In the field of robotics and precision engineering, the six-axis force sensor plays a critical role in measuring multi-dimensional forces and moments. As a researcher focused on advancing sensor technology, I propose an orthogonal parallel six-axis force sensor structure designed to enhance decoupling performance and measurement accuracy. This article presents a comprehensive mathematical model and a parameter optimization method based on a task condition function, aiming to minimize the force range of measuring limbs while meeting specific operational requirements. The optimization approach ensures that the sensor structure is tailored to real-world applications, such as robotic surface grinding, where dynamic force variations occur. By integrating theoretical analysis with practical constraints, this work contributes to the development of high-performance six-axis force sensors for industrial use.
The Stewart parallel mechanism has been widely adopted in six-axis force sensor design due to its high stiffness, strong load capacity, and minimal error accumulation. These attributes make it ideal for applications requiring precise force measurement. However, traditional sensors often suffer from force coupling between sensitive elements, which complicates data processing and reduces accuracy. Recent studies have explored various six-axis force sensor configurations, such as those using piezoelectric quartz elements or over-constrained parallel structures, to mitigate these issues. For instance, some researchers have developed elastic body structures with static and dynamic calibration, while others have focused on fault-tolerant designs to improve reliability. Despite these advances, optimizing the structural parameters of parallel six-axis force sensors remains challenging, as existing methods like isotropy criteria or task models may not fully adapt to specific working conditions. Therefore, I introduce a novel orthogonal parallel six-axis force sensor with horizontally and vertically arranged measuring branches, which simplifies decoupling and enhances practicality. The subsequent sections detail the sensor structure, mathematical modeling, parameter optimization based on a task condition function, and a practical example validating the method.
The orthogonal parallel six-axis force sensor consists of a fixed platform, a measuring platform, support columns, and six measuring branches. These branches are divided into two groups: three are horizontally oriented, and three are vertically oriented, arranged in a circumferentially symmetric pattern around the central axis of the fixed platform. Each measuring branch is equipped with strain gauges at its midpoint to detect axial forces, and both ends are connected via elastic spherical hinges to minimize friction and coupling effects. The horizontal branches attach to support columns on the platforms, ensuring that their axes do not intersect at a single point, thereby improving force distribution. Key structural parameters include the radius of the circle where the spherical hinge points of the vertical branches are located (denoted as \( a_1 \)), the radius of the inscribed circle of the triangle formed by the horizontal branches (denoted as \( a_2 \)), the vertical distance between the horizontal branch plane and the measuring platform (denoted as \( a_3 \)), and the length of each measuring branch (denoted as \( L \)). The measurement coordinate system O-XYZ has its origin at the geometric center of the measuring platform, with the X-axis lying in the platform plane perpendicular to one of the horizontal branches and the Z-axis pointing vertically upward. This orthogonal arrangement allows for decoupled measurement of forces and moments: for example, X and Y direction forces and Z direction moments are measured by the horizontal branches, while Z direction forces and X and Y direction moments are measured by the vertical branches. This design not only simplifies data processing but also enhances the overall performance of the six-axis force sensor.
To derive the mathematical model of the orthogonal parallel six-axis force sensor, I begin by establishing the static equilibrium equations using screw theory. Considering the measuring platform as the free body and neglecting gravitational and frictional effects, the equilibrium equation is given by:
$$ \sum_{i=1}^{n} f_i \$_i = \mathbf{F} + \epsilon \mathbf{M} $$
Here, \( \mathbf{F} \) and \( \mathbf{M} \) represent the external force and moment vectors applied to the measuring platform, respectively, \( f_i \) is the axial force in the i-th measuring branch, and \( \$_i \) is the unit line vector of the branch in the reference frame, expressed as \( \$_i = \mathbf{S}_i + \epsilon \mathbf{S}_{0i} \), where \( \mathbf{S}_i \) is the unit direction vector satisfying \( \mathbf{S}_i \cdot \mathbf{S}_i = 1 \), and \( \mathbf{S}_{0i} = \mathbf{r}_i \times \mathbf{S}_i \) with \( \mathbf{r}_i \) being the position vector of a point on the branch axis. Rewriting this in matrix form yields:
$$ \mathbf{F}_w = \mathbf{G}_6 \mathbf{f}_6 $$
In this equation, \( \mathbf{F}_w = [F_x, F_y, F_z, M_x, M_y, M_z]^T \) is the six-dimensional external force vector, \( \mathbf{f}_6 = [f_1, f_2, \dots, f_6]^T \) is the vector of axial forces in the measuring branches, and \( \mathbf{G}_6 \) is the (6×6) mapping matrix that relates branch forces to external forces. Based on the sensor geometry, \( \mathbf{G}_6 \) is constructed from the coordinates of the spherical hinge points. For the orthogonal parallel six-axis force sensor, the matrix is derived as:
$$ \mathbf{G}_6 = \begin{bmatrix}
0 & 0 & 0 & 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\
0 & 0 & 0 & -1 & \frac{1}{2} & \frac{1}{2} \\
1 & 1 & 1 & 0 & 0 & 0 \\
-\frac{\sqrt{3}}{2} a_1 & \frac{\sqrt{3}}{2} a_1 & 0 & -a_3 & \frac{a_3}{2} & \frac{a_3}{2} \\
-\frac{a_1}{2} & -\frac{a_1}{2} & a_1 & 0 & -\frac{\sqrt{3}}{2} a_3 & \frac{\sqrt{3}}{2} a_3 \\
0 & 0 & 0 & -a_2 & -a_2 & -a_2
\end{bmatrix} $$
The non-singularity of \( \mathbf{G}_6 \) ensures that the six-axis force sensor can uniquely determine external forces from branch forces. The inverse mapping is given by \( \mathbf{f}_6 = \mathbf{G}_6^{-1} \mathbf{F}_w \), where the inverse matrix \( \mathbf{G}_6^{-1} \) is computed as:
$$ \mathbf{G}_6^{-1} = \begin{bmatrix}
-\frac{a_3}{3a_1} & \frac{\sqrt{3} a_3}{3a_1} & \frac{1}{3} & -\frac{\sqrt{3}}{3a_1} & -\frac{1}{3a_1} & 0 \\
-\frac{a_3}{3a_1} & -\frac{\sqrt{3} a_3}{3a_1} & \frac{1}{3} & \frac{\sqrt{3}}{3a_1} & -\frac{1}{3a_1} & 0 \\
\frac{2a_3}{3a_1} & 0 & \frac{1}{3} & 0 & \frac{2}{3a_1} & 0 \\
0 & -\frac{2}{3} & 0 & 0 & 0 & -\frac{1}{3a_2} \\
\frac{\sqrt{3}}{3} & \frac{1}{3} & 0 & 0 & 0 & -\frac{1}{3a_2} \\
-\frac{\sqrt{3}}{3} & \frac{1}{3} & 0 & 0 & 0 & -\frac{1}{3a_2}
\end{bmatrix} $$
This mathematical model provides a linear relationship between external forces and branch forces, facilitating the design and optimization of the six-axis force sensor. The decoupling characteristics are evident from the structure of \( \mathbf{G}_6^{-1} \), where specific branches respond predominantly to certain force and moment components.
To optimize the parameters of the parallel six-axis force sensor, I propose a method based on a task condition function model. This approach ensures that the sensor meets the demands of specific applications while minimizing the force range of the measuring branches. The task condition function, denoted as \( \mathbf{F}_w(t) \), represents the time-varying six-dimensional external forces and moments experienced by the sensor during a work cycle of period \( T \). It is expressed as:
$$ \mathbf{F}_w(t) = [F_x(t), F_y(t), F_z(t), M_x(t), M_y(t), M_z(t)]^T $$
where each component is a bounded function defined over the interval \( [0, T] \). Substituting into the inverse mapping equation gives the branch force function:
$$ \mathbf{f}_w(t) = \mathbf{G}^+ \mathbf{F}_w(t) = [f_{1w}(t), \dots, f_{6w}(t)]^T $$
Here, \( \mathbf{G}^+ \) is the generalized inverse of the mapping matrix for the sensor structure. For each branch force function \( f_{iw}(t) \), the maximum absolute value over the work cycle is determined, denoted as \( f_m \). The required force range for the measuring branches is then \( M = [-f_m, f_m] \). The optimization goal is to find the structural parameters that minimize \( f_m \) while satisfying the task condition function and spatial constraints.
Let the structural parameters be represented by a vector \( \boldsymbol{\delta} = [\delta_1, \dots, \delta_k] \), such as \( [a_1, a_2, a_3] \) for the orthogonal parallel six-axis force sensor. Given feasible intervals for each parameter, denoted as \( \mathbf{N} = \{N_1, \dots, N_k\} \), and step sizes \( \boldsymbol{\lambda} = [\lambda_1, \dots, \lambda_k] \), the parameters are traversed within these intervals. For each combination, \( \mathbf{G}^+ \) is computed, and \( f_m \) is evaluated. The optimal parameters \( \boldsymbol{\delta}_0 \) correspond to the minimum \( f_m \), denoted as \( f_0 \), resulting in the minimal branch force range \( M_0 = [-f_0, f_0] \). This method ensures that the six-axis force sensor is optimized for specific tasks, enhancing its force and measurement performance.
The following table summarizes the key steps in the parameter optimization process for a general parallel six-axis force sensor:
| Step | Description | Mathematical Expression |
|---|---|---|
| 1 | Define task condition function | $$ \mathbf{F}_w(t) = [F_x(t), F_y(t), F_z(t), M_x(t), My(t), M_z(t)]^T $$ |
| 2 | Compute branch force function | $$ \mathbf{f}_w(t) = \mathbf{G}^+ \mathbf{F}_w(t) $$ |
| 3 | Find maximum absolute branch force | $$ f_m = \max_{i,t} |f_{iw}(t)| $$ |
| 4 | Set optimization objective | Minimize \( f_m \) over parameter space |
| 5 | Determine optimal parameters | $$ \boldsymbol{\delta}_0 = \arg \min f_m $$ |
| 6 | Obtain minimal branch force range | $$ M_0 = [-f_0, f_0] $$ |
To demonstrate the applicability of this optimization method, I consider a case study involving a six-degree-of-freedom robot performing surface grinding on sanitary ceramics. The work cycle period \( T \) is 23 seconds, and the task condition function \( \mathbf{F}_w(t) \) is derived from force and moment analyses at the robot’s end-effector. The external force components \( F_x(t) \), \( F_y(t) \), and \( F_z(t) \) (in Newtons) and moment components \( M_x(t) \), \( M_y(t) \), and \( M_z(t) \) (in Newton-millimeters) vary over time, as illustrated in the function curves. For the orthogonal parallel six-axis force sensor, the structural parameters include \( a_1 \), \( a_2 \), and \( a_3 \). To simplify, I set \( a_3 = 15 \, \text{mm} \) and define the parameter vector as \( \boldsymbol{\delta} = [a_1, a_2] \), with feasible intervals \( N_1 = [20 \, \text{mm}, 50 \, \text{mm}] \) and \( N_2 = [20 \, \text{mm}, 50 \, \text{mm}] \), and step sizes \( \lambda_1 = 1 \, \text{mm} \) and \( \lambda_2 = 1 \, \text{mm} \). For each parameter combination, \( \mathbf{G}_6^{-1} \) is computed, and the branch force function \( \mathbf{f}_{6w}(t) \) is evaluated to find \( f_m \).
The numerical results, obtained using MATLAB, show the relationship between structural parameters and the maximum axial force in the measuring branches. The optimal parameters are found to be \( a_1 = 50 \, \text{mm} \) and \( a_2 = 20 \, \text{mm} \), yielding a minimal branch force range of \( M_0 = [-12.4 \, \text{N}, 12.4 \, \text{N}] \). This indicates that the six-axis force sensor with these parameters can effectively handle the dynamic forces in the grinding task while minimizing the required force range for the branches.
The table below presents a subset of the parameter optimization results for the orthogonal parallel six-axis force sensor, highlighting how variations in \( a_1 \) and \( a_2 \) affect the maximum branch force \( f_m \):
| \( a_1 \) (mm) | \( a_2 \) (mm) | \( f_m \) (N) |
|---|---|---|
| 20 | 20 | 15.8 |
| 30 | 30 | 14.2 |
| 40 | 40 | 13.5 |
| 50 | 20 | 12.4 |
| 50 | 50 | 13.1 |
This optimization process underscores the importance of tailoring the six-axis force sensor design to specific tasks. By minimizing the branch force range, the sensor achieves better force distribution and measurement accuracy, which is crucial for applications like robotic grinding where precision is paramount.
In conclusion, I have developed an orthogonal parallel six-axis force sensor with a decoupled structure and presented a mathematical model based on screw theory. The parameter optimization method using a task condition function enables the design of sensors that meet practical requirements while minimizing the force range of measuring branches. The case study on sanitary ceramic surface grinding validates the method, showing that optimal parameters can be identified to enhance sensor performance. This approach is generalizable to various parallel six-axis force sensor configurations and contributes to the advancement of force measurement technology in robotics. Future work could explore dynamic optimization and real-time adaptation for varying task conditions.
The integration of mathematical modeling and practical optimization ensures that the six-axis force sensor is both theoretically sound and application-oriented. As robotics continues to evolve, such sensors will play a vital role in enabling precise force control in complex environments. The orthogonal parallel structure, combined with the proposed optimization framework, offers a robust solution for improving the performance and reliability of six-axis force sensors in industrial settings.