International Global Navigation Satellite Systems Society IGNSS Symposium 2009 Holiday Inn, Surfers Paradise, Qld, Australia 1 - 3 December, 2009 The effect of the temperature-correlated error of inertial MEMS sensors on the integration of GPS/INS Kedong Wang School of Astronautics, Beihang University, Beijing 100191, China Phone: 86-10-8233-9586, Fax: 86-10-8233-8798, Email: firstname.lastname@example.org Yong Li School of Surveying & SIS, UNSW, Sydney 2052 Phone: 61-2-9385-4173, Fax: 61-2-9313-7493, Email: email@example.com Chris Rizos School of Surveying & SIS, UNSW, Sydney 2052 Phone: 61-2-9385-4205, Fax: 61-2-9313-7493, Email: firstname.lastname@example.org ABSTRACT The performance of inertial MEMS sensors – gyroscopes and accelerometers – depends heavily on the operating temperature. That is, their accuracy will degrade dramatically with rising temperature. In this paper the errors of inertial sensors as a result of varying temperature will be first modelled. The accuracies obtained with constant and varying temperatures will be compared and analysed. Then, the performance of the loosely-coupled GPS/MEMS-INS will be studied by simulations. The measurements are the differences of the GPS and INS positions. Two integration schemes are studied. In the first scheme, the error models of the inertial sensors are temperature-varying while they are temperature-invariant in the second scheme. That is, the temperature is not considered in the second scheme, even though the errors of the inertial sensors are temperature-varying. In this paper, three methods, including (1) control of the device’s working temperature, (2) modelling and compensation of the inertial sensors’ temperature-correlated errors, and (3) real-time compensation of these errors within the integration filter, are analysed and compared. The last method is the focus of this paper. Its effectiveness is investigated through simulations. KEYWORDS: GPS/INS integration, MEMS inertial sensors, temperature compensation. INTRODUCTION The integration of GPS (Global Positioning System) and INS (Inertial Navigation System) based on inertial MEMS (Micro-Electro-Mechanical System) gyros and accelerometers has been addressed recently as a result of their increasing use due to their convenient small/miniature size and low cost. However, the MEMS sensors’ poor accuracy still -limits their application in many navigation applications. Although many efforts have been made to compensate for the MEMS sensor errors, it is still a challenge to adequately model and compensate for the temperature-correlated error. This paper discusses the influence of the temperature-correlated MEMS sensor error on the performance of GPS/INS. In simulation tests, the operating temperature of the MEMS sensors is modelled as an exponential function of time. The inertial sensor errors are modelled as a combination of bias, random walk and a first-order Markov process. It is assumed that the bias and the standard deviations of the random walk and the Markov process are temperature-correlated. The temperature- correlated and temperature-invariant errors are compared in the simulations. The results show that the temperature-correlated error significantly degrades the accuracy of the sensors. The random walk error has the greatest impact. Two approaches are currently used to mitigate the influence of the MEMS sensors’ temperature-correlated errors: (1) control the device’s working temperature, or (2) modelling and compensation of the temperature-related errors. In the latter approach, the outputs of the MEMS sensors are compensated for using a model of temperature-correlated errors. However the compensation accuracy heavily depends on the accuracy of the model. This paper explores an adaptive estimation method in which the temperature-correlated errors are augmented in an adaptive integration Kalman filter. In this way the temperature-correlated errors are estimated and compensated for in real time. MODELLING OF THE MEMS SENSORS It is usually accurate enough to model the MEMS sensors with a combination of the random constant, the random walk and the 1st-order Markov process errors (Xing Z and Demoz GE, 2008). That is: ⎧ε = ε c + ε w + ε m ⎪ ⎪ε c = 0 ⎨ε = σ u (t ) (1) ⎪ w w ⎪ε m = − βε ε m + 2 βε σ m u (t ) ⎩ m m where ε , ε c , ε w , and ε m are the total error, the random constant error, the random walk error and the 1st-order Markov process error respectively; σ c , σ w and σ m are the standard deviations of the random constant, the random walk and the 1st-order Markov process errors respectively; u (t ) is the Gaussian white noise with mean zero and standard deviation one; and βε is m the time constant of the 1st-order Markov process. If the working temperature of the MEMS sensor changes with time, one can assume that the standard deviations of the errors are temperature-varying, while the time constant of the 1st-order Markov process error is not temperature-correlated. One can further assume that the working temperature is an exponential function of time. Hence, the standard deviations of the errors can be modelled as: σ i = σ i0 + ri ΔT (1 − e − β t ) i (2) where i is either c , w , or m ; ri is the scale factor; ΔT is the temperature difference; σ i0 is the standard deviation of the error when t = 0 ; βi is the time constant of the varying temperature; and t is the time. The random constant, the random walk and the 1st-order Markov process errors can be discretised as: ⎧ ⎪ε ⎡( k + 1) T ⎤ = ε ( kT ) ⎪ c⎣ ⎦ c ⎪ε w ⎡( k + 1) T ⎤ = ε w ( kT ) + ζ w ( kT ) ⎪ ⎣ ⎦ ⎪ ⎨ε m ⎡( k + 1) T ⎤ = e m ε m ( kT ) + ζ m ( kT ) − βε T ⎣ ⎦ (3) ⎪ ⎪ T ⎢ 0 ⎣ ( ⎪ζ w ( kT ) = ∫ ⎡σ w 0 + rw ΔT 1 − e − β w ⎡( k +1)T −τ ⎤ ⎤ u ⎡( k + 1) T − τ ⎤ dτ ⎣ ) ⎦ ⎥ ⎣ ⎦ ⎦ ⎪ ⎪ζ m ( kT ) = ∫0 e ⎩ T −β τ εm ⎡ ⎣ ( 2 βε m ⎢σ m0 + rm ΔT 1 − e m ⎣ ) − β ⎡( k +1)T −τ ⎤ ⎤ ⎦ ⎥ u ⎡( k + 1) T − τ ⎤ dτ ⎦ ⎣ ⎦ (a) The random walk error. (b) The 1st-order Markov process error. Figure 1. The gyro errors with the constant and varying temperatures. where T is the sample period; and ζ w and ζ m are the discretised equivalent noises of the random walk and the 1st-order Markov process errors. Their variances can be calculated as: ⎧ ⎪qw ( kT ) = (σ w 0 + rw ΔT ) T − 2rw ΔT (σ w 0 + rw ΔT ) β 1 − e 2 1 ( ) − βwT e− k βw T ⎪ w ⎪ ⎪ + ( rw ΔT ) 2 1 2β w ( ) 1 − e −2 β w T e −2 k β w T ⎪ ⎨ βε m ⎡ ⎪ ( ) ⎪qm ( kT ) = (σ m0 + rm ΔT )2 1 − e−2 βε m T + ( rm ΔT )2 βε m − β m ⎣ ⎢ −2 ( β − β )T 1 − e ε m m ⎤ e −2( k +1) βmT ⎥ ⎦ ⎪ ⎪ βε m ⎡1 − e− ( 2 βε m − βm )T ⎤ e − ( k +1) βmT ⎪ −4rm ΔT (σ m0 + rm ΔT ) ⎩ 2 βε m − β m ⎢ ⎣ ⎥ ⎦ (4) Since the value of ε c is determined by its initial value, ε c is temperature- uncorrelated. Of course, if the temperature difference ΔT changes, the standard deviation of ε c , σ c , would vary accordingly. Therefore, the random constant is excluded in the following analysis. Figure 1 depicts the simulated results of the gyro random walk and 1st-order Markov process errors with the constant and varying temperatures when σ w 0 = 0.4°/ s /s , rw = 0.1°/ s / °C/s , σ m0 = 0.02°/s 2 , rm = 0.01°/s 2 / °C , 1 1 1 ΔT = 20°C , = = 150s , T = 1s , and with a total operating = 300s , βw βm βε m period of 1800s. The means of the random walk errors with the constant and varying temperatures are 6.63 °/ s and 38.19 °/ s . The standard deviations of the random walk errors with the constant and varying temperatures are 5.58 °/ s and 29.88 °/ s . The means of the 1st-order Markov process errors with the constant and varying temperatures are -0.0093 °/s and -0.0789 °/s . The standard deviations of the 1st-order Markov process errors with the constant and varying temperatures are 0.015 °/s and 0.154 °/s . According to the simulation results, the temperature-correlated accuracy of the gyro is degraded by almost one order of magnitude, requiring that more attention be paid to this if the operating temperature changes significantly. INTEGRATION OF GPS/INS The loosely-coupled GPS/INS based on the use of MEMS gyros and accelerators is studied. The differences between the GPS and INS positions are used as observations within the integration Kalman filter. The state and measurement equations in the local geographic frame are modelled as: ⎧x = Fx + w ⎨ ⎩z = Hx + v (5) where x = [ Δr Δr φ η ε ] , including the position error Δr , the velocity T error Δr , the psi-angle φ , the accelerator bias η and the gyro bias ε . The biases η and ε are modelled as 1st-order Markov processes only. The coefficients of the accelerometer bias ε are the same as those used in the former simulation, that is σ m0 = 5 × 10−4 g/s , rm = 2.5 × 10−4 g/s/ °C , and 1 = 150s . The others are the same as those of the gyros. The observations βη m in three directions are contaminated by the Gaussian white noise with mean zero and standard deviation 10m. The vehicle trajectory is a straight level line lasting for a period of 1024s. According to the earlier simulations, the inertial sensors’ accuracy will degrade significantly as the operating temperature changes, hence the influence of the temperature should be taken into account in the integration. Three methods can be implemented to mitigate the temperature effect. The first is the traditional temperature control method. It is effective, but it is not an optimal solution due to the large size and energy consumption of the temperature-controlling sub-system. The second is to compensate the inertial sensors’ outputs using temperature models. Its effectiveness depends on the precision of the temperature models. In the following simulations the third method, the so-called adaptive estimation method, is studied. That is, the inertial sensors’ temperature models are augmented as additional states to be estimated, and thus compensate for their biases in real time. To verify the effectiveness of the adaptive estimation method, the aforementioned GPS/INS integration system is simulated. The following scenarios are studied: (1) with or without temperature compensation; (2) with varying and constant temperatures; and (3) the temperature differences of ΔT = 20°C and ΔT = 50°C . Figure 2. The horizontal position errors of the temperature compensated and the temperature ignored scenarios. The first scenario is to compare two results – the one for the temperature being compensated for in real time; and the other where the temperature influence is ignored. Figure 2 depicts the horizontal position errors of the temperature compensated and the temperature ignored scenarios. It is assumed that the INS is accurately aligned before the integration. From Figure 2, it can be seen that the difference between the two results is not obvious during the first 450s. After that period the integration result with the temperature ignored tends to diverge, while the result with the temperature compensated still remains convergent. Hence, the temperature compensation is effective and necessary. However, it should also be noted that the horizontal errors of the temperature compensated case are very large in comparison with the observation’s accuracy level of 10m. Figure 3. The horizontal position errors of the temperature varying and the temperature constant scenarios. Figure 4. The horizontal position errors when ΔT = 20°C and ΔT = 50°C . The second scenario is to compare two results derived from the variable and constant temperatures. Differing from the first scenario where the integration result of the temperature ignored is not temperature compensated at all, in this scenario the inertial sensors are well compensated for both integration results. Figure 3 shows that the horizontal position errors with the variable and constant temperatures. Although the error of the result with the constant temperature is smaller than that of the result with the variable temperature, the difference between them is not as much as that in the first scenario. This result shows that the adaptive estimation method is effective for temperature compensation. The third scenario is to examine the effect of the temperature difference on the compensation. According to equations (2) – (4), the inertial sensors’ accuracy will be more and more seriously degraded with an increase of the temperature difference, which further influences on the integration solution. Figure 4 shows the horizontal position errors in the cases of ΔT = 20°C and ΔT = 50°C . Although the error of ΔT = 50°C is a little larger than the one for ΔT = 20°C , the difference between them is not so remarkable especially given that the two temperatures differ by as much as 30°C . In other words, the influence of the temperature difference is not so obvious. CONCLUDING REMARKS The temperature effect of the MEMS inertial sensors on their performance both as standalone sensors and as components of the loosely-coupled GPS/INS was studied using simulations. The inertial sensors’ accuracy can be degraded significantly by one order of magnitude with variable operating temperatures. The integration position accuracy keeps convergent if the so- called adaptive estimation method is applied. Otherwise, if the variant temperature is not considered in the integration, the position accuracy tends to diverge. The temperature difference does not influence the position accuracy so remarkably. However, the following points should be noted: (1) real data are needed to confirm these results; (2) GPS is working for the whole integration process; and (3) the operating time is as short as 1024s. In a practice application, the unavailability of GPS signals for several seconds could easily occur, as for example in ‘urban canyon’ environments. If these factors are taken into account, the varying temperature will influence the integration accuracy heavily. Hence, compensating for temperature in an adaptive sense is necessary and effective. Further work will include using real data to validate the effectiveness of the proposed temperature compensation method. ACKNOWLEDGEMENT The research is sponsored by the Specialised Research Fund for the Doctoral Program of China Higher Education under the grant No.20070006006. Part of this work was completed while the first author was a visitor at the University of New South Wales with the support of the China Scholarship Council. 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