In the field of robotics and industrial automation, the ability to accurately measure multi-dimensional forces and torques is crucial for applications such as aerospace, manufacturing, and precision tasks. Among various sensor designs, the six-axis force sensor stands out due to its capability to simultaneously detect three-dimensional forces and three-dimensional moments. This article focuses on the structural performance and parameter optimization of an orthogonal parallel six-axis force sensor, which offers advantages like high stiffness, strong load-bearing capacity, and minimal error accumulation. I will analyze the statics of this sensor, derive the mapping relationships between external forces and branch reactions, and propose a multi-objective optimization method based on practical working conditions. Through a case study of sanitary ceramic surface polishing, I will demonstrate how to optimize structural parameters to minimize the measurement range of branches and enhance stability, ensuring the sensor’s feasibility in real-world scenarios.
The orthogonal parallel six-axis force sensor consists of a force-measuring platform, a fixed platform, and six measuring branches arranged in an orthogonal configuration. Specifically, three branches are vertically distributed, forming an equilateral triangle, while the other three are horizontally placed, tangent to a circle with a specific radius. This design ensures partial decoupling, simplifying the force analysis. The coordinate system is set with the origin at the geometric center of the force-measuring platform, facilitating the derivation of static equilibrium equations. To illustrate the structure, consider the following representation:
Based on screw theory, the static equilibrium equation for the force-measuring platform under external six-dimensional forces can be expressed as:
$$ F_w = G f $$
where \( F_w \) represents the spatial external force and moment vector, \( G \) is the mapping coefficient matrix between the axial forces of the measuring branches and the external forces, and \( f \) denotes the axial force vector of the branches. The inverse mapping, which relates external forces to branch reactions, is given by:
$$ f = G^{-1} F_w $$
The matrix \( G^{-1} \) depends solely on structural parameters \( a_1 \), \( a_2 \), and \( a_3 \), which define the geometry of the platforms and branches. For instance, the positions of connection points on the fixed and force-measuring platforms can be described using trigonometric functions. Assuming \( a_3 = 15 \, \text{mm} \) for simplicity, the analysis focuses on optimizing \( a_1 \) and \( a_2 \). The explicit form of \( G^{-1} \) is derived as:
$$ G^{-1} = \begin{bmatrix}
-\frac{a_3}{3a_1} & \frac{\sqrt{3}a_3}{3a_1} & \frac{1}{3} & -\frac{\sqrt{3}}{3a_1} & -\frac{1}{3a_1} & 0 \\
-\frac{a_3}{3a_1} & -\frac{\sqrt{3}a_3}{3a_1} & \frac{1}{3} & \frac{\sqrt{3}}{3a_1} & -\frac{1}{3a_1} & 0 \\
\frac{2a_3}{3a_1} & 0 & \frac{1}{3} & 0 & \frac{2}{3a_1} & 0 \\
0 & -\frac{2}{3} & 0 & 0 & 0 & -\frac{1}{3a_2} \\
\frac{\sqrt{3}}{3} & \frac{1}{3} & 0 & 0 & 0 & -\frac{1}{3a_2} \\
-\frac{\sqrt{3}}{3} & \frac{1}{3} & 0 & 0 & 0 & -\frac{1}{3a_2}
\end{bmatrix} $$
This matrix shows that the sensor is not in a singular configuration, allowing unique determination of six-dimensional forces from branch axial forces. The orthogonal arrangement contributes to this property, making the six-axis force sensor robust for various applications.
To optimize the structural parameters, I propose a multi-objective approach that minimizes the measurement range of the branches while ensuring structural stability under specific working conditions. The optimization criteria are based on the axial reaction forces in the branches over a work cycle period \( T \). Let \( f(t) = G^{-1} F_w(t) \) represent the axial force vector as a function of time, where \( F_w(t) \) is the external force vector defined by the task. For each branch \( i \), the axial force \( f_i(t) \) is a bounded function, and its absolute value must satisfy:
$$ f_{\min}(t) \leq |f_i(t)| \leq f_{\max}(t) \quad \text{for} \quad i = 1, 2, \ldots, 6 $$
The maximum axial force for the \( i \)-th branch is computed as:
$$ f_{i,\max}(t_j) = |G^{-1}_{j1} F_{xm}| + |G^{-1}_{j2} F_{ym}| + |G^{-1}_{j3} F_{zm}| + |G^{-1}_{j4} M_{xm}| + |G^{-1}_{j5} M_{ym}| + |G^{-1}_{j6} M_{zm}| $$
where \( F_{xm}, F_{ym}, F_{zm} \) are the maximum external forces in the x, y, z directions, and \( M_{xm}, M_{ym}, M_{zm} \) are the maximum external moments. The first objective is to minimize the maximum axial force among all branches, denoted as \( f_A \), which corresponds to the smallest possible measurement range. The second objective is to minimize the variance of axial forces across branches, ensuring balanced loading and stability. Let \( \sigma \) be the standard deviation of axial forces per column in \( f \), and \( \sigma_B \) be the maximum value among these standard deviations. Minimizing \( \sigma_B \) leads to a more stable sensor structure.
For the optimization process, I define the parameter ranges and step sizes, then use numerical methods to evaluate \( f_A \) and \( \sigma_B \) over these ranges. The goal is to find parameters that satisfy both objectives simultaneously. This approach ensures that the six-axis force sensor is tailored to specific tasks, enhancing its practical utility.
As a case study, consider a six-degree-of-freedom robot performing sanitary ceramic surface polishing. The work cycle \( T = 23 \, \text{s} \) involves steps like dust blowing, external and internal surface polishing, and fine polishing. The external force and moment functions are defined as follows:
| Force/Moment | Expression | Time Interval (s) |
|---|---|---|
| \( F_x(t) \) | \( 0 \) | \( 0 \leq t < 17 \) |
| \( -20(t – 17) \) | \( 17 \leq t < 17.5 \) | |
| \( -10 \) | \( 17.5 \leq t < 22.5 \) | |
| \( 20t – 460 \) | \( 22.5 \leq t < 23 \) | |
| \( F_y(t) \) | \( 14t \) | \( 0 \leq t < 0.5 \) |
| \( 7 \) | \( 0.5 \leq t < 6.5 \) | |
| \( 7t – 32 \) | \( 6.5 \leq t < 7 \) | |
| \( 10 \) | \( 7 \leq t < 10.5 \) | |
| \( 180 – 17t \) | \( 10.5 \leq t < 11 \) | |
| \( -7 \) | \( 11 \leq t < 17 \) | |
| \( 14t – 245 \) | \( 17 \leq t < 17.5 \) | |
| \( 0 \) | \( 17.5 \leq t < 23 \) | |
| \( F_z(t) \) | \( 60t \) | \( 0 \leq t < 0.5 \) |
| \( 30 \) | \( 0.5 \leq t < 22.5 \) | |
| \( 1380 – 60t \) | \( 22.5 \leq t < 23 \) | |
| \( M_x(t) \) | \( -400t \) | \( 0 \leq t < 0.5 \) |
| \( -200 \) | \( 0.5 \leq t < 6.5 \) | |
| \( 100t – 850 \) | \( 6.5 \leq t < 7 \) | |
| \( -150 \) | \( 7 \leq t < 10.5 \) | |
| \( 700t – 7500 \) | \( 10.5 \leq t < 11 \) | |
| \( 200 \) | \( 11 \leq t < 17 \) | |
| \( 7000 – 400t \) | \( 17 \leq t < 17.5 \) | |
| \( 0 \) | \( 17.5 \leq t < 23 \) | |
| \( M_y(t) \) | \( 0 \) | \( 0 \leq t < 17 \) |
| \( 300(t – 17) \) | \( 17 \leq t < 17.5 \) | |
| \( 150 \) | \( 17.5 \leq t < 22.5 \) | |
| \( 6900 – 300t \) | \( 22.5 \leq t < 23 \) | |
| \( M_z(t) \) | \( 0 \) | \( 0 \leq t < 23 \) |
These functions define the operational conditions for the six-axis force sensor mounted on the robot’s end-effector. To optimize the parameters \( a_1 \) and \( a_2 \), I set the range for \( a_1 \) as [40, 70] mm and for \( a_2 \) as [20, 50] mm, with a step size of 1 mm. Using numerical computations, I evaluate \( f_A \) and \( \sigma_B \) across these ranges. The results indicate that for minimizing the branch measurement range, \( a_1 = 70 \, \text{mm} \) and \( a_2 = 20 \, \text{mm} \) are optimal, while for stability, \( a_1 = 70 \, \text{mm} \) and \( a_2 = 28 \, \text{mm} \) are best. A compromise gives \( a_1 = 70 \, \text{mm} \) and \( a_2 = 24 \, \text{mm} \), balancing both objectives.
The optimization process highlights the importance of task-specific design for six-axis force sensors. By minimizing the measurement range and ensuring force distribution stability, the sensor becomes more efficient and reliable. This method can be extended to other applications, such as assembly or medical robotics, where precise force control is essential. The orthogonal parallel structure of the six-axis force sensor, combined with this optimization approach, offers a practical solution for enhancing performance in diverse industrial settings.
In conclusion, the orthogonal parallel six-axis force sensor demonstrates significant potential through its structural design and parameter optimization. The derived mapping relations and multi-objective optimization criteria enable tailored sensor configurations for specific tasks, as validated by the ceramic polishing example. Future work could explore dynamic responses or integration with control systems to further improve the six-axis force sensor’s capabilities.