In recent years, the application of robot technology in high-precision automated manufacturing has expanded significantly, driven by advancements in automation and flexibility. However, the absolute positioning accuracy of industrial robots remains a critical challenge due to factors such as geometric errors, non-geometric errors, and multi-directional repeatability issues. These limitations hinder the widespread adoption of robot technology in fields like aerospace, automotive, and precision machining. To address this, we propose an offline compensation method based on Extended Joint Angle and Extreme Learning Machine (EJA-ELM), which enhances absolute positioning accuracy by accounting for joint reversal errors and other error sources without relying on complex kinematic models. This approach leverages the power of neural networks in robot technology to predict and compensate for position errors efficiently.
Robot technology often involves serial-chain manipulators where small errors in joints and links accumulate, leading to significant end-effector inaccuracies. Traditional compensation methods, such as kinematic model-based approaches, require detailed parameter identification and high controller access, limiting their practicality. In contrast, our EJA-ELM method treats the robot as a black box, using a neural network to map theoretical joint angles to position errors. By extending the joint space to include direction coefficients, we capture the effects of joint reversal errors, which are often overlooked in conventional methods. This innovation in robot technology allows for more accurate error prediction and compensation, as demonstrated through simulations and experiments on a Comau NJ-220-2.7 industrial robot.
The core of our method lies in the extended joint space, which incorporates joint angle direction coefficients to model the impact of gear backlash and other non-geometric factors. For a robot with n joints, the extended joint space is defined as $[\theta_1, \theta_2, \dots, \theta_n, \lambda_1, \lambda_2, \dots, \lambda_n]$, where $\theta_i$ represents the theoretical joint angle and $\lambda_i$ is the direction coefficient indicating forward or backward motion. The actual joint angle $\theta_i’$ is given by:
$$ \theta_i’ = \theta_i + \lambda_i b_i(\theta_i) $$
where $b_i(\theta_i)$ denotes the gear backlash error. The direction coefficient $\lambda_i$ is defined as:
$$ \lambda_i = \begin{cases}
1 & \text{for forward motion} \\
-1 & \text{for backward motion}
\end{cases} $$
This extension allows the neural network to learn the relationship between joint configurations and position errors more effectively, enhancing the robustness of robot technology in dynamic environments.
We employ an Extreme Learning Machine (ELM) neural network for error prediction due to its fast learning speed and minimal parameter tuning. The ELM structure consists of an input layer, a hidden layer, and an output layer. The input nodes correspond to the extended joint angles, while the output nodes represent the position errors in the X, Y, and Z directions. The total position error $e$ is computed as:
$$ e = \sqrt{e_x^2 + e_y^2 + e_z^2} $$
Given M training samples, each comprising the extended joint angles $Q_{aj}$ and the corresponding position errors $e_j$, the hidden layer output $h_i$ for the i-th neuron is:
$$ h_i = G(\alpha_i \cdot Q_{aj} + b_i) $$
where G is the activation function, $\alpha_i$ is the input weight, and $b_i$ is the bias. The output of the ELM is then:
$$ e_j = \sum_{i=1}^{k} \beta_i h_i(Q_{aj}) $$
In matrix form, this can be expressed as $H\beta = T$, where H is the hidden layer output matrix, $\beta$ is the output weight matrix, and T is the target error matrix. The output weights are determined using the Moore-Penrose generalized inverse:
$$ \beta = H^+ T $$
This training process enables the ELM to quickly adapt to the error patterns in robot technology, making it suitable for real-time applications.
For error compensation, we use a feedforward approach where the predicted position error is added to the theoretical position to generate a corrected command. The compensated position $P_m$ is calculated as:
$$ P_m = P_r + e $$
where $P_r$ is the theoretical position and e is the predicted error. This method ensures that the robot reaches the desired position with improved accuracy, leveraging the predictive capabilities of ELM in robot technology.
To validate our approach, we conducted simulations using MATLAB on a Comau NJ-220-2.7 robot model. The Denavit-Hartenberg (D-H) parameters for the robot are listed in the table below, along with introduced errors to simulate real-world conditions.
| Joint | a (mm) | d (mm) | α (rad) | θ (rad) | Δa (mm) | Δd (mm) | Δα (rad) | Δθ (rad) |
|---|---|---|---|---|---|---|---|---|
| 1 | 400 | 0 | -π/2 | 0 | -0.7 | -1.03 | -0.00003 | -0.0003 |
| 2 | 175 | 0 | 0 | -π/2 | -0.4 | -0.15 | -0.002 | 0 |
| 3 | 250 | 0 | -π/2 | 0 | 0.5 | -0.006 | 0.08 | 0 |
| 4 | 0 | 125.33 | π/2 | 0 | 0.5 | -0.1 | -0.08 | 0.0052 |
| 5 | 0 | 0 | -π/2 | 0 | -0.1 | 0.0003 | 0.05 | 0.0003 |
| 6 | 0 | 230 | 0 | π | -0.2 | 0.05 | -0.06 | -0.0105 |
We generated 500 sets of theoretical joint angles using Latin hypercube sampling and computed the corresponding theoretical and actual end-effector positions. The transformation matrix between consecutive links, considering errors, is given by:
$$ T_i^{i-1} = R_Z(\theta_i + \Delta\theta_i) D_Z(d_i + \Delta d_i) D_X(a_i + \Delta a_i) R_X(\alpha_i + \Delta\alpha_i) $$
where $R_Z$ and $R_X$ are rotation matrices, and $D_Z$ and $D_X$ are translation matrices. The overall transformation from the base to the end-effector is:
$$ T_N^0 = T_1^0 T_2^1 \dots T_N^{N-1} $$
We used 300 samples for training the ELM model with 40 hidden neurons and tested on the remaining 200 samples. The results showed a significant reduction in position errors after compensation, as summarized in the table below.
| Sample Point | Error Before Compensation (mm) | Error After Compensation (mm) |
|---|---|---|
| 1 | 1.371 | 0.127 |
| 2 | 1.177 | 0.081 |
| 3 | 1.559 | 0.288 |
| … | … | … |
| 200 | 1.248 | 0.115 |
| Mean | 1.383 | 0.169 |
| Standard Deviation | 0.106 | 0.088 |
The simulation demonstrated that the EJA-ELM method reduced the average position error from 1.383 mm to 0.169 mm, with the standard deviation decreasing from 0.106 mm to 0.088 mm. This confirms the effectiveness of our approach in enhancing the precision of robot technology.
Following the simulation, we conducted experimental validation on a physical Comau NJ-220-2.7 robot equipped with an API Radian Core laser tracker for high-precision measurement. The tool center point (TCP) was calibrated using a spherically mounted reflector (SMR), and the transformation matrices were established as follows:
$$ P_{tcp}^{EE} = T_{SMR}^{EE} P_{tcp}^{SMR} $$
$$ P_{tcp}^{SMR} = T_T^{SMR} P_{tcp}^T $$
where $P_{tcp}^{EE}$ is the TCP position in the end-effector frame, $P_{tcp}^{SMR}$ is in the SMR frame, and $P_{tcp}^T$ is in the tool frame. The base frame position was obtained using:
$$ P^B = T_{Laser}^B P^{Laser} $$
We collected 300 joint angle sets via Latin hypercube sampling, generated robot trajectories using offline programming, and measured actual TCP positions with the laser tracker. The ELM model was trained on this data, and compensation was applied to 150 test points. We compared our EJA-ELM method with three other approaches: Method 1 (ELM without joint direction), Method 2 (spatial interpolation-based), and Method 3 (kinematic and neural network hybrid). The results are presented in the table below.
| Method | Error Range (mm) | Mean Error (mm) | Standard Deviation (mm) |
|---|---|---|---|
| No Compensation | [0.0586, 1.6913] | 0.7961 | 0.3493 |
| EJA-ELM | [0.0027, 0.4329] | 0.1402 | 0.0932 |
| Method 1 | [0.0049, 1.1416] | 0.3545 | 0.2321 |
| Method 2 | [0.0026, 0.7503] | 0.2242 | 0.1641 |
| Method 3 | [0.0030, 0.4915] | 0.2007 | 0.1050 |
Our EJA-ELM method achieved the lowest mean error (0.1402 mm) and standard deviation (0.0932 mm), outperforming the other methods. This highlights the advantage of incorporating joint direction coefficients in error prediction for robot technology.
In conclusion, the EJA-ELM offline compensation method effectively improves the absolute positioning accuracy of industrial robots by addressing joint reversal errors and other sources of inaccuracy. The extended joint space enables better error modeling, while the ELM neural network provides efficient and accurate predictions. Simulations and experiments on a Comau robot validate the method’s superiority over existing approaches, with significant reductions in position errors. This advancement in robot technology paves the way for more reliable and precise robotic applications in high-precision manufacturing. Future work will focus on extending this method to dynamic machining conditions and further optimizing the neural network parameters for enhanced performance.
The integration of advanced neural networks like ELM into robot technology represents a significant step forward in automation. By continuously refining these methods, we can unlock new potentials for robots in complex tasks, ensuring that robot technology remains at the forefront of industrial innovation. The EJA-ELM approach not only compensates for errors but also contributes to the overall intelligence and adaptability of robotic systems, making it a valuable tool for the future of manufacturing and beyond.