In modern robotics and industrial automation, the six-axis force sensor plays a critical role in measuring multi-dimensional force and torque components simultaneously. This sensor detects forces and moments in three-dimensional space, specifically the force components (FX, FY, FZ) and moment components (MX, MY, MZ). Its applications span across various fields, including remote-controlled robots, robotic surgery, intelligent manipulators, precision assembly, automated grinding, and contour tracking. Additionally, it finds uses in aerospace, mechanical processing, and automotive industries. The dual-E elastomer structure is a common design for such sensors, offering high sensitivity, rigidity, and low inter-dimensional coupling. However, the measurement signals from a six-axis force sensor are often contaminated by complex noise interference during data acquisition, transmission, and processing. This noise arises from sources like thermal noise in strain gauges and shot noise in signal processing circuits, which can significantly degrade measurement accuracy and resolution. Therefore, effective noise reduction techniques are essential to enhance sensor performance.
This article focuses on the thin rectangular plate component of a dual-E elastomer six-axis force sensor, which is primarily responsible for detecting the moment component MZ. By analyzing the forced vibration of the plate and incorporating noise characteristics into the system state equations, an Additive White Gaussian Noise (AWGN) model is developed. The Kalman filter, an optimal estimation algorithm for linear Gaussian systems, is then applied to filter the noise. The derived AWGN-Kalman filter model is validated through simulations, demonstrating its effectiveness in improving signal quality for the six-axis force sensor.
The structure of the dual-E elastomer six-axis force sensor consists of several key components: a rectangular plate, an upper E-shaped membrane, a lower E-shaped membrane, a central force transmission ring, an inner force transmission ring, and an outer force transmission ring. The upper E-shaped membrane detects moment components MX and MY, while the lower E-shaped membrane detects force components FX, FY, and FZ. The thin rectangular plate, connected to the outer force transmission ring, is designed to sense the moment MZ. When subjected to a tangential force MZ, the rectangular plate undergoes bending deformation. The boundary conditions of the plate are modeled as fixed on one side and free on the other, with strain gauges positioned at specific coordinates to measure the strain responses.
To analyze the dynamic behavior of the thin rectangular plate, the forced vibration equation is derived based on thin plate theory. The free vibration mode shapes are expressed as:
$$W_{mn}(x, y) = \sin\left(\frac{\pi x}{a}\right) \sin\left(\frac{m \pi x}{a}\right) \left[ \cosh(\lambda_n y) + C_n \sinh(\lambda_n y) – \cos(\lambda_n y) – C_n \sin(\lambda_n y) \right],$$
where $\lambda_n$ and $C_n$ are frequency-dependent coefficients, and $m, n = 1, 2, 3, \ldots$. The natural frequencies $\omega_i$ are obtained by solving the characteristic equation, and the mode shapes $W_i(x, y)$ are determined accordingly. Under sinusoidal excitation, the transverse deflection $w(x, y, t)$ of the plate is given by:
$$w(x, y, t) = \sum_{i=1}^{\infty} W_i(x, y) T_i(t),$$
where $T_i(t)$ is the generalized coordinate for the i-th mode. The generalized force $F_i(t)$ for each mode is:
$$F_i(t) = \int_0^a \int_0^b P(x, y, t) W_i(x, y) \, dx \, dy = K_i \sin(\omega_q t),$$
with $P(x, y, t) = K \delta(x – a) \sin(\omega_q t)$ representing the sinusoidal excitation force applied at $x = a$. Here, $K$ is the amplitude, $\omega_q$ is the excitation frequency, and $K_i$ is the generalized force amplitude. The equation of motion for the generalized coordinate is:
$$\frac{d^2 T_i(t)}{dt^2} + \omega_i^2 T_i(t) = \frac{F_i(t)}{m},$$
where $m$ is the mass per unit area of the plate. Solving this yields the transverse deflection:
$$w(x, y, t) = \sum_{i=1}^{\infty} \frac{K_i W_i(x, y)}{m (\omega_i^2 – \omega_q^2)} \left( \sin(\omega_q t) – \frac{\omega_q}{\omega_i} \sin(\omega_i t) \right).$$
Using Kirchhoff’s plate theory, the strain $\epsilon(x, y, t)$ on the plate surface is related to the deflection by:
$$\epsilon(x, y, t) = -\frac{h}{2} \frac{\partial^2 w(x, y, t)}{\partial x^2},$$
where $h$ is the plate thickness. Substituting the deflection expression, the strain in the time domain is:
$$\epsilon(x, y, t) = \sum_{i=1}^{\infty} S_i \left( \sin(\omega_q t) – \frac{\omega_q}{\omega_i} \sin(\omega_i t) \right),$$
with $S_i$ being a coefficient dependent on the mode shape and excitation. For the strain gauges located at coordinates $(x_{m_j}, y_{m_j})$, the strain signal is measured and processed.
In practical six-axis force sensor systems, the acquired signals are affected by noise, primarily thermal noise from strain gauges and shot noise from signal processing circuits. These noise sources exhibit narrowband Gaussian characteristics when passed through bandpass filters centered at frequencies $\omega_i$ and $\omega_q$. The thermal noise $\eta(t)$ and shot noise $\xi(t)$ are modeled as:
$$\eta(t) = p_\eta(t) \cos[\omega_i t + \phi_\eta(t)],$$
$$\xi(t) = p_\xi(t) \cos[\omega_q t + \phi_\xi(t)],$$
where $p_\eta(t)$ and $p_\xi(t)$ are random envelopes, and $\phi_\eta(t)$ and $\phi_\xi(t)$ are random phases. These noises are zero-mean Gaussian white noises with variances $\sigma_\eta^2$ and $\sigma_\xi^2$, respectively. Incorporating these into the system state equation, the continuous-time state model for the thin rectangular plate becomes:
$$\dot{\epsilon}(t) = A_S \sum_{i=1}^{\infty} S_i \omega_i \cos(\omega_i t) + \frac{K \omega_q}{K} \sin(\omega_q t) + W_1(t) + W_2(t),$$
where $A_S$ is a mode coefficient typically in the range [1, 6], and $W_1(t)$ and $W_2(t)$ are additive noise terms given by:
$$W_1(t) = p_\eta(t) \cos[\omega_i t + \phi_\eta(t)] \cos(\omega_i t + \zeta),$$
$$W_2(t) = \Lambda \cos(\omega_q t) + p_\xi(t) \cos[\omega_q t + \phi_\xi(t)],$$
with $\Lambda = \sum_{i=1}^{\infty} S_i \omega_i^2 + A_S \omega_q \sum_{i=1}^{\infty} S_i$, $\Gamma = \sum_{i=1}^{\infty} S_i \omega_i$, and $\zeta = \arctan(A_S / \omega_i)$. Using the Box-Muller transform, it can be shown that $W_1(t)$ and $W_2(t)$ are Gaussian processes. Assuming they are white noise processes, the discrete-time state space model is formulated as:
$$\epsilon(k+1) = \Phi(k+1, k) \epsilon(k) + G(k+1, k) U(k) + \Psi(k+1, k) W_1(k) + \Omega(k+1, k) W_2(k),$$
where $\Phi(k+1, k) = \exp(A_S T_c)$ is the state transition matrix, $T_c$ is the sampling period, $U(k)$ is the control input (sinusoidal excitation), and $G$, $\Psi$, $\Omega$ are coefficients derived from the continuous-time model. The measurement equation is:
$$Z(k+1) = H \epsilon(k+1) + V(k+1),$$
where $H$ is the measurement matrix, and $V(k+1)$ is measurement noise (e.g., creep and flicker noise), modeled as zero-mean white noise with variance $R$. The AWGN-Kalman filter algorithm is then applied for optimal state estimation. The prediction step is:
$$\epsilon(k+1|k) = \Phi(k+1, k) \epsilon(k|k) + G(k+1, k) U(k),$$
$$P(k+1|k) = \Phi(k+1, k) P(k|k) \Phi^T(k+1, k) + \sigma_\eta^2 + \sigma_\xi^2,$$
where $P$ is the error covariance matrix. The update step is:
$$K(k+1) = P(k+1|k) H^T \left( H P(k+1|k) H^T + R \right)^{-1},$$
$$\epsilon(k+1|k+1) = \epsilon(k+1|k) + K(k+1) \left( Z(k+1) – H \epsilon(k+1|k) \right),$$
$$P(k+1|k+1) = \left( I – K(k+1) H \right) P(k+1|k).$$
This AWGN-Kalman filter provides an optimal estimate of the strain signal, effectively reducing noise and enhancing the resolution of the six-axis force sensor.
To validate the model, simulations were conducted using a third-order mode state space model. A sinusoidal excitation force $P(t) = 10 \sin(40\pi t)$ with frequency 20 Hz and amplitude 10 was applied to the six-axis force sensor. The sampling period was set to 5 ms, and 50 samples were taken. Two filtering algorithms were compared: Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF). The results demonstrate the effectiveness of the AWGN-Kalman filter in noise reduction. The UKF method outperformed EKF, with a significant reduction in root mean square error (RMSE).
| Filtering Algorithm | RMSE | Improvement of UKF over EKF |
|---|---|---|
| EKF | 0.0027434 | 62.98% |
| UKF | 0.0010155 |
The table above summarizes the performance metrics, showing that UKF achieves a 62.98% improvement in RMSE compared to EKF. This highlights the importance of selecting an appropriate filtering method for the six-axis force sensor applications.
In conclusion, the AWGN-Kalman filter model developed for the thin rectangular plate of a dual-E elastomer six-axis force sensor effectively addresses noise interference issues. By incorporating Gaussian noise characteristics into the system state equations and applying Kalman filtering, the model significantly improves signal quality and sensor resolution. The simulation results confirm that the UKF algorithm provides superior performance, making it a suitable choice for real-time noise reduction in six-axis force sensor systems. Future work could explore adaptive filtering techniques and real-time implementation on embedded systems for broader applications.
The six-axis force sensor is indispensable in advanced robotics and automation, and continuous improvements in signal processing, such as the AWGN-Kalman filter, will further enhance its accuracy and reliability. By leveraging mathematical models and optimal estimation theory, the performance of six-axis force sensors can be optimized for demanding environments.