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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

To address the bottleneck issues of an elastic-style six-axis force/torque sensor (six-axis force sensor), this work proposes a no-elastic piezoelectric six-axis force sensor. The operating principle of the piezoelectric six-axis force sensor is analyzed, and a structural model is constructed. The static-active design theory of the piezoelectric six-axis force sensor is established, including a static analytical/mathematical model and numerical simulation model (finite element model). A piezoelectric six-axis force sensor experimental prototype is developed according to the analytical mathematical model and numerical simulation model, and selected static characteristic parameters (including sensitivity, isotropic degree and cross-coupling) are tested using this model with three approaches. The measured results are in agreement with the analytical results from the static-active design method. Therefore, this study has successfully established a foundation for further research into the piezoelectric multi-axis force sensor and an overall design approach based on static characteristics.

A six-axis force sensor is a device designed for measuring external forces and collecting spatial force information from three force components (Fx, Fy, Fz) and three torque components (Mx, My, Mz). Such a device also detects the position information of the force functional point. These sensors play significant roles in space robot design, space station docking simulations, rocket engine thrust testing, rocket-assisted aerodynamic characteristics testing, the collection of real time center position information for a flexible seating system, machine health monitoring and other applications. According to the GB7665-87 national standard, the six-axis force sensor can be classified as either elastic style [

Currently, three bottleneck issues exist in the elastic-style six-axis force sensor, including a degree of structural complexity and difficulty in decoupling [

The ultimate goal for the static design of the piezoelectric six-axis force sensor is to realize an active design based on its static performance. This effort requires research into the isotropic characteristics that affect the measurement accuracy [

To meet the need for a piezoelectric six-axis force sensor active design based on its static characteristics, this work researches the active design method of the piezoelectric six-axis force sensor's static characteristics based on preliminary studies. Piezoelectric quartz is selected for the force sensing elements. The operating principle of the six-axis force sensor is analyzed, and an eight-point support structure based on a double quartz crystal chip group is proposed. Furthermore, a static analytical/mathematical model is built, a numerical finite element model of the piezoelectric six-axis force sensor is set up, and the active design theory of this type of sensor is established. The correctness of the active design theory is verified by the calibration results from the piezoelectric six-axis force sensor experimental prototype, and the conclusions of the study reveal the main factors that affect the six-axis force sensor's static characteristics.

According to the structural characteristics of the piezoelectric element inside the piezoelectric force sensor, the force sensor can be divided into two types, including the integral structure and disaggregated structure. The integral structure piezoelectric force sensor consists of an internal piezoelectric element in the form of a complete wafer or annular plates. The disaggregated structure of the piezoelectric force sensor includes a number of piezoelectric elements, which are evenly arranged according to specific rules. The integral structure can reduce the cross-sectional area of the sensor, but its number of measurement dimensions cannot exceed four, which is not suitable for large-size structures. Therefore, it is difficult to miniaturize this sensor using MEMS technology. Therefore, a piezoelectric element multi-point support structure should be used if the piezoelectric elements are expected to achieve multi-axis force measurements of more than four dimensions.

The experimental prototype of the piezoelectric six-axis force sensor is shown in _{x}_{y}_{z}_{x}_{y}_{z}_{X}_{Y}_{Z}_{X}_{Y}_{Z}

^{0}-crystals are distributed on the nodes of the X and Y axes and the quartz crystal chip groups distribution circle and are used for the measurement of _{x}_{y}_{z}^{0}-crystals are distributed on the other locations and are used for the measurement of _{z}_{x}_{y}

Due to the influence of the piezoelectric six-axis accelerometer structure, the layout of the quartz chip group, the quantity and production level (among other factors), the arrangement of the quantity of quartz crystal cells and the production level, the actual conditions do not fully meet the above assumption in practice. Therefore, the acceleration transfer coefficients of _{fx}_{fy}_{fz}_{mx}_{my}_{mz}

To simplify the analysis, the following assumptions are adopted. The rigidities of the quartz crystal chip groups are identical, with equal sensitivity and symmetric uniform distribution. The cover of the piezoelectric six-axis force sensor is a rigid body with the same stiffness in all directions, equal sensitivity and uniform distribution. The directions of _{z}_{x}_{y}_{x}_{y}_{z}

_{1}-X_{1}Y_{1}Z_{1} denotes the installation layout position coordinate system of the quartz crystal chip groups. The quartz crystal chip groups are arranged along the same circle with radius is R, the distance between quartz chip groups 2 and 4 is 2r = 1.414R, and the distance between the force and the surface of quartz crystal chip groups is h. The component forces acting on each quartz crystal chip groups can be expressed by

According to the Equations _{Qm}_{e} is the available cross-sectional area of electrode, d_{11} and d_{26} are the piezoelectric moduli of the quartz crystals:

As can be seen from Equations _{y}_{x}_{x}_{y}

In the process of obtaining a high-precision analytical/mathematical model of the piezoelectric six-axis force sensor from the structural model, the key difficulties lie in solving for the load transfer coefficients _{f}x_{f}y_{f}z_{m}x_{m}y_{and} _{m}z

_{0}–k_{11} are the equivalent rigidity of the load functional part of the upper cover; the sensitive part of upper cover; the inner and outer ring elastic modulus of upper cover; the shell, the inner and outer ring elastic modulus of the lower cover; the boss of lower cover; the inner tube; the quartz crystal chip groups, and other components. After the externally measured spatial six-axis force acts on the surface of the upper cover, the force is transferred from the upper boss of the upper cover down to the upper part of the lower cover through the following components: the outer ring, inner the ring elastic modulus and lower boss of the upper cover; the shell; the upper quartz crystal chip groups; the electrode pads; the lower quartz crystal chip groups; the inner tube; the inner and outer ring elastic modulus of lower cover and other components. According to the theory of series and parallel spring's equivalent stiffness [

For example, as torque _{x}_{i}_{i}_{i}_{i}_{i}_{x}_{m}x

To verify the effectiveness of the piezoelectric six-axis force sensor analytical mathematical model, ANSYS software is used to pre-assess the piezoelectric six-axis force sensor static characteristics. The analysis process primarily applies a modeling approach and load application method.

In the first step, the physical structural model of the piezoelectric six-axis force sensor is built with CAD software (

This approach includes the installation constraints and acting force/torque loading on the piezoelectric six-axis force sensor. The constraints set adheres to the sensor's installation status, and the preload force is applied through a section of the cover. The degree of freedom of the pedestal's mounting surface is zero. The measured force/moments are applied to the key point, which is established on the Z-axis and located in the same plane as the upper surface of the piezoelectric six-axis force sensor. Additionally, the key point and the upper surface of the sensor's cover are built in the rigid region. _{x}_{y}

_{x}_{y}_{x}_{y}_{y}_{x}

An experimental prototype of the piezoelectric six-axis force sensor was constructed to verify the validity of the piezoelectric six-axis force sensor analytical/mathematical model and numerical simulation model. A conclusion can be obtained by comparing the analytical model, numerical model and experimental calibration results of the piezoelectric six-axis force sensor with the same structure.

_{X} and F_{Y} direction input force and output voltage calibration curve for the piezoelectric six-axis force sensor. The curve shows that the experimental test results are consistent with the theoretical analysis results based on the analytical/mathematical model and the numerical simulation model:

According to the output charges _{ij}_{Q}_{ij}^{T}^{−1} of the piezoelectric six-axis force sensor can be constructed according to

In Equations (_{Qm}_{Qs}_{Qe}_{66} are the elements of matrix _{Qs}

_{Qm}_{Qs}_{Qe}_{Qm}_{Qs}_{Qe}

The sensitivities obtained from the analytical mathematical model and numerical simulation model are quite consistent; due to the influence of the equivalent methods, production precision and calibration accuracy, the sensitivity test results are lower than the design aim; due to the influence of charge amplifier's zero shift, in the _{z}

The cross-coupling between _{x}_{y}_{y}_{x}_{x}_{y}_{y}_{x}_{x}_{y}_{y}_{x}

The force isotropy is 0.6531, 0.6467 and 0.7098, and the moment isotropy is 0.7737, 0.7567 and 0.6233, as obtained from the analytical mathematical model, numerical simulation model and experimental prototype of the piezoelectric six-axis force sensor, respectively. The force/torque isotropy of the analytical mathematical model is broadly consistent with that of the numerical simulation model. Because of the influence of machining accuracy, deviations are observed among the experimental prototype and the analytical mathematical model and numerical simulation model of the piezoelectric six-axis force sensor. However, due to the test gear, the quasi-static charge amplifier affects the sensitivity of the piezoelectric force sensor measurement system, and therefore, the force/torque isotropy of the piezoelectric six-axis force sensor could be greatly improved by selection of better test gear with respect to the charge amplifier.

In this paper, we have investigated a novel six axis force/torque sensor based on double quartz crystal chip groups. Its analytical mathematical model and numerical simulation model are presented. The research conclusions can be drawn as follows:

The sensor's static performances (

Due to the influence of the six axis force/torque sensor's spatial structure, there are some cross coupling interferences take place in the _{y}_{x}_{x}_{y}

These sensors' analytical mathematical model which is derived using the material mechanics and theoretical mechanics, and numerical simulation model based on ANSYS is presented, are effective. We can realize the design of these sensors' static performance through the two models.

Due to the influence of the simulation method, production precision and calibration accuracy, the test accuracy of the sensor experimental prototype in the _{Z}_{Z}

This work is supported by the Fundamental Research Funds for the Central Universities (No. 1061120131205), the Natural Science Foundation Project of Chongqing China CSTC (No. CSTC2012JJA40024), and the National High Technology Research and Development Program of China (863 Program) (No. 2012AA040107).

The authors declare no conflict of interest.

Schematic diagram of piezoelectric 6-axis force sensor. (

Block diagram of the sensor's structure.

The static spring equivalent model of the piezoelectric six-axis force sensor.

Selected input and output simulation characteristics of the sensor: (_{X} direction, (_{Y} direction, (_{X}, (_{Y}.

Photo of the static calibration test system: (1) Static standard power source. (2) Vertical loading device, (3) Six-axis force sensor, (4) Transverse loading device, (5) Quasi-static charge amplifier, (6) Pre-processing circuit, (7) Acquisition software.

Selected input force and output voltage calibration curve of the sensor: (_{X} direction, (_{Y} direction.

Main structural parameters of six-axis force sensor model.

^{3}) | ||||||
---|---|---|---|---|---|---|

Cover | 6 | 15 | 47 | 1Cr18Ni9Ti | 2.1e11 | 7,900 |

Inner tube | 12 | 15 | 16 | 1Cr18Ni9Ti | 2.1e11 | 7,900 |

Shell | 12 | 46 | 50 | 1Cr18Ni9Ti | 2.1e11 | 7,900 |

Quartz crystal chip | 1 | - | 10 | SiO_{2} |
8.0e10 | 2,650 |

Piezoelectric six-axis force/torque sensor sensitivity.

| ||||||
---|---|---|---|---|---|---|

Sfx | Sfy | Sfz | Smx | Smy | Smz | |

Analytical model | 2.016 | 2.016 | 2.306 | 362.418 | 362.418 | 280.403 |

Numerical model | 2.052 | 2.071 | 2.237 | 389.593 | 389.540 | 294.831 |

Experimental prototype | 1.869 | 1.654 | 2.877 | 209.938 | 212.685 | 150.156 |