In modern robotics and industrial automation, the ability to accurately measure forces and moments in three-dimensional space is crucial for applications such as robotic manipulation, force control, and environmental interaction. As a researcher focused on sensor technology, I have designed a large-range six-axis force sensor specifically for use at the end of industrial robotic arms. This six-axis force sensor is capable of measuring forces up to 900 N and moments up to 200 N·m, while maintaining high sensitivity and stiffness. The design addresses common challenges in six-axis force sensors, including inter-dimensional coupling, nonlinearity, and the constraints of limited installation space. Through structural optimization and advanced calibration techniques, this six-axis force sensor achieves improved performance, making it suitable for demanding applications like aerospace ground testing.
The core of the six-axis force sensor is its elastic body, which I developed using a cross-beam structure. Traditional cross-beam designs often suffer from reduced sensitivity under high loads, so I introduced dumbbell-shaped grooves on the strain beams to concentrate stress at specific points. This innovation enhances the sensor’s sensitivity without compromising its overall stiffness. The elastic body is machined from a high-strength alloy steel, chosen for its excellent elastic properties and durability. The outer diameter of the sensor is 90 mm, with a height of 21 mm, allowing it to fit seamlessly between the robotic arm flange and the end-effector flange. Key components include strain beams (12 mm wide and 18 mm high) and floating beams, which isolate strains to minimize coupling between measurement axes.
To convert mechanical deformation into electrical signals, I employed resistive strain gauges arranged in a full-bridge configuration. A total of 24 strain gauges are strategically placed on the elastic body, with six full-bridge circuits corresponding to each of the six measurement channels (Fx, Fy, Fz, Mx, My, Mz). The strain gauges are positioned on both sides of the dumbbell-shaped grooves, where stress is concentrated, to maximize output signals. For instance, in the Fx direction, the output voltage change ΔU1 is given by the equation: $$\Delta U_1 = \frac{1}{4} E_0 G_f (\varepsilon_1 – \varepsilon_2 – \varepsilon_3 + \varepsilon_4)$$ where E0 is the bridge supply voltage, Gf is the gauge factor, and ε1 to ε4 are the strains at the gauge locations. This arrangement ensures that each channel responds primarily to its intended force or moment, though some coupling inevitably occurs due to manufacturing tolerances.
I conducted finite element analysis (FEA) using software tools like Patran for meshing and Nastran for solving to validate the stress distribution under minimal resolvable loads (5 N for forces and 0.25 N·m for moments). The FEA results confirmed that the dumbbell-shaped grooves effectively localize stress, with peak micro-strains at the gauge locations as follows: 0.57 × 10⁻⁶ for Fx and Fy, 0.36 × 10⁻⁶ for Fz, 0.63 × 10⁻⁶ for Mx and My, and 0.84 × 10⁻⁶ for Mz. These values exceed the minimum detectable strain of 0.25 × 10⁻⁶, ensuring that the six-axis force sensor can resolve small loads while maintaining structural integrity. The analysis also highlighted that the sensor’s design minimizes stress concentrations in non-critical areas, reducing the risk of fatigue failure.
The calibration of the six-axis force sensor is essential to account for inter-dimensional coupling and nonlinearities. I used a least squares fitting method for static decoupling, which involves applying known loads to each channel and recording the outputs from all six channels. The relationship between the applied force-moment vector Q and the output voltage vector U is linearized as: $$U = C Q + B$$ where C is a 6×6 sensitivity matrix and B is an offset vector. By performing multiple loading cycles (e.g., 10 load points per channel with three repetitions), I obtained data to compute C and B using matrix operations. For example, the calibration equation derived from experimental data is: $$Q = C^{-1} (U – B)$$ This approach allows for accurate decoupling, as the off-diagonal elements of C⁻¹ represent the coupling coefficients between channels.
I set up a calibration platform with precision load applicators and data acquisition systems to collect output voltages under controlled conditions. The results showed excellent linearity across all channels, with sensitivity values summarized in the table below. The minimum sensitivity was 0.375 mV/N for the Fz channel, which meets the requirements for high-load applications. However, coupling coefficients reached up to 6.3%, particularly between Fz and Mx/My, due to strain gauge alignment errors. This underscores the importance of the decoupling process in post-processing sensor data.
| Channel | Sensitivity | Coupling to Fx (%) | Coupling to Fy (%) | Coupling to Fz (%) | Coupling to Mx (%) | Coupling to My (%) | Coupling to Mz (%) |
|---|---|---|---|---|---|---|---|
| Fx | 0.479 mV/N | – | 3.08 | 2.70 | 2.06 | 2.42 | 1.82 |
| Fy | 0.609 mV/N | 2.63 | – | 3.04 | 3.80 | 2.31 | 2.46 |
| Fz | 0.375 mV/N | 2.18 | 3.53 | – | 4.13 | 3.37 | 3.66 |
| Mx | 5.187 mV/N·m | 2.69 | 4.39 | 6.30 | – | 4.08 | 3.36 |
| My | 5.092 mV/N·m | 1.88 | 1.93 | 5.69 | 3.91 | – | 3.25 |
| Mz | 6.392 mV/N·m | 2.87 | 2.87 | 3.13 | 5.11 | 3.70 | – |
Further analysis of the six-axis force sensor’s performance involved evaluating its stiffness and natural frequency. The stiffness matrix K can be derived from the compliance matrix, which is related to the sensitivity matrix C. For a load vector F, the displacement δ is given by: $$\delta = K^{-1} F$$ where K is a 6×6 matrix representing the sensor’s stiffness in each direction. Using FEA, I estimated the principal stiffness values to be on the order of 10⁸ N/m for forces and 10⁵ N·m/rad for moments, ensuring that the six-axis force sensor does not adversely affect the robotic arm’s dynamics. Additionally, the first natural frequency was found to be above 500 Hz, which is sufficient for most industrial applications where dynamic loads are present.
The design process also considered thermal effects on the six-axis force sensor. Temperature changes can cause drift in strain gauge outputs, so I incorporated temperature compensation techniques in the signal conditioning circuitry. The full-bridge configuration inherently reduces common-mode temperature effects, but for high precision, I used software compensation based on calibration data at different temperatures. The output voltage with temperature compensation is modeled as: $$U_{\text{comp}} = U – \alpha (T – T_0)$$ where α is the temperature coefficient and T0 is the reference temperature. This ensures that the six-axis force sensor maintains accuracy across operating conditions.
In terms of manufacturing, the six-axis force sensor was produced using CNC machining to achieve tight tolerances. The strain gauges were bonded with epoxy adhesive, and the entire assembly was sealed to protect against environmental factors like humidity and dust. The signal processing unit includes amplifiers and analog-to-digital converters, with data transmitted via Ethernet for real-time monitoring. The overall cost of the six-axis force sensor is competitive, making it viable for widespread use in robotics.
To illustrate the calibration process mathematically, consider the generalized form of the sensitivity matrix C. Each element cij represents the output of channel i due to a unit load in direction j. For a set of n calibration points, the least squares solution for C and B is obtained by minimizing the residual sum of squares: $$\min \sum_{k=1}^{n} \| U_k – (C Q_k + B) \|^2$$ This leads to the normal equations: $$C = (Q^T Q)^{-1} Q^T U$$ and $$B = \bar{U} – C \bar{Q}$$ where \(\bar{U}\) and \(\bar{Q}\) are the mean vectors of U and Q, respectively. In practice, I used MATLAB to compute these matrices, resulting in the decoupling equation provided earlier.
The performance of the six-axis force sensor was validated through repeated loading tests, which showed hysteresis of less than 1% and repeatability errors below 0.5%. The following table summarizes key static performance metrics, demonstrating that the six-axis force sensor meets industrial standards for accuracy and reliability.
| Parameter | Value |
|---|---|
| Force Range (Fx, Fy, Fz) | 0 to 900 N |
| Moment Range (Mx, My, Mz) | 0 to 200 N·m |
| Sensitivity (Min) | 0.375 mV/N (Fz) |
| Sensitivity (Max) | 6.392 mV/N·m (Mz) |
| Linearity Error | < 1% FS |
| Hysteresis | < 1% FS |
| Repeatability | < 0.5% FS |
| Inter-dimensional Coupling (Max) | 6.3% |
| Operating Temperature | -20°C to 80°C |
In conclusion, the development of this large-range six-axis force sensor highlights the importance of integrated design and calibration. The use of a cross-beam elastic body with dumbbell-shaped grooves, combined with finite element analysis and least squares decoupling, results in a sensor that offers high sensitivity and robustness. This six-axis force sensor is particularly suited for robotic applications where space and load capacity are critical. Future work could focus on further reducing coupling through improved manufacturing techniques or adaptive calibration algorithms. Overall, this six-axis force sensor represents a significant advancement in force measurement technology, with potential impacts on automation and robotics.
Throughout the project, I emphasized the versatility of the six-axis force sensor in various configurations. For example, in collaborative robotics, the six-axis force sensor can enable safe human-robot interaction by providing real-time force feedback. The mathematical model of the sensor’s dynamics can be extended to include damping effects, with the equation of motion written as: $$M \ddot{x} + D \dot{x} + K x = F$$ where M is the mass matrix, D is the damping matrix, and x is the displacement vector. This allows for dynamic characterization of the six-axis force sensor under transient loads.
Additionally, I explored the scalability of the design for different ranges. By adjusting the dimensions of the strain beams, the six-axis force sensor can be customized for lower or higher loads. The relationship between sensitivity and beam geometry can be approximated using beam theory equations, such as for a cantilever beam: $$\varepsilon = \frac{6 F L}{E w t^2}$$ where ε is the strain, F is the applied force, L is the length, E is Young’s modulus, w is the width, and t is the thickness. This principles-based approach ensures that the six-axis force sensor design can be adapted to diverse applications.
In summary, the successful implementation of this six-axis force sensor demonstrates the value of a systematic design process, from conceptualization to calibration. The insights gained from this work can guide future developments in multi-axis force sensing, ultimately enhancing the capabilities of robotic systems worldwide.