The precision and consistency of an industrial robot’s end effector are paramount for enabling advanced manufacturing tasks such as precision assembly, high-quality machining, and reliable material handling. Among the critical performance metrics, pose repeatability, which quantifies the dispersion of the actual attained poses when the robot repeatedly executes the same command, is a fundamental indicator of an end effector‘s capability to perform dependable and precise operations. While pose accuracy defines the deviation from a commanded pose, repeatability reflects the inherent stability and consistency of the robot’s mechanical and control systems. Consequently, the development of robust, accurate, and efficient methods for online pose repeatability testing is a significant challenge in robotics metrology.
Traditional methods for assessing robot pose often rely on complex, point-by-point measurement using laser trackers, coordinate measuring machines (CMMs), or vision systems with elaborate calibration procedures. These approaches, while accurate, can be time-consuming, operationally complex, and ill-suited for rapid, multi-point evaluation across a robot’s entire workspace or for continuous online monitoring. To address these limitations, this work presents a novel, integrated methodology for the efficient detection of end effector pose deviations, facilitating high-precision repeatability assessment.
The core of our system involves an optical pose reference fixture rigidly attached to the robot’s end effector. This fixture emits three mutually perpendicular crossed-line laser beams, effectively materializing the abstract pose of the end effector into tangible light patterns. Three independent image-sensing modules, each comprising a reception screen and a camera, capture these laser cross patterns. The minute changes in the position and orientation of these projected crosses on the screens encode the six-degree-of-freedom (6-DoF) pose variations of the end effector. By precisely extracting features from these images and applying sophisticated algorithms, we can reconstruct and evaluate the pose repeatability.
Theoretical Foundation: A Direction Cosine-Based Pose Repeatability Model
To establish the mathematical relationship between the measured image features and the physical pose changes of the end effector, we developed a theoretical model based on direction cosines and Euler angles. Let the coordinate system of the optical fixture at its initial, reference pose be defined as {A₀}, with its axes (X, Y, Z) aligned with the three laser beams. A fixed measurement coordinate system {Om} is established in the workspace.
When the robot re-executes a pose command, the fixture moves to a new pose, A₁. This transformation can be decomposed into a pure rotation from A₀ to an intermediate pose A′₁, followed by a pure translation from A′₁ to A₁. The rotation is described by Euler angles (Δα, Δθ, Δγ) around the X, Y, and Z axes, respectively. The corresponding rotation matrix R is given by the sequential multiplication of elemental rotation matrices:
$$ R(Z, Δγ) = \begin{bmatrix} \cosΔγ & -\sinΔγ & 0 \\ \sinΔγ & \cosΔγ & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
$$ R(Y, Δθ) = \begin{bmatrix} \cosΔθ & 0 & \sinΔθ \\ 0 & 1 & 0 \\ -\sinΔθ & 0 & \cosΔθ \end{bmatrix} $$
$$ R(X, Δα) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cosΔα & -\sinΔα \\ 0 & \sinΔα & \cosΔα \end{bmatrix} $$
$$ R = R(Z, Δγ) \cdot R(Y, Δθ) \cdot R(X, Δα) = \begin{bmatrix}
r_{11} & r_{12} & r_{13} \\
r_{21} & r_{22} & r_{23} \\
r_{31} & r_{32} & r_{33}
\end{bmatrix} $$
where the elements \( r_{ij} \) form the direction cosine matrix, representing the cosines of the angles between the rotated fixture axes and the original axes. The translational displacement is denoted by (Δx, Δy, Δz).
The laser beams project onto three reception planes. The change in the cross pattern’s center point coordinates (e.g., from \(k_0\) to \(k_1\)) and its orientation on these planes provides the raw measurement data. Through geometric analysis, the positional deviation components can be derived as functions of these measured image coordinate changes and the direction cosines. For instance, the displacement along the measurement system’s X-axis is given by two equivalent expressions, providing a means for verification:
$$ Δx = (k_{1x} – k_{0x}) + l_x \frac{r_{21}}{r_{11}} = (q_{1x} – q_{0x}) + l_z \frac{r_{23}}{r_{33}} $$
Similarly,
$$ Δy = (p_{1y} – p_{0y}) + l_y \frac{r_{12}}{r_{22}} = (q_{1y} – q_{0y}) + l_z \frac{r_{13}}{r_{33}} $$
$$ Δz = (k_{1z} – k_{0z}) + l_x \frac{r_{31}}{r_{11}} = (p_{1z} – p_{0z}) + l_y \frac{r_{32}}{r_{22}} $$
Here, \(l_x, l_y, l_z\) are constant distances from the fixture’s origin to the reception planes along the initial axes, obtained through system calibration. The terms \((k_{1x} – k_{0x})\), etc., are the directly measured shifts of the cross center in the image plane coordinates. The angles Δα, Δθ, Δγ are obtained by measuring the rotation of the cross lines in the images relative to the image coordinate axes. This model elegantly connects the observable image features to the 6-DoF pose change of the optical fixture, and consequently, the robot end effector.
Pose Deviation Detection Methodology
The practical implementation of the pose repeatability measurement involves a sequence of image processing and computational steps: image preprocessing, light stripe separation, centerline extraction, and finally, position and angle calculation.
Image Processing and Centerline Extraction via IGCF Algorithm
The raw grayscale image of the laser cross is first thresholded to create a binary image, separating the bright laser stripes from the background. Morphological operations are then applied to extract the contours of the horizontal and vertical light stripes. The key step is the precise extraction of the centerline of each light stripe, for which we developed an Improved Gaussian Curve Fitting (IGCF) algorithm.
The intensity profile of a laser stripe cross-section approximates a Gaussian distribution. The traditional Gaussian fitting method is susceptible to error from pixel saturation, where multiple consecutive pixels reach the maximum intensity value. The IGCF algorithm mitigates this by implementing a saturation-handling rule during the row/column-wise scan of the stripe image: if two or more consecutive saturated pixels are detected, all are excluded from the fit; if only one saturated pixel is found, it is included. After this step, the width of the pixel set used for fitting is intelligently reduced around a coarsely estimated center, ensuring a robust set of 5-9 pixels for the final 1D Gaussian fit:
$$ f(y) = A e^{-\frac{(y-y_0)^2}{2\sigma^2}} $$
where \(y_0\) is the extracted sub-pixel center location of the cross-section. This process is repeated for multiple cross-sections along the stripe. The collection of these center points is then fitted to a straight line using a robust Bisquare method to determine the final centerline equation for each stripe, effectively rejecting outliers.
Cross Center Location and Angle Calculation
The center of the cross pattern is defined as the intersection point of the fitted horizontal and vertical centerlines. Solving the two line equations yields the center coordinates with sub-pixel accuracy. The orientation (tilt) of the cross is obtained by calculating the angle of each centerline relative to the image axes. For increased robustness, the final reported angle is the average of the angles derived from the horizontal and vertical lines, leveraging an averaging effect to reduce random error.
High-Fidelity Error Compensation using the CSF-PPSO-ESN Algorithm
In an ideal system, the measured image shifts would have a perfectly linear relationship with the actual physical displacements and angles of the end effector. In reality, nonlinearities due to optical distortions, slight beam misalignments, and sensor imperfections introduce systematic errors. Therefore, a high-accuracy compensation model is essential. We designed a novel hybrid algorithm, CSF-PPSO-ESN, to establish and utilize the complex relationship between pose measurement values and their inherent errors across the entire workspace.
The process begins with a calibration procedure to gather ground-truth data. The robot’s end effector, equipped with the optical fixture, is moved to a series of known poses (using high-precision references like a laser interferometer and a theodolite). At each pose, the system’s raw measurement is recorded alongside the reference value, giving a set of measurement errors.
- Cubic Spline Fitting (CSF): The initial error model is built using cubic spline interpolation on the calibration data points \((X_i, \Delta S_i)\), where \(X\) is the measured value and \(\Delta S\) is the error. This provides a smooth, piecewise-polynomial function that can model nonlinear trends:
$$ \Delta S(X) = \sum_{i=0}^{3} P_i \cdot \Delta S_{i,3}(X) $$
where \(P_i\) are coefficients and \(\Delta S_{i,3}(X)\) are the cubic B-spline basis functions. - Pareto Particle Swarm Optimization (PPSO): Directly solving for the optimal spline coefficients can be trapped in local minima. We employ a multi-objective PSO variant based on Pareto optimality. Particles search the coefficient space, and non-dominated solutions are stored in an archive. A fitness function balancing the distance to the Pareto front and the local density of solutions guides the search, efficiently finding a globally optimal set of coefficients for the spline model.
- Echo State Network (ESN) Enhancement: To achieve real-time compensation with maximum accuracy beyond what the spline model offers, the calibrated data is further processed by an Echo State Network, a type of recurrent neural network with a fixed, randomly connected “reservoir.” The preprocessed measurement-error pairs are fed into the ESN, which learns the residual nonlinear mapping. The dynamic properties of the reservoir allow it to model complex temporal and spatial dependencies in the error data, leading to a superior final compensation model. The ESN’s state update and output are given by:
$$ \mathbf{U}(p+1) = a \cdot f_1(\mathbf{W}^{in}\mathbf{x}(p+1) + \mathbf{W}^{state}\mathbf{u}(p)) $$
$$ \mathbf{Y}(p+1) = f_2(\mathbf{W}^{out}(\mathbf{u}(p+1), \mathbf{x}(p+1))) $$
where \(\mathbf{U}\) is the reservoir state, \(\mathbf{Y}\) is the output (compensated error), \(\mathbf{W}^{in}\), \(\mathbf{W}^{state}\), \(\mathbf{W}^{out}\) are connection matrices, and \(a\) is a leakage rate.
The integrated CSF-PPSO-ESN algorithm leverages the precise local fitting of splines, the global optimization capability of PPSO, and the powerful nonlinear regression of ESNs to create a compensation model that dramatically enhances measurement accuracy across the robot’s operational volume.
Experimental Validation and Results
The proposed system and methodology were rigorously tested. The measurement system’s fundamental performance was first characterized through displacement and angle calibration experiments.
System Calibration
Displacement Calibration: The optical fixture was moved along a single axis using a high-precision translation stage. A laser interferometer provided the reference displacement. The raw system error was recorded and the CSF-PPSO-ESN compensation model was built. After compensation, the residual displacement measurement error was significantly reduced.
| Axis | Range (mm) | Error before Compensation (µm, ±) | Error after Compensation (µm, ±) |
|---|---|---|---|
| X | 0-70 | 80 | 1.5 |
| Y | 0-70 | 80 | 1.5 |
| Z | 0-70 | 80 | 1.5 |
Angle Calibration: Similarly, the fixture was rotated about each axis using a high-precision rotary stage, with a theodolite as the angle reference. The compensation algorithm was applied to the angular measurements.
| Rotation Axis | Range (deg) | Error before Compensation (arc-sec, ±) | Error after Compensation (arc-sec, ±) |
|---|---|---|---|
| X | ±5 | 40 | 2 |
| Y | ±5 | 40 | 2 |
| Z | ±5 | 40 | 2 |
End-Effector Pose Repeatability Test
The complete system was deployed to evaluate the pose repeatability of an ABB IRB 2600 industrial robot. The optical fixture was mounted on the robot’s end effector flange. Five measurement points (P1 to P5) were defined within the robot’s workspace. The robot was commanded to move to each point 30 times consecutively in a specific cycle under full load and maximum speed. At each attained pose, the system measured the 6-DoF pose of the end effector. The repeatability was calculated as the radius of the sphere (for position) and the dispersion (for angles) containing all measured points with a 95% confidence level. The results, compared against high-precision reference instruments, are summarized below.
| Measurement Point | Position Repeatability, RPl (mm) | Angle Repeatability, RPa (deg) | Angle Repeatability, RPb (deg) | Angle Repeatability, RPc (deg) |
|---|---|---|---|---|
| Measured | Reference | Measured | Reference | Measured | Reference | Measured | Reference | |
| P1 (Center) | 0.0371 | 0.0356 | 0.0037 | 0.0032 | 0.0036 | 0.0031 | 0.0040 | 0.0036 |
| P2 | 0.0327 | 0.0335 | 0.0031 | 0.0027 | 0.0036 | 0.0032 | 0.0038 | 0.0035 |
| P3 | 0.0315 | 0.0322 | 0.0034 | 0.0032 | 0.0035 | 0.0032 | 0.0025 | 0.0024 |
| P4 | 0.0274 | 0.0263 | 0.0035 | 0.0036 | 0.0031 | 0.0033 | 0.0034 | 0.0031 |
| P5 | 0.0245 | 0.0231 | 0.0024 | 0.0026 | 0.0029 | 0.0025 | 0.0023 | 0.0019 |
The experimental results demonstrate the high performance of the proposed system. After compensation using the CSF-PPSO-ESN algorithm, the system achieves a displacement measurement accuracy of ±1.5 µm and an angular measurement accuracy of ±2 arc-seconds. The pose repeatability measurements obtained by the system show excellent agreement with those from reference metrology instruments, validating the entire methodology. The system is capable of rapid, multi-point assessment of the end effector‘s pose stability, providing a practical solution for online monitoring and performance verification in industrial settings.
Conclusion
This work presents a comprehensive and effective solution for the online detection and assessment of industrial robot end effector pose repeatability. The foundation is a rigorous direction cosine-based mathematical model that translates measurements of laser cross patterns into 6-DoF pose data. The implementation relies on two key algorithmic innovations: the IGCF algorithm for robust, sub-pixel centerline extraction from laser stripes, and the hybrid CSF-PPSO-ESN algorithm for high-fidelity, real-time compensation of system-wide nonlinear errors. The integrated system proves capable of performing rapid, accurate, and multi-point evaluations across a robot’s workspace, achieving micron-level positional accuracy and arc-second level angular accuracy after compensation. This methodology provides a valuable new approach for the quantitative evaluation and ongoing monitoring of robot end effector performance, which is critical for ensuring quality and reliability in advanced, precision-driven manufacturing applications.