In the realm of advanced manufacturing and robotics, the precise measurement of forces and torques is critical for applications ranging from industrial automation to scientific research. The six-axis force sensor, capable of detecting three-dimensional forces and moments, plays a pivotal role in these domains. However, the performance of such sensors can be significantly affected by assembly-induced stresses during installation and encapsulation. These stresses, arising from factors like machining tolerances and misalignment, introduce non-measured loads that lead to initial deformations, zero drift, and potential structural damage. This study investigates the influence of assembly stress on a cylindrical multi-beam resistive strain-type six-axis force sensor, employing finite element simulations and experimental measurements to analyze the effects of assembly processes. Our findings provide insights into optimizing assembly techniques to minimize performance degradation, ensuring the reliability and accuracy of six-axis force sensors in practical applications.
The six-axis force sensor is designed with a cylindrical elastic body that incorporates multiple beams to facilitate strain measurements under various load conditions. The sensor’s nominal capacity is defined as $F_x = F_y = \pm 300 \text{N}$, $F_z = \pm 600 \text{N}$, and $M_x = M_y = M_z = \pm 25 \text{Nm}$. Strain gauges are strategically placed on the elastic body to form Wheatstone bridge circuits for each degree of freedom. Theoretical calculations indicate that under no load, the bridge output strain should be zero, while under full load, the expected strains are $\varepsilon_{F_x} = \varepsilon_{F_y} = 1073.4 \mu\varepsilon$, $\varepsilon_{F_z} = 1022.6 \mu\varepsilon$, $\varepsilon_{M_x} = \varepsilon_{M_y} = 2125.4 \mu\varepsilon$, and $\varepsilon_{M_z} = 1204.3 \mu\varepsilon$. The encapsulation structure includes upper and lower cover plates, with the lower plate serving both as a protective housing and a connection interface to the base. This connection is secured using eight M5 bolts, and two $\phi 4$ alignment pins ensure precise positioning. The assembly process involves applying a pre-tightening torque of 2.5 Nm followed by a final tightening torque of 5 Nm to each bolt in sequence.
To simulate the assembly process, finite element analysis (FEA) was conducted using Abaqus 6.14. The bolt connections were simplified as applied forces and moments, with the clamping force $F$ calculated using the empirical formula: $$ F = \frac{T}{kD} $$ where $T$ is the total torque, $k$ is the torque coefficient (set to 0.2), and $D$ is the bolt nominal diameter. The load distribution assumed that 50% of the torque is borne by the contact area between the lower cover plate and bolt end, 40% by the threaded portion of the elastic body, and the remainder as compressive forces. The materials for both the elastic body and cover plate were titanium alloy, with an elastic modulus $E = 107.8 \text{GPa}$, Poisson’s ratio $\nu = 0.34$, density $\rho = 4510 \text{kg/m}^3$, and yield strength $\geq 895 \text{MPa}$. Three assembly sequences were evaluated: Sequence A started near the pins and proceeded clockwise for pre-tightening and counterclockwise for tightening; Sequence B began at bolts far from the pins using a diagonal pattern; and Sequence C initiated near the pins with a diagonal order. Additionally, the effects of friction coefficient (0.1, 0.2, 0.3) and contact area size were examined to understand their impact on assembly stress.
Experimental validation involved assembling actual six-axis force sensors using the same sequences and measuring the output drift with a custom signal conditioning system. The sensors were subjected to incremental bolt tightening, and the zero drift in each axis was recorded to compare with simulation results. This combined approach allowed for a comprehensive analysis of how assembly parameters influence the performance of the six-axis force sensor.
The finite element simulations revealed that assembly sequences significantly affect the zero drift of the six-axis force sensor. For instance, under a friction coefficient of 0.2, Sequence A resulted in a final zero drift of $-44.9 \mu\varepsilon$ in the $F_x$ direction, while Sequence B and Sequence C showed values of $-53.8 \mu\varepsilon$ and $-40.4 \mu\varepsilon$, respectively. The $F_z$ direction exhibited the highest sensitivity, with drifts reaching up to 4.2% of the full-scale value in Sequence A. The moments $M_x$, $M_y$, and $M_z$ also displayed varying drift patterns, with Sequence C generally yielding the smallest deviations. The table below summarizes the percentage of full-scale zero drift for different assembly sequences under a friction coefficient of 0.2:
| Assembly Sequence | $F_x$ Drift (%) | $F_y$ Drift (%) | $F_z$ Drift (%) | $M_x$ Drift (%) | $M_y$ Drift (%) | $M_z$ Drift (%) |
|---|---|---|---|---|---|---|
| Sequence A | -4.2 | -0.2 | 4.2 | 0.6 | -0.2 | 0.3 |
| Sequence B | -5.0 | 0.5 | 3.1 | 0.4 | -0.2 | -0.1 |
| Sequence C | -3.8 | 0.4 | 2.1 | 0.2 | -0.2 | 0.1 |
Friction coefficient variations introduced complex behaviors in the six-axis force sensor response. Contrary to initial expectations, increasing the friction coefficient from 0.1 to 0.2 amplified the zero drift, but a further increase to 0.3 reduced it slightly, though not to the level of 0.1. This non-monotonic relationship suggests that friction influences the stress distribution in the elastic body in a non-linear manner, potentially due to the multi-beam geometry. For example, in Sequence C, the $F_z$ drift changed from 20.5 $\mu\varepsilon$ at $\mu=0.1$ to 21.8 $\mu\varepsilon$ at $\mu=0.2$, and then to 19.2 $\mu\varepsilon$ at $\mu=0.3$. The following equation models the effective strain $\varepsilon_{\text{eff}}$ due to friction: $$ \varepsilon_{\text{eff}} = \varepsilon_0 + \alpha \mu – \beta \mu^2 $$ where $\varepsilon_0$ is the base strain, and $\alpha$ and $\beta$ are constants derived from material properties. This highlights the need to carefully consider friction in contact surfaces during the assembly of a six-axis force sensor.
Reducing the contact area between the elastic body and the cover plate proved beneficial in minimizing assembly stress. Simulations showed that with a smaller contact area, the maximum zero drift across all sequences decreased from 5.8% to 5.0% of full-scale, while experimental measurements confirmed a reduction from 3.6% to 2.9%. This improvement can be attributed to the decreased influence of frictional forces, as the strain energy density $U$ in the contact region is proportional to the area $A$: $$ U = \frac{1}{2} \int_A \sigma \epsilon dA $$ where $\sigma$ is stress and $\epsilon$ is strain. By limiting the contact area, the cumulative effect of assembly-induced deformations is mitigated, enhancing the accuracy of the six-axis force sensor.
The experimental results aligned closely with the finite element predictions, though the measured zero drifts were generally smaller in magnitude. This discrepancy may stem from manufacturing tolerances and variations in actual friction coefficients compared to the idealized models. For instance, in Sequence C, the experimental $F_x$ drift was $-35.2 \mu\varepsilon$ versus $-40.4 \mu\varepsilon$ in simulation, indicating that real-world factors such as surface roughness and bolt pre-load variations play a role. The table below compares experimental and simulated zero drifts (in $\mu\varepsilon$) for Sequence C with a friction coefficient of 0.2:
| Direction | Simulated Drift | Experimental Drift |
|---|---|---|
| $F_x$ | -40.4 | -35.2 |
| $F_y$ | 4.3 | 3.8 |
| $F_z$ | 21.8 | 18.5 |
| $M_x$ | 8.5 | 7.1 |
| $M_y$ | -3.7 | -2.9 |
| $M_z$ | 0.8 | 0.6 |
Discussion of these findings emphasizes that the assembly process is a critical factor in the performance of six-axis force sensors. The optimal sequence, starting near pins and following a diagonal pattern, reduces asymmetric stress distributions that cause zero drift. Friction management is equally important, as it can either exacerbate or alleviate stress depending on the coefficient. The non-linear response to friction suggests that material selection and surface treatments could be optimized to control assembly stress. Furthermore, minimizing contact areas not only lowers frictional effects but also simplifies the alignment process, contributing to more consistent sensor outputs. These insights are vital for the design and application of six-axis force sensors in precision-dependent fields such as robotics and aerospace.
In conclusion, our investigation demonstrates that assembly stress significantly impacts the performance of six-axis force sensors, leading to zero drift that can compromise measurement accuracy. Through finite element analysis and experimental validation, we have identified key strategies to mitigate these effects: (1) initiating bolt loading near alignment pins and adhering to a diagonal sequence minimizes stress concentrations; (2) carefully controlling the friction coefficient on contact surfaces can reduce drift, with higher coefficients sometimes being advantageous due to complex mechanical interactions; and (3) reducing the contact area between components effectively decreases the influence of assembly-induced stresses. These recommendations provide a practical framework for enhancing the reliability of six-axis force sensors, ensuring their optimal performance in diverse industrial and scientific applications. Future work could explore dynamic loading conditions and long-term effects of assembly stress on sensor durability.