Stochastic Modeling of MEMS Inertial Sensors

Transcription

1 BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 10, No Sofia 010 Stochastic Moeling of MEMS Inertial Sensors Petko Petkov, Tsonyo Slavov Department of Automatics, Technical University of Sofia, 1756 Sofia s: ts Abstract: A etaile methoology that allows the eveloping of stochastic iscretetime moels of MEMS gyroscope an accelerometer noises is presente. The methoology is base on the frequency-omain an time-omain characteristics of the sensors noises an is illustrate for the case of Analog Devices tri-axis Inertial measurement Sensor ADIS It is shown that the gyro an accelerometer noises have similar secon-orer moels which are appropriate for usage in the evelopment of Kalman filters for navigation an control systems. Keywors: MEMS inertial sensors, Stochastic Error moels, Kalman filtering. 1. Introuction The Micro Electro Mechanical Systems (MEMS) inertial sensors have several applications in low-cost navigation an control systems [4, 5, 9]. A common isavantage of these sensors are the significant errors which accompany the corresponing measurements. This necessitates the evelopment of aequate error moels which may be use to achieve sufficient measurements accuracy with the ai of appropriate filtering. The inertial sensor errors consist of eterministic an stochastic parts. The eterministic part inclues constant biases, scale factors, axis nonorthogonality, axis misalignment an so on, which are remove from row measurements by the corresponing calibration techniques. The stochastic part contains ranom errors (noises) which cannot be remove from the measurements an shoul be moele as stochastic processes. 31

2 The MEMS gyroscope noise typically consists of the following terms: Bias instability. This is a stationary stochastic process which may be consiere as a low-orer zero-mean Gauss-Markov process. Angular ranom walk. This is an angular error process which is ue to white noise in angular rate. Rate ranom walk. This is a rate error ue to white noise in angular acceleration. Discretization error. This is an error representing the quantization noise. Similarly, the MEMS accelerometer noise may be represente as a sum of the following terms. Bias instability. Velocity ranom walk. This is a velocity error ue to white noise in acceleration. Acceleration ranom walk. This is an acceleration error ue to white noise in jerk. Discretization error. Several other noise terms are escribe in etail in [6]. Different techniques for builing moels of MEMS sensor noises are presente in [4, 7, 8], to name a few. Usually, they exploit the autocorrelation function of the noise in orer to obtain 1-st orer Gauss-Markov or higher orer Auto-Regressive moels. Note that it is esirable to keep the moel orer as low as possible since the moel is frequently use in the esign of Kalman filter to etermine optimal estimates base on the sensor measurements. The aim of this paper is to present a etaile methoology that allows the evelopment of stochastic iscrete-time moels of MEMS gyro an accelerometer noises. The methoology is base on the frequency-omain an time-omain characteristics of the sensors noises an is illustrate for the case of Analog Devices tri-axis Inertial measurement Sensor ADIS It is shown that the gyroscope an accelerometer noises have similar secon-orer moels which are appropriate for usage in the evelopment of Kalman filters for navigation an control systems. The units use in the paper conform to the units use in [] for comparison purposes.. Allan variance The Allan variance provies a means of ientifying various noise terms in the original ata set [1, 6]. Assume that a quantity θ(t) is measure at iscrete time moments t=kt 0, k=1,,, L. The average value of θ between times t k an t k + τ where τ = mt 0, is given by θ θ k m k Ω ˆ + ( τ ) = k τ where θ = θ kt ). 3 k ( 0

3 The Allan variance is efine as 1 1 (1) σ ( τ ) = ( Ωˆ Ωˆ ) = ( θ θ + θ ) k + m k k + m k + m k τ where is the ensemble average. The Allan variance is estimate as follows: L () m 1 σ ( τ ) = ( θ θ + θ ). k+ m k + m k τ ( L m) k = 1 The Allan variance is relate to the PSD of the measure quantity an is connecte with the parameters of the noise terms escribe in Section 1. This connection is exploite in the next sections to etermine the gyroscope an accelerometer noises moels. 3. Stochastic moels of gyroscopic sensors The stochastic iscrete-time moel of the gyro sensors is erive on the base of the gyro noise measure at rest. For this aim we use L = samples of one of the gyro outputs of ADIS16350 Inertial Measurement Sensor [] measure with frequency f s = 100 Hz uring the perio of s. After removing a small constant bias of eg/s. we foun that the noise mean square value is (to six igits) σ gyro_noise = eg/s. The centere output gyro noise is shown in Fig ω, eg/s t, s Fig. 1. Output gyro noise 33

4 10 y gy 10 1 Power, eg /s K /ω B /ω N Frequency, Hz Fig.. Power spectral ensity of gyro noise Allan variance of gyro noise 10 0 N/τ 1/ σ, eg/s (ln/π) 1/ B K/(τ/3) 1/ (s) τ, s Fig. 3. Allan variance of gyro noise To etermine the ifferent terms of the gyro noise we stuy its frequency omain an time omain characteristics. In the frequency omain we make use of the one sie power spectral ensity (PSD) of the gyro noise, etermine by the function pwelch from MATLAB an shown in Fig.. Taking into account the connection between the slopes of the PSD an the slopes of the spectral ensities of ifferent kins of gyro noises [6], we conclue that there are three noise 34

5 components, namely bias instability, angular ranom walk an rate ranom walk with mean square values B, N an K, respectively. Since the gyro has 14-bit resolution its iscretization error is negligible. To etermine the parameters B, N an K precisely, we use the Allan variance of the gyro noise, compute with overlapping estimates an shown in Fig. 3. The connection between parameters B, N an K an the Allan variance, taken from [6], is shown in the same figure. As a result one obtains the values B = eg/s, N = eg/s 1/, an K = eg/s 1/. The bias instability is usually escribe by the first orer lag (3) T x& ( t) + x( t) = v( t), where x() t is a Gauss-Markov process an vt () is an input white noise. Accoring to Fig. 3 we take T = 10 s. Since we are intereste to etermine a iscrete-time moel of the noise, we iscretize equation (3) to obtain (4) x( k + 1) = a x( k) + b η( k) 1 ( ) ΔT T where a = e = , b Δ = T T e 0 1 T = 0., 1 ( ) τ 5 τ = , Δ = 01 f s an η (k) is a iscrete-time white noise with unknown variance σ. η It is well known that for the time-invariant iscrete-time system x( k + 1) = A x( k) + B η( k) the state covariance matrix P is relate to the input covariance matrix Q by the matrix equation [3] T T (5) P = A PA + B QB. That is why the variances of the iscrete-time stochastic processes x( k ) an η ( k) in respect to the 1-st orer system (4) are relate by (6) Since σ = x B, we have that ( 1 a ) σ = b x σ η. a B σ = 1. η b As a result one fins σ η = Consier now how to etermine the moel of rate ranom walk, which will be enote by rrw. The effect of this noise on the gyro output may be represente by the response of an integrator to a white noise. The corresponing iscrete-time relationship is given by (7) rrw(z) = Kfilt(z)ω(z), where (8) ΔT filt( z ) = z 1 35

6 is the integrator iscrete-time transfer function an ω is a white noise with unknown variance σ. ω The value of σ is foun by approximating the low-frequency part of the gyro ω noise PSD by the PSD of rate ranom walk (7), as shown in Fig. 4. As a result we fin σ = 5. Finally, the angular ranom walk is moele as a white noise arw ω whose variance σ arw is etermine accoring to the expression (9) σ = σ σ σ. arw gyro_noise x yyw This gives σ arw = pp Gyro error PSD Rate Ranom Walk PSD Power, eg /s q y ( g ) Frequency, Hz Fig. 4. Rate ranom walk approximation Power, eg /s Frequency, Hz Fig. 5. Power spectral ensity of the gyro moel noise In this way the gyro noise is represente as (10) gyro_noise = x + arw + rrw, where x, rrw an arw are escribe by (4), (7), an (9), respectively.

7 As it is seen from Fig. 5, the PSD of the moel noise, generate accoring to (10), coincies well with the PSD of the measure noise, shown in Fig.. In Fig. 6 we show the measure gyro noise x means along with the noise x moel generate by the gyro noise moel. The closeness of both signals is estimate by the quantity ( x ( k) x ( k)) means moel k = 1 (11) fit = 1 100%. L ( x ( k)) L k = 1 In the given case one obtains fit = 96% which shows that the signals are quite close. means ω, eg/s T, s Fig. 6. Measure gyro noise an the gyro moel noise 4. Stochastic moels of accelerometers In eveloping a moel of the accelerometer noise we implement the same methoology as in the case of gyroscopes. This is ue to the fact that the PSD an Allan variance of the accelerometer noise have properties similar to the corresponing gyro characteristics. In the given case we use again samples of the accelerometer noise measure at rest. The mean square value of the noise is foun to be σ accellerometer_noise = , g. (Note that the acceleration values are measure in the earth gravitation unit (g)). 37

8 The PSD of the accelerometer noise is shown in Fig. 7 an the Allan variance is shown in Fig. 8. From Allan variance we fin that the accelerometer noise has the following parameters: Bias instability B = 0.35 mg; Velocity ranom walk. N = 0.94 (m/s/ h); Acceleration ranom walk K = 0.06 (m/s / h ); Time-constant T = 80 s p y 10-3 K /ω 10-4 Power, g, B /ω 10-7 N Frequency, Hz Fig. 7. Power spectral ensity of accelerometer noise 10 N/τ 1/ 10 3 σ, g K/(τ/3) 1/ (ln/π) 1/ B (s) τ, s Fig. 8. Allan variance of accelerometer noise As in the case of gyro noise, the bias instability is escribe by equation of the type (4), with coefficients a an b compute for the new value of the time constant T. The acceleration ranom walk is escribe by the equation (7) with the same 38

9 integrator transfer function an white noise mean square value σ ω = 5. The velocity ranom walk is taken as a white noise with variance σ vrw etermine by an equation of the type (9). This gives σ vrw = g. The final moel of the acceleration noise is obtaine by relationship of the type (10) Power spectral ensity of moel noise Power, g Frequency, Hz Fig. 9. Power spectral ensity of the accelerometer moel noise Axeleration, g T, s Fig. 10. Measure accelerometer noise an the accelerometer moel noise 39

10 The power spectral ensity of the acceleration noise generate accoring to the moel escribe is shown in Fig. 9. This PSD is close to the PSD compute from the measurements. In Fig. 10 we show the measure accelerometer noise along with the noise generate by the accelerometer noise moel. Using the expression (11) one obtains fit = 85% which shows that the signals are sufficiently close. Finally, it shoul be note that when the gyro an accelerometer noise moels are use in the esign of a Kalman filter instea of eg/s one shoul express the moel parameters in ra/s an instea of g, the corresponing parameters shoul be expresse in m/s. 5. Conclusions It is shown that the gyroscope an accelerometer noises pertaining to Analog Devices MEMS Inertial Measurement Sensor ADIS16350 have similar seconorer moels which may be etermine with sufficient accuracy by using the frequency-omain an time-omain characteristics of the noises. The error moels are implemente as programs in MATLAB an Simulink an are appropriate for usage in the evelopment of Kalman filters for navigation an control systems. Acknowlegment: This work is supporte by project 10ni048-8, fune by the Scientific Research Sector of the Technical University of Sofia. References 1. A l l a n, D. W. Statistics of Atomic Frequency Stanars. In: Proc. of the IEEE, 54, 1966, Analog Devices. ADIS16350/ADIS16355: High Precision Tri-Axial Inertial Sensor. http: 3. A n e r s o n, B. D. O., J. B. M o o r e. Optimal Filtering. Englewoo Cliffs, NJ, Prentice-Hall, F a r r e l, J. A. Aie Navigation: GPS with High Rate Sensors. New York, McGraw Hill, Grewal, M. S., L. R. Weill, A. P. Anrews. Global Positioning Systems, Inertial Navigation an Integration. n e. Wiley-Interscience, IEEESt , IEEE Stanar Specification Format Guie an Test Proceure for Single- Axis Interferometric Fiber Optic Gyros. IEEE Aerospace an Electronic Systems Society, N a s s a r, S. Improving the Inertial Navigation System (INS) Error Moel for INS an INS/DGPS Applications. PhD Dissertation, Department of Geomatics Engineering, University of Calgary, Alberta, Canaa, P a r k, M., Y. G a o. Error an Performance Analysis of MEMS-Base Inertial Sensors with a Low-Cost GPS Receiver. Sensors, 8, 008, Titterton, D., J.Weston. Strapown Inertial Navigation Technology. n e. The Institutionof Electrical Engineers, UK,