In modern industrial robotics, precise force control is critical for applications such as polishing, assembly, and material handling. Accurate real-time monitoring of contact forces ensures smoother operations, improved efficiency, and enhanced safety. To achieve this, six-axis force sensors are often integrated at the robot’s end-effector to measure three orthogonal forces and their corresponding moments. However, dynamic environments introduce significant challenges in sensor calibration, primarily due to inertial forces and gravitational effects that cause zero-point drift. Traditional static calibration methods fall short in flexible production lines where tool changes and varying installation angles are common. This paper addresses these limitations by proposing a dynamic zero-point calibration system that leverages a hybrid neural network model to compensate for both linear and nonlinear variations in real-time.
The core of our approach lies in combining theoretical models of inertial forces with neural networks to enhance calibration accuracy. We analyze the dynamic interactions between the robot’s Denavit-Hartenberg (DH) parameters, joint angles, and the end-effector’s mass and center of gravity. By integrating these factors into a hybrid model, we effectively decouple the sensor’s zero drift from the actual polishing forces. Our experiments demonstrate that this method reduces the relative mean deviation (RMD) of three-dimensional forces and moments to below 4%, outperforming conventional theoretical and BP neural network models. The hybrid model’s ability to handle complex, time-varying systems makes it particularly suitable for high-precision industrial tasks.
To illustrate the system’s configuration, consider the following representation of a typical setup involving a six-axis force sensor integrated into a robotic polishing system:
The dynamic calibration system accounts for gravitational and inertial forces acting on the six-axis force sensor. The robot’s kinematic chain, described by transformation matrices, plays a crucial role in determining these forces. For a six-degree-of-freedom serial robot, the transformation matrix from the base frame to the sensor frame is derived as follows. Let \( T_i^{i-1} \) represent the homogeneous transformation between consecutive joints, accounting for joint variable offsets \( \theta_i \). The overall transformation matrix \( T_H \) from the base frame \( O_0 \) to the sensor frame \( O_6 \) is given by:
$$ T_H = T_1^0 T_2^{O_1} T_3^2 T_4^3 T_5^4 T_6^5 = \begin{bmatrix} n_x & s_x & a_x & p_x \\ n_y & s_y & a_y & p_y \\ n_z & s_z & a_z & p_z \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Here, the upper-left 3×3 submatrix \( T_{Hj} \) represents the rotational component. The gravitational force vector in the base frame is \( F_G = [0, 0, -m_t g]^T \), where \( m_t \) is the mass of the polishing tool and \( g \) is gravitational acceleration. The force components \( F_{tg} = [F_{xtg}, F_{ytg}, F_{ztg}]^T \) due to gravity in the sensor frame are computed as:
$$ F_{tg} = T_{Hj}^T \cdot F_G = \begin{bmatrix} -m_t g \cdot n_z \\ -m_t g \cdot s_z \\ -m_t g \cdot a_z \end{bmatrix} $$
The corresponding moments \( M_{tg} = [M_{xtg}, M_{ytg}, M_{ztg}]^T \) are derived using the cross-product with the center of gravity vector \( P_{O6} = [x_{O6}, y_{O6}, z_{O6}, 1]^T \) in the sensor frame:
$$ M_{tg} = \begin{bmatrix} 0 & F_{ztg} & -F_{ytg} \\ -F_{ztg} & 0 & F_{xtg} \\ F_{ytg} & -F_{xtg} & 0 \end{bmatrix} \cdot \begin{bmatrix} x_{O6} \\ y_{O6} \\ z_{O6} \end{bmatrix} $$
In dynamic conditions, inertial forces must also be compensated. The acceleration vector \( \alpha_{O0} = [\alpha_x, \alpha_y, \alpha_z]^T \) of the tool’s center of gravity in the base frame is obtained from the second derivative of its position \( P_{O0} = T_H \cdot P_{O6} \). The inertial force components \( F_{ta} = [F_{xta}, F_{yta}, F_{zta}]^T \) in the sensor frame are:
$$ F_{ta} = m_t \cdot (T_{Hj}^T \cdot \alpha_{O0}) $$
Similarly, the inertial moments \( M_{ta} \) are calculated using the same cross-product method. The total compensation matrix \( FM_t \) combining gravitational and inertial effects is:
$$ FM_t = FM_{tg} + FM_{ta} $$
where \( FM_{tg} = [F_{tg}, M_{tg}]^T \) and \( FM_{ta} = [F_{ta}, M_{ta}]^T \). This theoretical model, however, suffers from practical errors due to unmeasured parameters like exact center of gravity and external disturbances from cables. To address this, we integrate a hybrid neural network that learns residual errors.
The hybrid neural network model combines the theoretical inertial compensation with a BP neural network to handle nonlinearities. The BP network has six input nodes (robot joint angles), two hidden layers with 50 neurons each, and six output nodes (compensated forces and moments). The inputs are normalized, and the network is trained using stochastic gradient descent over 400 samples with a time interval of 50 ms to capture dynamic changes. The hybrid model operates in parallel, where the theoretical component computes inertial forces, and the BP network adjusts for systematic errors. The structure ensures real-time processing, with compensation times as low as 0.2 ms per sample, suitable for high sampling rates.
To evaluate the model, we conducted dynamic calibration experiments using a six-axis robot equipped with a six-axis force sensor. The sensor’s specifications are summarized in the table below:
| Parameter | Fx | Fy | Fz | Mx | My | Mz |
|---|---|---|---|---|---|---|
| Range (N/N·m) | 580 | 580 | 1160 | 20 | 20 | 20 |
| Resolution (N/N·m) | 1/16 | 1/16 | 1/16 | 1/1600 | 1/1600 | 1/1600 |
| Uncertainty (%) | 1.75 | 1.25 | 1.00 | 1.25 | 1.75 | 1.00 |
We collected 500 data samples during robot motion without polishing, with 450 used for training and 50 for testing. The hybrid model was compared against a pure theoretical model and a standalone BP network. The results show that the hybrid model achieves superior accuracy, with force RMD below 3.72% and moment RMD below 3.19%. The following table compares the RMD values across models:
| Model | Fx RMD (%) | Fy RMD (%) | Fz RMD (%) | Mx RMD (%) | My RMD (%) | Mz RMD (%) |
|---|---|---|---|---|---|---|
| Theoretical | 29.72 | 18.12 | 8.15 | 30.45 | 32.10 | 12.35 |
| BP Network | 9.04 | 6.68 | 5.98 | 4.87 | 10.38 | 5.40 |
| Hybrid Model | 3.72 | 3.12 | 2.89 | 3.19 | 3.18 | 3.20 |
The hybrid model’s performance is further analyzed using absolute deviations. For forces, the maximum absolute deviation is 2.81 N, with an average of 0.90 N. For moments, the maximum absolute deviation is 0.21 N·m, with an average of 0.04 N·m. These values meet the industrial requirement of ±5 N force fluctuation tolerance. The hybrid network’s ability to reduce errors is attributed to its dual mechanism: the theoretical part handles linear gravitational and inertial effects, while the BP network compensates for nonlinearities and installation errors.
In terms of computational efficiency, the hybrid model processes each compensation in 0.2 ms, enabling real-time application at sampling rates up to 5 kHz. This exceeds typical industrial needs of 1 kHz, ensuring no latency in force monitoring. The model’s robustness is validated through repeated tests on varying paths, where it maintains consistent accuracy despite changes in robot dynamics. The integration of the six-axis force sensor with the hybrid network thus provides a reliable solution for dynamic environments.
In conclusion, our hybrid neural network approach effectively addresses the dynamic calibration challenges of six-axis force sensors in robotic systems. By merging theoretical models with data-driven learning, we achieve high precision in compensating for zero drift caused by gravity and inertia. The experimental results confirm that the hybrid model outperforms both theoretical and pure BP network models, with all force and moment RMD values remaining under 4%. This method enhances the accuracy of force sensing in applications like polishing, paving the way for more adaptive and flexible industrial automation. Future work will focus on extending the model to multi-sensor systems and optimizing network architecture for faster convergence.
The proposed system highlights the importance of dynamic compensation in six-axis force sensor applications. As robotics continues to evolve, integrating intelligent calibration methods will be crucial for achieving higher levels of precision and efficiency. Our hybrid model serves as a foundational step toward this goal, demonstrating the synergy between physical models and artificial intelligence in solving complex engineering problems.