Random Error Analysis of Inertial Sensors output Based on Allan Variance Shaochen Li1, a, Xiaojing Du2,b and Junyi Zhai3,c

Transcription

1 International Conerence on Civil, Transportation and Environment (ICCTE 06) Random Error Analysis o Inertial Sensors output Based on Allan Variance Shaochen Li, a, Xiaojing Du, and Junyi Zhai3,c School o Aerospace Engineering. Beijing Institute o Technology. Beijing China @qq.com Keywords: MEMS sensors; Allan variance; Mathematical simulations; actual test Astract:With characteristics o small volume, low cost and high reliaility, MEMS inertial sensors are developing into one o the mainstream sensors in inertial navigation system, and its output precision has a great eect on the overall system precision. In this paper, the random error o inertial sensors output are mainly discussed and identiied y Allan variance. To analysis the random error in the output o MEMS sensors, which contriute to improve the system precision, Mathematical simulations and actual tests are carried out. Introduction With the development o micro-electronic technology, the MEMS-IMU (Micro-Electronic Mechanical Inertial Measurement Unit) has made great changes in navigation System. In particularly, the emergence o micro-accelerometer and gyroscope enale the attitude and heading reerence system to e smaller and cheaper. To some extent, the output precision o inertial sensors determines the overall navigation system precision. Aected y production process and external environment, ractional error exists in the output o MEMS sensors, which can e roughly divided into deterministic error and random error. Deterministic error can e otained and compensated y mathematical modeling while random error is relatively diicult to identiy. In this paper, simulations and tests are presented to identiy random error in the output o inertial sensors. For the caliration o MIMU (micro inertial measurement unit) random error, quantity o researches have een done in drit testing and modeling such as estalishing ARMA model, adopting neural network and using wavelet transorm method. But these kinds o error models usually estalish the equations whose orders are too high to do real-time estimation at low cost []. Allan variance analysis is a method ased on time domain analysis. It is convenient to do detailed characterization and identiication o error inluence on system noise statistical characteristics. Random error analysis o MEMS The typical random error in output o MEMS inertial sensors mainly includes gauss white noise, random walk process, licker noise and the exponentially-correlated process []. Gauss white noise. Gauss white noise with zero mean widely exists in the sensor output, its signal amplitudes to gauss distriution with mean zero ( µ = 0) and variance, while its power spectral density is uniorm. The auto-correlation unction is () q (u ) = cov[ (t ), (t + u )] = δ (u ) In Eq., δ (u ) is Dirac unction. According to Wiener-Khinchine law, power spectral density (Power Spectral, Density, PSD) S ( ) has the ollowing relationship with variance : S ( ) = 06. The authors - Pulished y Atlantis Press () 345

2 Random Walk noise. Random walk noise ξ () t can e considered as a result o integration y zero mean gauss white noise () t whose autocorrelation unction is unction and variance are showed as ollows: q(, tu) = cov[ ξ(), t ξ( t+ u)] (3) = min( tt, + u) ξ t q t t var[ ()] = (,0) = (4) q ( µ ) δ ( µ ) =.The autocorrelation Random walk noise variance changes over time, so its power spectrum density unction can e seen as gauss white noise iltered y H( ) = jπ : ( ) = H( ) S ( ) (5) Knowing S ( ) =, ( ) = (6) ( π ) Flicker noise. Flicker noise ξ () t cannot e descried with speciic dierential equations, ut it can also e otained y gauss white noise iltered y H( ) = jπ. Assume the white noise autocorrelation unction is: q ( ) = δ( u ) (7) The PSD o licker noise is: ( ) = (8) π Exponentially-correlated noise. Exponentially correlated noise ξ e() t is a constant and zero mean random process, its autocorrelation unction is: u q ( u) = cov[ ξ (), t ξ ( t+ u)] = e (9) e e e The process can e descried in the time domain as: ξ& () t = ξ () t + () t (0) e e e In ormula(0), () t e e is zero-mean Gaussian White noise with autocorrelation unction o q ( u) = δ( u), Same to random walk process, exponentially-correlated noise can also e e e descried in the requency domain with Gaussian white noise iltered y H( ) = ( + j π ).The PSD o exponentially-correlated noise is: S ξe ( ) = e + ( π ) () Allan variance analysis Allan variance analysis which can identiy diversity error is an analytical method ased on time domain. Its principle and mathematical expressions are as ollows: Assuming that y k is the average value o a signal yt () within a period time o, then y k can e descried as: tk + yk = ytdt () t k () tk+ = tk +, k =,, K Allan variance o yt () can e descried as: 346

3 M ( yk+ yk) Ay() = lim (3) M M k= 0 With sampling time intervalt s, the signal yt () which lasts or a period time o T can e denoted as yk ( ) k =,,... NN ( = TT s ). Then yk ( ) can e divided into KK ( = N / n) clusters with n sequential data points. The average time or each cluster is = nts. The average value o cluster k + can e ound according to: n yk+ ( ) = ynk+ i (4) n i= The Allan variance estimation is: K ˆ Ay( ) = ( yk+ yk) (5) ( K ) k= According to reerence [], the error etween ˆ Ay ( ) and δ ( Av) can e descried as: δ ( Av) = (6) ( Nn ) Based on Eq.6, a conclusion can e drawn that the more data points and less numer o points per data set, the higher the quality o Allan variance estimates. Simulation and test Mathematical simulation. Between Allan variance o random signal and the corresponding PSD, a conversion relationship exists, which can e expressed as ollow: 4 sin ( π ) ( ) 4 S ( ) Ay = 0 y d (7) ( π ) Sustituting Eq., Eq.6, Eq.8, Eq. into Eq.7, then Gauss white noise: A ( ) = (8) Random Walk noise: A ξ ( ) = (9) 3 Flick noise: ln A ξ ( ) = (0) π Exponentially-correlated noise: () According to reerence [3], the random error o inertial sensors can e considered to e composed o gauss white noise and exponentially-correlated noise. The Allan variance o Gauss white noise and exponentially-correlated noise superposition is ( ) = ( ) + ( ) () Aξe A Aξe 347

4 Figure Gauss white noise and Exponentially-correlated noise superposition ( ), variance o Gaussian white noise, is inversely proportional to the time, and the variance A o exponentially-correlated noise, reaching its maximum at =.893, while =.893. Simulation shows that = 8.9s with 0s Aξ e Aξ e ( ) increases initially and then decreases with the growth o ( ) also reaches its maximum at Aξ e ( ) reaches its maximum and = at = and = The error o e e is within 0.8%. The result shows that Allan variance can eectively distinguish the random error o inertial sensors. Actual test. In this chapter, Allan variance is applied to analysis the actual output data o MIMU. MTI, a miniature inertial measurement unit with three-axis gyroscope and accelerometer as well as 3D magnetometers, is chosen to acquire the test data or 6 hours and a total o 5.76e6 amount. Take the x axis or example, Allan variance o rate gyroscope and acceleration in log () log plot are showed as ollows: A e Figure The x axis accelerometer output and Allan variance analysis Figure 3 The x axis gyroscope output and Allan variance analysis The results show that liners with -0.5 slope in log A () log plot can e otained when mean time is small, suggesting that the output o accelerometer and gyroscope contain Gaussian white noise. With Allan variance δ A( ) at =, can e acquired. The random error parameters o gyroscope and accelerometer are shown in Tale and Tale. Tale The error mode o gyroscope Gyros ARW Gauss Markov [ rad s ] [rad s ] [ s ] X Y Z

5 Tale The error mode o accelerometer Accelerometers VRW Gauss Markov [ ms s ] [ ms ] [ s ] X Y Z According to reerence [4], error model o the chosen MTI is as ollows: % = + +, % = + + (3) i i v, v are the white noises o gyroscope and accelerometer, and its intensity can e otained y tale., are the drit, including constant ias and irst-order mark-o noise. = o + e, = o + e (4) According to Eq.0, e and e can e expressed as: & e = e + (5) & = + e e, are the mark-o process correlated time o Gyroscope and accelerometer, which can e acquired rom tale as well as the driving noise = = Conclusions T T (6),. As the output precision o sensors makes great dierence in inertial system, in this paper, random error in the output o MEMS sensors is mainly identiied. Based on the characteristic analysis o the random error, mathematical simulation is proceeded to otain the typical random error curve and Allan variance-average time logarithm curve. With the attitude and heading reerence system MTI, actual test is carried out to otain the output random error parameters. The simulation and test results shows random error o inertial sensors output can e well identiied with Allan variance analysis, which provides the asis or urther research. Reerence [] Guangchun Li, He Kun Peng, jian-hui zeng, etc. The MEMS inertial device error modeling and compensation method review [J]. Journal o navigation and control, 009, 8 (). [] Karthik Naarayanan. Perormance Aanlysis o Attitude Determinnation Algorithms or Low Cost Attitude Heading Reerence Systems. ACM International Collegiate Programming Contest 00. [3] Zhao Sihao, Liu Mingquan, Feng Zhenming. MEMS inertial device error coeicient o Allan variance analysis method [J]. Chinese science, physics, mechanics, astronomy, 00 (5) : [4] Xiaoyang Lan. Small spacecrat attitude determine technology research[d]. Beijing institute o technology, 05. [5] Kastelan D R. Design and implementation o a GPS-aided inertial navigation system or a helicopter UAV[M]