In the field of robot technology, monitoring milling forces during robotic milling processes is crucial for assessing workpiece quality, tool condition, and overall system performance. However, direct measurement of these forces in large structural components using dynamometers is often impractical due to size constraints and cost. This paper introduces a fast identification method for robot milling forces based on acceleration signals, leveraging the relationship between acceleration and milling forces in robot systems. By exploiting the recursive nature of the system matrix and the specificity of regularization operators, the proposed method significantly reduces identification time while maintaining accuracy. Experimental validation through robotic milling tests confirms the effectiveness of this approach, with identification times reduced to under 15 ms. The integration of robot technology in this context enables efficient, non-contact force monitoring, paving the way for real-time process optimization and condition monitoring in industrial applications.
The milling force in robot technology serves as a key parameter for monitoring machining quality, tool wear, and dynamic behavior. Traditional methods, such as using table-mounted dynamometers, are limited by workpiece size, while rotary dynamometers attached between the tool and spindle are expensive and may affect robot stability. Indirect force identification through signals like current, displacement, or acceleration offers a cost-effective alternative. Among these, acceleration signals are particularly advantageous due to their wide frequency bandwidth and ease of acquisition in robot technology environments. This paper focuses on developing a rapid identification method that utilizes acceleration signals to estimate milling forces, addressing the need for high-speed computation in real-time applications.
The relationship between acceleration response and milling force in a robot system can be described by the convolution integral: $$ x(t) = \int_0^t h(t – \tau) f(\tau) d\tau $$ where \( x(t) \) is the acceleration, \( h(t) \) is the impulse response function of the robot system, and \( f(t) \) is the milling force. In discrete form, this becomes: $$ \mathbf{x} = \mathbf{H} \mathbf{f} + \mathbf{e} $$ Here, \( \mathbf{x} \) is the acceleration vector, \( \mathbf{H} \) is the system matrix derived from the impulse response, \( \mathbf{f} \) is the force vector, and \( \mathbf{e} \) represents measurement errors. To solve this ill-posed problem, Tikhonov regularization is applied, minimizing the objective function: $$ \min_{\mathbf{f}} \left( \| \mathbf{x} – \mathbf{H} \mathbf{f} \|_2^2 + \lambda \| \mathbf{f} \|_2^2 \right) $$ where \( \lambda \) is the regularization parameter. The solution is given by: $$ \mathbf{f}^* = (\mathbf{H}^T \mathbf{H} + \lambda \mathbf{I})^{-1} \mathbf{H}^T \mathbf{x} $$ This approach, however, can be computationally intensive. To enhance speed, we exploit the recursive structure of the system matrix \( \mathbf{H} \), which exhibits a Toeplitz-like property due to the time-invariance of the impulse response in robot technology systems. Specifically, each row of \( \mathbf{H} \) can be related to the previous one by a shift operator: $$ \mathbf{h}_{i+1} = \mathbf{h}_i \mathbf{S} $$ where \( \mathbf{S} \) is a shift matrix. This allows for efficient computation of the force estimate \( \mathbf{f}^* \) by reusing previous calculations, reducing redundant operations.
The fast identification method involves decomposing the regularized solution into incremental updates. For instance, at time step \( i+1 \), the force estimate can be computed as: $$ \mathbf{f}_{i+1}^* = \mathbf{h}_{i+1}^* \mathbf{x} \approx \mathbf{h}_i^* \mathbf{S} \mathbf{x} $$ where \( \mathbf{h}_i^* \) is the effective part of the regularized matrix row. This recursion minimizes computational load, enabling real-time force identification in robot technology applications. The regularization parameter \( \lambda \) is determined using generalized cross-validation: $$ \lambda = \arg \min \left\{ \frac{\| (\mathbf{H}^T \mathbf{H} + \lambda \mathbf{I})^{-1} \mathbf{x} \|^2}{\text{Tr}((\mathbf{H}^T \mathbf{H} + \lambda \mathbf{I})^{-1})} \right\} $$ ensuring optimal balance between accuracy and noise suppression.
To validate the method, experimental studies were conducted using a robotic milling setup. The impulse response function of the robot system was obtained through modal testing, where a hammer impact excited the system, and acceleration responses were measured. The frequency response function (FRF) was derived and transformed into the time domain to construct the system matrix \( \mathbf{H} \). The following table summarizes the modal testing parameters used in the robot technology framework:
| Parameter | Value |
|---|---|
| Sampling Frequency | 10,240 Hz |
| Hammer Sensitivity | 1.1 mV/N |
| Accelerometer Type | Triaxial |
The impulse response function \( h(t) \) was discretized to form \( \mathbf{H} \), and the regularized solution matrix \( \mathbf{H}^* \) was analyzed for its recursive properties. The characteristic plot of \( \mathbf{H}^* \) confirmed the shift-invariance, enabling the fast identification algorithm. Robotic milling tests were then performed on polyoxymethylene workpieces using a two-flute end mill. The milling parameters varied across tests, as shown in the table below, to evaluate the method under different conditions in robot technology:
| Test No. | Spindle Speed (rpm) | Cutting Width (mm) | Cutting Depth (mm) | Feed Rate (mm/min) |
|---|---|---|---|---|
| 1 | 3,000 | 3 | 0.7 | 130 |
| 2 | 3,000 | 3 | 1.0 | 130 |
| 3 | 4,500 | 3 | 1.0 | 130 |
Acceleration signals were measured during milling and processed using comb filtering to isolate relevant frequency components (e.g., spindle rotation harmonics). The fast identification method was applied to estimate milling forces, which were compared to reference forces measured by a dynamometer (low-pass filtered at 600 Hz). The results demonstrated close agreement between identified and measured forces, with error metrics quantified by peak error (PE) and root mean square error (RMSE): $$ \text{PE} = \frac{\max |f_{\text{identified}} – f_{\text{measured}}|}{\max |f_{\text{measured}}|} \times 100\% $$ $$ \text{RMSE} = \sqrt{\frac{1}{N} \sum_{i=1}^N (f_{\text{identified}} – f_{\text{measured}})^2} \times 100\% $$ The following table compares the performance of the standard Tikhonov regularization and the fast identification method in robot technology applications:
| Test No. | PE (Tikhonov) (%) | PE (Fast) (%) | RMSE (Tikhonov) (%) | RMSE (Fast) (%) | Time (Tikhonov) (s) | Time (Fast) (ms) |
|---|---|---|---|---|---|---|
| 1 | 23.31 | 23.20 | 12.63 | 16.12 | 38.9 | 11.6 |
| 2 | 19.24 | 19.55 | 10.76 | 12.39 | 38.7 | 12.3 |
| 3 | 12.95 | 12.39 | 6.00 | 4.07 | 38.2 | 12.2 |
The fast method achieved identification times under 15 ms, making it suitable for online monitoring in robot technology, whereas the standard approach took over 35 s. The slight increase in error for some tests is attributed to the approximation in the recursive computation, but the trade-off is justified by the significant speed improvement. This advancement in robot technology allows for real-time force monitoring without expensive sensors, facilitating adaptive control and optimization in machining processes. The method’s generality extends to other outputs like displacement, provided the system’s impulse response is known, and can be adapted to CNC machines if the transfer function matrix exhibits Toeplitz structure.
In conclusion, the proposed fast identification method for robot milling forces based on acceleration signals effectively balances accuracy and computational efficiency. By leveraging the recursive properties of the system matrix and Tikhonov regularization, it reduces identification time to under 15 ms, enabling real-time applications in robot technology. Experimental results confirm its robustness across varying milling conditions, with peak errors below 23.20% and RMSE below 16.12%. This approach underscores the potential of robot technology in enhancing manufacturing processes through innovative sensor fusion and algorithm optimization. Future work could focus on extending the method to multi-axis force identification and integrating it with machine learning for improved adaptability in dynamic environments.