In the rapidly evolving field of robot technology, industrial robots have become integral to modern manufacturing, offering unparalleled flexibility, automation, and cost-efficiency. As industries such as semiconductor dispensing and precision assembly demand higher accuracy, often requiring tolerances as tight as 10 micrometers, the repeatability of robot positioning has emerged as a critical challenge. I have observed that various error factors, particularly joint rotation errors, significantly impact the repeatability of industrial robots. These errors arise from gaps between joint holes and axes, leading to fluctuations in the initial points of trajectories and ultimately degrading positioning accuracy. In this article, I explore a non-model-based approach to predict and compensate for joint rotation errors, aiming to enhance the repeatability of robot technology in practical applications. By leveraging polynomial prediction models and dynamic weighting strategies, I address the limitations of traditional methods, such as complex modeling and instability under multiple error sources, providing a robust solution for improving robot performance.
Robot technology relies heavily on precise kinematic models to describe the motion of multi-link, strongly coupled open-chain structures. The forward kinematics of an industrial robot, such as the CCR06-900 model I studied, can be represented using the Denavit-Hartenberg (D-H) parameters. For instance, the transformation matrix between adjacent links is given by:
$$ \mathbf{T}_{i}^{i-1} = \begin{bmatrix}
\cos\theta_i & -\sin\theta_i \cos\alpha_i & \sin\theta_i \sin\alpha_i & a_i \cos\theta_i \\
\sin\theta_i & \cos\theta_i \cos\alpha_i & -\cos\theta_i \sin\alpha_i & a_i \sin\theta_i \\
0 & \sin\alpha_i & \cos\alpha_i & d_i \\
0 & 0 & 0 & 1
\end{bmatrix} $$
where $\theta_i$ is the joint angle, $d_i$ is the link offset, $a_i$ is the link length, and $\alpha_i$ is the twist angle. The overall end-effector pose relative to the base frame is computed by multiplying these matrices:
$$ \mathbf{T} = \mathbf{T}_0^1 \mathbf{T}_1^2 \mathbf{T}_2^3 \mathbf{T}_3^4 \mathbf{T}_4^5 \mathbf{T}_5^6 $$
Inverse kinematics, which determines joint angles from a given end-effector pose, is solved numerically. For example, the joint angles can be derived as:
$$ \theta_1 = \text{atan2}(p_y – d_6 a_y, p_x – d_6 a_x) $$
and similarly for other joints, ensuring minimal angular changes for practical robot technology applications. However, joint rotation errors, denoted as $\Delta\theta_i$, introduce deviations in the actual joint angles, leading to end-effector positioning errors. The differential positioning error model can be expressed as:
$$ \Delta\mathbf{P} = \sum_{i=1}^{6} \mathbf{J}_i \Delta\theta_i = \mathbf{J} \Delta\boldsymbol{\theta} $$
where $\Delta\mathbf{P}$ is the end-effector error vector, $\mathbf{J}$ is the Jacobian matrix, and $\Delta\boldsymbol{\theta}$ is the vector of joint errors. This model highlights how small joint inaccuracies accumulate, emphasizing the need for precise error compensation in advanced robot technology.
To address these issues, I developed a joint rotation error prediction model based on a non-model approach, which avoids the complexity of modeling individual error sources. By collecting actual trajectory data, I constructed a polynomial model to predict joint errors at the home point, where fluctuations occur due to backlash and other factors. The joint error matrix is defined as:
$$ \delta\boldsymbol{\theta} = \begin{bmatrix}
\delta\theta_1^1 & \delta\theta_2^1 & \delta\theta_3^1 & \delta\theta_4^1 & \delta\theta_5^1 & \delta\theta_6^1 \\
\delta\theta_1^2 & \delta\theta_2^2 & \delta\theta_3^2 & \delta\theta_4^2 & \delta\theta_5^2 & \delta\theta_6^2 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\delta\theta_1^j & \delta\theta_2^j & \delta\theta_3^j & \delta\theta_4^j & \delta\theta_5^j & \delta\theta_6^j
\end{bmatrix} $$
where $\delta\theta_i^j$ represents the rotation error of the $i$-th joint at the $j$-th sampling point. Using least-squares fitting, I derived a cubic polynomial model for each joint:
$$ \delta\theta_i = \theta_{i,\text{set}} – (k_{1i} x^3 + k_{2i} x^2 + k_{3i} x + \varepsilon_i) $$
Here, $\theta_{i,\text{set}}$ is the commanded joint angle, $x$ is the trajectory run count, $k_{1i}$, $k_{2i}$, and $k_{3i}$ are fitting coefficients, and $\varepsilon_i$ is a constant compensation term accounting for static errors. The compensated joint angle is then:
$$ \tilde{\theta}_i = \theta_{i,\text{real}} + \delta\theta_i $$
This prediction enables pre-compensation at the home point, reducing initial point volatility and enhancing repeatability in robot technology systems.
However, to improve model stability against random disturbances, I incorporated a dynamic weighting strategy. The weighted error prediction model for each joint becomes:
$$ \tilde{\theta}_i = \theta_{i,\text{real}} + \omega_i \delta\theta_i $$
where $\omega_i$ is a dynamic weight adjusted based on feedback from sampling points located 20% along the trajectory from the home point. The weight update rule is:
$$ \omega_{i,\text{new}} = \mu_i \omega_{i,\text{old}} = \frac{\sum_{n=1}^{n} \mu_{i,n}}{n} \omega_{i,\text{old}} $$
with $\mu_i$ representing the ratio of joint values between consecutive trajectories. This approach ensures adaptability to varying conditions, a crucial aspect of resilient robot technology.
To validate this method, I conducted experiments on a CCR06-900 industrial robot controlled via CODESYS software, with a FARO laser tracker measuring end-effector positions. The workspace was defined as a 300 mm × 400 mm × 400 mm volume, and tests included straight-line and circular trajectories under no-load conditions at 30% of the rated speed. The results compared three compensation methods: model-based compensation, standard least-squares, and the proposed weighted least-squares approach. The table below summarizes the repeatability errors before and after compensation for straight-line trajectories:
| Trajectory Type | Compensation Method | Max Error (mm) | Average Error (mm) |
|---|---|---|---|
| Straight Line | Uncompensated | 0.1121 | 0.0380 |
| Model-Based | 0.1186 | 0.0336 | |
| Least-Squares | 0.1039 | 0.0296 | |
| Weighted Least-Squares | 0.0569 | 0.0202 | |
| Circular Arc | Uncompensated | 0.1650 | 0.0545 |
| Model-Based | 0.1249 | 0.0447 | |
| Least-Squares | 0.1232 | 0.0374 | |
| Weighted Least-Squares | 0.1083 | 0.0291 |
The data clearly shows that the weighted least-squares method achieved the highest reduction in repeatability errors, with decreases of 46.8% for straight-line and 46.6% for circular trajectories, compared to 11.6% and 17.9% for model-based compensation. This underscores the effectiveness of the dynamic weighting strategy in enhancing robot technology reliability. Additionally, the standard deviations of the positioning errors were calculated to assess data dispersion:
| Trajectory Type | Uncompensated Std Dev (mm) | Model-Based Std Dev (mm) | Least-Squares Std Dev (mm) | Weighted Least-Squares Std Dev (mm) |
|---|---|---|---|---|
| Straight Line | 0.0204 | 0.0201 | 0.0195 | 0.0111 |
| Circular Arc | 0.0306 | 0.0254 | 0.0232 | 0.0178 |
The lower standard deviations after weighted compensation indicate more consistent positioning, further validating the method’s superiority in robot technology applications. For instance, the joint error distributions narrowed significantly post-compensation, as illustrated by the reduced ranges in experimental plots. In straight-line trajectories, joint 1 errors decreased from a spread of ±0.05° to ±0.02°, and similar improvements were observed across all joints. This demonstrates how predictive compensation mitigates the impact of joint rotation errors, a common issue in robot technology systems.
In conclusion, the integration of non-model-based prediction with dynamic weighting offers a practical solution for improving the repeatability of industrial robots. By focusing on joint rotation errors and home point fluctuations, this approach reduces complexity compared to traditional model-based methods while maintaining robustness under multiple error sources. The experimental results confirm significant enhancements in repeatability, making it a valuable contribution to the advancement of robot technology. As industries continue to push for higher precision, such methods will play a pivotal role in ensuring that robot technology meets the demands of modern manufacturing, from micro-assembly to automated processes. Future work could explore real-time adaptation and integration with other error sources to further elevate the capabilities of robot technology in diverse applications.