Modeling of Acceleration Influence on Hemispherical Resonator Gyro Forcing System

Acceleration adds negative effect to Hemispherical Resonator Gyro (HRG) output; therefore, it is important to model the influence and thenmake necessary compensations, accordingly. Based on the elastic thin-shell theory under the Kirchhoff-Love assumption, the acceleration influence on HRG forcing system is modeled and then schemes for incentive are suggested. Firstly, the dynamic model of resonator is introduced.Then, inertial load and electrostatic force are calculated to obtain the deformation of resonator. At last, schemes for pickoff incentive are proposed to weaken the effect of acceleration on HRG forcer.The simulation results illustrate that acceleration has negative effects on the exciting confidents of forcers and the proposed scheme can eliminate the acceleration influence on forcing system.


Introduction
The hemispherical resonator gyro (HRG) is a solid state gyroscope whose sensing property is based on the vibration structure.It has the features of high accuracy, long life span, inherent high reliability, natural radiation hardness, and no parts that can wear out, which makes it very suitable for applications in long-time working state such as in space vehicles [1][2][3][4].Particularly, over 125 spacecraft missions whose Space Inertial Reference Unit (SIRU) is based on HRGs have been launched successfully by NASA, Department of Defense (DOD) and so on.Moreover, it is reported that HRGs have already achieved 18 million h of continuous operation with 100 percent mission success in space application.
In the practical use of HRG, acceleration will have inevitable effects on the HRG, which also can increase the gyro drift.As a kind of high precision gyro, the drift caused by acceleration is not allowable to be ignored.So the research on the drift caused by the accelerations has important practical application value.
The HRG contains three primary functional components: the hemispherical resonator, the forcer, and the pickoff.According to the elastic thin-shell theory under the Kirchhoff-Love assumption [5], under the action of acceleration, the resonator will produce a large deformation, causing the change of relative position between resonator and the exciting electrode, which inevitably impacts on internal control system of HRG.Zhou et al. [6] provided some acceleration consequence analysis of HRG based on experiments.Zhbanov [7][8][9] discussed the effect of movability of the resonator center excited by an electrostatic field.Watson [10] studied on the driver geometry of skewed pick-off.Forcer is the "actuator" of the internal control system of HRG.The resonator's deformation will increase the errors in the excitation system, which is an important part of gyro's drift of HRG, so the influence of resonator deformation caused by acceleration on forcing system will be discussed in this paper.

Dynamic Model of Resonator.
Acquiring from the elastic thin-shell theory, under the Kirchhoff-Love assumption, shell satisfies the force balance equation, the geometric equation of deformation and displacement, and the elastic equation.Synthesizing the three equations, the differential equations of the shell's middle surface can be set in the domain of shell, under the given boundary conditions; the distribution of displacement field or the force field can be obtained by solving the differential equations within the domain.
In the process of modeling, the structure of an umbrellatype resonator whose radius is  and thickness is 2ℎ can be simplified as hemispherical and be modeled; then the hemispherical latitude line  and longitude line  constitute the principal coordinate system, as shown in Figure 1; the differential equation to establish the displacement components is shown as follows: where  is defined as a spherical point  that moves to   for the spherical deformation, deriving the displacement vector U, and the projections on the local frame  1  2 n are , V, , respectively, in which  is the normal displacement, shown in Figure 1;  is the radius of middle surface of spherical shell;  = 2ℎ/(1− 2 );  1 = ℎ/(1+);  = 2ℎ 3 /3(1− 2 );  1 = 2ℎ 3 /3(1 + );  is Young's modulus;  is Poisson ratio; , , and  are the projections of external load (the distributed force) on  1  2 n.
It can be seen from ( 1) that, under the function of additive loads, the equation meets the superposition principle.The external loads discussed in the paper include inertial load and the electrostatic force of exciter.
where a ℎ is the acceleration of resonator coordinate system relative to inertial coordinate system;   is the rotation angular rate of the local frame  1  2 n relative to the inertial coordinate system.Because  1  2 n is static relative to the resonator coordinate system,   is also the rotation angular rate of resonator coordinate system relative to inertial coordinate system which is described as  ℎ .
The types of components of the vectors in (2) in the local frame are expressed as where    ,    , and    are the acceleration components of material point of harmonic point relative to inertial coordinate system in the local frame, respectively;  ℎ ℎ ,  ℎ ℎ , and  ℎ ℎ are components of the acceleration of the resonator coordinate system relative to inertial coordinate system in the resonator coordinate system, respectively. ℎ ℎ ,  ℎ ℎ , and  ℎ ℎ are the components of angular rate of the resonator coordinate system relative to inertial coordinate system in the resonator coordinate system, respectively.
Because (1) satisfies the superposition principle, the inertial load can be divided into two conditions in this paper.
The first condition is the consideration of the function of external low-frequency acceleration a ℎ as well as the deformation of resonator  2 U/ 2 |  caused by a ℎ , namely, The second condition is the inconsideration of a ℎ in the state of second-order vibration of resonator; meanwhile (2) can be simplified as 2.3.Electrostatic Force.The structure of polar plate of actuator on external base of HRG is spherical sector, shown as Figure 2.   ,   are the latitude ranges of the spherical sector and   ,   are the longitude ranges of the sector.The frequency of the control alternating current voltage is the half of the second order resonance frequency of resonator.The expression of the AC voltage is: Because the capacitance clearance is in micron level, which is small relative to the resonator radius, so the spherical curvature can be ignored, the exciter can be treated as platetype capacitor, and the boundary effect is ignored at the same time.The expression of electric field force (surface force) acting to the resonator can be expressed as where Ignoring the constant of the electric field force, (7) can be formed as The effecting direction of electric field force is the normal direction of the outside surface of resonator.When resonator is under the condition of vibration, because the vibration contains the tangential motion, the normal direction of the outside surface of resonator varies constantly, leading to the existence of tangential component.While the amplitude of resonator is small compared to the radius, the tangential component can be negligible and the normal component of electric field force can be approximately taken as (9).

Multielectrode Incentive Features
3.1.Actuator Configuration Structure.The external base of HRG uniformly distributes 16 exciter, and the interval is 22.5 ∘ , presented as  1 ∼ 16 as shown in Figure 3.The longitude and latitude ranges of electrode are shown in Table 1.Typically, the exciter in position of wave amplitude of resonator actuator can be used in amplitude vibration and amplitude stability and electrode of node can be used for rate feedback.According to the symmetry of the second-order resonance state of resonator, 16 exciters can construct the following three kinds of structure forms.
(1)  1 at 0 ∘ of electrode axis is taken as exciter for amplitude stability, and  3 at 45 ∘ of electrode axis is taken as exciter for force feedback.As shown in Figure 3, this is the most basic form of testing structure, defined as Scheme a.
(2)  1 and  5 are paralleled as the exciter for amplitude stability. 2 and  6 are paralleled as exciter for force feedback, shown as Figure 3, which is defined as Scheme b.
( shown as Figure 3(c) and defined as Scheme c.

Exciter Characteristics.
As Figure 2 showed, in the condition of existence of electrode excitements, , , and  in components of external load in (1) can be expressed as Substituting it into (1), the vibration equation can be complied under the effect of single exciter of resonator.While the form of the analytical solution of the state equation of second-order harmonic vibration of resonator is where (), () are undetermined variables related to time and  2 ,  2 , and  2 are Rayleigh Functions, the form is as follows: When the dynamic equation and the analytical solution form are known, a method of solving the equation's approximate solution is to use Bubnov-Galerkin method.Eventually the undetermined variables () and () in (11) can be obtained that they meet the second-order differential equation.Substituting damping coefficient  2 into the equation, it can be formed as where When the excitement mode is taken as Scheme a, When the excitement mode is taken as Scheme b, When the excitement mode is taken as Scheme c, where Then, For further discussion of the exciting features, Scheme a is taken as an example to explain, ignoring the structure error of HRG.Substitute the data in Table 1 into the equation: Seen from the equation above, appropriate   ,   can be found to make  = 0; when  = 0 in (19), (19) can be simplified as When the vibration mode angle of resonator is locked at zero, the above two equations represent the rules for amplitude control and control link of the vibration mode angle.It can be seen that the two control loops are no coupling, but actually because of factors such as errors of form and position and deformation, the two loops are coupling, and the coupling relationship can be described by (19).In the process of gyro design, make sure to avoid coupling of two control loops to the greatest extent; namely, the aim  0 =  45 = 0 comes true and  45 ,  0 are constants.

Deformation Caused by Acceleration
As shown in Figure 2, take any point  within the scope of exciter area.Line  intersects with the middle surface after the deformation of resonator at point , where  =   ; suppose  =   , and the expression of capacitance clearance is In order to solve   in the above equation, the surface equation after deformation should be known.As shown in Figure 2, the material point  in the middle surface of resonator moves to point   after deformation, the displacement vector   corresponds to   and  corresponds to , and the deformation vector has the relationship as follows: And the component form in the resonator coordinate system where , V, and  include the deformation caused by acceleration  1 , V where   is the amplitude;  2 is the frequency of secondorder harmonic vibration of resonator;  is the time;  is vibration mode angle;  2 ,  2 , and  2 are Rayleigh Functions, the form of which is shown as follows: The form of deformation equation under the effect of resonator is where   is the amplitude of the deformation of resonator under the effect of acceleration;  is the orientation of the acceleration effect;  1 ,  1 , and  1 are Rayleigh Functions, the form of which is shown as follows: ) . (30)

Simulation Analysis
Under the effect of acceleration, a relative displacement occurs between the resonator and the excitation electrode in external base, which leads to the changes of the integral domain in (18) and the initial electrode clearance , making the change of exciting coefficients  0 ,  0 ,  45 , and  45 .Known from (19), when  0 ,  45 are not zeros, amplitude stability loop and rate loop couple, and the zero drift of gyro occurs, while the variation of the  0 ,  45 will cause changes in the gyro scale factor.Because of the complexity of the transcendental equation and double integral analytical solution, the method of numerical calculation is used to research the exciting coefficient errors, and the process of establishment of the simulation system is as follows.
(1) Deformation caused by acceleration:   is changing from 0.01 m to 0.1 m, the interval is 0.01 m, and  is changing at the range of 0 ∘ ∼360 ∘ ; then the interval is 10 ∘ .The deformations  1 , V 1 , and  1 caused by acceleration in (29) can be calculated.(2) Calculation of vibration deformation: given the amplitude  V of second-order resonance state of resonator and the vibration mode angle , using (27),  2 , V 2 , and  2 can be calculated.
(4) Calculation of capacitance clearance and integral range: (i) Quasi-Newton method is used to calculate the numerical solution of   at each grid point in (26) and corresponding range of latitude and longitude of exciting electrode after deformation; in (26) , V, and  contain deformations  1 , V 1 , and  1 caused by acceleration; (ii) the capacitance clearance at each point can be calculated by ( 23).
Analyzing the effect of acceleration on incentive system is the analysis of the relationship between   ,  in (26) and the exciting coefficients  0 ,  0 ,  45 , and  45 .The simulation system parameter is set as Table 2 showed; implement the simulation process and calculate the exciting coefficients and compare the solutions to the ideal value of the nodeformation circumstance; then the exiting coefficient errors can be acquired in Schemes a, b, and c, shown as Figures 4, 5, and 6.
It can be known from the harmonic analysis of the 10 curves shown in Figure 4 (only give the results because of more data) that the two times with the second-order harmonic coefficient of errors is equal to the amplitude of the exciting coefficient error in Scheme b, which proves that Scheme b can compensate for the first-order of the exciting coefficient error in Scheme a.In Scheme b, the constant component is amplified two times, but comparing with the first-order harmonic amplitude, the constant component is small, so it greatly weakens the exciting coefficient errors caused by acceleration in Scheme b.
It can be seen from Figure 6 that Scheme c compensates for the second-order harmonic component of Scheme b, and the constant component is amplified.While the scale of the
1,  1 and  2 , V 2 ,  2 when resonator steps into second-order state.The intersection between Line  and the middle surface of resonator meets the equation as follows: where θ, φ are the latitude and the longitude of the base polar plate in vibration pickup, θ ∈ [ V ,  V ], φ ∈ [ V ,  V ] are known variables, and   , , and  are unknown variables.The deformations  1 , V 1 , and  1 in (26) are shown in (29) and the deformations  2 , V 2 , and  2 in (29) are shown as follows: