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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/).

A variety of gyroscopes have been extensively studied due to their capability of precision detection of rotation rates and extensive applications in navigation, guidance and motion control. In this work, a new Hybrid Gyroscope (HG) which combines the traditional Dynamically Tuned Gyroscope (DTG) with silicon micromachined technology is investigated. The HG not only has the potentiality of achieving the same high precision as the traditional DTG, but also features a small size and low cost. The theoretical mechanism of the HG with a capacitance transducer and an electrostatic torquer is derived and the influence of the installation errors from the capacitance plate and the disc rotor module is investigated. A new tuning mechanism based on negative stiffness rather than the traditional dynamic tuning is proposed. The experimental results prove that the negative stiffness tuning is practicable and a tuning voltage of as high as 63 V is demonstrated. Due to the decreased installation error, the non-linearity of the scale factor is reduced significantly from 11.78% to 0.64%, as well as the asymmetry from 93.3% to 1.56% in the open loop condition. The rebalancing close-loop control is simulated and achieved experimentally, which proves that the fundamental principle of the HG is feasible.

As an important inertial sensor, the gyroscopes which are used to detect the rotation rate have been a hot researching topic since its emergence. A variety of gyroscopes, such as Traditional Mechanical Gyroscope (TMG), Electrostatic Suspended Gyroscope (ESG), Ring Laser Gyroscope (RLG), Fiber Optic Gyroscope (FOG), Dynamically Tuned Gyroscope (DTG), as well as Silicon Micro-Gyroscope (SMG), have been developed. With the merits of small volume, light weight, low cost, mass production, and high endurance to shock and vibration, SMGs have received great attention in the last twenty years [

The classical gyroscopes, such as the DTG, have high precision (the bias stability is usually around 0.001∼1°/h), but these gyroscopes are large in volume, heavy in weight, costly (they may cost tens of thousands of dollars), weak in endurance to shock and vibration, and complicated in process and assembly [

To meet these requirements, the Hybrid Gyroscope (HG) is a hot research topic. The research on the microelectromechanical hybrid gyroscope was first proposed by Jenkins

The HG structure is designed to verify the basic principle. The HG as shown in

Compared with the traditional DTG, the disc rotor module is adopted simultaneously, which greatly simplifies the rotor structure and the flexible joint in the traditional DTG. Besides, the mass and volume of the rotor with sheet structure are greatly reduced, and the impact resistance and the start-up characteristics are greatly improved. In addition, the silicon disc rotor module and glass upper/down plate which are fabricated in volume through the silicon micro-machining technology do not require assembly.

The disc rotor module of the HG is driven by the electrical motor to obtain a large momentum moment along the z axis. When a rotational rate along X-axis or Y-axis is input, the rotating axis of the disc rotor is deflected due to the Coriolis force, changing the capacitance between the disc rotor and the upper/down capacitor plates. This capacitance change can then be measured to calculate the input rotational rate. To maintain the equilibrium position of the disc rotor, the electrostatic feedback moment is exerted by electrostatic torquer to generate the revising and compensating effects, which is used to indirectly measure the input rotational rate of HG. Neglecting some secondary factors such as damping moments and second order harmonic moment, the motion equations of HG can be expressed as [_{e} + I_{e}/2, J_{e} is the moment of inertia of disc rotor around X-axis or Y-axis , and I_{e} is the moment of inertia of equilibrium ring around X-axis or Y-axis; ΔK is the remaining stiffness; γ and δ are the angular displacements in the rotation axis of the disc rotor along the X-axis and Y-axis respectively; _{y}_{x}_{x} and M_{y} are respectively the feedback moments of rotor along X-axis and Y-axis, c and β are viscous damping coefficient of the flexible torsion spring and orthogonal damping elasticity coefficient. From

The structure parameters are shown in _{1} and h_{2} are the thicknesses of disc rotor module and the inner/outer torsional spring, K_{p} is the torsional rigidity of the inner/outer torsion spring and K_{f} is the flexural rigidity of the inner/outer torsion spring. The disc rotor module of the HG is driven by the electrical motor to rotate rapidly. When a rotational rate along X-axis or Y-axis is input, the disc rotor is deflected around the drive axis and the equilibrium ring is driven to turn over by the disc rotor simultaneously, as shown in

The disc rotor module is the core part of the HG and directly determines the performance of the gyroscope. The disc rotor is simulated and optimized by ANSYS.

In order to simplify the process, a 4-inch P-type single crystalline silicon wafer with 210 um thickness is adopted. The photoresist (PR) is coated on the wafer to form the mask of disc rotor module. The standard Deep Reactive Ion Etching (DRIE) process with 20:1 aspect ratio is used for fabrication of disc rotor module. Six disc rotor modules can be processed in a 4-inch silicon wafer. Similarly, the capacitor plate can be fabricated first by laser cutting. Then a layer of metal is sputtered on the glass to form capacitor electrodes. The thickness of the capacitor plate is 1 mm.

The strength of the detection signal, which is an important parameter in the system simulation and interface circuit design, is directly determined by the capacitance transducer. Suppose the disc rotor turns a small angle δ around the Y axis when the Coriolis force is input, as shown in

Select the infinitesimal dA in the disc rotor, where θ is an angle from infinitesimal to X-axis, and r is the distance from the infinitesimal to center dot O. Therefore the capacitance between the infinitesimal dA and the down plate can be expressed as:

Thus:

Like the traditional DTG, the new HG will be an ideal two-degree-of-freedom gyroscope under ideal tuning conditions. When the angular velocity is presented, the disc rotor will precess due to the Coriolis effect. The entire system is consequently unstable and the closed-loop control must be employed. The electrostatic force derived from the capacitive torque of the new HG is used for the closed-loop control. The electrostatic torquer is small in size and light in weight with extremely small heat dissipation effects. Therefore the micro-miniaturization of HG is feasible in practice. The outer capacitance plate is both the capacitance transducer and the electrostatic torquer. When the feedback voltage U-V_{f} is exerted on the outer capacitance plate as shown in

Similarly, the feedback voltage U+V_{f} is exerted on the differential outer capacitance plate, then the consequent electrostatic torquer is

The total electrostatic torquer exerted on the disc rotor is
_{f}_{Uf}_{NUf}_{f}_{f} and torque, as well as the relationship between the bias voltage U and the negative stiffness effect can be determined, which is important to the design of closed-loop control system and negative stiffness dynamic tuning.

Since the the down/upper capacitance plate is extremely thin, it is susceptible to installation error which will affect the accuracy of the capacitance transducer and the capacitance torquer. Suppose the capacitance plate has an initial installed deflection angle δ_{0} along the positive deflection direction of the disc rotor as shown in _{0} + δ, according to

Similarly, when the disc rotor turns an angle δ around the negative direction, the deflection angle changes into δ_{0}-δ:

It can be concluded that the first item in

According to _{0} will result in the tuning failure and feedback instability.

In summary, the installation error of the capacitance plate will result in the asymmetry of scale factors, the static error torquer and the coupling effect between the feedback torquer and the negative stiffness torquer. The precision installation should be achieved to reduce the angle error δ_{0}.

Similarly, the disc rotor module is extremely thin and susceptible to installation errors, which causes the error of the capacitance transducer and the capacitance torquer. Suppose the disc rotor module have an initial installed deflection angle δ_{0}, and the external input of rotation rate is zero, as shown in

The change of the static capacitance is:

It is shown that even if there is no external rotation rate input, the capacitance transducer still has an output error signal. The frequency of the first item of the error output signal is the same as the rotation frequency of the rotor. The error output signal can be reduced by decreasing the deflection angle from the installation error of the disc rotor, δ_{0}. Similarly, when the feedback voltage V_{f} = 0, the electrostatic moment is:

It can be seen that even if the feedback voltage is zero, the electrostatic torquer still has an error output torquer and the frequency of the second item of the error output torquer is twice the rotation frequency of the rotor.

In summary, the installation error of disc rotor module will result in the output signal error in the rotation frequency of the rotor and the output torquer error in twice the rotation frequency of the rotor. The closed-loop control system should be designed to suppress the above errors.

Tuning is one of the important means to improve performance of the HG. According to _{p} is the positive torsional rigidity of the inner /outer torsion spring, K_{n} is the negative torque coefficient of inertia moment.

When ΔK = 0, the new HG will beome an ideal two-degree-freedom gyroscope whose mechanical sensitivity is equal to infinity, therefore, optimal performance can be obtained. The traditional DTG is usually tuned by precision machining technology to repeatedly adjust the positive torsional coefficient of the torsional spring K_{p} and the inertia moment of equilibrium ring I_{e} or change the rotation rate of the electrical motor _{p}. The traditional tuning technique by adjusting the parameters is shown in _{n} is much smaller than the torsional coefficient of the inner and outer torsion spring K_{p} due to the thin thickness (K_{n} < 10^{−3} K_{p}). Therefore, the tuning can not be achieved by the tradition implementation method and need be searched for new approaches.

This paper proposes a new tuning method by using the negative stiffness effect of the feedback moment, as shown in

Substituting _{f}V_{fA} and K_{f}V_{fB} are the feedback moments imposed on the disc rotor along X-axis and Y-axis separately.

Suppose Δ_{Uf}^{2} ≫ _{f}^{2}, then(the derivation can be seen in the

The preload bias voltage U used to tune can be calculated by the structural parameters in

The closed-loop control of HG must be achieved under the ideal tuning conditions. The performance of HG can be affected directly by the design of a closed-loop system. The block diagram of the rebalancing control loop system is shown in _{11}, P_{12}, P_{21}, P_{22}, G_{11}, G_{12}, G_{21}, G_{22} are defined in the _{y}_{x}_{1} are applied to a group of outer capacitance plates. At the same time, another group of high frequency carrier waves ±V_{2} with the frequency ω_{2} are applied to another group of capacitance plates. The two groups of sensed differential capacitances are modulated by the two carrier waves in different frequencies (only one interface model is shown in

Suppose ΔC1 = 2C_{s11} − 2C_{s12}, then the induced voltage of the disc rotor is:
_{CV}_{1} = (_{t}_{b}_{1}/(2_{s}_{11} + 2_{s}_{12} +_{t}_{b}_{f}_{t} and C_{b}.

In order to optimize the design of the rebalance control loop, the open loop and the closed-loop system is simulated by Matlab. The simulation parameters are shown in _{x}

At the same time, the simulation results of the closed-loop system are shown in _{x}

The HG is studied experimentally to verify the principle and the negative stiffness tuning mechanism. In order to simplify the experiment, a traditional hysteresis motor used in the DTG is adopted primarily. A miniature motor based on the micromachining technology will be redesigned to further reduce the volume in the future. The disc rotor module, the down capacitance plate, the motor and other parts are assembled together to form the HG prototype (due to the difficulty in the symmetrical installation of the upper and down plate, only the down plate is installed presently). The upper gasket and down gasket in

In the traditional DTG, the tuning point is estimated by the locus of the tip of the gyroscope. However, the tuning point in the new HG is judged by the open loop gain. According to the simulation results in ^{−6}. Since the disc-rotor is driven to rotate under the atmosphere with a large damping in the disc-rotor (the vacuum package is not realized).

In order to investigate the influence of installation errors, the scale factor for different installation errors are tested experimentally, as shown in

At the tuning voltage point, the closed-loop control is achieved basically by adjusting the loop gain and parameters. The closed-loop scale factor test is carried out, as shown in

In this work, a new dynamically tuned HG is investigated. The structure of the new HG which adopts the capacitance transducer and electrostatic torquer is designed. The crucial silicon disc-rotor module is simulated and fabricated on SOI wafers using a standard microfabrication process. The theoretical mechanism of the capacitance transducer and the electrostatic torquer is deduced and analyzed under ideal conditions. Simultaneously the influence of the installation errors of the capacitance plate and disc rotor module upon the performance of capacitance transducer and electrostatic torquer are investigated. A new tuning mechanism based on negative stiffness rather than the traditional dynamic tuning is proposed. The rebalancing closed-loop control scheme is designed, and the open loop tuning mechanism and the closed-loop system are verified by simulation. By assembling and adjusting the HG parts, the principle prototype is realized. The experiment results validate that the negative stiffness tuning is feasible. The comparison in two generation prototypes with different installation error shows that the non-linearity and the asymmetry of the scale factor are reduced significantly by decreasing the installation error. Finally, the rebalancing control loop is achieved, which proves that the fundamental principle of the HG is feasible. Future work includes further reducing installation error, redesigning the miniaturization motor and packaging HG in vacuum.

The authors gratefully acknowledge Fundamental Research Funds of Southeast University for Innovation Fund for the financial support through contract 3222000501.

The authors declare no conflict of interest.

In order to illuminate the relation between the input and output, the transfer function is derived. According the

So the Laplace transform is
^{2}^{4} + 2^{3} + (^{2} + 2^{2})^{2} + (2^{2} + ^{2}.

In addition, the tuning voltage of the negative stiffness is deduced. According to the

Substituting the above equations into the

The negative torque coefficient of inertia moment K_{n} is
_{z} is the inertia moment of equilibrium ring around Z-axis. R_{3} and R_{4} are the outside diameter and the inner diameter of equilibrium ring. h_{1} is the thickness of the equilibrium ring. ρ is the density of silicon.

The positive torsional coefficient of the inner and outer torsion spring, K_{p} is
_{2} is thickness of the torsional spring , G is shear modulus, 5.2 × 10^{10}. α is relational with h/a.

Suppose Δ_{Uf}

_{2}methodology

The schematic diagram of the HG. (

Schematic diagram of torsional motion of disc rotor module. (

The mode simulation of disc rotor module. (

The prototypes of disc rotor module and capacitor plates. (

Schematic diagram of disc rotor deflection.

Schematic diagram of the installation error of the capacitance plate.

Schematic diagram of the installation error of disc rotor module.

Traditional tuning technique by adjusting the parameters. (

Schematic diagram of the tuning by negative stiffness.

The curve of installation distance d with the tuning voltage U.

Block diagram of the rebalancing control loop.

The open-loop system simulation. (

The closed-loop system simulation. (

HG Prototype. (

Open-loop tuning experiment.

Scale factor comparison test in different installation error. (

The closed-loop scale factor test of the second generation with a small installation error.

The structure parameters.

R_{1} (mm) |
10 | R_{5} (mm) |
4.95 | K_{p} (N·m/rad) |
6.860 × 10^{−4} |

R_{2} (mm) |
7.05 | R_{6} (mm) |
2.5 | I_{e} (kg·m^{2}) |
6.4107 × 10^{−10} |

R_{3} (mm) |
6.95 | h_{1} (um) |
210 | J_{e} (kg·m^{2}) |
2.8928 × 10^{−9} |

R_{4} (mm) |
5.05 | l × w × h_{2} (um) |
1,000 × 48 × 210 | K_{f} (N·m/rad) |
0.0096 |

The first six modes of the disc rotor module.

Frequency (Hz) | 125 | 136 | 989 | 1,071 | 1,995 | 2,333 |

The simulation parameters.

r_{1} (mm) |
10 | 10000 | |

r_{2} (mm) |
8.1 | H(rad·kg·m^{2}/s) |
6.7355 × 10^{−6} |

d (um) | 60 | K_{N}(N·m/rad) |
1.41 × 10^{−7} |

V1(V) | 2.5 | U(V) (Tuning point) | 46.9 |

β | 5 × 10^{−9} |
c | 1 × 10^{−8} |

T_{1} |
0.1 | T_{2} |
0.002 |

T_{3} |
0.01 | T_{4} |
0.001 |

_{x} |
1 | _{y} |
0 |

Scale factor comparison test in different installation error.

The first generation with a large installation error (open-loop) | entire range | 0.0014 | 11.78% | 93.3% |

positive direction | 0.0007 | 2.05% | ||

negative direction | 0.00201 | 7.99% | ||

| ||||

The second generation with a small installation error (open-loop) | entire range | 0.05402 | 0.64% | 1.56% |

positive direction | 0.05405 | 0.62% | ||

negative direction | 0.05321 | 0.53% |