In the field of robotics and precision engineering, the demand for high-performance six-axis force sensors has grown significantly due to their ability to measure three-dimensional forces and moments simultaneously. These sensors are critical in applications such as industrial automation, aerospace, and biomedical devices, where accurate force feedback is essential. However, conventional six-axis force sensors often face limitations in handling large loads without increasing their structural size, which can restrict their usability in compact or high-load scenarios. To address this, we propose a novel 8/4-4 split-load parallel six-axis force sensor design that incorporates a central load-bearing beam to distribute external loads effectively, thereby increasing the sensor’s range without enlarging its footprint. This article presents a comprehensive mathematical model, decoupling algorithm, and simulation-based validation of this sensor, emphasizing the principles of load distribution and structural optimization.
The core innovation of our design lies in the integration of a Stewart platform-based structure with eight piezoelectric branches arranged in an 8/4-4 configuration, combined with a central beam that承担 a significant portion of the external generalized six-axis force. Traditional six-axis force sensors, such as those with cross-beam or cylindrical elastic bodies, often suffer from reduced accuracy under heavy loads or require bulky designs to maintain stiffness. In contrast, our approach leverages the isotropic properties of the 8/4-4 parallel arrangement, where piezoelectric ceramics act as sensing elements in each branch, while the central beam enhances load-bearing capacity. This not only improves the dynamic response but also ensures high decoupling precision. The mathematical formulation involves deriving the relationship between the external forces and the internal branch forces, incorporating stiffness matrices and Jacobian transformations to achieve real-time force measurement.
To elaborate, the split-load principle operates on the basis that when an external generalized six-axis force $ \mathbf{F} = [F_x, F_y, F_z, M_x, M_y, M_z]^T $ is applied to the sensor’s upper platform, it is divided into two components: a major portion $ \mathbf{F}_c $ borne by the central beam and a minor portion $ \mathbf{F}_s $ handled by the eight branches. This can be expressed as:
$$ \mathbf{F} = \mathbf{F}_c + \mathbf{F}_s $$
where $ \mathbf{F}_c $ and $ \mathbf{F}_s $ are derived from the deformations of the central beam and branches, respectively. The branches, modeled as two-force members with piezoelectric ceramics, generate measurable forces $ \mathbf{f} = [f_1, f_2, \dots, f_8]^T $, which are used to compute the external load via a decoupling algorithm. The central beam, with its high stiffness, reduces the stress on the branches, allowing the six-axis force sensor to handle larger loads while maintaining accuracy. This design is particularly advantageous in applications like heavy-duty robotics, where space constraints and high load requirements coexist.
The mathematical model begins with the geometry of the 8/4-4 split-load parallel six-axis force sensor. As illustrated in the figure, the sensor consists of an upper and lower platform, parallel to each other, with eight branches connected via flexible spherical joints at the upper platform and flexible Hooke’s joints at the lower platform. The central beam is aligned coaxially between the platforms. A reference coordinate system is established at the geometric center of the upper platform, with the Y-axis vertical, the X-axis bisecting the angle between vectors $ \mathbf{b}_1 $ and $ \mathbf{b}_2 $, and the Z-axis determined by the right-hand rule. Similarly, a coordinate system at the lower platform’s center has axes parallel to the reference frame. Key parameters include the distribution radii $ R_a $ for the upper platform, $ R_{B1} $ and $ R_{B2} $ for the lower platform’s inner and outer circles, the central beam’s cross-sectional radius $ r $, the height $ h $ between platforms, and angles $ \alpha_1 $, $ \alpha_2 $, $ \beta_1 $, $ \beta_2 $, $ \theta_1 $, and $ \theta_2 $ defining the joint orientations.
The force analysis for the Stewart structure portion relates the branch forces $ \mathbf{f} $ to the component $ \mathbf{F}_s $ through the Jacobian matrix $ \mathbf{J}_{10} $:
$$ \mathbf{F}_s = \mathbf{J}_{10} \cdot \mathbf{f} $$
where $ \mathbf{J}_{10} $ is defined as:
$$ \mathbf{J}_{10} = \begin{bmatrix}
\frac{\mathbf{b}_1 – \mathbf{B}_1}{\|\mathbf{b}_1 – \mathbf{B}_1\|} & \frac{\mathbf{b}_2 – \mathbf{B}_2}{\|\mathbf{b}_2 – \mathbf{B}_2\|} & \cdots & \frac{\mathbf{b}_8 – \mathbf{B}_8}{\|\mathbf{b}_8 – \mathbf{B}_8\|} \\
\frac{\mathbf{B}_1 \times \mathbf{b}_1}{\|\mathbf{b}_1 – \mathbf{B}_1\|} & \frac{\mathbf{B}_2 \times \mathbf{b}_2}{\|\mathbf{b}_2 – \mathbf{B}_2\|} & \cdots & \frac{\mathbf{B}_8 \times \mathbf{b}_8}{\|\mathbf{b}_8 – \mathbf{B}_8\|}
\end{bmatrix} $$
Here, $ \mathbf{b}_i $ and $ \mathbf{B}_i $ represent the position vectors of the joint centers on the upper and lower platforms, respectively. The deformation of the branches under load is small, allowing us to use a first-order linearization approach. The relationship between the infinitesimal displacement $ \Delta \mathbf{x} = [dx, dy, dz, \omega_x, \omega_y, \omega_z]^T $ of the upper platform and the branch deformations $ \Delta \mathbf{l} = [\Delta l_1, \Delta l_2, \dots, \Delta l_8]^T $ is given by:
$$ \Delta \mathbf{x} = (\mathbf{J}_{10}^T)^{-1} \cdot \Delta \mathbf{l} $$
Applying Hooke’s law, the branch deformation can be expressed as $ \Delta \mathbf{l} = \frac{\mathbf{f} \cdot l_0}{E_s \cdot A_s} $, where $ E_s $ is the elastic modulus of the branches, $ A_s $ is their equivalent cross-sectional area, and $ l_0 $ is the original length of the branches. Thus, the displacement becomes:
$$ \Delta \mathbf{x} = (\mathbf{J}_{10}^T)^{-1} \cdot \frac{l_0}{E_s \cdot A_s} \cdot \mathbf{f} $$
The force component $ \mathbf{F}_c $ carried by the central beam is then calculated using its stiffness matrix $ \mathbf{K}_c $:
$$ \mathbf{F}_c = \mathbf{K}_c \cdot \Delta \mathbf{x} = \mathbf{K}_c \cdot (\mathbf{J}_{10}^T)^{-1} \cdot \frac{l_0}{E_s \cdot A_s} \cdot \mathbf{f} $$
For a circular cross-section beam, the stiffness matrix $ \mathbf{K}_c $ accounts for bending, torsion, and shear effects, with a shear correction factor $ k = 0.9 $ to minimize errors in large-diameter scenarios. The matrix is formulated as:
$$ \mathbf{K}_c = \begin{bmatrix}
\frac{E_c A_c}{h} & 0 & 0 & 0 & 0 & 0 \\
0 & a_z & 0 & 0 & 0 & c_z \\
0 & 0 & a_y & 0 & -c_y & 0 \\
0 & 0 & 0 & \frac{G_c I_p}{h} & 0 & 0 \\
0 & 0 & -c_y & 0 & e_y & 0 \\
0 & c_z & 0 & 0 & 0 & e_z
\end{bmatrix} $$
where the terms are derived as follows:
$$ \phi_y = \frac{12 E_c I_z k}{G_c A_c h^2}, \quad \phi_z = \frac{12 E_c I_y k}{G_c A_c h^2} $$
$$ a_y = \frac{12 E_c I_y}{h^3 (1 + \phi_z)}, \quad a_z = \frac{12 E_c I_z}{h^3 (1 + \phi_y)} $$
$$ e_y = \frac{(4 + \phi_z) E_c I_y}{h (1 + \phi_z)}, \quad e_z = \frac{(4 + \phi_y) E_c I_z}{h (1 + \phi_y)} $$
$$ c_y = \frac{6 E_c I_y}{h^2 (1 + \phi_z)}, \quad c_z = \frac{6 E_c I_z}{h^2 (1 + \phi_y)} $$
In these equations, $ E_c $ is the elastic modulus of the central beam, $ G_c $ is its shear modulus, $ A_c $ is the cross-sectional area, $ I_p $ is the torsional moment of inertia, and $ I_y $ and $ I_z $ are the moments of inertia about the Y and Z axes, respectively. By defining $ \mathbf{J}_{20} = \mathbf{K}_c \cdot (\mathbf{J}_{10}^T)^{-1} \cdot \frac{l_0}{E_s \cdot A_s} $, we simplify the expression to $ \mathbf{F}_c = \mathbf{J}_{20} \cdot \mathbf{f} $. Combining this with the earlier equation for $ \mathbf{F}_s $, the total external load on the six-axis force sensor is:
$$ \mathbf{F} = \mathbf{J} \cdot \mathbf{f} $$
$$ \mathbf{J} = \mathbf{J}_{10} + \mathbf{K}_c \cdot (\mathbf{J}_{10}^T)^{-1} \cdot \frac{l_0}{E_s \cdot A_s} $$
Here, $ \mathbf{J} $ is the overall Jacobian matrix of the 8/4-4 split-load parallel six-axis force sensor, which enables the decoupling of the branch forces into the six-axis output. This decoupling algorithm is crucial for real-time applications, as it allows the sensor to provide accurate force and moment measurements even under varying load conditions.
To validate the mathematical model and decoupling algorithm, we conducted simulations using ADAMS software, where a virtual prototype of the six-axis force sensor was created. The branches were modeled as springs, and the central beam was represented as a massless beam element based on Timoshenko beam theory, which accurately captures axial, bending, and torsional effects. The structural parameters were set as follows: $ R_a = 0.1 \, \text{m} $, $ R_{B1} = 0.1527 \, \text{m} $, $ R_{B2} = 0.0578 \, \text{m} $, $ h = 0.041 \, \text{m} $, $ \theta_1 = -11^\circ $, $ \theta_2 = 20^\circ $, and $ r = 0.025 \, \text{m} $. The lower platform was fixed, and time-varying force and moment drives were applied to the upper platform’s center: $ F_x = F_y = F_z = 3000 \sin(4\pi t) \, \text{N} $ and $ M_x = M_y = M_z = 300 \sin(4\pi t) \, \text{N·m} $, with a simulation time of 2 seconds and a sampling frequency of 500 Hz.
The branch forces obtained from the simulation were processed using the decoupling algorithm in MATLAB, and the computed six-axis forces were compared to the applied loads in ADAMS. The results demonstrated high accuracy, with errors in force components not exceeding 0.037% and errors in moment components below 0.14%. Specifically, the errors for the X, Y, and Z direction forces were 0.037%, 0.023%, and 0.037%, respectively, while for moments, they were 0.14%, 0.098%, and 0.14%. Notably, the error curves for the X and Z directions coincided, indicating excellent isotropy in these directions, which is a key advantage of this six-axis force sensor design. The table below summarizes the error analysis:
| Direction | Force Error (%) | Moment Error (%) |
|---|---|---|
| X | 0.037 | 0.140 |
| Y | 0.023 | 0.098 |
| Z | 0.037 | 0.140 |
This validation confirms the effectiveness of the split-load approach in enhancing the performance of the six-axis force sensor. The integration of the central beam allows for a significant increase in load capacity without compromising the structural compactness, making it suitable for applications in heavy machinery and precision robotics. Moreover, the use of piezoelectric ceramics in the branches ensures high sensitivity and dynamic response, which is essential for real-time force feedback systems.
In conclusion, the 8/4-4 split-load parallel six-axis force sensor represents a significant advancement in force measurement technology. By combining the isotropic benefits of the Stewart platform with a central load-distributing beam, this design addresses the challenges of large-load applications while maintaining high decoupling accuracy. The mathematical model and decoupling algorithm derived here provide a robust framework for sensor calibration and implementation. Future work could focus on optimizing the material properties and geometric parameters to further reduce errors and expand the application range of this six-axis force sensor. Overall, this innovation holds promise for revolutionizing force sensing in industries requiring high precision and reliability under demanding conditions.