Calibration of a Hybrid Machine Tool from the Point of View of Positioning Accuracy

: The development of machine tools in the last twenty years includes, among other things, the application of mechanisms with a non-linear kinematic structure as the mechanical basis of machines. This results in significant improvements in kinematic characteristics and problems related to non-linear dependencies of the accuracy of the drive elements and the realization of movement in the machine’s external coordinates. The paper presents an approach to machine tool calibration based on the original O-X glide mechanism based on the ISO 230-4 standard with the mono-and bi-directional compensation of systematic errors and adaptation to the specifics of the mechanism’s kinematics. A machine tool prototype was designed and built for the research presented in the paper. The obtained results indicate the possibility of applying the existing recommendations and standards for testing the accuracy of machine tools with the need to correct the methodology by using linear and non-linear kinematic structures in machine tools.


Introduction
The development and improvement of machine tools since the mid-1950s has included research on improving the kinematic structure, components, and subsystems.This ensures the fulfillment of market requirements, which implies more accurate and efficient machining and the application of higher cutting speeds while reducing processing errors.
An important role in the creation of foundations for the maximum use of existing tools and the development of new machine tools is played by tests of inaccuracies that arise during exploitation and the definition of calibration procedures.Research has shown twenty-one errors in machine tools with three numerically controlled axes with a linear kinematic structure [1].However, it should be emphasized that there is no unequivocal systematization and definition of machine tool errors in the literature.The systematization of machine tool errors is most often performed according to their sources [2]: presented an overview of the measurement of quasi-static kinematic errors [3]; a very similar, modernized view has been given elsewhere [4].W. Gao et al. presented an overview of the most modern measurement methods, mathematical models, and compensation strategies for calibrating machine tools [5].The basics of error compensation and available methods for measuring machine tool geometric errors are summarized and updated in [3].In Refs.[1,6], the techniques were proposed and are based on artifacts.This research enables the control of machine tool error mapping and calibration by compensating them.A methodology for quantitatively evaluating the success of machine tool calibration methods was proposed by Bringmann et al. [7].
Improvement of the mechanical concept of machine tools in the previous period included applying non-linear kinematic structures (parallel and hybrid mechanisms) to improve the machines' exploitation characteristics.Therefore, research was initiated into the accuracy and calibration of such machine tools.The first paper on the accuracy of these machines analyzes the accuracy of mechanisms based on the Stewart platform or other types of parallel mechanisms, in which external sensors were used to measure the position and orientation of the final end-effectors.A kinematic model was used to evaluate the influence of parallel manipulator mechanism errors relative to the final end-effector's positioning accuracy.The paper [8] proposed a new method of identifying mechanism errors for a 3-PUU parallel mechanism.It is based on the inverse kinematics model and volumetric error measurement techniques using a 3D laser ball bar device.The experimental results combined with the optimization technique can determine the errors of the mechanism of the tested machine, i.e., the structural symmetry errors of the rods and joints.This method can help machine tool manufacturers to determine the basic length parameters (rod lengths) and fine-tune the kinematics (by determining the position of joints or rods) in the assembly phase.
Based on the results obtained from research on the impact of parallel kinematic structure on mechanical properties, the first commercial solutions of machine tools with parallel kinematics appeared at the end of the twentieth century.That solution enabled significant improvements in stiffness, accuracy, and movement dynamics.With the implementation of the new kinematic concept, it was observed that the application of parallel mechanisms in the development of machine tools, in addition to several advantages [9], also has certain disadvantages [10,11] that determine the area of their application (dimensions and shape of the workspace and specific control structure).
In order to make maximum use of the advantages of serial and parallel mechanisms and to reduce the impact of their shortcomings, hybrid, serial-parallel mechanisms have been used in the development of machine tools for many years.In this way, a good compromise solution that has proven to be applicable to the production of machine tools, manipulation systems, and industrial robots is provided.Examples of successful applications of hybrid mechanisms [12] include Tricept [13], Execon [13], Ecospeed [14], Geodetic Hexapod [15], and George V [16].
In machine tools with serial kinematics, the kinematic relationship between the executive part of the machine and the moving elements is linear.Therefore, we can call them machine tools with linear kinematics.In contrast, the kinematic relationships of machine tools with parallel or hybrid kinematics are significantly more complex and not linear, which is why they can be called machine tools with nonlinear kinematics [17].
In current research on the characteristics of machine tools with linear and non-linear (parallel and hybrid) kinematics, problems related to the accuracy of the machines [18], as well as the possibilities of their calibration [19], are often considered.Calibrating machine tools ensures maximum utilization of the machine's mechanical and control systems.The calibration of machine tools is a process that is carried out periodically in order to verify and improve the characteristics of machine tools [5].This process is carried out to reduce or eliminate errors due to thermal, static, dynamic loads, and control system errors.
Calibration of linear axes of machine tools includes the following: • Testing the positioning accuracy of the complete machine; Appl.Sci.2024, 14, 5275 3 of 17

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Identification of systematic and random errors; • Their removal or numerical compensation in the control system of the machine tool.
The fundamental problem of applying this procedure to machine tools with parallel or hybrid kinematics is the need to recalculate the errors on the internal coordinates of the machine defined by the nonlinear kinematics of the mechanism, even when defining the calibration parameters, and the test parameters (movement method, movement speed, stopping time) that are not explicitly defined by the standard adapt to the appropriate mechanism and its characteristics.
This paper presents part of the research conducted to determine the influence of machine tool calibration with an O-X glide kinematic structure on the positioning accuracy of the moving platform.The research was conducted to enable the compensation of errors through the calibration process by applying standardized techniques for identifying errors in the mechanical structure.In the paper, research was carried out according to the ISO 230-4 [20] standard, in static conditions, on a machine tool prototype that was made for research.In contrast, the standard parameters for which no selection recommendation was varied in a range corresponding to the exploitation values for a specific kinematic machine configuration.The results were processed according to the kinematic differences between the mechanical structure of machines with linear and non-linear kinematics.The calibration was carried out using the mono-and bi-directional compensation of systematic errors obtained by measuring accuracy.

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.

Inverse and Direct Kinematics
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 P , z P ); • r P1 and r 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 R1 , r R2 -vectors of the positions of reference points of the particular sliders (R 1 and R 2 ); • r u1 , r 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.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]: 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.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.
Expressed in coordinates: 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.
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": Appl.Sci.2024, 14, 5275 5 of 17 For cross mechanism configuration "X": Upon closer examination, we find that the expressions we have derived are structurally identical, with the only difference being the sign in front of the square root.This square root is a key component, representing the basic relationship between the internal and external coordinates of the mechanism, which is essential for configuring the control system.
The expression can generally describe the solution of direct kinematics: By transforming the previous expressions, it is determined on the basis of Equations ( 4) and ( 5) and the following expression is obtained: based on which, we derive x p as z is obtained by inserting expression x p into Equation (5): After the substitutions are introduced in the previous expressions, we derive Based on the previous equations, the following expressions are obtained: The obtained expressions were used to configure the control system of the machine tool prototype based on the O-X glide mechanism.

Realized Prototype
The supporting structure of the machine tool prototype based on the O-X mechanism is made of aluminum profiles with adjustments that allow the installation of IGUS linear axis sub-assemblies with NEMA 19 step motors.The moving platform is made of aluminum and designed in accordance with the schedule of mechanical loads that occur during processing.The connection elements between the platform and the linear axes are realized in the form of adjustable rods made of structural steel.Figure 2 shows the appearance of the realized prototype.

Realized Prototype
The supporting structure of the machine tool prototype based on the O-X me nism is made of aluminum profiles with adjustments that allow the installation of I linear axis sub-assemblies with NEMA 19 step motors.The moving platform is ma aluminum and designed in accordance with the schedule of mechanical loads that o during processing.The connection elements between the platform and the linear axe realized in the form of adjustable rods made of structural steel.Figure 2 shows th pearance of the realized prototype.For prototyping and testing the influence of the calibration process of indiv internal axes on positioning accuracy in external coordinates, a control system based PC with LinuxCNC open architecture control system, which enables control of mach with a non-linear kinematic structure, was used (Figure 3).For prototyping and testing the influence of the calibration process of individual internal axes on positioning accuracy in external coordinates, a control system based on a PC with LinuxCNC open architecture control system, which enables control of machines with a non-linear kinematic structure, was used (Figure 3).

Positioning Accuracy Test and Applications in Machine Tool Calibration
The positioning accuracy test procedure, which is necessary for the condition analysis and calibration of machine tools, is defined by numerous standards and recommendations.ISO 230-2 [22], VDI/DGQ 3441 [23], ANSI B5.54 [24], JIS 6333 [25], and others are currently used.All procedures involve the exposure of machine elements to programmed movement along the measured axis while simultaneously measuring the

Positioning Accuracy Test and Applications in Machine Tool Calibration
The positioning accuracy test procedure, which is necessary for the condition analysis and calibration of machine tools, is defined by numerous standards and recommendations.ISO 230-2 [22], VDI/DGQ 3441 [23], ANSI B5.54 [24], JIS 6333 [25], and others are currently used.All procedures involve the exposure of machine elements to programmed movement along the measured axis while simultaneously measuring the achieved displacement with the machine's measuring system and reference based on instrumentation, including a laser interferometer with a compensator for environmental conditions.In this way, it is possible to compare the values measured on the machine and forward them to the machine's control system with reference values.
The conducted research is based on the procedure defined by the ISO 230-2 standard with compensation for environmental conditions.The procedure includes placing the laser measuring head (1), interferometer (2), retroreflector (3), acquisitional device (4), compensator of external conditions (5), external condition measuring device (6), sensor of temperature of machine elements (7), and computer analyzing data (8), according to the scheme shown in Figure 4.The procedure for testing the positioning accuracy of machine tools implies th machine is programmed to perform an incremental movement along the measure in an automated cycle.Stopping the movement is conducted with a pause at the priate points according to the standard's requirements (in the positive and negat rection of movement from the start to the end programmed point).Realization of t procedure gives the values of xi.The results are presented tabularly and in the for diagram.According to the standard, the processing of measurement results includ calculation of the following statistical quantities.These are reversal error at a posi point "i" (Bi), reversal error of an axis (B), unidirectional positioning repeatability axis (R), unidirectional systematic positioning error of an axis (E+ and E−), bi-dire systematic positioning error of an axis (E), mean bi-directional positioning error axis (M), unidirectional positioning error of an axis (A+ and A−), and bi-directiona tioning error of an axis (A).The procedure for testing the positioning accuracy of machine tools implies that the machine is programmed to perform an incremental movement along the measured axis in an automated cycle.Stopping the movement is conducted with a pause at the appropriate points according to the standard's requirements (in the positive and negative direction of movement from the start to the end programmed point).Realization of the test procedure gives the values of x i .The results are presented tabularly and in the form of a diagram.According to the standard, the processing of measurement results includes the calculation of the following statistical quantities.These are reversal error at a position in point "i" (B i ), reversal error of an axis (B), unidirectional positioning repeatability of an axis (R), unidirectional systematic positioning error of an axis (E + and E − ), bi-directional systematic positioning error of an axis (E), mean bi-directional positioning error of an axis (M), unidirectional positioning error of an axis (A + and A − ), and bi-directional positioning error of an axis (A).
The obtained results can be displayed as a standardized diagram showing the mean and individual deviation values from the programmed position, which enables the axis to be calibrated.
The analysis of the obtained results in the form of tables and diagrams enables the calibration of individual machine tool axes by introducing predictive error values at individual machine locations.
Calibrating machine tools with non-linear kinematics represents an additional challenge since the error is measured experimentally and expressed in external machine coordinates (the position of the executive body of the machine in the MCS coordinate system).Compensation values are entered in internal coordinates (in individual axis coordinates).Bearing in mind that movement in the direction of the X axis is realized by constant movement in the direction of the u 1 and u 2 axes and the direction of the Z axis in opposite directions at the same speed, the compensation value is determined for each point of movement according to the expression obtained by forming the first differential of the equation for internal coordinates.This gives expressions for the errors at specific points u 1i and u 2i , corresponding to the measurement points x pi and z pi :

LinuxCNC and the Machine Calibration Methodology
The control system of the realized prototype is based on PC architecture and applies a modified kernel of the Linux operating system adapted for calculations of interpolation functions in real-time (real-time Linux) kernel and LinuxCNC 2.9.2 Debian 12 Bookworm RTAI management software.This type of control system was chosen for the prototype because of the open-source license, the possibility of defining non-linear kinematics, and the possibility of compensating systematic errors according to several criteria.The use of LinuxCNC primarily refers to the possibility of compensating for the backlash of the ballscrew spindle and systematic positioning errors according to the unidirectional and bidirectional methods.Error correction is realized by creating a file that introduces compensations for the indicated coordinates of the axis position (internal coordinates, i.e., slider position) in the machine coordinate system, eliminating systematic errors caused by the geometry of the spindle, guides, and assembly of machine elements.
There are two procedures for calibrating the measuring system: unidirectional, which enables the systematic errors at a certain point to be approximately brought to zero; and bidirectional, which enables separate calibration of the measuring system for movement in the positive and negative directions.The unidirectional method is used to adjust machine tools of low-and medium-accuracy classes.The bidirectional method, an additional function in most control systems, adjusts precision machine tools and coordinates measuring machines.
Table 1 shows the layout of the parameter file for bidirectional error compensation.The first column of the file represents the coordinates of the internal axis in the machine's coordinate system.In contrast, the other two represent the compensation values of the points when moving in the positive and negative directions.Position coordinates, as well as compensation values, are given in the table in millimeters.During calibration, it is possible to choose whether the compensations are specified as a compensation value or a real coordinate value for a given position.

Presentation of Tests and Results
Calibration of the prototype O-X glide machine tool was realized using measuring equipment for laser interferometry manufactured by HP (laser head 5500 C, display 5505 A, interferometer 10565 B retroreflector 10550 B, and compensator of external conditions 2044 A). Figure 5 shows the setup of the measuring equipment on the machine prototype during measurement, where (1) denotes the laser measuring head, (2) indicates the interferometer, and (3) signifies the retroreflector.

Presentation of Tests and Results
Calibration of the prototype O-X glide machine tool was realized using measuring equipment for laser interferometry manufactured by HP (laser head 5500 C, display 5505 A, interferometer 10565 B retroreflector 10550 B, and compensator of external conditions 2044 A). Figure 5 shows the setup of the measuring equipment on the machine prototype during measurement, where (1) denotes the laser measuring head, (2) indicates the interferometer, and (3) signifies the retroreflector.Data acquisition and processing were implemented using the Matlab r2024a programming environment, which included an application for compensating environmental conditions, calculating errors, and generating graphics created for research purposes.
Testing and calibration of the hybrid machine tool was carried out in three stages: • Testing the machine in an uncompensated state for all three external axes.After that, the characteristic error sizes were calculated, and graphs were created for all three axes; • Defining compensation values for the unidirectional compensation method, creating compensation files for all three internal axes.After compensating for the errors, the machine was tested after calibration; • Defining compensation parameters for movement in the positive and negative direction for all three axes (bidirectional compensations).
For the Y axis (serial axis), compensation values were expressed directly from the measurement results, while for X and Z (plane parallel mechanism segments), compensation parameters were calculated for each point according to Expression (11).
Figure 6a,b show graphs for the initial state and bidirectional error compensation for the Y axis.Data acquisition and processing were implemented using the Matlab r2024a programming environment, which included an application for compensating environmental conditions, calculating errors, and generating graphics created for research purposes.
Testing and calibration of the hybrid machine tool was carried out in three stages: • Testing the machine in an uncompensated state for all three external axes.After that, the characteristic error sizes were calculated, and graphs were created for all three axes; • Defining compensation values for the unidirectional compensation method, creating compensation files for all three internal axes.After compensating for the errors, the machine was tested after calibration; • Defining compensation parameters for movement in the positive and negative direction for all three axes (bidirectional compensations).
For the Y axis (serial axis), compensation values were expressed directly from the measurement results, while for X and Z (plane parallel mechanism segments), compensation parameters were calculated for each point according to Expression (11).The presentation of numerical values of characteristic quantities before the compensation process as well as after unidirectional and bidirectional calibration is shown in Table 2.

Measurement Uncertainty of Positioning Accuracy Measurement
Following the ISO 230 standard, analysis of the test conditions and used measuring equipment was performed for the conducted tests, and in accordance with the ISO 230-9:2005 [26] standard, the measurement uncertainty was presented.Measuring equipment is set to meet all manufacturer's regulations and minimize unwanted sources of errors such as misalignment error, Brayan error (generalized Abbe error), dead path error, etc.
During the measurement, the laboratory temperature was in the range of 23 • C ± 1 • C. The measurement time did not last longer than 15 min for an individual axis.The instruments were not previously calibrated, so the uncertainty values provided by the equipment manufacturer were used.The instrument was aligned within 1 mm of the axis under test by aligning the return beam of the laser beam.The repeatability of the setting was in increments of 50 mm, and the deflection gradients were a maximum of 50 µm/m.The environmental variation error was taken from recommendations [27].The laser interferometer was positioned so that the path error could be minimized.Table 1 shows the obtained uncertainty values for the farthest measurement point for all axes.The laser measurement system used includes automatic compensation of the environment and the coefficient of expansion.u M,Device and u E,Device were considered zero.For steel, a thermal expansion coefficient of α = 0.012 µm/mm • C was used, and the uncertainty range was set to ∆α = 0.002 µm/mm • C, following ISO recommendations.
For testing, a helium-neon laser interferometer with automatic Zeeman-split twofrequency output, model number HP 5500C, was used.Table 3 presents the influential parameters of measurement uncertainty for the equipment used.All values of measurement uncertainties for the observed measurements are within the limits defined by the standard.

Discussion
By analyzing the obtained results, the construction and exploitation characteristics of the prototype, the efficiency of the calibration process in the specific case, and the applicability of the adopted calibration methodology to machines with non-linear kinematics can be seen.
Analysis of test graphs during the initial measurement (without calibration) and after compensating errors on all three axes indicate that the Y axis of the machine tool prototype has a significant contribution of random error in the measurement.This can be explained by the construction solution of the prototype, which includes the traversal Y axis with the drive sub-assembly on one side of the gantry.Therefore, the calibration effects are the least pronounced for this axis, and the error value is the highest.The remaining two spatial axes (X and Z) created by the parallel mechanism performed very well from the point of view of random errors, bearing in mind that the prototype has an open steering coupling.
As in the case of serial mechanisms, the test indicates the necessity of calibrating machine tools with non-linear kinematics.This is indicated by the reduction values of the two-way system error along the axis (E) and the two-way positioning error along the axis (A).The results are shown in Table 5.The obtained results indicate the efficiency of the process of calibrating machine tools as well as the possibilities realized by the application of unidirectional and bidirectional calibration for all examined axes.
Calibrating machine tools with non-linear kinematics implies the process of defining compensation values, which includes the conversion of external coordinates into internal coordinates, as well as calculating the contribution of the internal axis to the error that occurs at the top of the tool during movement.This process includes the application of expressions determined by the inverse kinematics of the mechanism and the modification of the calibration procedure following the specifics of the specific type of mechanism.In all calibration stages, measurements of current positioning accuracy values according to ISO 230 were performed three times.Control measurements showed that only two measurements (before and after the introduction of compensations) were needed to achieve the full effects of the process.This confirms the hypothesis that the calibration process of machine tools with non-linear kinematics can be realized with the same efficiency level as for machines with a linear kinematic structure.
During positioning accuracy measurements, as per ISO 230-2, the platform's speed was constant.Since the measurements were made on a machine tool with non-linear kinematics, the slide speeds were not linearly related to the platform speed, resulting in variable slide speeds during the measurement.This variable speed can affect the resulting compensation factors.During this positioning accuracy check procedure, of the two machine axes that are created with a parallel mechanism (X and Z), the X axis behaved kinematically identically or at least very similarly to the movements that are performed when checking the accuracy of machine tools with linear kinematics, since in this case the two sliders move at the same constant speed as the platform moves.On the other hand, the influence of the movement of the platform along the Z axis at constant speed on the sliders is non-linear, and different speeds and accelerations occur.This is the cause of the appearance of an additional error in the total positioning error along the mentioned axis.As a result, the calibration process had less efficiency (less error reduction).Further tests should include the influence of differences in kinematic and dynamic movement parameters on the mechanical components of mechanisms and include those results in defining compensatory parameters.Additionally, a more profound observation of such influences (observing uneven movements when measuring and measuring two or more axes) should include tests with circular interpolation using the so-called Ballbar device.

Conclusions
Improvements in machine tools in the 21st century are carried out in several directions.One of the most important is improving their mechanical structure through the search for solutions that are adaptable to specific application areas.By applying reconfigurable building principles and non-linear kinematic forms, solutions are realized where, in addition to an adjustable workspace, variable kinematic and dynamic characteristics of machine tools are obtained with a whole series of advantages and disadvantages compared to machine tools with serial kinematics.In such conditions, the analysis of the accuracy of the realized movements and the definition of the methodology of its determination following the valid standards represents an essential step towards a unique process of calibrating machines that considers the differences in the kinematic structure.Testing the accuracy of positioning and calibration of the hybrid machine tool described in the paper is a step in this direction.It represents an introduction to the formation of a unique methodology for testing the volumetric accuracy of machine tools, which takes into account the machine tool's kinematic characteristics, in addition to the components' characteristics.
Based on the obtained results, it can be concluded that the application of bidirectional error compensation on all three external axes of the O-X slide of the machine tool provides the complete elimination of discrete damage to the lead screw spindle and sliders as well as the gap in the nut.By analyzing the systematic errors that occur with machine tools, applying this error compensation method eliminates the need to decompose errors into individual errors and eliminate them with discrete values.
Positioning accuracy results obtained after the calibration process fully correspond to machine tools with open-loop control and applied stepper motors.They can even be considered very good, except for the Y axis, where the error is somewhat more significant due to construction reasons (on one side of the gantry is the driving element, and on the other is the sliding, driven element).
The research conducted on the results presented in the discussion indicates the influence of the stopping time, the feedrate movement speed, and the movement method during measurement.The standards determine all these influential parameters but without recommendations on when and how to choose them according to the machines' kinematic structure, purpose, and working conditions, which directly impact the calibration process.In addition, the results of such research create prerequisites for the application of Edge computing technologies to compensate for errors in real time, which is a continuation of the presented research.

Figure 1 .
Figure 1.Kinematic model of the O-X glide parallel mechanism.

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P-position of the moving platform expressed in external coordinates of the mechanism (xP, zP); • rP1 and rP2-vectors of moving platform, which defines the orientation of platform and distance between the end of rods and position of point P (P1 and P2); • rR1, rR2-vectors of the positions of reference points of the particular sliders (R1 and R2); • ru1, ru2-vectors of the slide of the parallel mechanism in relation to the axis reference point, internal coordinates of the mechanism; • l1, l2-vectors of rods of constant length; • d-distance between sliders.

Figure 1 .
Figure 1.Kinematic model of the O-X glide parallel mechanism.

FigureFigure 6 .
Figure 6a,b show graphs for the initial state and bidirectional error compensation for the Y axis.Appl.Sci.2024, 14, x FOR PEER REVIEW 11 of 18

FigureFigure 6 .
Figure 7a,b show the graphs for the initial state and bidirectional error compensation for the X axis.

Figure
Figure 7a,b show the graphs for the initial state and bidirectional error compensation for the X axis.

Figure 6 .
Figure 6.Initial condition (a) and test results after error compensation (b) for Y axis.

FigureFigure 7 .
Figure 7a,b show the graphs for the initial state and bidirectional error compensation for the X axis.

Figure 7 .
Figure 7. Initial state (a) and test results after error compensation (b) for the X axis.

Figure
Figure 8a,b show graphs for the initial state and bidirectional error compensation for the Z axis.

Figure 7 .
Figure 7. Initial state (a) and test results after error compensation (b) for the X axis.

Figure
Figure 8a,b show graphs for the initial state and bidirectional error compensation for the Z axis.

Figure 7 .
Figure 7. Initial state (a) and test results after error compensation (b) for the X axis.

FigureFigure 8 .
Figure 8a,b show graphs for the initial state and bidirectional error compensation for the Z axis.

Table 1 .
Values of bidirectional error compensation.

Table 2 .
Results of the calibration process of the O-X glide mechanism.

Table 3 .
Table of influential parameters of measurement uncertainty.

Table 3 .
Cont.Uncertainty parameters were also determined based on the measurement length.The obtained values are presented in Table4.

Table 4 .
Uncertainty values of the measurement parameters.

Table 5 .
Analysis of the success of the O-X glide mechanism calibration process.