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This paper presents an adaptive control algorithm for realizing a vibratory angle measuring gyroscope so that rotation angle can be directly measured without integration of angular rate, thus eliminating the accumulation of numerical integration errors. The proposed control algorithm uses a trajectory following approach and the reference trajectory is generated by an ideal angle measuring gyroscope driven by the estimate of angular rate and the auxiliary sinusoidal input so that the persistent excitation condition is satisfied. The developed control algorithm can compensate for all types of fabrication imperfections such as coupled damping and stiffness, and mismatched stiffness and un-equal damping term in an on-line fashion. The simulation results show the feasibility and effectiveness of the developed control algorithm that is capable of directly measuring rotation angle without the integration of angular rate.

MEMS vibratory gyroscopes are typically designed to measure the angular rate [

MEMS vibratory gyroscopes can conceptually operate in the rotation angle measurement mode, where integration is done mechanically rather than electronically [

If a MEMS angle measuring gyroscope can be developed, it will open up new market opportunities and applications in the area of small, low-cost and medium-performance inertial devices. For example, the angle measuring gyroscope can directly measure the yaw angle which is not affected by external interference such as magnetic disturbances. The angle measuring gyroscope can also be combined with regular rate gyroscopes, accelerometers and/or magnetometers to improve accuracy and robustness of attitude measurements. Moreover, three-axis angle measuring gyroscope constitutes an attitude reference system (ARS) which can greatly reduce the size and cost of current ARS.

Although a MEMS angle measuring gyroscope is a promising sensor, it is not developed yet, and even in the literature, very few control algorithms have been reported for realizing angle measuring gyroscopes [

Along with the efforts of robust structure design to the fabrication imperfections, active compensations are required to null out the remaining imperfections and environment variations during the operation. Friedland and Hutton [

In this paper, we present a new adaptive control algorithm for realizing angle measuring gyroscopes. Compared to the previous works [

The dynamics of an ideal vibratory angle measuring gyroscope is defined as follows:
_{0} in both axes, which is oscillating on a rotating gyro frame with a constant angular rate Ω_{z}_{1} axis of the inertial frame, then the solution of

The rotation angle (_{z}t

A physical angle measuring gyroscope can be implemented by the 2-DOF mass-spring-damper system whose proof mass is suspended by spring flexure anchored at the gyro frame. Considering fabrication imperfections and damping, a realistic model of a z-axis gyroscope is described as follows:
_{xx}_{yy}_{x}_{y}_{xy}_{xy}_{x}_{y}_{1} and _{2} axis of the gyro frame, respectively. The coupled damping and frequency terms, called quadrature errors, comes mainly from asymmetries in suspension structure and misalignment of sensors and actuators. Therefore, the control problem of angle measuring gyroscope is to determine control laws for _{x}_{y}

In this section, we present an adaptive controller to realize an angle measuring gyroscope. The basic idea of the adaptive control approach is to treat both the angular rate and the fabrication imperfections as unknown gyroscope parameters, and these are estimated using a parameter adaptation algorithm (PAA). In adaptive control problems, the persistent excitation condition is an important factor to estimate the unknown parameters correctly. To solve this problem, a trajectory following approach is used. The reference trajectory that the gyroscope must follow should be generated such that the persistent excitation condition is met.

A reference trajectory may be generated by an ideal angle measuring gyroscope (1). However, the dynamics of an ideal gyroscope is not sufficiently exciting for parameter identification and moreover the angular rate Ω_{z}_{z}_{xm}_{ym}_{xm}_{ym}_{0} sin(_{f}t_{0} ≠ _{f}

Now, let us rewrite the dynamics of non-ideal Gyroscope (3) and the reference Gyroscope Model (4) as follows:

Then the control problem is formalized as follows: given the

Defining the trajectory error as _{p}_{m}_{0} will be subsequently defined, and

Considering the following Lyapunov function candidate:
_{(·)} are positive constants and

If _{0} is chosen to be:

With control laws (7), (11) and PAA (13), the trajectory error _{p}_{p}_{p}

For convenience, the regressor _{m}_{m}

The error equations are:

Since the time derivative of the Lyapunov function is negative semi-definite, all elements of the Lyapunov function are bounded. Taking one more derivative of

Because _{p}_{p}_{p}_{p}

Since _{p}_{p}_{p}^{T}_{m}_{m}^{T}_{m}_{m}

Again, the integral of _{p}_{p}_{p}_{p}_{p}^{T}_{m}_{m}

With control laws (7), (11) and PAA (13), if the Gyroscope (5) is controlled to follow the reference Model (6) and _{0} ≠ _{f}

By persistent excitation of _{m}_{m}_{1}, _{1}and _{0} such that:

Since the Lyapunov derivative is negative semi definite, it can be argued that

Since ^{T}_{m}_{m}

Because the trajectory _{m}_{0} ≠ _{f}_{1}, _{1} > 0 and _{0} can always be found such that (15) is satisfied. According to the Theorems 1 and 2, trajectories _{m}_{m}_{m}

Note that control laws (7), (11) and PAA (13) are driven by reference model signals _{m}_{m}_{p}_{v}_{1}, _{2}}. To complete the modification, the velocity term

With the proposed observer, the velocity measurement based adaptive control structure is not modified, and the analytic results of stability and persistent excitation condition are preserved, which can be proved in a similar way as the one found in reference [

The response of _{m}

If the line of oscillation of the proof mass with amplitude _{1} axis of the inertial frame, then the steady-state response of the

The rotation angle (_{z}t_{0}_{0} is initial procession angle and LPF denotes a low-pass filter.

The bandwidth of the proposed controlled gyroscope is defined by the cutoff frequency of the low-pass filter used in the rotation angle calculation process. Considering typical bandwidth of angular rate is a few hundred Hz, the cutoff frequency of the low-pass filter can be chosen. The driving frequency _{f}_{0} of the ideal angle measuring gyroscope such that the difference of two frequencies is bigger than the cutoff frequency of the low-pass filter for successful separating rotation angle signal from the response to the auxiliary sinusoidal input. However, the driving frequency should not be too far apart from the reference frequency and also the magnitude of the auxiliary sinusoidal input _{0} should not be chosen too small such that the magnitude

A simulation study is conducted to evaluate the proposed control scheme using the design data of the MEMS gyroscope model shown in [_{0} = 4.17^{−26} m^{2}. The auxiliary inputs are designed to be f_{xm} = f_{vm} = 343.24 sin(2π × 2.9 × 10^{3}t)m/s^{2}. The gyroscope parameters in the model and the numerical values for the controller in the simulations are summarized in _{0} (s).

According to the plots, all estimation errors quickly converge to zero, and the estimate of angular rate also converges to its true value. Therefore dynamics of controlled gyroscope follows that of the ideal reference gyroscope. In these simulations, it is assumed that the gyroscope experiences step input angular rate of 100 deg/s at 0.6 s after the gyroscope is turned on. _{m}

The estimates of angle response to step and sinusoidal input angular rates of 100 deg/s are shown in

According to the plots, the error bound of the angle estimation is 0.7 deg. Considering about 0.05 s delay introduced in the angle estimate due to the low pass filter in demodulation process, if the estimated angle is shifted 0.05 s and compared to the input angle, the angle estimation accuracy with the error bound of 0.2 deg can be achieved under the presence of noise. These simulation studies show the feasibility and effectiveness of the developed algorithm that is capable of directly measuring rotation angle without integration of angular rate.

This paper presents an adaptive control algorithm for realizing vibratory angle measuring gyroscope so that rotation angle can be directly measured without integration of angular rate, thus eliminating the accumulation of numerical integration errors. The proposed control algorithm uses a trajectory following approach. The reference trajectory that the gyroscope must follow is generated by an ideal angle measuring gyroscope driven by the estimate of angular rate and auxiliary sinusoidal input. The role of auxiliary sinusoidal input is to increase the complexity of the internal dynamics of the gyroscope so that its response is persistently exciting. In such a way, the developed control algorithm can identify and compensate for all fabrication imperfections and environmental variations in an on-line fashion. The stability of proposed adaptive controlled gyroscope is rigorously proved. An adaptive observer is also designed to avoid direct measurement of the velocity of the proof mass, since normally velocity sensing circuitry produces a larger noise than position sensing. The simulation studies show that the proposed control algorithm realizes angle measuring gyroscope operation successfully.

This work was supported by the Korea Research Foundation Grant (KRF-2009-013-D00022).

Proposed adaptive control scheme.

Velocity of proof mass.

Time response of parameter estimation.

Trajectory of proof mass in x-y plane.

Time response of rotation angle estimate to the 100 deg/s step input.

Time response of rotation angle estimate to the 100 deg/s sinusoidal input at 10 Hz.

Non-dimensional values of the control parameters.

Value | |
---|---|

_{0} = 1, _{x}_{y}_{xy}_{xx}^{−4}, _{yy}^{−4},_{xy}^{−4} | |

_{1} = 1, γ_{R} = 1/5,_{D} = 1/100, γ_{Ω} = 1/50 | |