Adaptive Mismatch Compensation for Rate Integrating Vibratory ...

Proceedings of the ASME 2014 Dynamic Systems and Control Conference DSCC2014 October 22-24, 2014, San Antonio, TX, USA

DSCC2014-6053

ADAPTIVE MISMATCH COMPENSATION FOR RATE INTEGRATING VIBRATORY GYROSCOPES WITH IMPROVED CONVERGENCE RATE

Fu Zhang∗, Ehsan Keikha, Behrooz Shahsavari, Roberto Horowitz

ABSTRACT This paper presents an online adaptive algorithm to compensate damping and stiffness frequency mismatches in rate integrating Coriolis Vibratory Gyroscopes (CVGs). The proposed adaptive compensator consists of a least square estimator that estimates the damping and frequency mismatches, and an online compensator that corrects the mismatches. In order to improve the adaptive compensator’s convergence rate, we introduce a calibration phase where we identify relations between the unknown parameters (i.e. mismatches, rotation rate and rotation angle ). Calibration results show that the unknown parameters lie on a hyperplane. When the gyro is in operation, we project parameters estimated from the least square estimator onto the hyperplane. The projection will reduce the degrees of freedom in parameter estimates, thus guaranteeing persistence of excitation and improving convergence rate. Simulation results show that utilization of the projection method will drastically improve convergence rate of the least square estimator and improve gyro performance.

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Department of Mechanical Engineering University of California at Berkeley Berkeley, California 94720

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FIGURE 1. A two dimensional harmonic oscillator model. m is the gyro effective mass, Ω is the external input rotation rate, ω1 and ω2 are the two resonance frequencies respectively along two stiffness axes, τ1 and τ2 are the two time constants respectively along two damping axes, θω is the misalignment between the stiffness axes and readout axes and θτ is the misalignment between the damping axes and readout axes.

INTRODUCTION Coriolis Vibratory Gyroscopes (CVGs) are sensing devices that measure either the rotation rate or rotation angle of the base where they are mounted. CVGs are usually modeled as two-dimensional harmonic oscillators [1]. As shown in Fig.1, the readout axis, stiffness axis and damping axis of a CVG are

usually misaligned due to fabrication imperfections. By projecting the spring and damping forces onto the readout axes, we obtain the following governing equations of the

∗ Address

all correspondence to this author. Email: [email protected]

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Copyright © 2014 by ASME

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readout x and y signals :

˙ + 2 x˙ + Δ( 1 )(x˙ cos 2θτ + y˙ sin 2θτ ) x¨ − 2κΩy˙ − κ Ωy τ τ + (ω 2 − κ 2 Ω2 )x − ωΔω(x cos 2θω + y sin 2θω ) =

fx (1) m

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˙ + 2 y˙ − Δ( 1 )(−x˙ sin 2θτ + y˙ cos 2θτ ) y¨ + 2κΩx˙ + κ Ωx τ τ fy 2 2 2 (2) + (ω − κ Ω )y − ωΔω(x sin 2θω − y cos 2θω ) = m

T = ³ N:dt

where κ is the gain factor, Ω is the input ration rate, fx , fy are the drive forces respectively being applied along x and y axes, ω 2 +ω 2

FIGURE 2.

ω 2 −ω 2

ω 2 = 1 2 2 is the gyro resonance frequency, ωΔω = 1 2 2 is the stiffness frequency mismatch, τ2 = τ11 + τ12 is the gyro damping ratio and Δ( τ1 ) = τ11 − τ12 is the damping mismatch. Term 2κΩy˙ and 2κΩx˙ are called Coriolis accelerations, which produce a precession rate on the gyro’s principal axis of vibration. The Coriolis acceleration-induced precession rate or angle is normally measured. [2] presented an algorithm that directly measures the rotation rate without moving the gyro’s principal axis of vibration. Due to its simplicity and low computational cost, it has been widely used in low-cost commercial applications. However, this algorithm requires precisely calibrating the gyro’s parameters ahead of operation, since the rate measurement precision is extremely sensitive to the calibrated parameters, especially when the gyro is underdamped (i.e. small τ1 ). In addition, Signal to Noise Ratio (SNR) will be degraded dramatically as ω1 deviates presented in [2] does not directly from ω2 . Finally, the scheme measure the rotation angle 0t Ω(τ)dτ. As a consequence, this algorithm barely meets the performance requirements of most applications where rotation angles need to be precisely measured. A more sophisticated algorithm is therefore required to directly measure rotation angles. One popular way of achieving this goal is to control CVGs to vibrate in the way depicted by Fig.2, where the CVG’s principal axis of vibration is allowed to precess. The precession rate is proportional to the rotation rate by the gain factor κ. The control system maintains the CVG’s oscillations at the frequency of ω, with a desired energy level a and zero quadrature motion. Demodulation techniques are then exploited to separate the precession motion from oscillatory mot tion. The demodulated precession angle (i.e. −κ Ω(τ)dτ) 0 makes the external rotation angle (i.e. 0t Ω(τ)dτ) immediately available. κ is calibrated ahead of gyro operation. In order to achieve this control goal, [3] proposed a control scheme based on the method of averaging. The so-called method of averaging control scheme in [3] decomposes the control goal into three subgoals [4]: a phase-locked loop (PLL), an energy control loop and a quadrature control loop. The phaselocked loop locks on to the oscillation frequency ω, demodu-

Desired trajectory of CVGs.

lates the precession angle, the energy level and the quadrature motion from the measured position signals x and y. The demodulated precession angle is proportional to external rotation angle. The demodulated energy level and quadrature are respectively fed into the energy control loop, which attempts to maintain the vibration energy at a desired level, and the quadrature control loop, which is designed to eliminate quadrature motion. Both the energy and the quadrature feedback control loops utilize a basic Proportion-Integration (PI) controller to achieve their respective subgoals. The PI controller takes its controlled variable ( i.e. energy level or quadrature motion) and computes the corresponding control action. Then the two control actions are synthesized into drive forces fx and fy , thus forming a closed loop control system. Because the major components of the method of averaging control scheme are easy to implement, the resulting control algorithm has very low computational cost. As a result, it has been widely used in applications where rotation angles need to be directly measured. However, [5] showed that this algorithm requires the controlled gyro to be well fabricated in order to achieve the required performance. Very small gyro mismatches can cause several fatal problems like residual quadrature motion, precession at a wrong rate, or even failure to precess at all. To eliminate the affection of damping and stiffness frequency mismatches, several approaches have been proposed. The first one is to eliminate the damping and stiffness frequency mismatches during the manufacturing process to a level that does not significantly affect the gyro performance. However, such manufacturing process is rather sophisticated, requiring iterative polishing and extremely fine fabrication. Such a sophisticated process will drastically increase the manufacturing cost. The second approach is to compensate for the mismatch via electrostatic spring softening and trimming during the calibration phase [6]. However, these forms of calibration are time consuming, require human involvement and cannot be performed while the gyro is in operation. The third option is compensating for the mismatch by improving control algorithms. In [7] and [8], adaptive controllers

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REVIEW AND ANALYSIS OF ADAPTIVE MISMATCH COMPENSATOR The three control loops in the method of averaging control scheme are the energy control loop, the quadrature control loop and the phase lock loop. Analysis in [5] showed that the energy control loop and the phase lock loop are barely affected by the mismatch, require no compensation action. However, the mismatch dramatically degrades the performance of the quadrature control loop. In addition, the gyro’s principal axis of vibration fails to precess at the correct rate (i.e.−κΩ) due to the presence of mismatches. Therefore, [5] proposed two independent adaptive mismatch compensators: one is in the quadrature control loop and the other one is in the angular precession loop. Simulation results in [5] showed that the proposed compensation scheme can effectively suppress the quadrature motion and precess the gyro’s principal axis of vibration at the correct rate. However, the scheme presented in [5] does not eliminate the DC bias in angle measurement. In this section, we will discuss in detail the affection of mismatches on the precession of the gyro’s principal axis of vibration and the design of the adaptive compensation scheme presented in [5].

were proposed and derived using the Lyapunov direct method. Simulation results were presented to show its efficacy when the control scheme was implemented in continuous time and in a deterministic environment. However, if the controller is implemented in discrete time with a relatively low sampling rate, its performance will be degraded dramatically. Moreover, the control schemes presented in [7] and [8] need to access velocities, which are not measurable in most practical gyroscopes. On top of these work, [5] proposed a more practical compensation algorithm to compensate the damping and stiffness frequency mismatches. The proposed compensator can be running as an add-on feed-forward compensator to the control scheme based on the method of averaging. The PI controllers in the method of averaging provide the baseline feedback action with sufficient robustness, while the adaptive compensator is dedicated to compensating the damping and stiffness frequency mismatches in order to achieve a sufficient level of performance. The combination of these two schemes can effectively compensate gyro mismatches while stabilizing the system. Simulation results showed that the proposed compensator can drastically suppress gyro quadrature motion and the gyro can precess at the correct rate ( i.e. −κΩ). However, in the absence of an artificially induced precession, the parameter estimates have a low convergence rate and as a consequence, the compensator has a rather long transient response. As a result, the gyro precession angle has a considerable DC bias from the real external rotation angle.

Dynamics of Angular Precession Loop The angular precession dynamics is given by [3]: f qs 1 1 √ θ˙ = −κΩ + Δ sin 2(θ − θτ ) − 2 τ 2ω E

This paper attempts to increase the parameter error convergence rate and to reduce the transient time in the adaptive compensator presented by [5] by introducing an offline calibration phase ahead of gyro operation. During the calibration phase, the adaptive mismatch compensator estimates the damping and stiffness frequency mismatches under a given and known external rotation rate. Iterating this process over different external rotation rates, we can identify the hyperplane, which relates the estimated mismatches with the external rotation rate. When the gyroscope is in operation, the adaptive mismatch compensator estimates all the unknown parameters (i.e. mismatches and rotation rate) using least square method. The estimated parameters are then projected onto the hyperplane that was estimated during the calibration phase. This projection reduces the degrees of freedom (DOF) in parameter estimates, thus improving the convergence rate.

(3)

where θ is the angular precession of the gyro’s principal axis of vibration, as depicted in Fig.2, Ω is the external rotation rate, fqs is the corresponding control force and E is the vibration energy level. In the conventional method of averaging, fqs is set to zero. Under ideal conditions, θ˙ = −κΩ when fqs = 0. As a consequence, the gyro’s principal axis of vibration is allowed to precessfreely at the rate of −κΩ. However, the damp ing mismatch Δ τ1 affects the angular precession through term 1 1 2 Δ τ sin 2(θ − θτ ), where θτ is the angular misalignment between the damping axes and the readout axes. It is worth of mentioning that both θ and E can be measured by demodulating the sensed signals x and y. The discretized dynamics are therefore

This paper is organized as follows: we first review the adaptive compensator proposed in [5] and analyze its convergence rate and performance. Then we introduce an offline calibration algorithm that identifies the relation between mismatches and external rotation rate. Subsequently, we incorporate the results of the offline calibrations into online adaptation algorithms. Finally, a simulation study is presented, which validates the efficacy of the proposed algorithm in significantly improving the parameter error convergence rate and gyro’s transient response.

θ (k + 1) = θ (k) − κΩts + aθ cos 2θ (k) + bθ sin 2θ (k) + fθ (4) where ts is the sampling time, aθ = − 12 Δ τ1 ts sin 2θτ , bθ = 1 1 2 Δ τ ts cos 2θτ are the damping mismatch coefficients to be estimated. The feed-forward compensation action to be designed f is fθ = − 2ωq√s E .

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In our problem, it is easy to see that PE = 3. Thus, the least square estimator will converge when the number of estimated ˆ aˆθ and bˆ θ will parameters are no more than 3. Therefore, the Ω, converge to their real values. However, since the persistence of excitation is no more than the number of estimated parameters, the parameters converge at the rate of κΩ, which is rather small in most applications. As a result, the parameters converge extremely slowly, causing a long transient time. The long transient response will result in large DC bias in the angle measurement. In next section, we will find offline relations between the unknown parameters during the calibration phase. The found relations can be used to reduce the number of parameters to be estimated by the adaptive mismatch compensator, thus improving its convergence rate.

Design of Adaptive Mismatch Compensator in Angular Precession Loop As shown in Eqn (4), the effect of the damping mismatch on the angular precession dynamics is reflected as a sinusoidal disturbance dθ (k) = aθ cos θ (k) + bθ sin θ (k), where the damping mismatch coefficients are usually unknown. We can however use the least square estimation method to estimate the mismatch coefficients aθ and bθ . Let Θ = [Ω, aθ , bθ ]T be the unknown parameter vector to be T ˆ ˆ estimated, Θ(k) = Ω(k), aˆθ (k), bˆ θ (k) be the parameter estimate at step k and Ψ(k) := [−κts , cos 2θ (k − 1), sin 2θ (k − 1)]T be the known regressor vector. The adaptive mismatch compensator presented in [5] is shown in Algorithm 1.

Algorithm 1 Mismatch Estimation and Compensation 1: while 1 do ˆ 2: θˆ (k + 1) ← θ (k) − κ Ω(k)t s 3: θ (k + 1) ← Demod(x, y) 4: eo (k + 1) ← θ (k + 1) − θˆ (k + 1) 5: Ψ(k + 1) ← [−κts , cos 2θ (k), sin 2θ (k)]T ˆ + 1) ← RLSE(Θ(k), ˆ Θ(k eo (k + 1), Ψ(k + 1)) 6: 7: fθ (k + 1) ← −aˆθ (k + 1) cos 2θ (k + 1) − bˆ θ (k + 1) sin 2θ (k + 1) 8: k ← k+1 9: end while

OFFLINE CALIBRATION PHASE During the calibration phase, we want to identify relations between the unknown parameters aθ , bθ and Ω. This can be achieved by calibrating the gyro on a rotation table with adjustable rotation rate, using Algorithm 2.

Algorithm 2 Mismatch Calibration Require: Calibration Table RateTable 1: while i < length(RateTable) do 2: Set the rate of the rotation table Ω ← RateTable[i] 3: Run Algorithm 1 using known Ω. 4: Record (Ω, aˆθ , bˆ θ ) once converged 5: i ← i+1 6: end while

where the function Demod demodulates the precession angle from measured signals x, y and the function RLSE is a standard recursive least square estimation routine. Convergence Analysis of The Adaptive Mismatch Compensator By simple derivations, it can be shown that the adaptive compensation algorithm described in previous subsection yields

˜ − 1)T Ψ(k) eo (k) = Θ(k

Performing Algorithm 2, we obtain the offline relations between aθ , bθ and Ω shown in Fig.3 and Fig.4. These offline relations are not a coincidence. Rather, they are a consequence of discrete time discretization of the gyro dynamics equations of motion. A full analysis of this effect is beyond the scope of this paper. Fitting the measured data using first order polynomials yields

(5)

˜ − 1) = Θ − Θ(k ˆ − 1) is the parameter estimation error, where Θ(k which follows the same form as the A-priori error in recursive least square estimation problems. Therefore, we can analyze its convergence rate by checking the rank of its associated persistence of excitation matrix, which is defined as

aθ = a1 Ω + b1 bθ = a2 Ω + b2

(7)

A·Θ = b

(9)

(8)

Or equivalently PE = rank

∑Nk=0 Ψ(k)Ψ(k) N→∞ N lim

T

(6)

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Adaptive Mismatch Compensator With Hard Projection The adaptive mismatch compensator in [5] uses a standard least square method to estimate the unknown parameter vector Θ. The analysis from previous section showed that this standard least square method converges very slowly in our problem. However, since the unknown parameter vector lies on the hyperplane defined by Eqn (9), we can project the parameter estimate vector ˆ Θ(k) onto this hyperplane, as described below. Definition: Projecting a vector y onto a hyperplane defined by {x|Ax = b} means solving the following optimization problem

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1 xo = arg min x − y22 x 2 s.t. Ax − b = 0

FIGURE 3. Converged aˆθ (k) versus rotation rate Ω. Simulation setup is shown in Appendix A.

where AA∗ is assumed to be invertible. Applying the Karush-Kuhn-Tucker (KKT) conditions to the above optimization problem yields

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where K = I − A∗ (AA∗ )−1 A and c = A∗ (AA∗ )−1 b. K and c can be computed during the calibration phase, in order to decrease the online computation burden. Since K is a 3 by 3 matrix while c is a 3 by 1 vector, the projection only uses very few computations.

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FIGURE 4. Converged bˆ θ (k) versus rotation rate Ω. Simulation setup is shown in Appendix A.

Algorithm 3 Mismatch Estimation and Compensation With Hard Projection Require: K and c 1: while 1 do ˆ θˆ (k + 1) ← θ (k) − κ Ω(k)t 2: s 3: θ (k + 1) ← Demod(x, y) 4: eo (k + 1) ← θ (k + 1) − θˆ (k + 1) 5: Ψ(k + 1) ← [−κts , cos 2θ (k), sin 2θ (k)]T ˆ ˆ o (k + 1) ← RLSE(Θ(k), 6: eo (k + 1), Ψ(k + 1)) Θ 7: fθ (k + 1) ← −aˆθ (k + 1) cos 2θ (k + 1) − bˆ θ (k + 1) sin 2θ (k + 1) ˆ + 1) ← K · Θ ˆ o (k + 1) + c Θ(k 8: 9: k ← k+1 10: end while

a1 −1 0 −b1 T where Θ = [Ω, aθ , bθ ] ; A = ;b = ; Eqn (9) a2 0 −1 −b2 defines a linear constraint on the unknown parameter vector Θ. This constraint can later be used in the online adaptive compensator to improve its convergence rate.

ADAPTIVE MISMATCH COMPENSATOR WITH IMPROVED CONVERGENCE RATE In this section, we will use the linear constraint that was obtained during the calibration phase on the unknown parameter vector Θ to improve the adaptive mismatch compensator’s convergence rate. Two types of projection methods will be proposed: hard projection and soft projection. The hard projection method directly projects the parameter estimate from the least square estimator onto the hyperplane defined by the linear constraint. The soft projection method provides a mechanism to trade off the offline calibrations and the online estimations. Both the hard projection and the soft projection are capable of improving the adaptive mismatch compensator’s convergence rate.

The resulting adaptive mismatch compensator with projection is shown in Algorithm 3. Since a projection operator is nonexpansive, hooking it onto a least square estimator does not alter the convergence properties of the least square estimator. However, since the action of the projection is to reduce the degrees of

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freedom (DOF) in the least square estimator, it does improve the parameter estimate convergence rate. It is worth of mentioning that, when the hyperplane where the unknown parameter vector lies is precisely known, the effect of the above projection method is to reduce overparameterization. It is possible to incorporate the hyperplane constraint directly into the RLS problem formulations by reparameterizing the parameter vector and the regressor vector at a lower dimension. However, when the hyperplane is not known accurately due to calibration errors, the two-step RLS/projection can be easily extended to a soft projection method, as discussed in next section.

When the offline calibration is completely accurate, then we set γ → +∞. xo = lim (I + γA∗ A)−1 (y + γA∗ b) γ→+∞

which is essentially the hard projection solution because when γ = +∞, in order to prevent the objective function from being infinity, r(Ax − b) = 0, as a result, Ax − b = 0 has to hold. When the offline calibration is reasonable but not completely accurate, which is the most usual case in practice. We set γ ∈ (0, +∞). The more accurate the offline calibration is, the larger γ we use. To save computations, let K = (I + γA∗ A)−1 ; c = γ · (I + γA∗ A)−1 A∗ b, which can be computed offline during the calibration phase. Then

Adaptive Mismatch Compensator With Soft Projection One potential risk of using the hard projection method is that the calibrated hyperplane may not be accurately known due to calibration errors. In this case, brutally projecting the parameter estimate vector onto an inaccurate hyperplane helps the convergence rate but may introduce estimation error, if the true parameter vector Θ is not exactly on the hyperplane, the parameter ˆ estimate vector Θ(k) will never converge to Θ, but its projection on the hyperplane. This will introduce steady errors in parameter estimates and therefore degrade the gyro performance. This risk can be mitigated by modifying the projection operator by introducing the constraint as a penalty additive term in the objective function with a user defined weight. We will call this modification a soft projection. Definition: Softly projecting a vector y onto a hyperplane defined by {x|Ax = b} means solving the following optimization problem

xo = K y + c

(14)

The resulting adaptive mismatch compensator with soft projection is described by Algorithm.4.

Algorithm 4 Mismatch Estimation and Compensation With Soft Projection Require: K and c 1: while 1 do ˆ θˆ (k + 1) ← θ (k) − κ Ω(k)t 2: s 3: θ (k + 1) ← Demod(x, y) 4: eo (k + 1) ← θ (k + 1) − θˆ (k + 1) 5: Ψ(k + 1) ← [−κts , cos 2θ (k), sin 2θ (k)]T ˆ ˆ o (k + 1) ← RLSE(Θ(k), 6: eo (k + 1), Ψ(k + 1)) Θ 7: fθ (k + 1) ← −aˆθ (k + 1) cos 2θ (k + 1) − bˆ θ (k + 1) sin 2θ (k + 1) ˆ + 1) ← K · Θ ˆ o (k + 1) + c Θ(k 8: 9: k ← k+1 10: end while

1 xo = arg min x − y22 + γ · r(Ax − b) x 2 where AA∗ is assumed to be invertible, r(·) is a non-negative convex function and γ is an adjustable constant that weights the online estimation and offline calibration. The soft projection is also known as a proximity operator, denoted as xo = proxr (y). This is an unconstrained convex optimization problem. In our problem, a simple choice for r(·) is r(x) = 12 x22 , which is convex, non-negative and differentiable. Setting the differentiation of the objective function to zero yields xo = (I + γA∗ A)−1 (y + γA∗ b)

(13)

Since a soft projection operator is non-expansive. as is the case of the hard projection operator, the soft projection will not alter the converge properties of the RLS method but it will improve the converge rate. γ can also be time varying. At the beginning, we set a large γ to improve the convergence rate. As the parameter estimate vecˆ tor Θ(k) approaches its true value Θ, we decrease γ to increase the significance of the online adaptations. As γ goes to zero, the ˆ parameter estimate Θ(k) will converge to Θ. This however requires more computations since K , c have to be updated in real time as γ varies.

(12)

By adjusting γ, we place different degrees of faith on the accuracy of hyperplane identified during the calibration phase. When the offline calibration is too inaccurate, we set γ = 0. Then xo = y, which means the estimated parameter directly comes from the least square estimation.

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FIGURE 5. Simulated aˆθ response, with the hard projection on (dashdot red) and off (dashed blue)

FIGURE 7. Simulated response of rate estimation error, with the hard projection on (dash-dot red) and off (dashed blue)

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FIGURE 8. Simulated precession angle response, with the hard projection on (dash-dot red) and off (dashed blue), when compared with the ground true rotation angle (dotted black)

SIMULATION RESULTS In order to verify their efficacy, the proposed adaptive mismatch compensator with hard/soft projection are tested on a simulated gyro model with parameters provided in Appendix A.

see that the the compensator with projection has maximal angle measurement error of 0.005 degree.

Efficacy Validation of Hard Projection Fig.5, Fig.6 and Fig.7 show the comparison of parameter (mismatches aθ , bθ and external rotation rate Ω ) estimation, of adaptive mismatch compensator with and without the hard projection. It can be seen that both converge to the same values. However, with projection, the compensator converges much faster. Fig.8 shows the precession angle response. It can be seen that due to its low convergence rate, the adaptive mismatch compensator with no projection has angle measurement error of 45 degrees. By using projection in the compensator, its convergence rate is drastically improved. As a result, the precession angle can precisely follow the ground true rotation angle. A finer plot of the angle measurement error is shown in Fig.9, where we can

Efficacy Validation of Soft Projection In order to verify its efficacy, the adaptive mismatch compensator with the proposed soft projection method is tested on the simulated gyroscope as well. Fig.10, Fig.11 and Fig.12 show responses of parameter estimations with different penalty weights (i.e. γ). We can see that the estimated parameters converge to the same values in both cases. By placing less faith (i.e. smaller γ) on offline calibrations, the compensator has lower convergence rate but gains more robustness to calibration errors. Lower convergence rate causes more bias in angle measurement, as shown in Fig.13. But both cases are better (i.e. smaller angle bias) than the one with no projection.

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FIGURE 9. Simulated angle measurement error of the adaptive mismatch compensator with projection

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Simulated response of rate measurement error.

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FIGURE 13. FIGURE 10.

known angular rate. The identified hyperplane provides extra constraints on the unknown parameter vector and therefore reduces their degrees of freedom when they are later estimated using a RLS algorithm. Projection method was utilized to incorporate the constraints with the RLS algorithm. Simulation results showed that the projection method drastically improved the convergence rate of the compensator and eliminated the DC bias in angle measurement. As an alternative to the hard projection, we proposed a soft projection RLS algorithm by penalizing the constraint in the least square objective function. We believe this a good way to combine offline calibrations with online estimations. Moreover, the penalty weight (i.e.γ) provides a mechanism to adjust how much influence of the offline calibration should have on the online estimation. Simulation results showed that the soft projection also improves the convergence rate and angle measurement precision. Future work can be done in several aspects including implementing the proposed algorithm on real gyroscopes, analyzing the performance of the projection method when the rotation rate Ω is time varying and generalizing the projection method to nonlinear constraints.

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Simulated bˆ θ response.

CONCLUSION In this paper, we proposed a projection methodology to improve the convergence rate of the adaptive mismatch compensator proposed in [5]. The first step is identifying a hyperplane where the unknown parameter vector lies during a calibration phase in which the gyro is placed on a rate table rotating with

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ACKNOWLEDGMENT This research is supported by the DARPA MRIG program, through a subcontract from Honeywell Aerospace Advanced Technology. We would like to thank Dr. Burgess R. Johnson and all our other collaborators from Honeywell for their gyroscope modeling data and continuous and helpful feedback on control design development.

REFERENCES [1] Lynch, D. D., 1998. “Coriolis Vibratory Gyro”. Symposium Gyro Technology, Stuttgart, Germany. [2] Acar, C., and Shkel, A., 2008. MEMS Vibratory Gyroscopes: Structural Approaches to Improve Robustness. MEMS Reference Shelf. Springer, Chap. 6, pp. 143–185. [3] Lynch, D. D., 1995. “Vibratory Gyro Analysis by The Method of Averaging”. 2nd St. Petersburg Int. Conf. on Gyroscopic Technology and Navigation, St. Petersburg, Russia, pp. 26–34. [4] Lynch, D. D., and Matthews, A., 1996. “Dual-Mode Hemisphericcal Resonator Gyro Operation Characteristics”. 3rd Saint Petersburg International Conference on Integrated Navigation Systems, Saint Petersburg, Russia, pp. 37–44. [5] Zhang, F., Keikha, E., Shahsavari, B., and Horowitz, R., 2014. “Adaptive Mismatch Compensation for Vibratory Gyroscopes”. The 1st IEEE International Symposium on Inertial Sensors and Systems, pp. 67–70. [6] Cao, H., Li, H., Ni, Y., and Pan, H., 2014. “Electrostatic force correction for the imperfections of the MEMS gyroscope structure”. The 1st IEEE International Symposium on Inertial Sensors and Systems, pp. 71–74. [7] Park, S., and Horowitz, R., 2001. “Adaptive control for Zaxis MEMS gyroscopes”. Vol. 2 of Proceedings of the 2001, American Control Conference, IEEE, pp. 1223–1228. [8] Leland, R. P., 2006. “Adaptive control of a MEMS gyroscope using Lyapunov methods”. Vol. 14, Control Systems Technology, IEEE Transactions on, pp. 278–283.

Appendix A: SIMULATION SETUP The Gyro model used through this paper has below parameters: resonance frequency ω = 10, 000Hz; sampling rate fs = 98, 000Hz; frequency mismatch Δω = 5Hz; time constant τ = 2sec; damping mismatch Δ( τ1 ) = 0.001sec−1 ; gain factor κ = 1; input rotation rate Ω = 1Hz

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