Method of Compensating for Instrumental Uncertainty in Measurements Using a Coordinate Measuring Arm
Due to the influence of dynamic factors in various measurement configurations, the degree of uncertainty in measurements using a Coordinate Measuring Arm (CMA) is directly related to the measurement configuration. However, existing models for compensating CMA errors do not account dynamic factors, w...
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General Energy Institute of the National Academy of Sciences of Ukraine
2024
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| author | Zaporozhets, Artur Kataiev, Denys |
| author_facet | Zaporozhets, Artur Kataiev, Denys |
| author_institution_txt_mv | [
{
"author": "Artur Zaporozhets",
"institution": null
},
{
"author": "Denys Kataiev",
"institution": null
}
] |
| author_sort | Zaporozhets, Artur |
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| collection | OJS |
| datestamp_date | 2026-07-18T12:57:48Z |
| description | Due to the influence of dynamic factors in various measurement configurations, the degree of uncertainty in measurements using a Coordinate Measuring Arm (CMA) is directly related to the measurement configuration. However, existing models for compensating CMA errors do not account dynamic factors, which impose certain limits for improving the accuracy of CMAs. To solve this issue, a method for residual error correction based on a polynomial model for single-point measurements was proposed. The influence of the CMA configuration on the residual probe error was analyzed. To enhance accuracy, a polynomial model has been developed by studying the relationship between the rotation angles of the CMA's moving elements and the probe coordinates in a cylindrical coordinate system. Experimental results demonstrate that the residual error correction method significantly compensates for instrumental uncertainty, greatly improving the accuracy of measurements using CMAs. |
| doi_str_mv | 10.15407/srenergy2024.01.045 |
| first_indexed | 2026-03-24T02:03:12Z |
| format | Article |
| fulltext |
Системні дослідження в енергетиці. 2024. 1(76) 45
ІНФОРМАЦІЙНО-ВИМІРЮВАЛЬНІ ТЕХНОЛОГІЇ,
МОНІТОРИНГ ТА ДІАГНОСТИКА В ЕНЕРГЕТИЦІ
_____________________________________________________________________________
ISSN 2786-7102 (Online), ISSN 2786-7633 (Print)
https://doi.org/10.15407/srenergy2024.01.045
UDC 531.7:62-2:629.7 (043.3)
Artur Zaporozhets, Dr. Sci. (Engin.), Senior Researcher, https://orcid.org/00000-0002-0704-4116
Denys Kataiev*, https://orcid.org/ 0000-0002-2383-3123
General Energy Institute of NAS of Ukraine, 172, Antonovycha St., Kyiv, 03150, Ukraine
*Corresponding author: kataev_d.a@ukr.net
_______________________________________________________________________________________
METHOD OF COMPENSATING FOR INSTRUMENTAL UNCERTAINTY IN MEASUREMENTS
USING A COORDINATE MEASURING ARM
Abstract. Due to the influence of dynamic factors in various measurement configurations, the degree of
uncertainty in measurements using a Coordinate Measuring Arm (CMA) is directly related to the measurement
configuration. However, existing models for compensating CMA errors do not account dynamic factors, which
impose certain limits for improving the accuracy of CMAs. To solve this issue, a method for residual error
correction based on a polynomial model for single-point measurements was proposed. The influence of the
CMA configuration on the residual probe error was analyzed. To enhance accuracy, a polynomial model has
been developed by studying the relationship between the rotation angles of the CMA's moving elements and the
probe coordinates in a cylindrical coordinate system. Experimental results demonstrate that the residual error
correction method significantly compensates for instrumental uncertainty, greatly improving the accuracy of
measurements using CMAs.
Keywords: coordinate measuring arm, measurement error, coordinate measurements, calculation method,
single-point residual correction, compensation.
1. Introduction
Development of scientific methods and standardization of procedures for assessing the accuracy of
measurements using coordinate measuring arms (CMAs) is of utmost importance to enhance the quality and
efficiency of production in the energy industry and various other sectors. These devices have become indispensable
in the manufacturing of energy-related equipment due to their simple design, easy transportability, and high
operational flexibility, as extensively discussed in references [1–2]. It has been demonstrated [3] that the
introduction of CMAs into the energy sector allows for the modernization of metrological support by eliminating
the need for designing and producing complex and metal-intensive measuring equipment, thus enabling the reverse
engineering of components. Scientific research and methods for compensating measurement errors using CMAs
have become an integral part of the energy equipment manufacturing process, aiming to ensure the highest quality
and safety.
The precision of measurements made with CMAs is influenced by both static and dynamic factors. Static
factors pertain to predictable errors arising from deviations between actual and theoretical parameters and can be
mitigated through error compensation methods such as structure parameter modeling [5], calibration, and error
compensation. On the other hand, dynamic factors are associated with errors arising from elastic deformation,
inertia, weight in connections, thermal deformation, gaps between connections, and rebound. To compensate for
these dynamic factors, it is necessary to explore the nonlinear relationships between various errors affecting
measurement accuracy and consider their influence on the instrumental component of measurement uncertainty.
Системні дослідження в енергетиці. 2024. 1(76) 46
However, isolating individual errors can be challenging, necessitating the consideration of all factors together. In
the case of CMAs, configuration is determined by motion parameters, including the lengths of struts and encoder
values in different connections. Typically, dynamic factors of CMAs can be linked to the configuration defined
by encoder values in all connections [8, 9].
If encoder readings are known, compensating for dynamic factor errors in different CMA configurations
can be determined through system modeling [10]. Guided by these findings, the relationship between the bending
angles of the arm and probe coordinates has been investigated. To improve the effectiveness of residual error
correction, a triangular measurement arm model has been explored in reference [11], taking into account the
peculiarities of CMAs with an adjacent orthogonal joint design. Additionally, the impact of compensating dynamic
factors on the instrumental component of measurement uncertainty has been considered, contributing to the
enhancement of measurement quality in various applications, particularly in the production of energy-related
equipment.
2. Methods and material
A method for correcting the residual error of Coordinate Measuring Arms (CMAs) based on a single-point
position configuration has been developed. As an example, a 6-element measuring system is used (Figure 1(a)),
the configuration of which is determined by the angles of all connecting elements of the system (Figure 1(b)), and
the position matrix P can be expressed by formula (1):
1
n
jj
P A
, (1)
where 𝐴(𝜃𝑗) is the Denavit-Hartenberg parameter matrix for the j-th connecting element, determined by the angle
value 𝜃𝑗 of this connecting element [15].
a) b)
Fig. 1. Configuration of the 6-element measuring system
Since the terminal connecting elements of the CMA, which include the fifth and sixth connecting elements,
have small length and mass, their influence on the system can be ignored [16]. The primary quality elements of
the CMA are the structural elements from the second to the fourth connecting element, which are essential
components leading to a certain regularity in the accuracy of the CMA in various positional configurations.
In conclusion, the relationship between the angles of the connecting elements (𝜃1, 𝜃2, 𝜃3) and the probe
coordinates exists in the residual error correction [17].
Investigation of the relationships between the angles of the CMA segment connections and the probe
coordinates.
Системні дослідження в енергетиці. 2024. 1(76) 47
As shown in Figure 1b, a set of configurations is established when the position of the CMA probe with the
first four connecting elements (𝜃1, 𝜃2, 𝜃3) is determined. In theory, the set of configurations can be located on the
boundary of a circle referred to as the configuration circle [18].
Fig. 2. CMA configuration circle (diagram of CMA element placement)
The coordinates of the connection point О5 (𝑥𝑂5
, 𝑦𝑂5
, 𝑧𝑂5
) can be calculated using the inverse solution of
the Denavit-Hartenberg parameter matrix, as described in [12]. With the position of О5 (𝑥𝑂5
, 𝑦𝑂5
, 𝑧𝑂5
) you can
calculate the coordinates of О2 (𝑥𝑂2
, 𝑦𝑂2
, 𝑧𝑂2
), the lengths L1, L2, and the distance between О2 and О5, denoted as
L, using formula (2) [19]:
2 5 2 5 2 5
2 2 2
O O O O O OL x x y y z z . (2)
The value of the angle 𝜃4 for the fourth connection can be calculated using formula (3):
2 2 2
1 2
4
1 2
cos .
2
L L L
L L
(3)
According to formulas (2) and (3), the position of the triangle Δ О2 О4 О5 in the coordinate system О1
(x, y, z) can be determined using the angle 𝜃3 of the third connecting element and the angle 𝜃𝑙 between the Z-axis
and the linear segment О2 О5[20]. As shown in Figure 1b, the angle 𝜃𝑙 can be calculated using formulas (4), (5)
and (6):
arccos ,l
ab
a b
(4)
2
,i j Oa O O z k (5)
5 2 5 2 5 2
,O O O O O Ob x x i y y j z z k (6)
where a is the vector in the direction of the Z-axis; b is the vector in the direction of the segment О2 О5.
Typically, the Z-axis is perpendicular to the horizontal plane.
Let's consider the dynamic deformation of the CMA as parameters Δz, Δs і Δh. As shown in Figure 3, Δz
represents the deformation along the Z-coordinate of О5; Δs represents the tangential deformation of the О5
coordinate in the XY plane, and Δh represents the radial deformation of the О5 in the XY plane [21]. (Δz, Δs, Δh)
can be referred to as residual errors in the cylindrical coordinate system of the CMA, which are discrepancies
between theoretical and practical values [22].
Системні дослідження в енергетиці. 2024. 1(76) 48
a) b)
Fig. 3. Front view of deformation (a), Top view of deformation (b)
So, the position configuration depends on 𝜃3, 𝜃4 and 𝜃𝑙; and the mapping relationship between (𝜃3, 𝜃4,
𝜃𝑙) and (Δz, Δs, Δh) can be expressed using (7):
3 4, , , , .l z s h (7)
To calculate the residual correction, the polynomial equation (8) can be used to approximate (7):
0 1 3 2 3 3 3
2
2 2 2
3 4 4 3
0
,
i i i
i i i i i i
z z z l z l
i
z k k k k
0 1 3 2 3 3 3
2
2 2 2
3 4 4 3
0
,
i i i
i i i i i i
s s s l s l
i
s k k k k
0 1 3 2 3 3 3
2
2 2 2
3 4 4 3
0
,
i i i
i i i i i i
h h h l h l
i
h k k k k
(8)
where 𝑘𝑧𝑗
, 𝑘𝑠𝑗
, 𝑘ℎ𝑗
are the coefficients of the polynomial [23].
The method for constructing a residual correction model consists of the following stages:
1. Data Collection: In the measurement space of the CMA, the horizontal plane of the base is taken as a
reference, and several positions within the range of 20–80 % from the radial direction of CMA
measurements (Figure 3) are selected. In these locations, data is collected from a single point using the
method illustrated in Figure 4. Data for 𝜃3 and 𝜃4, which are the values of the CMA's connections О3 and
О4 can be obtained from encoders. The probe coordinates and L can be calculated using equations (1) and
(4), respectively [24].
2. Calculation of Residual Values: The average values (̅x, y̅, z̅) and the coordinates of a single point (xi, yi,
zi) are used as initial data [25]. Then (Δzi, Δsi, Δhi) can be expressed using formula (9):
1 ,iz z z
2 2 sin ,i i i is x y (9)
2 2
2 2 cos , arctan arctan .i
i i i i i
i
yy
h x y x y
xx
Системні дослідження в енергетиці. 2024. 1(76) 49
Fig. 4. Determining the single-point position
Fig. 5. Sequence of positioning for determining single-point position
3. According to (8), the residual correction equations can be established along with (𝜃3, 𝜃4, 𝜃𝑙) and (Δzi,
Δsi, Δhi). The polynomial coefficients (kzj, ksj, khj) can be calculated using the residual correction
equations.
4. Compensation Calculation: To enhance the accuracy of the CMA, the compensation values (Δxi, Δzi, Δyi)
for the coordinate (xi, yi, zi) of the probe should be computed using the method of residual value correction.
According to the relationship between the cylindrical coordinate system and Cartesian coordinates, the
compensation values can be calculated using the equation.
cos sin 0 ,
cos sin 0
i i i i i
i i i i i
x h s s
x h s s
cos sin 0
cos sin 0
i i i i i
i i i i i
y s h h
y s h h
(10)
i ix x x i ix x x (11)
( 0);i iy y y x ( 0);i iy y y x ;i iz z z ,i iz z z
Системні дослідження в енергетиці. 2024. 1(76) 50
where
2 2
1
arctan arcsin .i i
i i
y s
x x y
Then, the root means square deviation (RMSD) can be calculated as (12) and (13):
2 2 2
i i a i a i ax x y y z z , (12)
2
1
i
S
n
, (13)
where n is the number of single-point measurements; (xi, yi, zi) are the coordinates for each measurement; and; (xa,
ya, za) is the mean value of measurements over n repetitions [26].
The algorithm for calculating the residual correction method is shown in Figure 6.
Fig. 6. Algorithm for calculation using the residual error correction method
The application of this method significantly reduces the instrumental component of uncertainty in
measurements using CMA.
3. Results
Using computer modeling, correction of residual errors in single-point measurements was carried out. The
structural parameters of the 6-element measurement system are provided in Table 1.
Table 1. Denavit-Hartenberg parameters for the 6-element measurement system
Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 Joint 6
li(мм) 0 -90 90 0 0 155.479
di(мм) 0 0 -650 0 -435 65.931
ai(rad) -1.57 1.57 -1.57 1.57 1.57 0
θi(rad) 0 0.001 0.002 0.003 0.005 0.04
Ten single-point measurements were conducted using the method illustrated in Figure 6. The obtained
results are provided in Table 2.
Системні дослідження в енергетиці. 2024. 1(76) 51
Table 2. Results of single-point measurements before applying the method
№ 1 2 3 4 5 6 7 8 9 10
x 0.23 0.18 0.18 0.08 0.05 -0.12 -0.15 -0.31 -0.08 -0.09
y 0.61 0.55 0.42 0.36 0.12 -0.42 -0.48 -0.42 -0.38 -0.28
z -0.26 -0.27 -0.33 -0.35 -0.29 -0.06 0.19 0.39 0.47 0.52
Data obtained after applying the single-point residual correction method are provided in Table 3.
Table 3. Results of single-point measurements after applying the method
№ 1 2 3 4 5 6 7 8 9 10
x 0.09 0.01 0.02 -0.05 0.01 0.02 0.01 -0.17 0.05 0.05
y 0.09 0.05 -0.02 -0.02 -0.12 -0.09 -0.06 0.02 0.10 0.18
z 0.04 0.04 -0.01 -0.01 0.04 0.04 -0.12 -0.05 0.01 0.17
A comparative analysis of the results of computer modeling for measurements conducted before and after
applying the single-point residual correction method is provided in Figure 7 (a, b).
a)
b)
Fig. 7. Computer modeling before applying the single-point residual correction method (a), Computer modeling after
applying the single-point residual correction method (b)
As a result, the uncertainty index of the instrumental error in measurements before applying the single-point
residual correction method is 0.59, and after applying the method, it is 0.14. Therefore, the proposed method of
residual value correction significantly compensates for the instrumental component of uncertainty, thus greatly
improving the accuracy of measurements using CMA.
4. Conclusion
The article presents a method aimed at improving the accuracy of measurements using Coordinate
Measuring Arms (CMAs). This method is based on the analysis of dynamic factors that affect the accuracy of
CMA measurements and considers their influence on measurement results. The impact of dynamic factors on
measurement accuracy has been investigated, and a method for compensating for residual errors has been
proposed. The relationship between the angles of inclination of CMA's moving elements and the probe coordinates
Системні дослідження в енергетиці. 2024. 1(76) 52
in a cylindrical coordinate system has been studied. Through systematic modeling, correlations between the angles
of inclination of moving elements and errors affecting measurement accuracy have been established.
Experimental results have shown that the proposed method significantly enhances the accuracy of CMA
measurements by reducing the instrumental component of uncertainty. Therefore, the application of the method
for compensating for the instrumental component of uncertainty in measurements using CMAs represents an
important step in improving measurement accuracy in various fields, including the production of energy
equipment. This method can be used to reduce the influence of dynamic factors and enhance the quality and
reliability of measurements using Coordinate Measuring Arms.
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МЕТОД КОМПЕНСАЦІЇ ІНСТРУМЕНТАЛЬНОЇ СКЛАДОВОЇ НЕВИЗНАЧЕНОСТІ
ПРИ ВИМІРЮВАННЯХ ІЗ ЗАСТОСУВАННЯМ КООРДИНАТНО-ВИМІРЮВАЛЬНОЇ
РУКИ
Артур Запорожець, д-р техн. наук, ст. досл., https://orcid.org/00000-0002-0704-4116
Денис Катаєв*, https://orcid.org/0000-0002-2383-3123
Інститут загальної енергетики НАН України, вул. Антоновича, 172, м. Київ, 03150, Україна
*Автор-кореспондент: kataev_d.a@ukr.net
Анотація. У зв’язку з впливом динамічних факторів у різних конфігураціях вимірювання ступінь
невизначеності при вимірюваннях за допомогою координатно-вимірювальної руки (КВР) безпосередньо
пов’язаний з конфігурацією вимірювання. Однак існуючі моделі компенсації похибок КВР не враховують
динамічні фактори, що встановлює певні межі підвищення точності КВР. Для вирішення цієї проблеми
було запропоновано метод коригування залишкової похибки на основі поліноміальної моделі для
одноточкових вимірювань. Проаналізовано вплив пози конфігурації КВР на залишкову помилку зонда.
Для підвищення точності пропонується поліноміальна модель, що була визначена шляхом вивчення
зв’язку між кутами повороту рухомих елементів КВР та координатами зонда в циліндричній системі
координат. Експериментальні результати показують, що метод корекції залишкових значень дозволяє
істотно компенсувати інструментальну складову невизначеності, що суттєво покращує точність
вимірювань за допомогою КВР.
Keywords: координатно-вимірювальна рука, похибка вимірювання, координатні вимірювання, метод
розрахунку, одноточкова корекція залишкових значень, компенсація.
Надійшла до редколегії: 08.11.2023
https://doi.org/10.3390/machines11090919
https://doi.org/10.3390/machines11090919
https://doi.org/10.1117/12.2217638
https://doi.org/10.1117/12.2217638
https://doi.org/10.1117/12.2217638
https://doi.org/10.1117/12.2217638
https://doi.org/10.1117/12.2217638
https://doi.org/10.1117/12.2217638
https://doi.org/10.1117/12.2217638
https://doi.org/10.1117/12.2217638
https://doi.org/10.1117/12.2217638
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| id | systemreorg-article-824 |
| institution | System Research in Energy |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-19T01:22:29Z |
| publishDate | 2024 |
| publisher | General Energy Institute of the National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | systemreorg/02/eaea929ffb74848ae3051a1725430502.pdf |
| spelling | systemreorg-article-8242026-07-18T12:57:48Z Method of Compensating for Instrumental Uncertainty in Measurements Using a Coordinate Measuring Arm Метод компенсації інструментальної складової невизначеності при вимірюваннях із застосуванням координатно-вимірювальної руки Zaporozhets, Artur Kataiev, Denys coordinate measuring arm, measurement error, coordinate measurements, calculation method, single-point residual correction, compensation. координатно-вимірювальна рука, похибка вимірювання, координатні вимірювання, метод розрахунку, одноточкова корекція залишкових значень, компенсація. Due to the influence of dynamic factors in various measurement configurations, the degree of uncertainty in measurements using a Coordinate Measuring Arm (CMA) is directly related to the measurement configuration. However, existing models for compensating CMA errors do not account dynamic factors, which impose certain limits for improving the accuracy of CMAs. To solve this issue, a method for residual error correction based on a polynomial model for single-point measurements was proposed. The influence of the CMA configuration on the residual probe error was analyzed. To enhance accuracy, a polynomial model has been developed by studying the relationship between the rotation angles of the CMA's moving elements and the probe coordinates in a cylindrical coordinate system. Experimental results demonstrate that the residual error correction method significantly compensates for instrumental uncertainty, greatly improving the accuracy of measurements using CMAs. У зв’язку з впливом динамічних факторів у різних конфігураціях вимірювання ступінь невизначеності при вимірюваннях за допомогою координатно-вимірювальної руки (КВР) безпосередньо пов’язаний з конфігурацією вимірювання. Однак існуючі моделі компенсації похибок КВР не враховують динамічні фактори, що встановлює певні межі підвищення точності КВР. Для вирішення цієї проблеми було запропоновано метод коригування залишкової похибки на основі поліноміальної моделі для одноточкових вимірювань. Проаналізовано вплив пози конфігурації КВР на залишкову помилку зонда. Для підвищення точності пропонується поліноміальна модель, що була визначена шляхом вивчення зв’язку між кутами повороту рухомих елементів КВР та координатами зонда в циліндричній системі координат. Експериментальні результати показують, що метод корекції залишкових значень дозволяє істотно компенсувати інструментальну складову невизначеності, що суттєво покращує точність вимірювань за допомогою КВР. General Energy Institute of the National Academy of Sciences of Ukraine 2024-02-16 Article Article application/pdf https://systemre.org/index.php/journal/article/view/824 10.15407/srenergy2024.01.045 System Research in Energy; No. 1 (76) (2024): System Research in Energy; 45-53 Системні дослідження в енергетиці; № 1 (76) (2024): Системні дослідження в енергетиці; 45-53 2786-7102 2786-7633 en https://systemre.org/index.php/journal/article/view/824/730 Copyright (c) 2024 Artur Zaporozhets, Denys Kataiev https://creativecommons.org/publicdomain/zero/1.0 |
| spellingShingle | coordinate measuring arm measurement error coordinate measurements calculation method single-point residual correction compensation. Zaporozhets, Artur Kataiev, Denys Method of Compensating for Instrumental Uncertainty in Measurements Using a Coordinate Measuring Arm |
| title | Method of Compensating for Instrumental Uncertainty in Measurements Using a Coordinate Measuring Arm |
| title_alt | Метод компенсації інструментальної складової невизначеності при вимірюваннях із застосуванням координатно-вимірювальної руки |
| title_full | Method of Compensating for Instrumental Uncertainty in Measurements Using a Coordinate Measuring Arm |
| title_fullStr | Method of Compensating for Instrumental Uncertainty in Measurements Using a Coordinate Measuring Arm |
| title_full_unstemmed | Method of Compensating for Instrumental Uncertainty in Measurements Using a Coordinate Measuring Arm |
| title_short | Method of Compensating for Instrumental Uncertainty in Measurements Using a Coordinate Measuring Arm |
| title_sort | method of compensating for instrumental uncertainty in measurements using a coordinate measuring arm |
| topic | coordinate measuring arm measurement error coordinate measurements calculation method single-point residual correction compensation. |
| topic_facet | coordinate measuring arm measurement error coordinate measurements calculation method single-point residual correction compensation. координатно-вимірювальна рука похибка вимірювання координатні вимірювання метод розрахунку одноточкова корекція залишкових значень компенсація. |
| url | https://systemre.org/index.php/journal/article/view/824 |
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