A Gyroscope Control System for Unknown Proof Mass and Interface Circuit Errors

Transcription

1 Aerican Control Conference Marriott Waterfront, Baltiore, MD, USA June 3-July, ThB8. A Gyroscope Control Syste for Unknown Proof Mass and Interface Circuit Errors Chien-Yu Chi and Tsung-Lin Chen Abstract This paper presents a novel approach that can copensate errors resulting fro the iperfections of echanical structures and interface circuits for MEMS gyroscope systes. The echanical structure errors discussed in this paper contribute to unknown proof ass, spring constants, daping coefficients, and existence of cross-axis resilient/daping forces. The interface circuit errors include: isatch of differential capacitors, parasitic capacitance, offset voltage of operation aplifiers, and circuit noise. Different fro ost of existing researches, the proposed ethod has the following features: () the echanical structure iperfections and interface circuit errors are copensated siultaneously using control techniques; () the ass of the proof ass can be unknown. This approach is verified on two types of gyroscope designs by nuerical siulations. Siulation results indicate that, under those iperfections, the proposed ethod can obtain angular rate within 8 illiseconds. I. INTRODUCTION A MEMS vibratory gyroscope can be briefly grouped into three subsystes, as shown in Fig.. The Mechanical structure ainly consists of a proof ass suspended in a rigid frae. This ass converts the Coriolis force (angular rate inforation) into displaceents along designated directions. These displaceents are then converted into another physical quantity for the ease of sensor easureents (eg, capacitance variations, resistance variation, etc.). The Interface circuits converts these easureents into voltage outputs for the ease of signal processing. The Control algorith processes these signals for the feedback control of vibratory ass and for calculating the angular rate. The iperfections existed in each subsyste would significantly degrade the sensing accuracy of angular rates. Fig.. Block diagra of a vibratory MEMS gyroscope syste The authors are with Departent of Mechanical Engineering, National Chiao Tung University, Hsin-Chu 356, Taiwan (R.O.C.) chienyu.e93g@nctu.edu.tw (C.-Y. Chi); tsunglin@ail.nctu.edu.tw (T.-L. Chen) The echanical structure iperfections ainly coe fro the structure designs and fabrication errors. It is noral to have % % diension variations and residual fil stress fro MEMS fabrication process such as lithography, etching, fil deposition, and etc [][]. All these errors cause the fabricated gyroscope dynaics (ass, spring constants, daping coefficients) deviated fro their designated values. Even worse, they induce cross-axis resilient force and crossaxis daping force, which lead to the serious quadrature error in gyroscope systes []. Solutions to echanical structure iperfections include: advanced icroachining processes, coplicated echanical structure designs, posticroachining [3]-[5], and etc. In a word, these iperfections are often iniized by expensive tooling processes. The reactance sensing schee is attractive to MEMS devices because neither additional processing steps nor aterials are required for the fabrication process. When adapting this sensing technique, charge aplifiers are often used as an interface circuit to convert capacitance variations into voltage signals [6]. The iperfections in this circuit includes: offset voltage of operational aplifiers, parasitic capacitances, circuit noises, bias abiguity, and etc. [7]. Several ethods have been proposed to deal with those probles including: auto-zeroing, chopper stabilizations, switched capacitor, correlated double sapling, dynaic eleent atching, and etc. [6][7]. Those approaches are effective and have been widely used. However, they are coplicated in circuit designs. In literatures, we have not found a paper using control algoriths to copensate iperfections for charge aplifiers, except our preliinary work [8]. In ost existing MEMS systes, the iperfections fro echanical structures and interface circuits were iniized physically and individually. The disadvantage of that is costly. Since late 99s, any feedback control ethods have been proposed to copensate the effect resulting fro the iperfections in MEMS gyroscopes. These approaches iprove the gyroscope perforance without expensive icroachining processes and intensive calibration work [8]- [], and thus could be proising for the ass production of MEMS gyroscopes. Unfortunately, in those reports, the interface circuits were all assued to be ideal and the ass of the proof ass ust be known. In this paper, we proposed a control algorith that can copensate the effect fro iperfections of both echanical structures and interface circuits. Different fro existing approaches and our previous work [8], the ass of the proof ass does not need to known beforehand, which is expect to further reduce the calibration work for MEMS vibratory //$6. AACC 343

2 gyroscopes. The proposed ethod is developed fro the state estiation techniques. The estiation properties are discussed in detail. Lastly, two types of frequently used gyroscope designs are used to deonstrate the feasibility of the proposed ethod. A. Gyroscope Dynaics II. SYSTEM MODELING Two types of echanical structures are frequently used in MEMS vibratory gyroscope designs. In Fig. (a), a single proof ass is suspended in a rigid frae by four flexures. Due to its syetrical design, the proof ass can ove in two axes. This gyroscope design is referred to as the singleass gyroscope in this paper. As shown in any existing papers [], [9]-[], the dynaic equations of this gyroscope can be written as: ẍ + d xx ẋ + d xy ẏ + k xx x + k xy y = u x + Ω z ẏ ÿ + d xy ẋ + d yy ẏ + k xy x + k yy y = u y Ω z ẋ() where is the ass of the proof ass; d xx, d yy, k xx, k yy are daping coefficients and spring constants along two principal axes; Ω z is the angular rate to be easured; d xy and k xy are the cross-axis daping coefficient and spring constant; u x and u y are the control input along x and y axis, respectively. This dynaic equation is often noralized by the ass to obtain the following equation. ẍ + d xx ẋ + d xy ẏ + k xx + k xy = u x + Ω zẏ ÿ + d xy ẋ + d yy ẏ + k xy + k yy = u y Ω zẋ() On the other hand, Fig. (b) shows a echanically decoupled gyroscope design that the ass is constrained to ove along x-axis, while the ass is free to ove along both axes. This gyroscope design is referred to as the decoupled gyroscope in this paper. The dynaics of this gyroscope can be obtained by cobining the results shown in [], which are: ( + )ẍ + d xx ẋ + k xx x + k xy y = u x + Ω z ẏ ÿ + d yy ẏ + k yx x + k yy y = u y Ω z ẋ (3) Siilarly, the dynaics in (3) can be noralized by the ass: ẍ + d xx ẋ + k xx x + k xy y = u x + Ω z ẏ ÿ + d yy ẏ + k yx x + k yy y = u y Ω z ẋ (4) where = + and =. It is noted that two cross-axis spring constants k xy, k yx are chosen to be different due to the asyetric echanical structure. The iperfections of echanical structures ainly contribute to the existence of the d xy and k xy and uncertain values of all spring constants, daping coefficients and the ass. Therefore, including angular rates, there are eight Fig.. Two types of vibratory gyroscope designs. (a)a single-ass gyroscope design; (b)a decoupled gyroscope design. unknown paraeters in a single-ass gyroscope design, while there are nine unknown paraeters in a decoupled gyroscope design. B. Capacitive Position Sensing The capacitive position sensing is one of the ost popular ethods to easure the position of the proof ass because it can be fairly accurate and easily ipleented with MEMS technologies. Depending on the echanis of its varying capacitance design, the capacitive position sensing can be divided into the cob drive schee and parallel plate schee. Here, the cob drive schee is used as an exaple to illustrate the concept. As shown in Fig. 3, C o and C o are the capacitances when the proof ass is at its noinal position; C and C are the capacitance variations induced by otions of the proof ass. Due to fabrication iperfections, C o ay not equal to C o, neither do C and C. For exaple, if the noinal position is shifted by a distance d fro the neutral position of the structure, the above capacitances can be calculated as (5). C o = Nǫ W Z (x + d) C o = Nǫ W Z (x d) C = C = Nǫ W Z x (5) where N is the nuber of cob fingers; ǫ is the perittivity; W and Z are the height and gap of cob fingers; x is the overlapped length between fingers when the proof ass is at its neutral position. 344

3 Fig. 3. Cob drive design for the lateral position sensing. (a)noinal position; (b)capacitance variation due to the displaceent (x). C. Interface Circuits Charge aplifiers are often used to convert capacitance variations into voltage signals. And ostly because of the offset voltage of the operational aplifier and soe serious low frequency noise like flicker noise, odulation techniques are often used to work with the aplifier [7]. Here, the charge aplifier with odulation techniques is analyzed for possible circuit errors. A differential capacitor pair fro the capacitive position sensing design and a charge aplifier with odulation techniques are shown in Fig. 4. C P is the parasitic capacitance; V os is the offset voltage of the operation aplifier and V n is the noise fro sensing circuits which ainly consists of theral noise, flicker noise, Brownian noise, and etc. charge sensing) and no input bias current for the operation aplifier (which is pretty uch true for aplifiers ade of MOS technology), the output voltage V o can be obtained as: V o = t i C dτ + V + V n = ( C + C + C o C o ) V s +ν (C o + C o + C C + C P + ) V +V n (6) where V n is the noise at around the odulation frequency and its standard deviation is saller than that of V n ; ν can be very sall depending on the design of the low pass filter shown in Fig. 4. If the operational aplifier functions properly in this feedback configuration, V equals to V os due to the virtual ground effect. By cobining (6) and (5), the output voltage of the interface circuit is: V o = V s Nǫ W Z x + α + νβ + V n ( α = Nǫ W ) Z d Vs C ( f β = Nǫ W ) Z x Vos + C P + (7) Since the values of fabrication error d, parasitic capacitance C P, and bias voltage V os are unknown, the values of α and β are unknowns. Furtherore, α is an unknown constant; β can ether be an unknown constant or a drift depending on V os is drifting or not. Fro (7), the output voltages of the sensing circuit is linearly proportional to the displaceent of the proof ass. However, it is noisy and deviated by biases or drifts. To adapted to the proposed control ethod, we define a new variable Φ: Φ = α + νβ (8) III. STATE OBSERVER DESIGN AND FEEDBACK CONTROL To copensate the effect fro both echanical structure iperfections and circuit errors using state estiation techniques, the syste dynaics and interface circuit are odeled together. To save the length of this paper, the following derivation is done for the single-ass gyroscope design only. Thus, by cobining () and (7), a gyroscope syste can be described by the following equations: Ẋ = f(x) + BU + N s Z = HX + N (9) Fig. 4. A scheatic plot of the charge aplifier with odulation techniques. When capacitance variation is induced, charges are squeezed out of differential capacitors and flow into R f,, and C p. Assuing that the current i R is negligible (R f is often chosen to be large so that it would not destruct the where Z is the output vector of a syste; X is the state vector of a syste; and are two bias signals (drifts) at the circuit outputs; N odels the noise, while N s odels the Brownian otions of the echanical structures; X, f(x), B, U, and H are shown at the top of the next page. It is noted that, except the scale factor in circuit output, every syste paraeters are assued to be unknown. Besides, 345

4 [ X = [ x ẋ y ẏ Ω z k xx k yy k xy d xx d yy d xy ], B = ẋ kxx kxy dxx ẋ dxy ẏ + Ωzẏ f(x)= H = ẏ kxy k yy d xy ( ) V N x xǫ Wx Z x ẋ d yy ẏ Ωzẋ. 4 ( ) V N y yǫ Wy Z y [ ] [ ] [ ] ux nx, U=, N=, u y n y [ ] 8. ] T, three new states (,, Ω z ) and syste paraeters are all assued to be constant for now. A. State Observer Construction With the syste equations shown in (9), a state observer can be constructed as the following: ˆX = f( ˆX) + BU + LH(X ˆX) Ẑ = H ˆX () where L is the observer gain and can be chosen fro various nonlinear observer algoriths. In this paper, the extended Kalan filtering (EKF) [3] is used for its effectiveness in noise reduction. When applying EKF to this syste, the continuous-tie dynaics in (9) are converted into discrete-tie difference equations ( f) first. And, the so-called prediction equations in the EKF can be calculated by the following steps: ˆX k+ = f(k, ˆX k ) + BU k P k+ = A k P k A T k () A k = f k / X X= ˆXk where the subscript k denotes the values obtained at the k th sapling tie; P k ( E[( ˆX k X k )( ˆX k X k ) T ]) is the state covariance atrix. The ion equations in the EKF are: L k+ = P k+ HT ( HP k+ HT + R k+ ) P k+ = (I L k+ H)P k+ ˆX k+ = ˆX k+ + L k+ () ( ( Z k+ Ẑ k, ˆX )) k+ where R is the covariance atrix of the easureent noise. In the above derivations, all syste paraeters are assued to be constant. However, a functional gyroscope needs to easure tie-varying angular rates. Also, the bias voltage of aplifiers could be drifting. To cope with these probles, the fading eory technique [3] can be adopted to work with the EKF, as shown in [8]. B. Feedback Control for Gyroscope Syste Since all syste paraeters and dynaics are d in real tie, the d states can be used to ipleent feedback controls for gyroscopes. Aong various controller designs, we choose the one which keeps the total energy transferred between two axes the sae. This control ethod is chosen because it enforces the feedback syste to operate at the resonant frequency of the original syste. Thus, the control input can be less. To ipleent this ethod, the control input is designed as follows: [ ] ˆdxx ˆx + ˆdxy ŷ + U = x (3) ˆd xy ˆx + ˆdyy ŷ + y where x and y can be chosen as proper bounded signals to avoid interfering the syste stability. Once the values of d states and paraeters converge to their values, the trajectory of proof ass can be described by the following equations: ẍ + k xx + k xy = Ω zẏ + x ÿ + k xy + k yy = Ω zẋ + y IV. OBSERVABILITY ANALYSIS (4) Since the proposed ethod uses state estiation techniques to syste states and unknown paraeters, the feasibility of the estiation can be exained by the rank of the observability atrix. The observability atrix of a nonlinear syste is obtained by the following: W o [ ] Z Ż Z (5) X For this feedback control gyroscope syste, the observability atrix (W o ) is calculated and has the following forat: [ ] [Wss ] W o = 6 6 [] 6 8 (6) [] 8 6 [W kd ] After tedious derivations, the above W ss and W kd atrices can be greatly siplified to the following: W ss = (7) k xx k xy k xy k yy The W kd is shown at the top of the next page. For W ss, as long as k xx k yy k xy, its rank is six and thus the associated six states are globally observable. Siilarly, it 346

5 W kd = ÿ ẋ ẏ ẍ ÿ ẍ ẏ ẋ ÿ ẍ y (3) ẍ ÿ (3) (3) x (3) ÿ ẍ (3) (3) y (4) (3) (3) (4) (4) (3) x (4) (3) (3) (4) (4) (3) y (5) (4) (4) (5) (5) (4) x (5) (4) (4) (5) (5) (4) can be shown that the rank of W kd is eight if the oscillation of the proof ass contain ore than one frequency and a proper choice of. V. SIMULATIONS In the following siulations, the proof ass is assued to be actuated around.5 µ, 3 khz. The circuit outputs are assued to be DC-biased and containated by white noise. Those bias values are 5 V for x-axis and V for y-axis. The standard deviation of the noise is.5 V. The paraeters of gyroscope odel are listed in Table I. And, the initial guess of those paraeters are % to % off fro their values. Two control signals x and y are both chosen to be sin(π 3t). The sapling rate of the control algorith is MHz. TABLE I SYSTEM PARAMETERS AND THEIR VALUES OF A SINGLE-MASS Paraeters Ω z k xx k yy k xy d xx d yy d xy VIBRATORY GYROSCOPE Values.8 7 kg rad/sec N/ 95.9 N/.779 N/.8 6 N s/.8 6 N s/ N s/ 5 V V Fig. 5 shows the circuit output voltages of the gyroscope syste, which correspond to the position of the proof ass along x-axis and y-axis. They are dc-biased and containated by white noise. Using the proposed ethod, the d proof ass position and velocity are shown in Fig. 6. According to siulation results, the d values quickly converge to their values. Figure 7 shows the d values of unknown syste paraeters, which includes two bias signals, seven syste paraeters and one angular rate. The d values converge to their values within 8 illi-seconds. The other siulations are shown for the decoupled gyroscope design, which paraeters are listed in Table II. According to the siulation results shown in Fig. 8 and Fig. 9. All the d state values converge to their values within 5 illi-seconds. VI. CONCLUSIONS In this paper, the effect of echanical structure iperfections were accounted as unknown proof ass, spring Vox (V) Voy (V) Sensing circuit outputs Fig. 5. Circuits outputs for the x-axis(upper) and y-axis(lower) position sensing. The resulting biases for x-axis and y-axis are 5 V and V; and the sensor noise at both axes are white with standard deviation of.5 V. x position (µ) x velocity (µ/s) y position (µ) y velocity (µ/s) x 3 x x x 3 x x 3 Fig. 6. Estiation of position and velocity of proof ass along x-axis and y-axis. The d state values and real state values are alost identical. The deonstration is for a traditional single-ass gyroscope design. constants, and daping coefficients of a dynaic syste. The effect of interface circuit iperfections were accounted as unknown signal drifts and easureent noises. The control input is designed to consist of state feedback and one bounded signal. By doing so, the proof ass trajectory is regulated to oscillate at ore than one frequency. And, these unknown paraeters, along with syste dynaics (position and velocity), can be ly d using state estiation techniques. Besides, the estiation properties can be exained by observability atrices. 347

6 Ω z kxy 4 3 kxx x 6 dxx kyy x 6 dyy 3 Ω z kxy.8 kxx kyx kyy dxx x 6 dxy 8 x x x 6 dyy x 7.5 x Fig. 7. Estiation of other unknown syste paraeters (states). The d state values converge to their values within. seconds. The deonstration is for a traditional single-ass gyroscope design. Fig. 9. Estiation of other unknown syste paraeters (states). The d state values converge to their values within. seconds. The deonstration is for a echanically decoupled gyroscope design. TABLE II SYSTEM PARAMETERS AND THEIR VALUES FOR MECHANICALLY x position (µ) y position (µ) x velocity (µ/s) y velocity (µ/s).5.5 DECOUPLED VIBRATORY GYROSCOPE Paraeters Ω z k xx k yy k xy k yx d xx d yy Values.4 7 kg.9 7 kg rad/sec 8.7 N/ N/.87 N/. N/.3 6 N s/.4 6 N s/ 8 V V x 3 x x x 3 x x 3 Fig. 8. Estiation of position and velocity of proof ass along x-axis and y-axis. The d state values and real state values are alost identical. The deonstration is for a echanically decoupled gyroscope design. In siulations, two types of gyroscope designs, singleass and decoupled, are both utilized to verify the concept. In the single-ass design, there are unknown paraeters and four syste states, while there are unknown paraeters and four syste states in the decoupled design. The extended Kalan filter are utilized to those unknown values, including angular rates, in real tie. The conversion tie of angular rates estiation is less than 8 illiseconds. REFERENCES [] A. Shkel, R. T. Howe, and R. Horowitz, Modeling and Siulation of Microachined Gyroscopes in the Presence of Iperfection, Internatinal Conference On Modelling and Siulation of Microsystes, pp , 999. [] M.S. Weinberg and A. Kourepenis, Error sources in in-plane silicon tuning-fork MEMS gyroscopes, Journal of Microelectroechanical Systes, Vol. 5, No. 3, pp , 6. [3] H. Kawai, K.-I. Atsuchi, M. Taura, and K. Ohwada, High-resolution icrogyroscope using vibratory otion adjustent technology, Sensors and Actuators A, 9, pp ,. [4] S.E. Alper and T. Akin, A syetric surface icroachined gyroscope with decoupled oscillation odes, Sensors and Actuators A, 97-98, pp ,. [5] S. Gunthner, M. Egretzberger, A. Kugi, K. Kapser, B. Hartann, U. Schid, and H. Seidel, Copensation of parasitic effects for a silicon tuning fork gyroscope, IEEE Sensors Journal, Vol. 6, No. 3, pp , 6. [6] N. Wongkoet, Position sensing for electrostatic icropositioners, University of California at Berkeley, Ph.D. thesis, 998. [7] C.C. Enz and G.C. Tees, Circuit techniques for reducing the effects of op-ap iperfections: auto-zeroing, correlated double sapling, and chopper stabilization, Proceedings of the IEEE, Vol. 84, Issue., pp , 996. [8] C.-Y. Chi and T.-L. Chen, Copensation of iperfections for Vibratory Gyroscope Systes Using State Observers, Sensors & Transducers Journal (ISSN ), Vol. 6, Special Issue on Modern Sensing Technologies II, pp. 8-45, 9. [9] S. Park and R. Horowitz, Adaptive Control for Z-Axis MEMS Gyroscopes, Proceedings of the Aerican Control Conference (ACC), Arlington, VA, USA, Jun. 5th-7th, pp. 3-8,. [] R. P. Leland, Adaptive control of a MEMS gyroscope using Lyapunov ethods, IEEE Transactions on Control Systes Technology, vol. 4, no., pp , 6. [] L. Dong, Q. Zheng, and Z. Gao, On control syste design for the conventional ode of operation of vibrational gyroscopes, IEEE Sensors Journal, vol. 8, no., pp , 8. [] C. Acar and A. Shkel, Nonresonant Microachined Gyroscopes With Structural Mode-Decoupling, IEEE Sensors Journal, Vol. 3, No. 4, pp , 3. [3] Y. Bar-Shalo, X.R. Li, and T. Kirubarajan, Estiation with applications to tracking and navigation, John Wiley & Sons,