In the field of space technology, the demand for large-aperture optical systems has driven the development of on-orbit assembly techniques for telescopes. Ground-based assembly experiments are crucial to validate these techniques, requiring precise force feedback during the manipulation of primary mirror modules. To address this, we designed a high-sensitivity large-range six-axis force sensor that balances key factors such as range, stiffness, and sensitivity. This sensor is integrated between a robotic arm and an end-effector to enable force-adaptive control, preventing excessive stress on delicate components. The design process involved mathematical modeling of a classical cross-beam structure, structural optimization for enhanced sensitivity, finite element validation, and experimental calibration. The resulting six-axis force sensor demonstrates excellent performance, with linear repeatability errors below 0.33% FS, force channel sensitivity exceeding 0.83 mV/V, and moment channel sensitivity above 2.6 mV/V. It has been successfully deployed in ground experiments for on-orbit telescope assembly, showcasing its reliability in real-world applications.
The six-axis force sensor measures three force components (Fx, Fy, Fz) and three moment components (Mx, My, Mz) in a spatial coordinate system. Strain-gauge-based sensors are widely used due to their stability and reliability. However, designing such sensors involves trade-offs between stiffness and sensitivity—increasing sensitivity often requires greater elastic deformation, which reduces stiffness. Our approach focuses on optimizing the elastic body structure to achieve high sensitivity without significantly compromising stiffness, tailored for the specific requirements of telescope assembly experiments. The following sections detail the mathematical modeling, design improvements, finite element analysis, calibration, and application of the six-axis force sensor.
Mathematical Modeling of the Classical Cross-Beam Elastic Body
The classical cross-beam structure for a six-axis force sensor includes fixed blocks (Q1–Q4), floating blocks (P1–P4), strain beams (S1–S4), floating beams (B1–B8), and a loading block (O). Assuming that deformation occurs only in the strain beams and floating beams, while other components are rigid, we derive mathematical expressions for strain and stiffness under individual loadings. The model is symmetric about the X and Y axes, simplifying analysis to one quadrant. For instance, under Fz loading, the strain at a distance x from the loading block on the lower surface of a strain beam is given by:
$$ \varepsilon_{Fz} = \frac{2F_z G h_2 \delta^3 l_1 (l_1 – 2x) + 3F_z E I_2 l_2 (2l_1 + b_3 – 2x)}{8E I_2 (6E I_2 l_2 + 4G h_2 \delta^3 l_1)} h_1 $$
where E is the elastic modulus, G is the shear modulus, I1 and I2 are moments of inertia for floating and strain beams, respectively, and other parameters define geometric dimensions. The stiffness under Fz is derived from the deflection at the beam end:
$$ k_{Fz} = \frac{F_z}{\omega_{Fz}} = \frac{1}{\frac{l_2^3}{96E I_1} + \frac{2G h_2 \delta^3 l_1^4 + 3E I_2 l_2 l_1^2 (4l_1 + 3b_3)}{24E I_2 (6E I_2 l_2 + 4G h_2 \delta^3 l_1)}} $$
Similarly, for Fy loading, the strain and stiffness are:
$$ \varepsilon_{Fy} = \frac{b_1}{2E I_3} \left( \frac{K}{l_1^2} – \frac{I_3 l_1 l_2^3 + I_4 b_3 l_2^3}{I_3 l_2^3 + 8I_3 l_1 l_2^2 + 12I_3 l_1 l_2 l_3 + 6I_3 l_1 l_3^2} – x \right) F_{4Fy} $$
$$ k_{Fy} = \frac{12E (12A_2 I_3 I_4 l_2 + 4A_1 A_2 I_3 K + 24A_1 I_3 I_4 l_1 + A_1 A_2 I_4 l_2^3)}{72I_3 I_4 l_2^2 + 3A_2 I_4 l_2^4 + 24A_1 I_3 l_2 K + A_1 A_2 l_2^3 K} $$
where K is a composite parameter. For Mz loading, the strain and stiffness are:
$$ \varepsilon_{Mz} = \frac{b_1 (l_1 – J – x)}{2E I_3 (4l_1 + 2b_4 – 4J)} M_z $$
$$ k_{Mz} = \frac{E I_3 (6l_3^2 + 12l_2 l_3 + 8l_2^2) (8l_1 + 4b_4 – 8J)}{(l_2^2 – 2J l_2) (6l_3^2 + 12l_2 l_3 + 8l_2^2) – (b_3 + J) l_2^3} $$
And for My loading:
$$ \varepsilon_{My} = \frac{h_1 (l_1 – Q_1 – x)}{2E I_2 (2l_1 + b_4 + 2Q_2)} M_y $$
$$ k_{My} = \frac{12E I_1 I_2 (2l_1 + b_4) (2l_1 + b_4 + 2Q_2)}{Q_2 (8l_1^3 + I_2 l_2^3 + 12Q_1 l_1^2)} $$
These equations highlight that sensitivity can be enhanced by reducing the distance x from the strain gauge to the loading block, decreasing the moment of inertia of strain beams, or increasing the length of floating beams. However, these changes may reduce stiffness. To balance this, we propose localized optimization by grooving at strain gauge positions to concentrate stress, thereby increasing sensitivity without大幅牺牲 stiffness. The improved design uses full-bridge circuits for strain measurement, with 24 strain gauges (R1–R24) allocated to measure the six force and moment components.
Sensitivity Enhancement Strategy
Based on the mathematical model, we identified three primary methods to increase the sensitivity of the six-axis force sensor: (1) minimizing the distance between the strain gauge attachment point and the loading block, (2) reducing the moment of inertia of the strain beams, and (3) optimizing the floating beam dimensions. To implement these without significantly compromising stiffness, we introduced grooves at the strain gauge locations on the strain beams. This local reduction in inertia concentrates stress at the measurement points, enhancing strain output while maintaining overall structural integrity. The optimized cross-beam elastic body model incorporates these grooves and uses full-bridge Wheatstone circuits for each channel, ensuring high signal-to-noise ratio and decoupling capabilities. The strain gauges are arranged as follows: R1–R4 for Fx, R5–R8 for Fy, R9–R12 for Fz, R13–R16 for Mx, R17–R20 for My, and R21–R24 for Mz. This configuration leverages the symmetric structure to minimize cross-talk and improve measurement accuracy.
The relationship between strain output and applied loads can be summarized using a compliance matrix. For the classical model, the ideal compliance matrix is diagonal, indicating no cross-talk. However, practical imperfections introduce coupling, which we address through structural adjustments and calibration. The compliance matrix C relates the output strain vector ε to the input load vector F:
$$ \mathbf{\varepsilon} = \mathbf{C} \mathbf{F} $$
where C is a 6×6 matrix. For our optimized design, the compliance matrix is derived from finite element analysis and experimental data, as discussed in later sections.
Structural Design of the Elastic Body
The six-axis force sensor is designed to fit within the constrained space of the telescope assembly system, with arc-shaped fixed and loading blocks to facilitate integration. Safety pin holes are incorporated to prevent plastic deformation. The material selected is high-strength alloy steel 40Cr, with parameters optimized for the required range and sensitivity: l1 = 16 mm, l2 = 9 mm, l3 = 20 mm, b1 = 12 mm, δ = 3 mm, b3 = 6 mm, and h1 = h2 = h3 = 18 mm. The elastic body features four strain beams and eight floating beams, arranged symmetrically to ensure decoupling. The grooves at strain gauge positions have a depth of 2 mm and width of 4 mm, strategically placed to maximize strain concentration. The overall dimensions are compact, with a diameter of 150 mm and height of 50 mm, allowing easy installation between the robotic arm flange and end-effector.
The fixed blocks are connected to the robotic arm via bolts, while the loading block interfaces with the end-effector. The safety pins, made of hardened steel, engage with corresponding holes to limit excessive deformation under overload conditions. This design ensures that the six-axis force sensor can withstand the maximum loads encountered during mirror handling—up to 3000 N for forces and 200 N·m for moments—while providing high sensitivity for precise force control.
Finite Element Analysis and Validation
We conducted finite element analysis (FEA) using MSC Patran for meshing and MSC Nastran for solving. The model was constrained at fixed block bolt holes, and unit loads (1 N for forces, 1 N·m for moments) were applied separately at the loading block via MPC nodes. The material properties were set as E = 206 GPa and Poisson’s ratio = 0.28. The strain distribution and deformation under individual loads are shown in the strain cloud diagrams, which confirm that maximum strain occurs at the grooved regions, validating our sensitivity enhancement approach.
The compliance matrix from FEA is:
$$ \mathbf{A} = \begin{bmatrix}
0.42 & 0 & 0 & 0 & 1.32 & 0 \\
0 & 0.42 & 0 & -1.32 & 0 & 0 \\
0 & 0.02 & 0.3 & 0.28 & 0 & 0 \\
0 & 0 & 0 & 14.78 & 0 & 0.4 \\
0 & 0 & 0 & 0 & 14.78 & 0.4 \\
0 & 0 & 0 & 0 & 0 & 18.7
\end{bmatrix} $$
This matrix indicates the strain output per unit load, with non-diagonal elements representing structural cross-talk. For example, Fy loading causes a strain response in Mx, with a coupling of 8.9%. The sensitivity and cross-talk derived from FEA are summarized in Table 1.
| Channel | Sensitivity (mV/V) | Cross-Talk (%) |
|---|---|---|
| Fx | 1.26 | 0 (Fy), 0 (Fz), 0 (Mx), 8.9 (My), 0 (Mz) |
| Fy | 1.26 | 0 (Fx), 0 (Fz), 8.9 (Mx), 0 (My), 0 (Mz) |
| Fz | 0.9 | 0 (Fx), 4.8 (Fy), 1.9 (Mx), 0 (My), 0 (Mz) |
| Mx | 2.96 | 0 (Fx), 0 (Fy), 0 (Fz), 0 (My), 2.1 (Mz) |
| My | 2.96 | 0 (Fx), 0 (Fy), 0 (Fz), 0 (Mx), 2.1 (Mz) |
| Mz | 3.74 | 0 (Fx), 0 (Fy), 0 (Fz), 0 (Mx), 0 (My) |
The FEA results show that the designed six-axis force sensor achieves high sensitivity, with force channels above 0.9 mV/V and moment channels above 2.96 mV/V. The cross-talk is primarily due to asymmetries from safety pins and mounting features, which will be compensated through calibration.
Calibration and Performance Evaluation
After manufacturing the six-axis force sensor, we performed calibration using a dedicated platform. Each channel was loaded separately with 10 points uniformly distributed across its range (3000 N for forces, 200 N·m for moments), and three loading-unloading cycles were conducted. The output voltages from the full-bridge circuits were recorded, and repeatability errors were calculated as the maximum deviation in outputs for the same load. The results, shown in Table 2, indicate excellent repeatability, with errors below 0.33% FS.
| Channel | Repeatability Error (% FS) |
|---|---|
| Fx | 0.16 |
| Fy | 0.17 |
| Fz | 0.29 |
| Mx | 0.23 |
| My | 0.22 |
| Mz | 0.33 |
The calibration curves for each channel are linear, as shown in the loading graphs. Using least squares fitting, we derived the compliance matrix A and offset vector B:
$$ \mathbf{A} = \begin{bmatrix}
0.3929 & 0.0099 & -0.0146 & -0.7537 & 1.4177 & 0.9700 \\
-0.0176 & 0.3729 & 0.0206 & -1.5810 & -0.7737 & -0.0686 \\
0 & -0.0209 & 0.2777 & 0.2368 & -0.2708 & 0.1206 \\
0.0162 & 0.0227 & -0.0162 & 13.0140 & -0.4072 & -0.9571 \\
0.0242 & 0.0200 & -0.0109 & 0.2769 & 13.8309 & 0.3802 \\
0.0235 & 0.0242 & -0.0041 & -0.2399 & 0.9682 & 18.1606
\end{bmatrix} $$
$$ \mathbf{B} = \begin{bmatrix}
4.0863 \\
18.9326 \\
3.2672 \\
-7.1947 \\
28.0749 \\
9.7840
\end{bmatrix} $$
The calibration equation is:
$$ \begin{bmatrix} F_x \\ F_y \\ F_z \\ M_x \\ M_y \\ M_z \end{bmatrix} = \begin{bmatrix}
2.5668 & 0.0881 & 0.1252 & 0.1595 & -0.2740 & -0.1234 \\
0.0956 & 2.6465 & -0.1666 & 0.3276 & 0.1333 & 0.0205 \\
0.0161 & 0.2083 & 3.5814 & -0.0381 & -0.0607 & -0.0246 \\
-0.0035 & -0.0046 & 0.0046 & 0.0761 & -0.0019 & 0.0042 \\
-0.0044 & -0.0036 & 0.0027 & -0.0023 & -0.0725 & -0.0014 \\
-0.0037 & -0.0038 & 0.0011 & 0.0002 & 0.0040 & 0.0552
\end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \\ V_3 \\ V_4 \\ V_5 \\ V_6 \end{bmatrix} – \begin{bmatrix} 4.0863 \\ 18.9326 \\ 3.2672 \\ -7.1947 \\ 28.0749 \\ 9.7840 \end{bmatrix} $$
From this, the sensitivity and direct coupling errors are calculated and listed in Table 3. The force channel sensitivity is above 0.83 mV/V, and the moment channel sensitivity exceeds 2.60 mV/V. After decoupling using the calibration matrix, the residual cross-talk is reduced to below 1.2%, meeting the project requirements.
| Channel | Sensitivity (mV/V) | Direct Coupling Error (%) |
|---|---|---|
| Fx | 1.18 | 2.65 (Fy), 5.26 (Fz), 5.79 (Mx), 10.25 (My), 5.34 (Mz) |
| Fy | 1.12 | 4.48 (Fx), 7.42 (Fz), 12.14 (Mx), 5.59 (My), 0.38 (Mz) |
| Fz | 0.83 | 0 (Fx), 5.61 (Fy), 1.82 (Mx), 1.96 (My), 0.66 (Mz) |
| Mx | 2.60 | 4.12 (Fx), 6.09 (Fy), 5.83 (Fz), 2.94 (My), 5.27 (Mz) |
| My | 2.77 | 6.16 (Fx), 5.36 (Fy), 3.93 (Fz), 2.13 (Mx), 2.09 (Mz) |
| Mz | 3.63 | 5.98 (Fx), 6.49 (Fy), 1.48 (Fz), 1.84 (Mx), 0.7 (My) |
The slight discrepancies between FEA and experimental results are attributed to manufacturing tolerances, strain gauge misalignment, and loading inaccuracies. Nevertheless, the six-axis force sensor performs reliably after calibration.
Application in On-Orbit Telescope Ground Experiments
The six-axis force sensor was integrated into a ground-based telescope assembly system, consisting of a robotic arm, vision camera, grasping mechanism, mirror modules, and control systems. During experiments, the sensor provided real-time force feedback for automated mirror grasping, transportation, and installation. The process involved three phases: grasping (0–600 s), transportation (600–1380 s), and installation (1380–1680 s). In the grasping phase, the sensor detected forces during insertion and tightening, allowing the robotic arm to adjust its posture for force-adaptive control. During transportation, the output remained stable, indicating smooth motion. In the installation phase, the sensor guided the mirror into a conical positioning hole, with force feedback enabling precise alignment.
The force measurements during these phases are plotted in a time-domain graph, showing variations corresponding to different operations. For example, peak forces occurred during grasping and installation, while transportation exhibited minimal fluctuations. This demonstrates the sensor’s ability to monitor interactions and prevent collisions, ensuring safe handling of delicate components. The successful completion of the ground experiments validates the six-axis force sensor’s design and its suitability for space assembly applications.
Conclusion
We have designed and validated a high-sensitivity large-range six-axis force sensor for space telescope assembly experiments. Through mathematical modeling, we identified optimization strategies to enhance sensitivity without大幅 compromising stiffness. The elastic body features localized grooving and full-bridge circuits, achieving force sensitivity above 0.83 mV/V and moment sensitivity above 2.6 mV/V. Finite element analysis and calibration confirmed the design’s performance, with repeatability errors below 0.33% FS and residual cross-talk under 1.2% after decoupling. The sensor has been successfully applied in ground experiments, providing reliable force feedback for robotic manipulation. This work demonstrates a balanced approach to six-axis force sensor design, addressing the challenges of range, stiffness, and sensitivity in constrained environments.