In the field of robotics and intelligent machinery, the six-axis force sensor plays a pivotal role in detecting three-dimensional force and moment information, enabling precise control and feedback in applications such as industrial automation, prosthetics, and aerospace. However, a significant challenge in the design of these sensors is the inherent coupling effects within the elastic body, where a load applied in one direction produces unwanted outputs in other directions. This coupling complicates the decoupling process, often addressed through static calibration methods. Yet, in real-world scenarios, six-axis force sensors are frequently subjected to dynamic loads, necessitating a focus on dynamic decoupling to ensure accurate real-time force measurements. In this study, we investigate the dynamic behavior of the upper E-type membrane, a key component in a dual E-type six-axis force sensor, by developing a mechanical model based on thin plate vibration and bending theories. We employ analytical methods to derive expressions for dynamic strain output under Z-direction dynamic loading, validate our findings through simulations, and emphasize the importance of dynamic decoupling for enhancing sensor performance.
The dual E-type six-axis force sensor comprises several structural elements, including rectangular thin plates, upper and lower E-type membranes, and rigid components like the central pillar. The upper E-type membrane, our primary focus, is modeled as a homogeneous, circular thin plate with specific boundary conditions: free at the outer edge and fixed at the inner edge. This simplification allows us to apply thin plate theory effectively. When a dynamic load is applied in the Z-direction, it is transmitted through the outer force ring to the inner force ring, eventually acting on the outer periphery of the upper E-type membrane. The resulting vibrations are critical for understanding the sensor’s dynamic response. To illustrate the structural configuration, consider the following representation:
The upper E-type membrane’s material and geometric properties are essential for our analysis. We summarize these parameters in the table below, which includes values for elastic modulus, density, Poisson’s ratio, and dimensions. These parameters are used throughout our derivations and simulations to ensure consistency and accuracy in modeling the six-axis force sensor’s behavior.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Elastic Modulus | E | 72 | GPa |
| Density | ρ | 2.18 × 103 | kg/m³ |
| Poisson’s Ratio | μ | 0.33 | – |
| Outer Diameter | D1 | 0.05 | m |
| Inner Diameter | d1 | 0.0075 | m |
| Thickness | h1 | 0.002 | m |
In the force analysis of the six-axis force sensor, when a dynamic load FZ is applied along the Z-axis, the upper E-type membrane undergoes forced bending vibrations. We model this using the classical thin plate theory in polar coordinates, where the governing differential equation for forced vibration is expressed as:
$$ D \nabla^4 w + \rho h \frac{\partial^2 w}{\partial t^2} = f_1(r, \phi, t) $$
Here, \( D \) represents the flexural rigidity of the plate, defined as \( D = \frac{E h^3}{12(1-\mu^2)} \), \( \nabla^4 \) is the biharmonic operator, \( \rho \) is the density, \( h \) is the thickness, and \( f_1(r, \phi, t) \) is the external dynamic load per unit area. The displacement \( w(r, \phi, t) \) denotes the transverse deflection of the plate. To solve this equation, we expand the load and displacement in terms of the natural mode shapes of the circular plate. The mode shapes satisfy the free vibration equation and are determined based on the boundary conditions. For a circular plate with a free outer edge and fixed inner edge, the mode shapes can be derived using Bessel functions. The external load is expressed as a series of these mode shapes:
$$ f_1(r, \phi, t) = \sum_{m=1}^{\infty} \sum_{n=0}^{\infty} F_{mn}(t) \Psi_{mn}(r, \phi) $$
where \( \Psi_{mn}(r, \phi) \) are the orthonormal mode shapes, and \( F_{mn}(t) \) are the generalized forces. Similarly, the displacement is written as:
$$ w(r, \phi, t) = \sum_{m=1}^{\infty} \sum_{n=0}^{\infty} q_{mn}(t) \Psi_{mn}(r, \phi) $$
Substituting these into the governing equation and applying orthogonality conditions, we obtain a set of ordinary differential equations for the generalized coordinates \( q_{mn}(t) \):
$$ \ddot{q}_{mn}(t) + \omega_{mn}^2 q_{mn}(t) = \frac{1}{\rho h} F_{mn}(t) $$
where \( \omega_{mn} \) are the natural frequencies of the plate. For the six-axis force sensor, the dynamic load is often harmonic, such as \( F_{mn}(t) = K \sin(\Omega t) \), where \( K \) is the amplitude and \( \Omega \) is the excitation frequency. The solution for \( q_{mn}(t) \) under zero initial conditions is:
$$ q_{mn}(t) = \frac{K}{\rho h (\omega_{mn}^2 – \Omega^2)} \left( \sin(\Omega t) – \frac{\Omega}{\omega_{mn}} \sin(\omega_{mn} t) \right) $$
However, in practical applications of the six-axis force sensor, the excitation frequency \( \Omega \) is much lower than the fundamental natural frequency \( \omega_{11} \), allowing us to simplify the response by considering only the first few modes. For this analysis, we focus on the first mode to reduce computational complexity while maintaining accuracy. The dynamic deflection of the upper E-type membrane is then approximated as:
$$ w(r, \phi, t) \approx q_{11}(t) \Psi_{11}(r, \phi) $$
where \( \Psi_{11}(r, \phi) \) is the first mode shape. The natural frequencies and mode shapes for the upper E-type membrane are listed in the table below, which we use in our simulations for the six-axis force sensor.
| Mode (m) | Frequency (Hz) | Mode Shape \( \Psi_{mn}(r, \phi) \) |
|---|---|---|
| 1 | 666 | \( J_0(\lambda_{11} r) + C_{11} I_0(\lambda_{11} r) \) |
| 2 | 5362 | \( J_1(\lambda_{21} r) \cos(\phi) + C_{21} I_1(\lambda_{21} r) \cos(\phi) \) |
| 3 | 16262 | \( J_2(\lambda_{32} r) \cos(2\phi) + C_{32} I_2(\lambda_{32} r) \cos(2\phi) \) |
Using the first mode approximation, the dynamic deflection becomes:
$$ w(r, t) = \frac{K \Psi_{11}(r)}{\rho h (\omega_{11}^2 – \Omega^2)} \sin(\Omega t) $$
Substituting the parameter values from the first table, we compute the specific expression for the upper E-type membrane in the six-axis force sensor. For instance, with \( K = 1 \), \( \rho = 2180 \, \text{kg/m}^3 \), \( h = 0.002 \, \text{m} \), \( \omega_{11} = 2\pi \times 666 \, \text{rad/s} \), and \( \Psi_{11}(r) \) derived from Bessel functions, the deflection simplifies to a function of radial position and time. This expression allows us to analyze the strain output, which is critical for sensor calibration in the six-axis force sensor.
To derive the dynamic strain output, we refer to the strain-displacement relations in thin plate theory. For a circular plate under bending, the radial strain \( \epsilon_r \) and circumferential strain \( \epsilon_\phi \) are given by:
$$ \epsilon_r = -z \frac{\partial^2 w}{\partial r^2}, \quad \epsilon_\phi = -z \left( \frac{1}{r} \frac{\partial w}{\partial r} + \frac{1}{r^2} \frac{\partial^2 w}{\partial \phi^2} \right) $$
where \( z \) is the distance from the mid-plane. Since strain gauges are attached to the surface of the upper E-type membrane in the six-axis force sensor, we evaluate these strains at \( z = h/2 \). Given that the load is axisymmetric for the Z-direction force, the derivatives with respect to \( \phi \) vanish, simplifying the expressions to:
$$ \epsilon_r = -\frac{h}{2} \frac{\partial^2 w}{\partial r^2}, \quad \epsilon_\phi = -\frac{h}{2} \frac{1}{r} \frac{\partial w}{\partial r} $$
Substituting the dynamic deflection \( w(r, t) \) into these equations, we obtain the time-dependent radial and circumferential strains. For example, using the first mode shape \( \Psi_{11}(r) \), the strains become:
$$ \epsilon_r(r, t) = -\frac{h}{2} q_{11}(t) \frac{\partial^2 \Psi_{11}(r)}{\partial r^2}, \quad \epsilon_\phi(r, t) = -\frac{h}{2} q_{11}(t) \frac{1}{r} \frac{\partial \Psi_{11}(r)}{\partial r} $$
These expressions highlight how the dynamic strain in the six-axis force sensor varies with position and time, directly influencing the sensor’s output signals. To validate our analytical model, we conducted simulation experiments using MATLAB, considering excitation frequencies of 20 Hz, 35 Hz, and 50 Hz—typical for robotic applications. The simulations involved computing the radial and circumferential strains over the plate surface and plotting them against time and radial position. The results showed that the vibration frequency of the upper E-type membrane increases with the excitation frequency, consistent with theoretical predictions for the six-axis force sensor. For instance, at 20 Hz, the strain outputs exhibited slower oscillations, while at 50 Hz, the oscillations were more rapid, confirming the dynamic coupling effects.
In conclusion, our analysis of the upper E-type membrane in a six-axis force sensor demonstrates the effectiveness of thin plate theory in modeling dynamic decoupling. By deriving the dynamic deflection and strain outputs under Z-direction loading, we provide a theoretical foundation for improving the accuracy of six-axis force sensors in dynamic environments. The simulations corroborate our analytical results, showing that the simplified first-mode approximation suffices for practical applications where excitation frequencies are below the fundamental natural frequency. This work underscores the importance of dynamic decoupling in six-axis force sensor design and offers insights for future optimizations, such as incorporating higher modes for enhanced precision. Ultimately, advancing the dynamic analysis of six-axis force sensors will contribute to more reliable and responsive force sensing in intelligent systems.