The effect of the temperature-correlated error of inertial MEMS

Document Sample
The effect of the temperature-correlated error of inertial MEMS Powered By Docstoc
					                                   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: wangkd@buaa.edu.cn
                                        Yong Li
                      School of Surveying & SIS, UNSW, Sydney 2052
          Phone: 61-2-9385-4173, Fax: 61-2-9313-7493, Email: yong.li@unsw.edu.au
                                     Chris Rizos
                      School of Surveying & SIS, UNSW, Sydney 2052
          Phone: 61-2-9385-4205, Fax: 61-2-9313-7493, Email: c.rizos@unsw.edu.au


                                     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.

                              REFERENCES

[1] Bekkeng JK (2009) Calibration of a novel MEMS inertial reference unit.
IEEE Transactions on Instrumentation and Measurement, 58(6): 1967-1974.
[2] Xing Z, Demoz GE (2008) Modeling and bounding low cost inertial
sensor errors. Proceedings of IEEE/ION Position, Location and Navigation
Symposium, May 5-8, 1122-1132.
[3] Hao W, Tian W (2005) Modeling the random drift of micro-machined
gyroscope with neural network. Neural Processing Letters, 22: 235-247.
[4] Walid AH (2005) Accuracy enhancement of integrated MEMS-
IMU/GPS systems for land vehicular navigation applications. UCGE reports
Number 20207 (PhD thesis), University of Calgary, Calgary, Canada, 46-73.

				
DOCUMENT INFO
Shared By:
Categories:
Stats:
views:17
posted:3/28/2010
language:English
pages:7
Description: The effect of the temperature-correlated error of inertial MEMS