GPS/INS Integreted Using Adaptive Kalman Filter

GPS/INS Integreted Using Adaptive Kalman Filter

Ashim Pradhananga

Information Technology and Engineering

Tianjin University of Technology and Education (TUTE) Tianjin, China

Prof. Zhang Xing Hui/ Asst. Prof. Li Shi Xin Information Technology and Engineering / School of Electronic Engineering

Tianjin University of Technology and Education (TUTE) Tianjin, China

Abstract –Inertial navigation is a self-contained navigation which are used to track the position and orientation of an object relative to a known starting point, orientation and velocity, like GPS. The INS can calculate the position of the aircraft without any help of other sources. Due, to a large numbers of error occurred by the sensors which leads to an unacceptable results. Therefore, GPS has been used to in INS using a Kalman filter.

Where INS, Kalman Filter are widely used to increase the accuracy and reliability of the navigation solution. Usually, Direct Kalman filter is used for among quantities like velocity,position and attitude are among the state variable of the filter, which allows them to be estimate directly.

KeywordsGPS/INS; Kalman Filter; Adaptive Kalman Filter; GPS; INS

1. INTRODUCTION

   Most of these navigation systems use no longer only the Global Positioning System (GPS) but also an inertial navigation system (INS).Together the two systems complement each other and permit improved navigation accuracy and reliability especially when GPS is degraded or interrupted for example because of buildings or tunnels and for this application the Kalman filter provides the basis.

   Most INS errors are attributed to the inertial sensors (instrument errors). These are the residual errors exhibited by

   of problems, it has been subject of extensive research.

   The performance of the Kalman lter is crucial to overall system performance, especially when low-cost sensors are integrated. This paper focus on integrated GPS and Inertial Navigation System.

2. NAVIGATION EQUATIONS

   The navigation equation is important for Kalman filter and the algorithm. Where INS consists of the 3-axis gyroscope from where we can have pitch and yaw rates about the body axes. It also has 3-axis accelerometers which give the accelerations along the three body axes. There are two basic inertial mechanisms which are used to derive the Euler angles from the rate gyros, viz. stable platform and strap-down INS. The acceleration values from the accelerometers are then corrected for rotation of the earth and gravity to give the velocity and position of the aircraft.

   The orientation of an aircraft with respect to a xed inertial frame of axes is given by three Euler angles. The relationship between the angular rates of roll, pitch and yaw, p, q, r (measured by the body mounted gyro), the Euler angles, , , and their relation are shown in figure:

   the installed gyros and accelerometers following calibration of

   1 sin tan

   cos tan p

   the INS such error leads to error in the velocity and position of an object. So unaided INS leads to error grow with time which

   0

   cos

   • sin

   q

   (1)

   is undesirable this is the reason we usually integrate INS with GPS. The primary role of GPS is to provide highly accurate

   0

   sin sec

   cos sec r

   position and velocity world-wide, based on range and range-

   rate measurements. GPS can be implemented in navigation as

   a xing aid by being a part of an integrated navigation system,

   for example GPS/INS.

   It constitutes a tool for correcting the predicted INS trajectory with GPS measurements. Also the determination of a reference orbit for these GPS satellites and correcting it with the data from the GPS control stations is a very important application of the Kalman filter. But these are only two examples of the wide variety of fields where Kalman filtering plays an important role. The application areas span from

   To determine the attitude at a given time we can derive the Euler angles using initial condition from equation (1) considering for the pitch angles around ±90 the error becomes unbounded as tan tends to innity. Where it depends on the parameters (e0, e1, e2, e3). Once we have calculated the time history of the four parameters, we can calculate the Euler angles using the following equations:

   1 3 0 2

   aerospace, to marine navigation, demographic modeling, weather science, manufacturing and many others. Because the Kalman filter is very effective and useful for such a large class

   sin1 2(e e e e )

   (2)

   cos1

   0

   1

   2

   3

   e2 e2 e2 e2

   sign2(e e

   • e e )

     (3)

     Now on this way we can calculate the rate of latitude ,

     14(e1e3 e0e2 )2

     2 3 0 1

     longitude and height H

     cos

     1

     e2 e2 e2 e2

     sign2(e1e2 e0e3 )

     (4)

     R

     VN

     e

     VE

     Re cos

     H VD

     (11)

     0 1 2 3

     14(e1e3 e0e2 )2

     We now have with us the attitude of the aircraft. To calculate the position we use the accelerations given by the accelerometers.

     Now,

     The INS program now takes 6 states from this time history.After we obtain the row, pitch, yaw (viz. p, q, r, ax, ay, az) from the gyros and accelerometers. Then the program

     integrates and calculates the Euler angles are calculated using equations 2-4 Now the accelerations from the accelerometers

     U ax Vr Wq g sin

     V ay Ur Wp g cos sin

     W az Uq Vp g cos cos

     Now the actual angular rate is given by

     p p

     (5)

     (6)

     (7)

     are used to calculate U ,V , W given by equations 5-7, which are then integrated to get the values of U , V , W. Now to calculate the velocity components in the body frame we have to convert the components into the navigation frame where we use the DCM matrix as in equation 8 and calculate velocities in equation 10. These velocities are then integrated to get the position (X, Y, and Z) along the axes. Hence, we can finally calculate the latitude, longitude and the height can be calculated using equation 11.

     Similarly, the GPS give the latitude, longitude and altitude of the current location of the receiver through which we can calculate the Flong and Flat in equation 12-13. Considering that earth has a semi major axis equatorial radius (a=6,378,137m) and with polar radius of (b= 6,356,752.3142m) where the distance corresponding to a 10 change in longitude Flong and

     latitude Flat for a specific latitude and height h.

     q q

     DCM 1

     (8)

     a2

     r

     Where,

     r m

     Flong 1800

     a2 cos2

     b2

     sin2

   • h cos

     (12)

     DCM is the direction cosine matrix or the transformation matrix which is given by

     Flat 0

     2

     a 2b2

     2 2

     • h

   2

   (13)

   cos cos

   cos sin

   • sin

   180

   a cos

   b

   sin

   DCM sin sin cos sin cos

   sin cos cos sin sin

   sin sin sin cos cos sin sin cos cos sin

   (9)

   cos cos

   Hence, the latitude and longitude at the current location (2,

   µ2) can be calculated from the latitude and longitude from the previous location (1, µ1) in the following manner

   To Calculate the Velocities along with north, east and the

   downward velocity is given by the equation (9)

   X

   F

   2

   lat

   1

   (14)

   N

   X

   V

   U

   V

   E

   Y

   DCM T

   (10)

   2

   Y

   F 1

   (15)

   V

   V

   W

   lon

   Z

   D

   VN NorthVelocity

   V

   E

   Whereas, EastVeloci ty

   VD DownwardVelocity

   Where, X and Y are the changes in position along North direction and East direction on the earth, respectively.

3. KALMAN FILTER MOUDLE

   Kalman filter is the best procedure for combining noisy sensor output to estimate the state of a system with uncertain dynamics. The noisy sensor output include output from the

   GPS and INS which include the position, velocity, attitude and attitude rate of the object and uncertain dynamics caused by a human error or any medium. The Kalman filter used to estimate the error introduce into the unaided INS system due to the gyro and accelerometer. In this model the error from the

   1. Covariance Scaling.

      The covariance scaling method was used for improving the stochastic modeling of deferential pseudo-range GPS. The predicted covariance, P(k+x)1 , is articially scaled by the

      INS state vector

      x k

      and the measured value of the state

      factor Sk>1 to apply more weight to the measurements:

      vector from the GPS forms a measurement vector Z. The

      P() S

      P()T Q

      (16)

      GPS/INS integrate system configuration and the error are

      k k k 1

      k 1

      wither compensated by the two mechanism as feedback mechanism or feedback mechanism as shown in figure I and II.

      True position, velocity

      INS

      CORRECTED OUTPUT

      GPS

      KALMAN

      Deferent techniques can be used in estimating Sk. For example, a priori methods can be used in alignment of low cost IMUs, where it is known that the inertial sensor errors will be larger before the system has been aligned.

   2. Adaptive Kalman Filter.

   The principle of the Adaptive Kalman Filter (AKF) is to make the Kalman lter residuals consistent with their theoretical co- variances (Mehra, 1972). An estimate of the covariance of the innovation residual is obtained by averaging the previous residual sequence over a window length N:

   FILTER

   1 ()

   () T

   j

   C( )

   vk

   N Vj

   v

   (17)

   Where: v (-1) = z – H x (-) the innovation residual from the

   j k k k

   Kalman lter. The estimated measurement noise is computed

   Fig I: Feed forward aided INS

   by comparing the theoretical covariance (H P (-) HT +R ) with

   the estimated covariance to give :

   k k k k

   Rk C

   • H P() H T

   (18)

   v

   ( ) ( k

   k k k

   INS

   CORREC CTED OUTPUT

   The estimated process noise can also be to give:

   1 k

   T ()

   () T

   k k

   Qk N

   x j x j Pk

   KALMAN FILTER

   j k N 1

   Pk 1

   (19)

   GPS

   Fig II: Feed backward aided INS

   Where, xk = x (-)-x (+).This is known as a residual based estimate. Equation (19) can be written in terms of the innovation sequence by making the following substitution for the covariance of the state corrections.

   1

   T

4. A D A P T I V E K A L M A N F I L T E R I N G A L G O R I T H M S

   N

   C xk

   x j x j j k N 1

   Kk

   k

   k

   Cv( ) K T

   k

   (20)

   A number of adaptive Kalman ltering techniques exist to achieve the criteria described in adapting the stochastic properties of the Kalman Filter. Most techniques use the innovation sequence as the basis for adapting the measurement noise covariance, Rk, and the process noise covariance, Qk. These methods use a windowing function on the most recent innovations. Correct identication of the window size also needs to be identied to obtain the correct balance between lter adaptivity and stability. Multiple Model Kalman ltering is also described as a potential method for lter adaptation.

5. RESULT

   The graphs for attitude computed and corrected by the adaptive Kalman lter are given in Figure III. We cannot expect the Kalman lter to correct the attitude, velocity and positions given by the INS perfectly. This corrected attitude forms a part of the integration loop in the whole system. In figure III, the Red line denotes the output obtained by adaptive kalman filter where as the blue line is the output obtained by kalman filter respectively. As can be seen from the graph, the error of adaptive kalman filter was smaller than the error of kalman filter.

   Figure III: longitude, latitude and velocity error obtained by the Kalman Filter and Adaptive Kalman Filter

   In the results, the X-axis shows the time period and the Y-axis shows the estimated error of the longitude, the latitude, the eastward velocity and the northward velocity.

6. CONCLUSION

This paper has shown that how adaptive Kalman lter can be congured. The innovation and residual sequences provide a useful performance indicator that can be used to adaptively tune the stochastic information used in the lter. This paper has shown the use of adaptive ltering techniques to improve the error. The integration of INS and GPS is generally implemented through a adaptive Kalman flter. This filter showed a major improvement over the kalman filter. The performance of the adaptive Kalman filter, for most of the navigation parameters used in this study, is improved by almost 50% or more when compared to that of the kalman filter.

In this paper, an adaptive Kalman flter, based on the flter innovation sequence, is introduced as an alternative for integrating INS/GPS systems.

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