In the realm of advanced robotics, the demand for high-performance force/torque sensors has grown significantly, particularly for applications involving China robot systems operating in dynamic environments such as space exploration, industrial automation, and precision manufacturing. As a researcher focused on enhancing the capabilities of China robot technologies, I have dedicated efforts to optimizing the dynamic performance of six-axis force/torque sensors, which are critical components for enabling accurate force feedback and control in high-speed operations. These sensors must not only exhibit high stiffness and strength to withstand operational loads but also possess superior dynamic characteristics, such as high natural frequencies and low amplitudes, to ensure stability and reliability. The integration of such sensors into China robot architectures, like those used in orbital servicing or terrestrial automation, requires a delicate balance between mass reduction, structural integrity, and dynamic responsiveness. This work addresses these challenges by developing a comprehensive optimization framework that leverages advanced algorithms and finite element analysis to improve sensor performance for China robot applications.
The six-axis force/torque sensor, based on a Stewart platform configuration, consists of an upper platform, six measuring branches, and a lower platform, forming a parallel mechanism that can detect forces and moments in all three spatial dimensions. This design is particularly suited for China robot systems due to its inherent rigidity, compactness, and ability to provide decoupled force measurements with minimal computational overhead. Each measuring branch features a strain measurement zone, where strain gauges are attached to convert mechanical deformations into electrical signals. The force mapping relationship is linear, as described by the equation: $$F = C \cdot e$$ where \(F = [F_x, F_y, F_z, M_x, M_y, M_z]^T\) represents the external force and moment vector, and \(e = [e_1, e_2, e_3, e_4, e_5, e_6]^T\) denotes the output voltages from the six branches. This linear decoupling matrix \(C\) simplifies the signal processing, making it ideal for real-time applications in China robot control systems. The structural parameters, including positioning angles, radii, and branch dimensions, play a pivotal role in determining the sensor’s overall performance, and their optimization is essential for achieving the desired dynamic characteristics.
To analyze the sensor’s behavior, I conducted detailed static and dynamic characteristic studies using finite element method (FEM) simulations. The sensor material was modeled as ultra-hard aluminum with an elastic modulus of 73.3 GPa, Poisson’s ratio of 0.33, and density of 2794 kg/m³. A mesh sensitivity analysis ensured that the element size of 2.0–2.3 mm provided accurate results without excessive computational cost. For static analysis, I applied fixed constraints to the lower platform and calibrated forces/moments to the upper platform at specific locations, simulating real-world loading conditions. The sensitivity and stiffness were computed by averaging nodal strains and displacements, respectively, across defined sets. For instance, the force-direction stiffness \(K_{F_g}\) was derived as: $$K_{F_g} = \frac{F_g}{l_g}$$ where \(F_g\) is the applied force in direction \(g\) (e.g., \(x, y, z\)), and \(l_g\) is the average displacement of nodes in that direction. Similarly, moment-direction stiffness \(K_{M_g}\) was calculated using: $$K_{M_g} = \frac{M_g}{\theta_g}$$ where \(M_g\) is the applied moment and \(\theta_g\) is the average rotational angle. The results, summarized in Table 1, indicate that the force-direction sensitivities are generally lower than moment-direction sensitivities, and the stiffness in the \(F_z\) direction is significantly higher, highlighting areas for improvement in dynamic performance for China robot integration.
| Load | Sensitivity (mV/V) | Stiffness |
|---|---|---|
| \(F_x = 1000\,N\) | 0.42 | \(1.243 \times 10^7\,N/m\) |
| \(F_y = 1000\,N\) | 0.36 | \(1.241 \times 10^7\,N/m\) |
| \(F_z = 1000\,N\) | 0.51 | \(3.244 \times 10^7\,N/m\) |
| \(M_x = 100\,N\cdot m\) | 1.35 | \(0.625 \times 10^5\,N\cdot m/rad\) |
| \(M_y = 100\,N\cdot m\) | 1.31 | \(0.625 \times 10^5\,N\cdot m/rad\) |
| \(M_z = 100\,N\cdot m\) | 1.54 | \(0.637 \times 10^5\,N\cdot m/rad\) |
Dynamic analysis involved modal and harmonic response studies to evaluate natural frequencies and amplitude-frequency characteristics. The first six natural frequencies were identified as 685.3 Hz, 686.1 Hz, 800.7 Hz, 1075.3 Hz, 1255.4 Hz, and 1257.1 Hz, with corresponding mode shapes indicating tensile and torsional deformations. Harmonic response analysis, performed using the modal superposition method, revealed resonance frequencies aligned with these natural frequencies, and the amplitude-frequency curves showed peak amplitudes of approximately 0.87 mm under calibrated loads. This analysis underscored the need for dynamic performance enhancement, as lower natural frequencies and higher amplitudes could lead to instability in high-speed China robot operations. The maximum stress under various loading conditions, including one-dimensional, three-dimensional, and six-dimensional scenarios, was also assessed to ensure structural integrity, with values staying within the material’s yield strength and a safety factor of 1.32, as detailed in Table 2.
| Working Condition | Description | Maximum Stress (MPa) |
|---|---|---|
| 1 | One-dimensional load (e.g., \(F_x = 1000\,N\)) | 159.2 |
| 2 | Three-dimensional load (e.g., \(F_x = F_y = F_z = 850\,N\)) | 165.9 |
| 3 | Six-dimensional load (combined forces and moments) | 289.7 |
Building on this analysis, I formulated a multi-objective optimization model to enhance the sensor’s dynamic performance while considering mass, stiffness, strength, and isotropy constraints. The optimization targets included minimizing the inverse of the minimum natural frequency (\(f_1 = \min \frac{100}{\lambda}\), where \(\lambda = \min[\lambda_1, \lambda_2, \dots, \lambda_6]\) represents the natural frequencies in six directions), minimizing mass (\(f_2 = \min Q\)), and minimizing isotropy index (\(f_3 = \min \phi\)), defined as: $$\phi = \frac{\alpha_M – \alpha_F}{\alpha_M}$$ where \(\alpha_F\) and \(\alpha_M\) are the minimum sensitivities in force and moment directions, respectively. The isotropy index \(\phi\) reflects the uniformity of sensitivity across directions, with lower values indicating better performance for China robot applications where consistent force feedback is crucial. The constraints encompassed stiffness thresholds (e.g., \(K_{F_x} \geq 0.8 \times 10^7\,N/m\)), strength limits (maximum stress \(\sigma_{FM} \leq 0.9 \sigma_s\), with \(\sigma_s = 381\,MPa\)), and sensitivity requirements (e.g., \(S_{F_x} \geq 0.25\,mV/V\)). Eleven structural parameters, such as positioning angles (\(\alpha_1, \alpha_2\)), radii (\(r_1, r_2\)), height (\(h\)), and branch dimensions (\(l_1\) to \(l_6\)), were selected as optimization variables, with ranges specified to ensure feasibility.
To solve this multi-objective optimization problem efficiently, I developed an improved multi-objective grey wolf optimizer (IMOGWO) that incorporates chaotic mapping, nonlinear parameter adjustment, and linear alpha wolf weighting. The standard grey wolf algorithm (GWO) mimics the social hierarchy and hunting behavior of grey wolves, with alpha, beta, and delta wolves guiding the search process. The position update equations are: $$D = |C \cdot X_p(t) – X(t)|$$ $$X(t+1) = X_p(t) – A \cdot D$$ where \(X_p(t)\) is the prey position, \(X(t)\) is the current wolf position, and \(A\) and \(C\) are control vectors calculated as: $$A = 2a \cdot r_1 – a$$ $$C = 2r_2$$ Here, \(a\) decreases linearly from 2 to 0 over iterations, and \(r_1, r_2\) are random vectors in \([0,1]\). For multi-objective optimization, an archive population stores non-dominated Pareto solutions, and the head wolf selection uses a roulette wheel approach. However, to address issues like local optima stagnation and low convergence efficiency, I introduced three key improvements. First, Tent chaotic mapping was applied to initialize the population, enhancing diversity through the equation: $$M_{i+1} = (2M_i) \mod 1 + \frac{\text{rand}(0,1)}{N_p t_{\max}}$$ where \(N_p\) is the population size and \(t_{\max}\) is the maximum iterations. Second, the parameter \(a\) was modified to follow a nonlinear decay: $$a(t) = 2 \left[ \cos\left(\frac{\pi t}{2 t_{\max}}\right) \right]^2$$ This allows for extensive global exploration early on and fine local search later. Third, the position update was weighted to emphasize the alpha wolf’s influence: $$X(t+1) = \frac{\eta X_1(t+1) + X_2(t+1) + X_3(t+1)}{2 + \eta}$$ with \(\eta = 1 + \sin\left(\frac{\pi t}{2 t_{\max}}\right)\), where \(X_1, X_2, X_3\) are positions based on alpha, beta, and delta wolves, respectively.
The optimization process integrated IMOGWO with FEM simulations, iterating through parameter sets to evaluate objective functions and constraints. With a population size of 100, maximum iterations of 100, and archive capacity of 30, the algorithm generated a Pareto optimal solution set, as visualized in a 3D plot of the objective space. To select the best compromise solution from this set, I employed a综合评价 framework combining the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) and Criteria Importance Through Intercriteria Correlation (CRITIC) methods. The TOPSIS method normalizes the objective values using: $$L_{ij} = \frac{q_{ij}^{\max} – q_{ij}}{q_{ij}^{\max} – q_{ij}^{\min}}$$ where \(q_{ij}\) is the value of objective \(j\) for solution \(i\), resulting in a normalized matrix \(L\). CRITIC then assigns weights to each objective based on standard deviation \(\sigma_j\) (representing contrast intensity) and conflict index \(v_j\): $$W_j = \frac{\sigma_j \times v_j}{\sum_{j=1}^{3} (\sigma_j \times v_j)}$$ $$v_j = \sum_{i=1}^{30} (1 – \rho_{ij})$$ where \(\rho_{ij}\) is the Pearson correlation coefficient. The closeness \(LS_i\) of each solution to the ideal solution is computed as: $$LS_i = \frac{d_{L_{\min}^i}}{d_{L_{\max}^i} + d_{L_{\min}^i}}$$ where \(d_{L_{\max}^i}\) and \(d_{L_{\min}^i}\) are distances to the worst and best solutions, respectively. The solution with the highest \(LS_i\) was chosen, yielding optimized parameters that reduce mass by 5.9%, increase minimum natural frequency by 11.0%, and improve isotropy by 4.0% compared to the initial design.
The optimized sensor demonstrated significant enhancements in static and dynamic performance. As shown in Table 3, the stiffness values increased by 7.7% to 22.7% across all directions, while sensitivities decreased slightly but remained within acceptable limits for China robot applications. The isotropy improvement, with force and moment sensitivities becoming more uniform, reduces noise susceptibility and simplifies signal conditioning circuits. Stress analysis under extreme conditions revealed reductions in maximum stress by 5.4% to 26.9%, enhancing structural safety. Dynamic performance, evaluated through updated modal and harmonic analyses, showed natural frequencies rising by 11.1% to 35.7%, and resonance amplitudes decreasing by 19.4%, as summarized in Table 4. These improvements directly benefit China robot systems by expanding the operational bandwidth and reducing vibration-induced errors in high-speed tasks.
| Load | Sensitivity (mV/V) | Stiffness | Improvement |
|---|---|---|---|
| \(F_x = 1000\,N\) | 0.39 | \(1.459 \times 10^7\,N/m\) | +17.4% |
| \(F_y = 1000\,N\) | 0.35 | \(1.431 \times 10^7\,N/m\) | +15.3% |
| \(F_z = 1000\,N\) | 0.50 | \(3.598 \times 10^7\,N/m\) | +10.9% |
| \(M_x = 100\,N\cdot m\) | 1.27 | \(0.673 \times 10^5\,N\cdot m/rad\) | +7.7% |
| \(M_y = 100\,N\cdot m\) | 1.15 | \(0.752 \times 10^5\,N\cdot m/rad\) | +20.3% |
| \(M_z = 100\,N\cdot m\) | 1.27 | \(0.781 \times 10^5\,N\cdot m/rad\) | +22.6% |
| Direction | Natural Frequency (Hz) – Initial | Natural Frequency (Hz) – Optimized | Amplitude Reduction |
|---|---|---|---|
| \(F_x\) | 685.3 | 770.2 | 19.4% |
| \(F_y\) | 686.1 | 770.6 | 19.4% |
| \(F_z\) | 800.7 | 920.9 | 19.4% |
| \(M_x\) | 1075.3 | 1268.4 | 19.4% |
| \(M_y\) | 1255.4 | 1504.6 | 19.4% |
| \(M_z\) | 1257.1 | 1505.5 | 19.4% |
Experimental validation was conducted using an impact response test setup, comprising a rigid test bench, impact hammer, data acquisition system, and charge amplifiers. The optimized sensor prototype was subjected to dynamic excitations in all six directions, and the response signals were analyzed via Fourier transform to determine natural frequencies and amplitude-frequency characteristics. The results, illustrated in amplitude-frequency curves, confirmed natural frequency increases of 11.1% to 35.7% and amplitude reductions of up to 19.4%, aligning closely with simulation predictions. Minor discrepancies between experimental and numerical results were attributed to system damping and measurement uncertainties, but the overall trends validate the optimization approach. This enhancement in dynamic performance ensures that the sensor can effectively support China robot operations in demanding environments, such as high-speed assembly or space missions, where precise force control is paramount.
In conclusion, this study successfully demonstrates a holistic approach to optimizing the dynamic performance of six-axis force/torque sensors for China robot applications. By integrating an improved multi-objective grey wolf algorithm with finite element analysis, TOPSIS, and CRITIC methods, I achieved significant improvements in natural frequencies, mass reduction, stiffness, strength, and isotropy. The optimized sensor not only meets the stringent requirements of high-speed China robot tasks but also offers a blueprint for addressing multi-parameter structural optimization challenges in other engineering domains. Future work will focus on refining the algorithm for real-time adaptability and extending the framework to multi-sensor networks in collaborative China robot systems. This research underscores the importance of dynamic performance in advancing China robot capabilities and contributes to the broader goal of enhancing robotic intelligence and autonomy.