechanical property on the rock’s complex surface properties under test and on the test conditions under which the hardness is determined.

During the hardness test, the test material’s surface is mechanically loaded by the pressure of a foreign body, and the result of this action is quantified as the hardness value. It follows that the hardness of the rock is a technical-mechanical property that has a conventional character. This conventionality is determined by the defined test procedure and test conditions.

Hardness and strength represent the properties of the rock characterizing their resistance to external forces’ mechanical action. Simultaneously, the strength expresses the resistance of the whole body against the breaking of the whole volume by dividing the rock into parts. Hardness represents the body’s resistance to the action of concentrated forces, not sufficient for complete disintegration, bu causing partial local failure or deformation.

Andesite rock was selected to assess the stability of the drilling process. During its rotary disintegration, there was a significant instability of the process. Rotational disintegration of andesite was a motivating factor for this study.

Table 2. Physical and mechanical properties of the rocks.

Physical Properties of Rocks					Mechanical Properties of Rocks
Rock	Specific Mass	Volume Mass	Porosity	Absorbability	Resistance to Simple Compression	Resistance to Crossfeed Tension
	$\mathit{r}$(kgm${}^{-\mathbf{3}}$)	$\mathit{r}$(kgm${}^{-\mathbf{3}}$)	$\mathit{p}$(%)	$\mathit{n}$(%)	(MPa)	(MPa)
Andesite	2720	2700	0.07	0.67	282	16
Limestone	3000	2960	1.33	0.1	225	13
Granite	2680	2650	1.12	0.14	196	7.2

,

Table 3. Detailed mechanical properties of the rocks.

Rock	Sekant Reshaping Modulus ${\mathit{M}}_{\mathit{s}}$ (MPa)	Tangent Reshaping Modulus ${\mathit{M}}_{\mathit{t}}$ (MPa)	Elastic Modulus E (MPa)	Poisson Number $\mathit{\mu }$
Andesite	84,607	84,314	84,608	0.21
Limestone	43,725	70,294	56,471	0.13
Granite	54,020	61,275	57,059	0.2

and

Table 4. Detailed technological properties of the rocks.

Rock	Class	Degree of Rock Strength	Rock Type	Strength Factor f
Andesite	IIIa.	strong	strong andesites	8
Limestone	III.	strong	very strong limestones	10
Granite	II.	very strong	strong granites	15

show the significant geomechanical properties that are observed during the rotary disintegration of the rocks. In the present issue, this is understood as uncontrollable process input parameters. Their fundamental influence is a separate subject of research and study. Direct research of the dependence between hardness and control parameters of rotational rock disintegration has not been carried out. This promising idea will be the subject of future research.

For the drilling process analysis, control charts were selected by measurement and if the basic values are not determined. The statistical characteristics as the arithmetic mean, variation range, standard deviation, median, moving variation range, individual values, and moving arithmetic means were determined, and the corresponding control charts were constructed for them. The most commonly used pair of control charts by measurement is the control diagram $\overline{x}-R$. Charts are used to monitor a continuous dynamic process with a relatively large amount of data. Its advantage is greater sensitivity to the existence of isolated outliers within a logical subgroup. In the first step, the range of the logical subgroup n was chosen. In this case, the range from 2 to 9 is most often chosen, with the recommended number being $n=5$. In the next step, the number of logical subgroups m (the recommended minimum number is 20 to 25 subgroups), and the time interval was determined. Since the process of rock disintegration by rotary drilling is fast and short-lived, a seconds time interval of measuring variables was chosen, from which samples were selected for the logical subgroup.

The sample arithmetic mean ${\overline{x}}_i$ and the sample range of variation $R_i$ were calculated for the i-th subgroup from the measured values. These parameters are calculated as the following:

$${\overline{x}}_i=\frac{1}{n}\sum _{j=1}^n{x}_{ij},$$

(1)

$$R_i=x_{i,max}-x_{i,min},$$

(2)

for $i=1,2,...,m$ and $j=1,2,...,n$, where n is the range of logical subgroup, m is the number of subgroups, $x_{i,max}$ is maximal, and $x_{i,min}$ is the minimum measured value in the i-th subgroup.

Finally, the overall arithmetic mean $\overline{\overline{x}}$ and the average range $\overline{R}$ to which it applies

$$\overline{\overline{x}}=\frac{1}{m}\sum _{i=1}^m{\overline{x}}_i,$$

(3)

$$\overline{R}=\frac{1}{m}\sum _{i=1}^m{R}_i,$$

(4)

were calculated.

The control chart contains the upper (UCL), lower control limit (LCL) and center line (CL). The regulatory limits for the $\overline{x}$-chart were calculated according to the following equations:

$$\mathrm{C}{\mathrm{L}}_{\overline{x}}=\overline{\overline{x}},$$

(5)

$$\mathrm{UC}{\mathrm{L}}_{\overline{x}}=\overline{\overline{x}}+A_2\overline{R},$$

(6)

$$\mathrm{LC}{\mathrm{L}}_{\overline{x}}=\overline{\overline{x}}-A_2\overline{R}.$$

(7)

The regulatory limits for the R-chart were calculated according to the equations:

$$\mathrm{C}{\mathrm{L}}_R=\overline{R},$$

(8)

$$\mathrm{UC}{\mathrm{L}}_R=D_4\overline{R},$$

(9)

$$\mathrm{LC}{\mathrm{L}}_R=D_3\overline{R}.$$

(10)

Since the value of the average range of variation evaluated in the R-chart is used to calculate the control limits of the $\overline{x}$-chart, the analysis of the control chart is first performed, which characterizes the variability of the process, i.e., the R-chart. This chart was obtained by plotting the values $R_i$ of the selective variation ranges of the individual subgroups and the control limits. From the chart analysis, it can be monitored whether the regulated variable does not signal the effect of definable causes of variability. The occurrence of points outside the regulatory limits and the existence of non-random groupings of points are considered to be a signal of the action of definable causes. If non-accidental causes of variability occur, the process is considered to be statistically unmanageable/unstable/not under control. If no signals of the action of definable causes appear, the control chart analysis characterizing the production process’s position, i.e., the $\overline{x}$-chart, is continued. Its evaluation is analogous to the R-chart. If signals of the action of identifiable causes are detected in practice, it is important to identify these causes and take corrective measures to eliminate them. Since the control chart for the arithmetic mean has already been calculated, it is possible to proceed with the control chart’s calculation and construction for the standard deviation [46,47,48,49].

The s-standard deviation control chart is the most accurate of the control charts by measurement. It is suitable for larger ranges of logical subgroups $\left(2\le n\le 25\right)$ and where it is possible to process data on a computer. When selecting a subgroup range, it is recommended to select $n>9$. The sample standard deviation is used to describe the variability of the technological process. The procedure for constructing a control chart s is similar to that for a control chart $\overline{x}-R$. In each subgroup, the sample arithmetic mean ${\overline{x}}_i$ and sample standard deviation $s_i$ were calculated. It is applied that

$${\overline{x}}_i=\frac{1}{n}\sum _{j=1}^n{x}_{ij},$$

(11)

$$s_i=\sqrt{\frac{1}{n-1}\sum _{j=1}^n{\left(x_{ij}-{\overline{x}}_i\right)}^2},$$

(12)

for $i=1,2,...,m$, $j=1,2,...,n$, where m is the number of subgroups and n represents the range of the subgroup.

The overall arithmetic mean $\overline{\overline{x}}$ and average of sampling standard deviations $\overline{s}$ were calculated according to equations

$$\overline{\overline{x}}=\frac{1}{m}\sum _{i=1}^m{\overline{x}}_i,$$

(13)

$$\overline{s}=\frac{1}{m}\sum _{i=1}^m{s}_i.$$

(14)

The control limits for the s-chart can be calculated according to the following equations:

$$\mathrm{C}{\mathrm{L}}_s=\overline{s},$$

(15)

$$\mathrm{UC}{\mathrm{L}}_s=B_4\overline{s},$$

(16)

$$\mathrm{LC}{\mathrm{L}}_s=B_3\overline{s},$$

(17)

where $A_2$, $D_3$, $D_4$,$B_3$, and $B_4$ are constants that depend on a range of subgroups and presented are in tables (see

Table 5. Conversion coefficients for calculation of UCL, LCL in CC (n = 10 in ISO 8258).

n	2	3	4	5	6	7	8	9	10
A	2.121	1.732	1.500	1.342	1.225	1.134	1.061	1.800	0.949
A${}_2$	1.880	1.023	0.729	0.577	0.483	0.419	0.373	0.337	0.308
A${}_3$	2.659	1.954	1.628	1.427	1.287	1.182	1.099	1.032	0.975
A${}_4$	1.880	1.190	0.800	0.690	0.550	0.510	0.430	0.410	0.360
B${}_3$	0	0	0	0	0.030	0.118	0.185	0.239	0.284
B${}_4$	3.267	2.568	2.266	2.089	1.970	1.882	1.815	1.761	1.716
B${}_5$	0	0	0	0	0.029	0.113	0.179	0.232	0.276
B${}_6$	2.606	2.276	2.088	1.964	1.874	1.806	1.751	1.707	1.669
D${}_1$	0	0	0	0	0	0.204	0.388	0.547	0.687
D${}_2$	3.686	4.358	4.698	4.918	5.078	5.204	5.306	5.393	5.469
D${}_3$	0	0	0	0	0	0.076	0.136	0.184	0.223
D${}_4$	3.267	2.574	2.282	2.104	2.004	1.924	1.864	1.816	1.777
C${}_4$	0.798	0.886	0.921	0.940	0.952	0.959	0.965	0.969	0.973
d${}_2$	1.128	1.693	2.059	2.326	2.534	2.704	2.847	2.970	3.078

). When evaluating the process control via the s-chart, the procedure was the same as when evaluating the previous control charts.

The following important control chart, which was constructed for the drilling technological process’s needs, is the chart for the median $\tilde{x}$. The chart for the median $\tilde{x}$ is, on the one hand, easy to use and does not require more complex calculations; on the other hand, it is less sensitive than the chart for the sample average. The control chart for the median shows the range of the technological process and gives a continuous view of the process’s measured variables’ fluctuations. The procedure for constructing a control chart for the median $\tilde{x}$ is similar to the other charts. It is commonly used for logical subgroups with a range of 10 or less than 10, with odd ranges of subgroups being more preferred. In each subgroup, sample characteristics were calculated, i.e., the sampling median ${\tilde{x}}_i$ and the sampling range of variation $R_i$, which has already been calculated. The total median mean $\overline{\tilde{x}}$ of the subgroups is calculated according to the equation

$$\overline{\tilde{x}}=\frac{1}{m}\sum _{i=1}^m{\overline{\tilde{x}}}_i,$$

(18)

The control limits for the $\tilde{x}$-chart were calculated according to the following equations:

$$\mathrm{C}{\mathrm{L}}_{\tilde{x}}=\overline{\tilde{x}},$$

(19)

$$\mathrm{UC}{\mathrm{L}}_{\tilde{x}}=\overline{\tilde{x}}+A_4\overline{R},$$

(20)

$$\mathrm{LC}{\mathrm{L}}_{\tilde{x}}=\overline{\tilde{x}}-A_4\overline{R}.$$

(21)

Control chart for individual values X and moving variation range $R_{MR}$ is suitable for high homogeneity in a subgroup or where it is impossible to obtain a large number of values in a short time (e.g., destructive test). There are special cases of control charts in which the size of the logical subgroup is $n=1$. To describe the variability, the so-called moving variation range. It is the absolute value of the difference between a pair of measurements that follow each other in a series of measurements. The following equation gives the moving variation range

$$R_{M{R}_i}=\left|x_i-x_{i-1}\right|\mathrm{for}i=1,2,...,m.$$

(22)

The overall arithmetic mean $\overline{\overline{x}}$ and average moving range ${\overline{R}}_k$ were calculated according to the equations:

$$\overline{\overline{x}}=\frac{1}{m}\sum _{i=1}^m{\overline{x}}_i,$$

(23)

$${\overline{R}}_{MR}=\frac{1}{m}\sum _{i=1}^m{R}_{M{R}_i}.$$

(24)

Control limits for $R_{MR}$-chart were calculated according to the following equations:

$$\mathrm{CL}={\overline{R}}_{{}_{MR}},$$

(25)

$$\mathrm{UC}{\mathrm{L}}_{R_{MR}}=D_4{\overline{R}}_{{}_{MR}},$$

(26)

$$\mathrm{LC}{\mathrm{L}}_{R_{MR}}=D_3{\overline{R}}_{{}_{MR}}.$$

(27)

The control limits for the individual values of the X-chart were calculated according to the equations:

$$\mathrm{C}{\mathrm{L}}_X=\overline{x},$$

(28)

$$\mathrm{UC}{\mathrm{L}}_X=\overline{x}+\frac{3}{d_2}\overline{R},$$

(29)

$$\mathrm{LC}{\mathrm{L}}_X=\overline{x}-\frac{3}{d_2}\overline{R}.$$

(30)

Subsequently, it is possible to create a control chart for moving averages ${\overline{X}}_{MA}$. This control chart is often used in practice. The original measured values are replaced by the arithmetic means’ values, which were calculated from the original values. Applying the moving averages control chart makes it possible to exclude the random and seasonal components and estimate the trend and cyclical components [50,51,52,53].

The control limit values for the moving average chart ${\overline{X}}_{MA}$ are calculated according to the equations:

$$\mathrm{C}{\mathrm{L}}_{{\overline{X}}_{MA}}={\overline{X}}_{MA},$$

(31)

$$\mathrm{UC}{\mathrm{L}}_{{\overline{X}}_{MA}}=\overline{X}+D_4{\overline{X}}_{MA},$$

(32)

$$\mathrm{LC}{\mathrm{L}}_{{\overline{X}}_{MA}}=\overline{X}-D_3{\overline{X}}_{MA},$$

(33)

where $A_4$, $d_2$, $D_3$, and $D_4$ are constants for $n=2$ (see Table 5).

3. Experimental Results and Discussion

The basic assumptions of the most frequently used control charts are the normal distribution of measured data. The basic assumptions of the most commonly used control charts are the normal distribution of measured data.

In practice, control charts are often used without verifying the normality of the input data. Even in such cases, the control chart is suitable for practical use. The normality of the measured input data from the drilling process was verified by descriptive statistics and histograms of technological parameters. The constructed histogram analysis was focused mainly on the centering of the histogram, which characterizes the mean value of the observed technological feature, then on the width, which characterizes the variability of values and its shape (form), which allows revealing some definable causes of variability. During the histogram analysis, the minimum and maximum values were determined. From them, the variability of the investigated technological feature, the center of the distribution, the number of peaks, the symmetry, the kurtosis, and skewness of the distribution. The following graphs show the time behaviors, histograms, and statistical characteristics calculated in the tables (see

Table 6. Statistical characteristics of revolutions’ speed.

Characteristics	Value
Mean	744
Standard Error	4.0
Median	751
Mode	811
Standard Deviation	58
Sample Variance	3359
Kurtosis	−1.00
Skewness	0.00
Range	205
Minimum	648
Maximum	853
Sum	163,688
Count	220
Largest (1)	853
Smallest (1)	648

,

Table 7. Statistical characteristics of measured pressure force.

Characteristics	Value
Mean	7547
Standard Error	32
Median	7483
Mode	7459
Standard Deviation	471
Sample Variance	221,970
Kurtosis	1
Skewness	0
Range	2666
Minimum	6066
Maximum	8732
Sum	1,660,295
Count	220
Largest (1)	8732
Smallest (1)	6066

and

Table 8. Statistical characteristics of the vibration signal of acceleration.

Characteristics	Value
Mean	−0.0281
Standard Error	0.0048
Median	−0.0317
Mode	0.0078
Standard Deviation	0.0713
Sample Variance	0.0051
Kurtosis	1.2310
Skewness	−0.3348
Range	0.5093
Minimum	−0.3603
Maximum	0.1489
Sum	−6.1856
Count	220
Largest (1)	0.1489
Smallest (1)	−0.3603

). Figure 7a–c shows the original measured behavior of technological parameters during the disintegration of rocks by rotary drilling.

Subsequently, histograms were constructed. Figure 8a shows a histogram of revolutions with a flat shape. These histograms are usually characterized by the data collected under variable conditions or indicating a poor process setup. It indicates a change in position during measurement. The change of position is also proved by the original measured time course of the revolutions n (rpm) (see Figure 7a).

The bell-shaped histograms (see Figure 8b,c) show the normal distributions of the measurable quality characters. Histograms of the frequencies of pressure force and vibrations of acceleration shown in Figure 8b,c are histograms with partially isolated peaks, the separated spikes, respectively. It indicates the evaluation of an inhomogeneous measured quality character. The time behaviors are shown in Figure 7b,c. The histograms of the measured parameters thus obtained have a high significance. They plot values from a certain interval divided into subintervals of a certain width, which follow each other. Histograms were supplemented by statistical characteristics.

The normal distribution is understood as a mathematical model of the empirical distribution of values, which can be used to analyze, explain, and implement results obtained from various measured technological parameters under comparable conditions. Deviations from the actual value are affected only by chance. A typical theoretical distribution is the Gaussian normal distribution. Its graphic representation is a bell-shaped Gaussian curve.

Gaussian curves of the normal distribution were constructed and plotted in histograms for input and output technological parameters to support the normal distribution hypothesis. The references that confirm the hypothesis of a normal distribution are as follows: [54,55,56,57,58].

Statistical characteristics at the selected confidence level were calculated for the reliability of the evaluated variables and the conclusions’ accuracy. For engineering tasks, it is recommended in the standard STN ISO 8258 The control diagrams assume a theoretical probability of p = 99.73 % and uncertainty of $\alpha =0.27$%. As exceeding regulatory limits is considered an exceptional phenomenon in terms of probability, the process must be intervened immediately in the event of its occurrence.

Subsequently, control charts were designed and constructed according to a simple algorithm shown in Figure 9.

The experimental drilling work aimed to obtain data on the permissible variability and dynamics of possible changes in revolutions and pressure force in interaction with the drilling tool’s vibrations. Control charts were applied to the measured operating mode with the speed of revolutions $n=800\left(\mathrm{rpm}\right)$ and pressure force F$=8000\left(\mathrm{N}\right)$. The experiment’s duration was $T=55\left(\mathrm{s}\right)$, with 220 values recorded at regular intervals (4 values per second). The drilled rock was andesite. In the calculations and construction of control charts (CC) by measuring for the variables of revolutions n (rpm), pressure force F (N), and vibrations of the drilling process, it was considered that the basic values are not determined.

The R-diagram for the variation revolutions’ range $n=800\left(\mathrm{rpm}\right)$ (see Figure 10a) was constructed as the first control chart. The number of logical subgroups $m=44$ and the subgroup range was $n=5$ measurements, and it was assumed that the technological parameter has a normal distribution.

According to Equations (5)–(7), the central line CL and control limits UCL, LCL were calculated for the R-diagram. The values of the coefficients $D_3=0$ and $D_4=2.104$ were obtained from Table 5 according to the STN ISO 8252 standard for $n=5$ measurement. The values of the center line and the control limits for the revolutions of R-control chart are CL = 131, UCL = 277 and LCL = 0. The diagram in Figure 10a shows that all values are between the control limits and the variability of the process is under statistical control.

Table 9. Values of the center line and control limits for the revolutions.

Revolutions n (rpm)
Process Quality Criteria	$\mathit{R}$	$\overline{\mathit{x}}$	$\mathit{s}$	$\overline{\tilde{\mathit{x}}}$	${\overline{\mathit{R}}}_{\mathit{MR}}$	$\mathit{X}$	${\overline{\mathit{X}}}_{\mathit{MA}}$
CL	131	744	53	744	98	744	744
UCL	277	819	111	830	206	870	800
LCL	0	668	0	658	0	617	687
							>UCL
							6–8 (800–803)
							51–52 (808)
							63 (805)
							75–76 (804)
							120–122 (802–805)
							132–133 (804–805)
							154 (805)
							219–220 (813)
Violation
							<LCL
							13–19 (672–665)
							29–32 (682–673)
							42–45(675–665)
							83 (682)
							94(681)
							139 (680)
							169( 686)
							179–182( 679–670)
							213–214 (671)

shows the base values of central line. Presented are values of revolutions outside the range of the upper and lower control limits.

The second constructed control charts according to (8)–(10) was the $\overline{x}$-diagram for the arithmetic mean. The value of the coefficient $A_2=0.577$ was obtained from Table 5 of the STN ISO 8252 standard for $n=5$. measurements, depending on the range of subgroups. The values of the central line of the control limits for the revolutions of the $\overline{x}$-control diagram are CL = 744, UCL = 819 and LCL = 668 (see Figure 10b). The $\overline{x}$-diagram shows that the sample arithmetic means are not outside the regulatory limits. However, it should be noted that the course of the control chart shows a periodic dependence of the evaluated technological parameter. Non-random causes of variability also influence the technological process. Nevertheless, the process is under statistical control. Subsequently, the control chart for the standard deviation s and the median $\overline{\tilde{x}}$ were constructed. The values of coefficient $B_3=0$, $B_4=2.089$, and $A_4=0.69$ for s and $\overline{\tilde{x}}$-diagram was obtained from the tables for $n=5$ measurements. The centerline and control limits for the s-diagram were calculated according to Equations (15)–(17). It applies in CL = 53, UCL = 111, and LCL = 0 (see Figure 10c). The centerline and control limits for the $\overline{\tilde{x}}$-control diagram were calculated according to Equations (19)–(21). The values are as follows CL = 744, UCL = 830 and LCL = 658 (see Figure 10d). There are several values in the chart close to the lower/upper control limit, so it would be appropriate to focus on this subgroup’s values and determine the cause of such great variability. The technological process is based on s, $\overline{\tilde{x}}$ charts under statistical control. However, the property of the periodicity parameter reappears. Subsequently, control charts were constructed for the moving range ${\overline{R}}_{MR}$, individual values X, and moving averages ${\overline{X}}_{MA}$ if the basic values are not determined (see Figure 11).

Values of coefficients $D_3=0$, $D_4=3.267$ were obtained from (Table 5) for $n=2$. The value of the central line and the control limits of the moving range ${\overline{R}}_{MR}$ for revolutions are CL = 98, UCL = 206, LCL = 0 (see Figure 11a), for individual values X, i.e., CL = 744, UCL = 870, LCL = 617 (see Figure 11b), and moving averages ${\overline{X}}_{MA}$, i.e., CL = 744, UCL = 800, LCL = 687 (see Figure 11c).

It is clear from the control charts that not all calculated points lie within the control limits. This was fundamentally reflected in the moving average diagram ${\overline{X}}_{MA}$ (see Figure 11c). Several values are outside the range of the upper and lower control limits (see Table 9). It is possible to read the strong periodicity of the parameter from the control charts, and therefore the variability of the process is not under statistical control. It is, therefore, necessary to look for the cause of the instability of the analyzed parameter. Similarly, the second technological parameter, i.e., the pressure force, was analyzed. Control charts were constructed and subsequently analyzed (see Figure 12).

The values for the R-control chart are CL = 988, UCL = 2087, LCL = 0. One subgroup is outside the upper limit, and there are several cases of non-random groupings (see Figure 12a). It was stated that the technological parameter pressure force is not under statistical control. The following charts also confirm the judgment. The values for the $\overline{x}$-control chart are CL = 7537, UCL = 8107, LCL = 6968 (see Figure 12b). Two subgroups are outside the upper limit, but several subgroups create a decreasing and increasing time trend. It is indicated that the parameter is not under statistical control. A similar partial conclusion can be drawn from the s and $\overline{\tilde{x}}$-control charts. The values for the s-control chart are CL = 399, UCL = 833, LCL = 0 (Figure 12c) and for $\overline{\tilde{x}}$-control chart are CL = 7588, UCL = 8260, LCL = 6915 (see Figure 12d). The technological parameter shows strong variability; it is necessary to look for its cause.

The following values were obtained from the control charts for the moving range ${\overline{R}}_{MR}$, individual values X., moving averages ${\overline{X}}_{MA}$: the value of the center line and the control limits of the sliding range for the pressure are CL = 411, UCL = 870, LCL = 0 (see Figure 13a); for individual values they are CL = 7547, UCL = 8078, LCL = 7015 (see Figure 13b) and for moving averages they are CL = 7546, UCL = 7784, LCL = 7309 (see Figure 13c).

The values of pressure force that exceeded the control limits are in

Table 10. Values of the center line and control limits for the pressure force.

Pressure Force F (N)
Process Quality Criteria	$\mathit{R}$	$\overline{\mathit{x}}$	$\mathit{s}$	$\overline{\tilde{\mathit{x}}}$	${\overline{\mathit{R}}}_{\mathit{MR}}$	$\mathit{X}$	${\overline{\mathit{X}}}_{\mathit{MA}}$
CL	988	7537	399	7588	411	7547	7546
UCL	2087	8107	833	8260	870	8078	7784
LCL	0	6968	0	6915	0	7015	7309
Violation	>UCL	>UCL	>UCL	>UCL	>UCL	>UCL	>UCL
	39 (2523)	36 (8123)	39 (1013)	36 (8470)	88–89 (1023)	29–44 (8184–8245)	1–8 (7850–8153)
		37 (8139		37 (8530)	147–148 (1178–882)	77–86 (8150–8173)	28–48 (7780–8560)
					215–217 (1011)	135–138 (8134–8161)	78–89 (7880–8410)
						207–214 (8138–8732)	136–142 (7875–8155)
							206–218 (7880–8410)
						<LCL
						58 (6960)	<LCL
						147–162 (6411–7007)	54–62 (7278–7073)
						189–196 (6865–6995)	96–99 (7297–7240)
							112–113 (7280–7271)
							147–167 (7228–6277)
							186–201 (7275–7015)

. It is clear from the charts that the process is not stable/not statistically mastered. The charts for individual values and moving averages show many samples outside the defined limits. It was stated that it is necessary to urgently find the causes of instability of the input technological parameter, i.e., the pressure force. After analyzing the process input parameters, the control charts of the output parameter vibration signal of acceleration were presented. The resulting control charts are shown in Figure 14.

The values for R-control chart are CL = 0.161, UCL = 0.342, and LCL = 0. One subgroup is outside the upper limit, and there are several cases of non-random groupings (see Figure 14a). It was stated that the technological parameter vibration signal of acceleration is not under statistical control. There are several cases of non-random groupings of subgroups. The following charts confirm this.

The values for $\overline{x}$-control chart are CL = 0.033, UCL = 0.342, and LCL = 0.126 (see Figure 14b). The subgroups are not outside the upper limit or the lower limit. This suggests that the parameter is under statistical control, but it is only a partial conclusion of one control diagram. A similar partial conclusion can be stated from the s and $\overline{\tilde{x}}$ control charts. The values for s-control diagram are CL = 0.065, UCL = 0.136, LCL = 0 (see Figure 14c) and for $\overline{\tilde{x}}$-control chart are CL = 0.038, UCL = 0.075, LCL = 0.151 (see Figure 14d). The technological parameter vibration signal shows strong variability, from which it was concluded that the process is not statistically controlled and is influenced by non-random causes.

The value of the center line and the control limits of the moving range for the vibration signal of acceleration are CL = 0.163, UCL = 0.343, LCL = 0 (see Figure 15a), for individual values are CL = −0.028, UCL = 0.181, LCL = −0.236 (see Figure 15b) and for moving averages are CL = −0.028, UCL = 0.066, and LCL = −0.122 (see Figure 15c). The values of the vibration acceleration signal that exceeded the control limits are shown in

Table 11. Values of the center line and control limits for the vibration signal acceleration.

Vibration-Acceleration
Process Quality Criteria	$\mathit{R}$	$\overline{\mathit{x}}$	$\mathit{s}$	$\overline{\tilde{\mathit{x}}}$	${\overline{\mathit{R}}}_{\mathit{MR}}$	$\mathit{X}$	${\overline{\mathit{X}}}_{\mathit{MA}}$
CL	0.161	−0.033	0.065	−0.038	0.163	−0.028	−0.028
UCL	0.342	0.0694	0.136	−0.151	0.343	0.181	0.066
LCL	0	−0.126	0	0.75	0	−0.236	−0.122
Violation	>UCL		>UCL		>UCL	>UCL	>UCL
	27 (0.44)		27 (0.165)		107–108 (0.365)	107 (−0.361)	107 (−0.135)
					109–111 (0.423)		108 (−0.133)
					108–110 (0.422)

.

It is clear from the control charts that most of the calculated points lie within the control limits. However, the samples show a rapidly decreasing and increasing time trend.

Variability is not under statistical control. It can be stated that the technological parameter vibration signal of drilling acceleration is not in a statistically controlled (stable) state. It is obvious that the input and output technological parameters of the drilling process are not in a statistically stable state. It should be noted that the course of control charts shows strong periodic dependence and time trends. There are several reasons why the drilling process is not in a statistically mastered state for technological parameters. One possible reason is that the operator at the control computer responded late at the experiment’s beginning. The second cause may be the rocks’ geomechanical properties (i.e., the sudden formation of cracks and fissures during the drilling process). A third possible cause may be that the drilling process is very dynamic, and several parts of the drilling rig are underwater; others are exposed to heavy dust. Other reasons are unknown and are the subject of the following research. A prerequisite for bringing the drilling process under control is the identification of identifiable causes in the process and the implementation of corrective measures. However, due to the nature of the experiments and the wear of the drilling rig at the time of drilling, the course is sufficient to ensure the measurement of the monitored control variables.

Industrial practice values the control charts because of their preventive nature. They help prevent unnecessary adjustment of equipment, production lines if the technological process is stable. In addition, they provide ongoing diagnostic data on process variability and make it possible to obtain information on monitored processes’ capability. In industrial practice, control charts contribute to optimizing economic aspects (i.e., to the overall reduction of control costs). Production quality control using control charts is used especially in repeated processes when under relatively stable production conditions (i.e., technology, material, technical parameters of machines and equipment) there are other influences (i.e., small deviations from prescribed technology, small deviations in material quality, small inaccuracy of setting parameters of machines and equipment). Significant non-exceedance of effects will manifest itself within certain "reasonable" limits as a random effect.

Shewhart emphasized the empirical usefulness of the control chart for recognizing deviations from the state in which the technological process is statistically mastered.

Measurement control charts are suitable for several reasons. Most technological processes and the above-mentioned drilling process, with their inputs and outputs, provide measurable features so that their wide use is possible. In the practical application, it was considered that the behavior of the technological process of drilling could be analyzed regardless of the technical specification. The charts reflect the technological process of drilling and give an image of what the process itself is capable of.

In industrial practice, the stabilization of the process also contributes to shortening the time difference between an unstable technological process and corrective intervention. The behavior of the measured values and the limits determined on the basis of the control characteristics give a sufficiently reliable visual image of the course of the measured values themselves. Thus, it is possible to evaluate the measured data in real time and provide the measured variables’ current state.

By their action, the sought systematic causes cause a fundamental change in the variability or position of the process, whereby the process ceases to be in a stable state.

4. Conclusions

In general, control charts are considered a proven means of improving the quality parameters of production and preventing errors - discrepancies in production.

Industrial practice values the control charts because of their preventive nature. They help prevent unnecessary adjustment of equipment, production lines if the technological process is stable. In addition, they provide ongoing diagnostic data on process variability and make it possible to obtain information on monitored processes’ capability. In industrial practice, control charts contribute to optimizing economic aspects (i.e., to the overall reduction of control costs). Production quality control using control charts is used especially in repeated processes when under relatively stable production conditions (i.e., technology, material, technical parameters of machines and equipment) there are other influences (i.e., small deviations from prescribed technology, small deviations in material quality, small inaccuracy of setting parameters of machines and equipment). Significant non-exceedance of effects will manifest itself within certain "reasonable" limits as a random effect.

Shewhart emphasized the empirical usefulness of the control chart for recognizing deviations from the state in which the technological process is statistically mastered.

Measurement control charts are suitable for several reasons. Most technological processes and the above-mentioned drilling process, with their inputs and outputs, provide measurable features so that their wide use is possible. In the practical application, it was considered that the behavior of the technological process of drilling could be analyzed regardless of the technical specification. The charts reflect the technological process of drilling and give an image of what the process itself is capable of.

In industrial practice, the stabilization of the process also contributes to shortening the time difference between an unstable technological process and corrective intervention. The behavior of the measured values and the limits determined on the basis of the control characteristics give a sufficiently reliable visual image of the course of the measured values themselves. Thus, it is possible to evaluate the measured data in real time and provide the measured variables’ current state.

By their action, the sought systematic causes cause a fundamental change in the variability or position of the process, whereby the process ceases to be in a stable state.

From the control charts, it can be stated that the foreseeable systematic causes of instability of the drilling process are probably caused by wear of the drilling tool, high flushing water pressure, insufficient fixing of the drilled rock sample. The impact of these causes can be reduced by early intervention, but it is not possible to completely eliminate their occurrence without technological changes.

Unforeseeable systematic causes cause significant changes in the technological process. They can be prevented. Typical such causes in the drilling process are the breakage of a drilling tool, the fall of a diamond from a drill bit, an improperly chosen working mode with respect to the rock being disintegrated, an uncontrollable oscillation of the drilling equipment, lack of knowledge of the drilling process and the human factor.

It was stated that the R-chart reveals undesirable fluctuations within the subgroups of technological parameters and is an indicator of the magnitude of variability of the observed technological drilling process.

The individual $\overline{x}$, s, $\overline{\tilde{x}}$-control charts show that the technological drilling process is off-center and unstable for the input parameters as revolutions, pressure force, and output parameter vibration signal. At revolutions and vibration signal, the periodic character of the measured quantity is clearly manifested. In the pressure force, inappropriate time trends appear. It is clear that the charts reveal undesirable fluctuations between subgroups in terms of their averages, standard deviations, and medians.

The control charts of the moving range, individual values and moving average show that the drilling process is not in a statistically controlled state for all measured process parameters. Suppose the technological process of drilling becomes stable in its parameters. In that case, the control charts system guarantees that the statistical characteristics of individual subgroups will change randomly and rarely lie outside the control limits. Likewise, time trends or aggregations of data should not appear in the drilling process, except in random cases.

Based on the analyzed control charts, it is necessary to take corrective measures, supplement the technology with new measuring devices, eliminate measurement errors, find systematic causes, and design new optimal experimental measurements.

Corrective measures will lead to an increase in the efficiency of rock drilling and the stabilization and optimization of the technological process as a complex unit. There will be a significant impact from an economic, environmental, and economic point of view.