Assessing path accuracy in industrial robots via ballbar technology

In the evolving landscape of Industry 4.0, the integration of industrial robots (IR) into production lines not only improves operational efficiency but also supports advanced manufacturing techniques. While IR excel in applications demanding high process speed and repeatability (e.g. assembly, pick-and-place, packaging, etc.), they face considerable limitations in tasks requiring high accuracy during pose reaching or path following Iglesias et al. (2015). Accuracy requirements in robotics vary significantly depending on the application, ranging from 0.1 mm in medical and precision assembly tasks (Song et al., 2009) to tolerances around 0.5–1 mm in less sensitive assemblies (Li et al., 2022) or heavy equipment welding, which may tolerate higher margins. Precision-driven fields such as machining demand accuracies of 0.01 mm or finer, where challenges are intensified by significant process forces and the relatively limited structural stiffness of most IR with open kinematic chains, as discussed in Wang et al. (2020), Verl et al. (2019) and Cen and Melkote (2017).

The IR accuracy errors, which vary significantly across different brands (see e.g. the comparison reported in Slamani et al., 2015) arise from various factors. According to Lehmann et al. (2013), these factors can be categorized into manufacturing-related (such as the characteristics and tolerances of links and joint components), environmental (such as temperature-induced thermal drift), installation, robot control software and sensors limitations. From a functional standpoint, the majority of the errors come from the robot’s joints, where servomechanisms, usually consisting of servomotors and speed reducers, are installed. As well documented in Slamani et al. (2012a), Jiang et al. (2024) and Xing et al. (2024), the complex dynamic behavior of these actuation modules is influenced by numerous operational parameters, such as speed, applied load, oil temperature and wear, which is challenging to quantify without disassembling the robot. This multitude of parameters, many of which are difficult to retrieve on commercial IR, poses significant challenges for the development of reliable models of the robot positioning errors via solely analytical or simulation approaches (Zhang et al., 2021). Therefore, experimental campaigns become essential to enrich behavioral models and obtain precise error mappings. Given that similar IR (in type and size) from various vendors use different mechanical solutions for motion transmission, the resulting error model applies uniquely to the single robot, which also inevitably undergoes behavioral changes over its life cycle due to wear.

In this context, there is an evident need for standardized and robust experimental procedures that can possibly be applied to any serial robot to rapidly assess its accuracy performance and enable the implementation of online (closed-loop) or offline (predictive, model-based) compensation strategies, thereby enhancing the robotic process efficiency (Ye et al., 2022; Wang et al., 2022). Despite the extensive research work conducted on IR motion accuracy over the past few decades, as evident by the dense literature available, there is still no uniformity in testing procedures, which limits the utilization and comparison of outcomes from different studies and hampers their application at industrial level.

In most industrial settings, robot motion is imposed via offline programming resorting to the standard motion instructions, namely, point-to-point, linear and circular movements (with some controllers also allowing splines). While thorough investigation has been conducted on the IR accuracy when performing the first two movement types, less attention has been dedicated to circular movements. This is also confirmed by the reference standard (ISO 9283), which predominantly focuses on linear movements within the test cube during motion tests. Measurements are commonly performed with laser trackers or vision systems, which are highly effective for any robotic motion type (Wang et al., 2020; Reun et al., 2022). In the specific case of linear movements, laser interferometers show higher performance (Kuric et al., 2020), whereas circular movements can be approached also via the use of ballbars (Slamani et al., 2015; Garnier and Subrin, 2022). The latter are commonly used to access the motion performance in the field of machine tools and have not yet been fully adopted in robotics. They consist of a linear transducer that captures variations in length between its two ends, one fixed and the other guided in a circular movement. In this way, the output data reflects the deviations recorded in the path radius. On the other hand, the rotation angle during the circular traveling is not provided by the ballbar and must be obtained from other feedback sources to properly distribute the data over the circular domain during post-processing. Despite this limitation, ballbars offer excellent measuring capabilities (accuracy and sampling rate up to 1 μm and 1000 Hz) and are more cost-effective compared to laser trackers (€15k vs €70k). In addition, ballbar measurements require a basic set of light-weight portable tools and minimal effort for preparation and setup.

Commercial software (e.g. the one by Renishaw) exist for conducting ballbar measurements and related data processing (Kuric et al., 2020), allowing the automatic diagnosis and error identification in Cartesian Computer Numerical Control (CNC) machines. These software tools generate detailed reports based on the ISO 230 standard. However, they are ineffective with most IR as these typically involve six interpolated axes to achieve a circular movement, unlike the two perpendicular axes used in Cartesian machines, making proper diagnosis (i.e. correlating the measured path errors with the contribution from each joint) difficult. In addition, the low sampling rate of the commercial software (30 Hz vs the ballbar’s maximum of 1000 Hz) limits the detection of high-dynamic servomechanism errors during robot testing. Previous studies have already explored the use of ballbars for IR characterization, primarily focusing on result analysis while also providing a solid foundation for future applications (see e.g. the path compensation algorithm proposed in Khaled et al., 2021). However, the used test methodologies are often poorly described. The lack of dedicated testing procedures for path accuracy analysis on IR is also evident in standards like ISO 230 and ISO 9283, highlighting the need for new engineering methods and tools. Such resources are essential for conducting thorough experimental investigations with ballbars on IR and achieving optimal measurement quality.

Building upon the previous considerations and with the aim to fill the identified gaps, the present paper provides the following novel contributions:

• Definition of a generalized, robust and easy-to-use method to assess the path accuracy in serial IR using ballbar technology, covering aspects such as installation, setup, measurement, test procedures, triggering and data synchronization during post-processing. The reported method is validated on high payload KUKA IR installed within a robotic deburring cell, and all measurements are carried out with a Renishaw QC20-W ballbar.

• Perform an experimental assessment of the KUKA robot accuracy performance by testing circular movements under different working conditions (speed, payload, working plane, circle radius, robot configuration). A statistical analysis is performed on the collected results to identity the most influencing parameters.

• Provide practical considerations to correlate the overall error committed at the end-effector with the contributions of the first three robot joints.

The collected results, along with the custom acquisition software developed for this study, are organized and shared as an open research data set to support future advancements.

The rest of the paper is structured as follows: Section 2 describes the experimental setup, Section 3 details the proposed procedure for assessing the robot accuracy, whereas Section 4 presents and discusses the results of the experimental campaign conducted on the KUKA robot. The concluding remarks are given in Section 5.

This section describes the equipment used for the entire set of experiments, conducted within a flexible robotic cell installed at the University laboratory. The setup includes the following equipment, as schematized in Figure 1:

• KUKA KR210 R2700 Prime serial IR having a total mass of 1111 kg, a maximum payload of 210 kg and a maximum reach of 2700 mm. The IR is controlled by a KRC4 cabinet, featuring the software KSS version 8.3.25. The robot motion is programmed with standard offline approaches (PTP, LIN and CIRC instructions for point-to-point, linear and circular movements, respectively), with a script produced on the lab PC within the KUKA WorkVisual software and then downloaded on the KRC4 computer, which is in charge of the entire robot control and supervision. During the experiments, the KUKA tracing function is also enabled to sample position data every 12 ms (for signal synchronization purposes, as explained in Section 3). The robot position is calculated by the kinematic model embedded in the controller and the feedback recorded in the servomotors encoders. Alternatively, if available, the KUKA Robot Sensor Interface package can be used to achieve a sampling period of 4 ms. However, in this case, the motion is typically not programmed with standard motion commands but rather streamed to the controller as a series of points with an external trajectory generator.

• Renishaw QC20-W ballbar linear metrology system with a measurement range of ±1 mm and an accuracy of 1 μm. One end of the ballbar is fixed to the magnetic support mounted on a tripod, while the other end is attached to the robot’s end-effector via a metal plate and a tool cup provided in the Renishaw’s kit (shown in the enlarged view of Figure 1). The test radius can be varied as different calibrated bars are available in the kit. Data is sampled in real time at a rate up to 1000 Hz, temporarily stored into the ballbar’s internal buffer, and subsequently streamed via Bluetooth to the lab PC at 28.5 Hz. In this setup, measurements are managed using a custom application written in C# exploiting the Renishaw’s Application Programming Interface (API) to ensure the highest sampling rate and obtain preprocessed raw data the form of an array containing the sequence of data points with their associated timestamps.

In addition, extra equipment can be considered if available, such as a force/torque sensor attached to the robot’s flange and a laser tracker. These would enhance the experiment preparation by respectively providing accurate estimation of the moved payload and of the Tool Center Point (TCP) to be set in the robot controller (further details on this are given in Section 3.2 and in Slamani et al., 2024). In this research work, an ATI 9105-NET-Omega191 transducer and a FARO Vantage laser tracker have been used. It shall be remarked that payload and TCP information could also have been estimated, although with less accuracy, from well-defined Computer Aided Design 3D models.

The ballbar can measure only a single degree of freedom, specifically the length to which it extends. Outside its measuring range, the instrument will output a constant feedback equal to the stroke limits (1 mm or −1 mm). According to the ISO 230, detailing the test practices for machine tools, the accuracy assessment is commonly executed by performing circular paths on planes parallel to the base coordinate system (XY, XZ and YZ), shown in Figure 1. Nevertheless, when adopting the ballbar for robotic applications, extra checks are necessary before starting the assessment. At first, the path errors committed by the robot must be within the ballbar measuring range. Such condition may not be always met as many heavy-duty IR present higher errors, especially if no ad hoc control modules are implemented to mitigate the major inaccuracies (e.g. the Absolute Accuracy package by KUKA). Then, with reference to Figure 2, the assumption that the measured displacement effectively describes the in-plane motion error is based on the following hypotheses:

• The radius of the nominal circle (r_n) is significantly larger than the measured quantity (e_tot).

• The error in the direction orthogonal to the plane (e_o) is of a comparable order of magnitude to the error incurred in the plane (e_p).

The first statement can be considered valid as the maximum error detectable by the ballbar is 1 mm, and the nominal radius of the experiment is at least 150 mm. As for the second statement, in presence of slight unwanted out-of-plane motions when executing an ideally planar circular path, the difference between e_tot and e_p must be low to validate the experiment. According to Figure 2, such difference is defined as:

which, after applying the Taylor expansion to the right term and considering $rn+ep≈rn$⁠, becomes:

In the worst case, i.e. by assuming e_o = 1 mm and r_n = 150 mm, δ is approximately 3.3 μm. This value is on the same order of magnitude as the instrument’s nominal accuracy (1 μm) and significantly below the accuracy level of IR, leading to consider $etot≈ep$⁠.

As previously mentioned, the ballbar lacks real-time communication capabilities and the sampled data is stored locally and then transmitted to the PC via Bluetooth at a low and nondeterministic rate. This limitation presents a challenge in synchronizing the start and end of the robot’s movement with the corresponding data recording by the ballbar. Practically, the ballbar must begin recording before the robot starts moving and continue recording after the robot has completed its movement. Consequently, specific routines must be used during post-processing to synchronize the start and end of the readings and accurately compute errors (details about this are provided in Section 3.4).

At last, the stock measuring software provided by Renishaw for simplified use in industrial environments proved to be inadequate for conducting a rigorous accuracy study on IR. The application does not enforce the highest possible sampling rate during recording, and the output data is presented in the form of precompiled report based on ISO 230 for the diagnosis of Cartesian machine tools. Previous researches (see e.g. Marwitz et al., 2022; Zhou et al., 2021) overcame these limitations by proposing hardware modifications to the ballbar. Conversely, in the present work, novel engineering methods and tools are introduced to fully exploit the commercial ballbar for robotic accuracy studies. The proposed method is detailed in Section 3 and the custom made acquisition software is shared with the community in the supplementary material section.

This section outlines the method developed for assessing the path accuracy of IR using ballbar technology. Although the proposed framework is validated on the setup described in Section 2, it is broadly applicable to any articulated robot. The process includes defining performance indexes based on ISO 230 and ISO 9283, identifying the new TCP on the robot, conducting the ballbar test and post-processing and synchronizing the data. Each of these steps will be discussed in detail in the following sections

When addressing motion accuracy studies on IR, the primary reference standard is ISO 9283. However, many of the indexes proposed by ISO 9283 are incompatible with the characteristics of the ballbar. Therefore, ISO 230 has also been adopted. Originally designed for machine tools, the ISO 230 is well-suited for use with the ballbar as it considers circular paths. According to this standard, a distinction is made between the nominal path and the actual path, i.e. respectively the path defined during the programming phase and the one measured. Referring to Figure 3, the following indexes are defined:

• Circular error G: minimum radial distance of two concentric circles enveloping the actual path. It can be applied to a single clockwise (CW) or counter-clockwise (CCW) measured path.

• Bi-directional circular error G(b): same as G but with the concentric circles enveloping two measured paths, one executed CW and the other CCW.

• Radial error F: maximal deviation between the actual path and the nominal path, expressed as maximal positive (F_max) and maximal negative (F_min) for both directions of rotation.

• Mean bi-directional radial error D: difference between the radius of the least squares circle calculated with data from both CW and CCW paths and the radius of the nominal path (r_n).

Among them, F provides a clear and direct measure of how accurately a robot follows its intended path. This metric can be directly correlated with the path accuracy index proposed by ISO 9283 and defined as follows:

where the subscript symbols n and a indicate the nominal and actual points within the m-dimensional vector of samples of a circular path. In the case of multiple repetitions of the same experiment, the actual values measured for the i-th point becomes the average calculated among the different records. Since the experiments are executed with a ballbar in polar system, AT_p can be determined as the maximum absolute value between F_max and F_min.

Prior to running the test, the new TCP, which coincides with the center of the moving sphere of the ballbar (see Figure 1), must be determined and entered into the robot controller to ensure accurate motion planning. For this purpose, one could either exploit virtual 3D models (i.e. measuring the distances in the assembled setup) or proceed experimentally, e.g. using a laser tracker system. In this work, the latter approach is chosen as it provides a more accurate estimation of the TCP given the manufacturing and installation tolerances that are not typically captured into virtual replicas. Specifically, a modified version of the procedure proposed in Ferrarini et al. (2024) is followed, involving four sequential steps. Since the TCP is defined in the robot control system with respect to the robot’s flange frame, the first three steps of the procedure are dedicated to identifying this frame. The final step focuses on determining the TCP and computing the distance vector. The procedure, illustrated in Figure 4, can be summarized as follows:

• A spherically mounted retro-reflector (SMR) is attached to the metal plate (Probe 1 in Figure 4). The robot is initialized with the last three joints set to zero and the laser tracker is positioned to correctly reach the SMR.

• Step 1: Joint-5 is rotated to determine the Y-axis of the flange frame. The laser tracker collects a point cloud, which is used to fit a circle and thus identify the normal vector passing through its center point.

• Step 2: Joint-5 is reset to zero, and joint-6 is rotated to determine the Z-axis in the same manner.

• Step 3: After installing an SMR on the flange plane (Probe 2 in Figure 4), joint-6 is rotated again to identify such plane.

• The X-axis is derived from the cross product of the Y- and Z-axes. An additional cross-product between axes may be required to ensure perfect orthogonality.

• The origin of the flange reference frame is found by intersecting the flange plane with the Z-axis.

• The ballbar is manually positioned with its extremity sphere matching the tool cup as described in Section 2. Next, a probe adapter from the FARO accessory kit is attached, with one side coupled to the ballbar sphere and the other to an SMR (Probe 3 in Figure 4).

• Step 4: The probe adapter is manually moved around its center (the ballbar sphere, representing the TCP) and a cloud of points is recorded with the laser tracker.

• The center of the measured spherical surface is determined with least-squares fitting and represents the new TCP.

• the distance vector from the flange frame is then computed.

As said, the obtained vector is used to update the tool data during robot programming.

The experiment involves executing circular paths on the three planes defined by the robot’s base frame shown in Figure 1. Complete circles are achievable solely in the XY plane, while in the XZ and YZ would inevitably cause collisions with the tripod supporting the fixed extremity of the ballbar. Consequently, tests in these planes consist of circular arcs spanning 220$°$ each. In addition, to travel the entire range of interest with constant speed, extra angular ranges are introduced to account for the transient effects of the robot motion, which are then filtered out during post-processing, as explained in Section 3.4. In the XY plane, the extra motions involve a complete circle, while in the XZ and YZ planes are limited to 10$°$⁠.

All the experiments are carried out according to the following steps:

• The robot is first manually guided until its new TCP matches a sphere mounted on the magnetic support fixed on the tripod. Once the sphere is engaged, the point is saved on the controller as it will become the center of the tested circular paths during the offline programming (Figure 5).

• The robot is moved away by a distance equal to the selected ballbar radius (r_n) and the ballbar is manually mounted.

• The complete robot program is downloaded on the controller.

• The ballbar recording is started via the custom acquisition software prior to initiate any movement.

• The robot starts executing the planned motions and recording the TCP position in Cartesian coordinates (tracing feature turned on, with sampling every 12 ms). This data will be used in postprocessing to accurately map the ballbar signal over the studied polar domain. To facilitate the synchronization among the acquired robot and ballbar signals, a preliminary radial displacement of 1.5 mm (0.5 mm beyond the ballbar’s measuring range, which will set the output to 1) is performed forward and then backward (with 3 s of pause in between) before starting the circle. This trigger movement, shown in Figure 5, will be easily distinguished in the result vectors.

At the end of each experiment, two files are generated: a txt file with all the ballbar readings and a r64 file with the KUKA tracing data. These files are essential in the postprocessing phase for extracting the information needed to calculate the indexes detailed in Section 3.1.

The collected data set, consisting of ballbar and robot tracing signals, is processed to effectively quantify the robot’s path accuracy. It is worth noting that commercial ballbars typically acquire data at frequencies up to 1000 Hz, exceeding the tracing capabilities of most IR, as discussed in Section 2. While a higher frequency of the robot tracing would improve the analysis of joint-related errors, particularly in setups embedding secondary encoders in the robot joints (as in Mesmer et al., 2022), it does not significantly affect the path accuracy evaluation, which is primarily determined by the high-quality data from the ballbar in the following analysis.

To correlate signals from different sources, the procedure outlined in the flowchart in Figure 6 has been implemented. In particular, for each j-th test, the nominal radius (r_n) is summed to the ballbar data (originally represented by a series of displacements in the range ±1 mm), while the following elaborations are made on the robot tracing:

• Upsampling: To facilitate comparison and synchronization with the ballbar data, additional points are added to the robot signal using a modified Akima interpolation algorithm. The upsampling factor is set to 12, based on the ratio of the original sampling rates of the acquisition systems. It is important to note that this operation does not affect the ballbar data used for performance evaluations.

• Coordinate transformation and conversion: by default, the TCP position is calculated within the robot controller with respect to the robot base frame. Therefore, the position vector is first translated and referred to the center of the nominal circle. Afterwards, a simple Cartesian to polar coordinate conversion is performed, providing both radial and angular information. Naturally, the joint angular positions remain unvaried.

To temporally synchronize the obtained arrays in the polar domain, the trigger movement previously described (see Figure 5) could be exploited. Specifically, by searching the instant where the radius crosses the $rn+1$ mm threshold (either in the forward or backward segments), the indexes k₁ and k₂ are obtained on the robot and the ballbar series, respectively, as visible in Figure 7. The signals can then be aligned by shifting the ballbar array elements of $k2−k1$⁠. At this point, the relevant part of the ballbar data, namely, the central record related to the effective circular movement, is extracted by identifying the start and end angles (k₃ and k₄ indexes) in the robot tracing data. The performance are then evaluated based on Section 3.1 and stored into the j-th row of the results matrix.

The collected results undergo statistical analysis to identify the input parameters that impact each performance index. Subsets of vectors are extracted from the results matrix (e.g. all tests executed at specific speeds, spanning across all other operational parameters) and compared. The statistical test aims to determine whether varying a single parameter (e.g. increasing speed from 25 to 50 mm/s) leads to significant differences in the robot accuracy. For this purpose, a nonparametric paired sign test is used as in Ferrarini et al. (2024). The null hypothesis considered is that the median difference between two conditions is zero. To determine the significance of these hypotheses, the p-value is calculated for each comparison. Specifically, if the p-value is less than 0.05, the null hypothesis is rejected, indicating a statistically significant difference between the conditions.

The measured circular paths are plotted in the polar domain and further analyzed to correlate the observed local effects with angular position errors associated with the robot joints, as discussed in Section 4.

A parametric study has been conducted to assess the robot’s path accuracy across various conditions relevant to machining operations. The multidimensional domain explored is tailored to the specific tasks performed by the robot within a deburring cell. For this scenario, the robot operates with a Schunk PGN-plus 380 / 2 gripper to position the parts in alignment with the spindle. The parameters studied include:

Considering that performing all possible combinations of the parameters would result in 576 tests, the path accuracy assessment has been divided into two separate studies. The first study evaluates all the parameters except for the payload (fixed at 30 kg) and the robot configuration. The center of the circle is located at point $P=(X700,Y−1600,Z835)$⁠. To explore the entire robot workspace, the tests are repeated in the four quadrants by changing only the angle of the first robot joint, i.e. by simply mirroring P with respect to the X- and Y-axes without altering the robot kinematic configuration. The four quadrants are encountered by considering a positive rotation around the Z-axis, starting from the X-axis. This approach results in 192 measured circles. The second study focuses on assessing the impact of varying the payload and the configuration on the robot’s accuracy. In this case, the experiments are conducted in both CW and CCW directions, but only on the XY plane. The tests use a radius of r_n = 150 mm, a speed of 25 mm/s, two different payloads and six configurations (from C₁ to C₆), resulting in a total of 24 circles.

Although a higher number of tests would be beneficial to the experimental study, it is essential to recognize that the reported methods are specifically tailored for industrial applications. Conducting these experiments requires stopping production, which imposes practical constraints on the number of tests that can be performed. Therefore, it is crucial to carefully define the test domain and select parameters and levels that strike a balance between minimizing tests and producing robust results to support meaningful conclusions.

This section presents and discusses the primary outcomes of the experimental campaign conducted on the KUKA KR210 R2700 Prime robot. For a detailed summary of the results, the reader should refer to the repository linked in the supplementary material section, which contains the complete data set and all processing scripts. To highlight the dependency of the considered accuracy performance indexes from each operating parameter, a statistical analysis has been performed as explained in Section 3.4. The obtained results can be seen in Table 1, where only the significant cases (i.e. those having null hypothesis rejected) are reported. A subset of experiments have been selected and plotted in Figures 10–16 to facilitate comparisons. The information retrieved from the robot tracing is here used also to monitor the joint behavior during the circular path execution and thus to correlate the cumulative path error with the joint-related contributions. Overall, the following considerations can be drawn:

• Almost all indexes exhibit moderate improvements when executing larger circles, as notable from Table 1. An interesting effect is observed with the index D, which exhibits a dependency on speed, as clearly visible in Figure 9. This can be attributed to the path approximation algorithms within the robot controller. Indeed, as also documented in Slamani et al. (2012b), higher speeds along the same circular path cause the trajectory to shift inward, reducing the D index due to the approximation of arcs by line segments. In addition, a notable effect of the radius on the D index is observed, but only at high speeds (200 mm/s), as shown in Figure 10.

• Circles executed on the vertical XZ and YZ planes show higher accuracy levels compared to those on the horizontal XY plane (see Figure 11). The only exception is the G index, which reaches its maximum level (i.e. worst condition) on the YZ plane. It shall however being remarked that higher payloads would inevitably lead to increased path errors along the Z-direction, as documented in Ferrarini et al. (2024).

• As previously observed in Slamani et al. (2012a) and Slamani and Bonev (2013) and clearly depicted in Figure 12, increasing speed tends to progressively shrink the circles, resulting in poorer values of D, F_min and AT_p. This trend aligns also with the findings from Ferrarini et al. (2024), where higher speeds negatively impacted the accuracy of IR during linear path execution.

• The payload negatively affects the robot accuracy on XZ and YZ planes due to the gravitational contribution (see also point 2). This may lead to path errors in the order of few millimeters as the payload approaches its nominal value. In the present study, the tests are conducted exclusively on the XY plane as the ballbar measuring range is limited to 1 mm. Nevertheless, significant deterioration in accuracy is observed when increasing the payload from 30 to 80 kg, as evident from both Table 1 and Figure 13. In addition, the impact of varying the robot kinematic configuration (inevitably altering its stiffness properties, see detailed discussion in Marwitz et al., 2022) can be observed in Figure 14, where the same circle is executed at six different poses.

• The impact of joint backlash on the executed circular paths is evident in all plots as a noticeable jump in the radius (Runan et al., 2023). Figure 15 clearly illustrates that these peaks occur when the robot axes reverse their motion. The plot indicates that axes 2, 3, 5 and axes 1, 6 experience this inversion simultaneously due to their synchronization during the execution of circular paths with a constrained orientation of the end-effector. Specifically, the sum of axes 2, 3 and 5, as well as the difference between axes 1 and 6, remain constant throughout the entire circular motion.

• The tests conducted across the four quadrants (by varying the position of the first robot joint) show no significant changes in the accuracy indexes. However, Figure 16 reveals noticeable differences in the amplitude of the backlash-related peaks. This variation is attributed to the nonuniform geometric condition of the gear units inside the joint reducers, which may result from manufacturing errors and uneven wear during the IR lifespan since the gears are not uniformly used throughout their working range (Guida et al., 2022). This condition is commonly observed in IR performing repetitive tasks, where they operate within the same limited angular range during nearly every working cycle.

In addition to the backlash, which represents a major source of path error when executing paths involving several joint inversions, other joint-related effects can be observed from the ballbar measurements. As extensively discussed in Bilancia et al. (2025) and Yang et al. (2021), the transmission error affecting the servo reducers in the robot joints arises from several factors. These occur a specific number of times per revolution, corresponding to the normalized frequency of the related physical phenomena within the reducer mechanism. The KUKA robot under consideration is equipped two-stage rotating vector (RV) reducers, produced by Nabtesco, in the first three joints. Their installation and operating principle within the robot system are shown in Figure 17, whereas the characteristics of their internal reduction mechanisms are detailed in Table 2. These have been determined by correlating the data and formulas provided in the producer’s catalog (Nabtesco, 2024) with the installation configuration observed on the robot and depicted in the proposed schematic.

In these reducer mechanisms, the transmission error is significantly influenced by the meshing between the cycloidal wheel and the reducer pins in the second reduction stage. Based on Table 2 and the findings from Bilancia et al. (2025), these interactions occur 58 times per revolution (output shaft) for joint 1, and 52 times for joints 2 and 3, along with their associated harmonics. Essentially, they result from the discontinuities caused by the pins’ engagement, occurring a number of times per revolution equal to the number of pins (Slamani and Bonev, 2013). Given that complete joint revolutions are generally not demanded during the execution of circular robotic paths, the effects associated with the cycloidal gears are expected to manifest fewer times within the measured circles, depending on the angle traversed by each joint. To facilitate the correlation between ballbar measurements and these reducer errors, the recorded radial error e_p has been analyzed in the frequency domain using a fast Fourier transform, with the frequencies normalized as follows:

where f is the frequency returned by the fast Fourier transform (expressed in Hz), r_n is the nominal radius (expressed in mm) and v is the speed (expressed in mm/s) at which the test is performed. This approach highlights how frequently each path error contribution occurs over the entire circle. Tests conducted at different operating speeds can thus be compared, as the position-related joint errors tend to overlap, as clearly visible in Figure 18. In this case, the expected pin engagements for each joint over the circle can be estimated from the robot tracing. Specifically, by examining the angular position function of the first three joints, the total angular distance traveled is found to be 41.6$°$⁠, 57.4$°$ and 67.4$°$⁠, corresponding to 0.12, 0.16 and 0.19 revolutions, respectively. These are obtained by summing both forward and backward joint motions, thereby considering the entire range of travel. The number of engagements is then calculated by scaling the value of z₆ from Table 2 (representing the number of engagements within one joint revolution) by these factors, resulting in 6.7, 8.3 and 9.7 for joints 1, 2 and 3, respectively. By rounding to the nearest integer values and accounting for some uncertainties due to joint reversals during the circular path execution, the greatest pin-related peaks are expected to manifest in the initial part of the spectrum, i.e. within $f/f0=10$⁠, as it can be verified in Figure 18.

This paper reports on efficient engineering methods for assessing the circular path accuracy of IR using ballbars. The study details the development of robust and efficient experimental procedures to adapt ballbars, traditionally used for calibrating CNC machines, for application in robotic systems. After an in-depth description of the hardware and software setup, the proposed framework is tested and validated within a deburring robotic cell featuring a KUKA KR210 R2700 Prime robot. The experiments are conducted on the three planes under varying operating conditions, including changes in circle radius, travel speed, payload and robot configuration. During post-processing, statistical analyses are performed to identify and examine correlations between input parameters and path accuracy indexes, following the ISO 9283 and ISO 230 standards. By synchronizing ballbar measurements with data obtained through the robot tracing function, the direct effects of joint backlash on the path deviations are observed. In addition, spectral analysis of ballbar data collected at different speeds showed the impact of the reducers transmission errors. The experimental findings confirm the effectiveness of the proposed approach, which is applicable to any articulated IR used in precision fields, i.e. with errors that fall within the ballbar measurement range. For serial configurations, the method supports the development of compensations models as it allows to correlate the impact of joint errors on the executed path. In addition, for Cartesian systems, it also detects geometric inaccuracies such as axis misalignments, straightness and orthogonality errors.

Future research could focus on expanding model-based compensation strategies, along with their integration into existing and next-generation open controllers. This effort aims to address inherent limitations of robotic joint servomechanisms, such as the use of basic encoders and the absence of secondary encoders, while also extending the empirical models derived from this study to enhance path accuracy for applications such as precision assembly, inspection and machining. To facilitate future comparisons and advancements, the complete data set and custom acquisition software have been shared with the research community.

The supplementary material for this article can be found at:https://data.mendeley.com/datasets/hjknbkw2gw/1