In the field of robotics and aerospace engineering, the demand for high-capacity multi-dimensional force/torque sensors has grown significantly. Traditional six-axis force sensors often face challenges such as limited load capacity, significant cross-coupling, and reduced stability under heavy loads. To address these issues, I have developed a novel over-constrained orthogonal parallel six-axis force sensor. This design incorporates redundant measuring branches in a spatial orthogonal configuration, enhancing structural stiffness and load-bearing capacity while minimizing the impact of joint friction on measurement accuracy. In this article, I will detail the sensor’s structure, derive its measurement models under both ideal and practical conditions, and present the results of static calibration experiments.
The sensor structure consists of a base plate, a force-sensing plate, and twelve measuring branches connecting them. Four branches are distributed on the bottom surface of the force-sensing plate, while the remaining eight are arranged on the four side surfaces, forming an orthogonal parallel over-constrained mechanism. Each horizontal branch interfaces with the force-sensing plate through spherical contacts and contains a uniaxial force sensor, fixed to the base. Vertical branches connect via nested structures incorporating sensor heads and steel balls. Kinematically, each branch can be modeled as a “prismatic-spherical-spherical (PSS)” chain, resulting in a 12-PSS parallel mechanism. Key dimensions include distances between specific branches, such as a between branches 7 and 8 (or 11 and 12), b between branches 5 and 6 (or 9 and 10), c between branches 1 and 4 (or 2 and 3), d between branches 1 and 2 (or 3 and 4), and the width e and length f of the force-sensing plate.
To establish the ideal measurement model, I applied screw theory to derive the first-order static influence coefficient matrix. The generalized external force vector F acting on the sensor is related to the axial forces f in the measuring branches by the equation:
$$ \mathbf{F} = \mathbf{G} \mathbf{f} $$
where F = [f_x, f_y, f_z, m_x, m_y, m_z]^T represents the six-dimensional external force and torque, and f = [f_1, f_2, …, f_12]^T is the vector of branch forces. The matrix G is the static influence coefficient matrix, expressed as:
$$ \mathbf{G} = \begin{bmatrix}
\mathbf{S}_1 & \mathbf{S}_2 & \cdots & \mathbf{S}_{12} \\
\mathbf{S}_{01} & \mathbf{S}_{02} & \cdots & \mathbf{S}_{012}
\end{bmatrix} $$
Here, S_i = [a_i – A_i, a_i – A_i] and S_{0i} = [A_i \times a_i, a_i \times A_i] for each branch i. By substituting the sensor’s dimensional parameters, the ideal G matrix is derived as:
$$ \mathbf{G} = \begin{bmatrix}
0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -\frac{d}{2} & \frac{d}{2} & \frac{d}{2} & -\frac{d}{2} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \frac{c}{2} & -\frac{c}{2} & \frac{c}{2} & -\frac{c}{2} & 0 & 0 & 0 & 0 \\
-\frac{b}{2} & \frac{b}{2} & \frac{b}{2} & -\frac{b}{2} & -\frac{a}{2} & \frac{a}{2} & -\frac{a}{2} & \frac{a}{2} & -\frac{b}{2} & \frac{b}{2} & \frac{b}{2} & -\frac{b}{2}
\end{bmatrix} $$
However, in practical applications, factors such as initial pre-tightening forces and varying branch stiffnesses affect the measurement accuracy. To account for these, I developed an enhanced model by considering the static equilibrium equations and displacement compatibility conditions. The generalized force F causes deformations in the branches, which include contributions from pre-tightening forces and external loads. The deformation in the Z-direction for each branch is given by δ_zi = Δu_zai + Δu_zbi, where Δu_zai is the deformation due to pre-tightening and Δu_zbi is from the external force. Similarly, horizontal deformations are influenced by translations and rotations of the force-sensing plate.
The static equilibrium equations are:
$$ \sum \mathbf{f}_i + \mathbf{F} = 0 $$
$$ \sum (\mathbf{M}_i + \mathbf{m}_i) + \mathbf{M} = 0 $$
where f_i and m_i are the force and moment vectors from branch i. The moment components depend on branch forces and geometric parameters, such as ball radii r (for horizontal branches) and R (for vertical branches). For example, the moment vector m_i for specific branches can be expressed as:
$$ \mathbf{m}_i = \begin{bmatrix}
R f_{y1} + R f_{y2} + R f_{y3} + R f_{y4} + r f_{z7} + r f_{z8} + \cdots \\
-R f_{x1} – R f_{x2} – R f_{x3} – R f_{x4} – r f_{z6} – r f_{z9} – \cdots \\
r f_{y6} + r f_{x7} + r f_{y8} + r f_{x9} + r f_{y10} + r f_{x12}
\end{bmatrix} $$
To resolve the hyperstatic nature of the system (with 30 unknown forces and only 6 equilibrium equations), I introduced displacement compatibility equations. Treating the force-sensing plate as rigid, the deformed positions of branch endpoints must lie on a plane defined by three points. This leads to matrix equations of the form:
$$ \mathbf{C}_Z \boldsymbol{\delta} = 0 $$
where C_Z is a coefficient matrix derived from geometric constraints, and δ is the deformation vector. Combining these with force-deformation relationships (e.g., f_i = k_i Δu_i for branch stiffness k_i), the enhanced model is formulated as:
$$ \mathbf{F}^* = \mathbf{G}^* \mathbf{f}^* $$
Here, F* includes external forces and terms related to pre-tightening forces, G* is the modified influence matrix incorporating stiffness values, and f* is the extended force vector. For instance, using parameters a = 66 mm, b = 96 mm, c = 66 mm, d = 96 mm, e = 224 mm, f = 254 mm, and branch forces ranging from 0 to 5000 N, numerical simulations showed deviations up to 8% in force measurements and 2.3% in torque measurements compared to the ideal model. Additionally, pre-tightening forces introduced coupling between branches, such as branches 6 and 9, with errors reaching 7% of full scale, highlighting the need for careful pre-load adjustment in calibration.
To validate the model, I designed and manufactured a prototype of the over-constrained orthogonal parallel six-axis force sensor. The sensor features twelve measuring branches with uniaxial force sensors, and the loading platform includes horizontal and vertical loading units equipped with wheel-rail sensors for applying controlled six-dimensional forces. Calibration experiments involved applying sequential loads in different directions while recording branch outputs.
The calibration process uses the equation:
$$ \mathbf{F} = \mathbf{G} \mathbf{V} + \mathbf{E} $$
where F is the applied force/torque matrix, G is the calibration matrix, V is the output matrix from the branches, and E is the error matrix. Based on experimental data, the calibrated G matrix was computed as:
$$ \mathbf{G} = \begin{bmatrix}
-0.0485 & -0.0487 & -0.0551 & -0.0118 & 0.3525 & 1.1994 & 1.2941 & 0.1102 & -1.2454 & -1.5670 & 0.5988 & 1.1969 \\
0.1862 & 0.7056 & -0.3279 & -0.1159 & -1.9744 & -2.3651 & 1.5409 & 0.3959 & -2.8296 & -2.6644 & 2.9549 & 1.7804 \\
-0.7601 & -1.1923 & 1.0290 & 0.7382 & -2.0540 & 0.3262 & 0.6173 & -0.6231 & 0.8851 & -2.6221 & -1.1500 & 0.2077 \\
0.1010 & 0.0395 & -0.0601 & 0.0396 & 0.2450 & -0.2499 & -0.0949 & 0.0060 & 0.3170 & 0.3126 & -0.1147 & -0.4357 \\
-0.0867 & 0.1035 & 0.0702 & -0.0594 & 0.1457 & 0.0344 & -0.2119 & 0.0839 & -0.1065 & 0.0008 & 0.0706 & 0.2452 \\
-0.0084 & -0.0018 & 0.0013 & -0.0275 & 0.2005 & 0.2142 & -0.3682 & 0.1275 & -0.3528 & 0.0307 & 0.0468 & 0.6045
\end{bmatrix} $$
The error matrix E, which includes Type I (diagonal) and Type II (off-diagonal) errors, was found to be:
$$ \mathbf{E} = \begin{bmatrix}
0.0059 & 0.0021 & 0.0056 & 0.0290 & 0.1170 & 0.1391 \\
0.0168 & 0.0028 & 0.0042 & 0.0620 & 0.1223 & 0.0577 \\
0.0177 & 0.0382 & 0.0009 & 0.0940 & 0.0519 & 0.0269 \\
0.0009 & 0.0077 & 0.0011 & 0.0002 & 0.0013 & 0.0008 \\
0.0010 & 0.0002 & 0.0003 & 0.0082 & 0.0089 & 0.0018 \\
0.0034 & 0.0004 & 0.0003 & 0.0011 & 0.0281 & 0.0137
\end{bmatrix} $$
This results in Type I errors of 0.59% for F_x, 0.28% for F_y, 0.09% for F_z, 0.02% for M_x, 0.89% for M_y, and 1.37% for M_z, with the maximum error in M_z measurement. The analysis indicates that discrepancies in branch stiffness and pre-tightening forces are primary sources of error, emphasizing the importance of precise manufacturing and calibration for this six-axis force sensor.
In conclusion, I have presented a comprehensive study on a novel over-constrained orthogonal parallel six-axis force sensor, from theoretical modeling to experimental validation. The sensor’s design effectively improves load capacity and reduces friction effects, while the derived models account for practical factors like pre-tightening and stiffness variations. Calibration results demonstrate acceptable accuracy, with errors primarily attributable to mechanical imperfections. This work lays a foundation for further development and application of heavy-duty six-axis force sensors in demanding fields such as aerospace and robotics.
| Parameter | Value | Description |
|---|---|---|
| Number of Branches | 12 | Orthogonal PSS configuration |
| Dimensions (a, b, c, d, e, f) | 66 mm, 96 mm, 66 mm, 96 mm, 224 mm, 254 mm | Key geometric parameters |
| Max Force Load | 5000 N | Per branch in calibration |
| Type I Error (F_x) | 0.59% | Force in X-direction |
| Type I Error (F_y) | 0.28% | Force in Y-direction |
| Type I Error (F_z) | 0.09% | Force in Z-direction |
| Type I Error (M_x) | 0.02% | Torque about X-axis |
| Type I Error (M_y) | 0.89% | Torque about Y-axis |
| Type I Error (M_z) | 1.37% | Torque about Z-axis |
The development of this six-axis force sensor highlights the potential of over-constrained parallel mechanisms in achieving high stiffness and accuracy for multi-dimensional force measurement. Future work will focus on optimizing branch stiffness uniformity and pre-load management to further enhance performance. The integration of such sensors into robotic systems could revolutionize tasks requiring precise force feedback, from assembly operations to space exploration.