International Conference on Engineering Education July 21-25, 2003, Valencia, Spain. 1

Integration of Control and DSP for Vehicle Navigation System with GPS/INS/GSM Modem Linking

Author:

Jium-Ming Lin, Chung-Hwa University, Phone: 886-35186483, Fax: 886-35186507 jmlin@chu.edu.tw

Abstract — The purpose of this paper is to teach the graduate students of Institute of Aeronautics and Astronautics to integrate the knowledge from the courses such as : stochastic control , navigation and communication , to study the performance of the vehicle navigation system of INS aided with GPS, i.e., the positioning outputs of GPS are taken as the measurements to calibrate the INS error parameters in the Kalman filter, such as the navigation error covariance matrices of positions, velocities, heading angle and angular speed, as well as the biases, drift rates and scale factors of the rate gyros and accelerometers. The integrated system can preserve the advantages and avoid the disadvantages of both systems. Not only the normal operating but also the lost tracking conditions of GPS are performed and analyzed. This is scarcely discussed in the other papers .The results show that the longer the GPS locks , the better the errors of INS are calibrated, thus the navigation system of INS aided with GPS is better than those systems with only GPS or INS . In addition, the vehicle monitoring system is also realized by adding the GSM modems to the vehicle as well as the base station, respectively. Thus the base station can monitor the vehicle via GSM modem linking as well as make Differential GPS (DGPS) correction to raise the performance of the vehicle positioning accuracy. Index Terms — GPS, INS, Kalman Filter, GSM Modem.

INTRODUCTION The advantage of the traditional Inertial Navigation Systems (INS) is that the data rate is high, but the accuracy of which would be degraded as the times go by [1]. In general, there are two methods to improve the performance. The first one is to increase the accuracy of both the inertial sensors and the computer. The second one is to calibrate the error parameters by using the other navigation information [2], such that the diverging rate can be reduced.

The Global Positioning System (GPS) is one of the navigation methods right now. The GPS receiver can decode the navigation messages from the satellites, solve the navigation equations [3]-[8] and provide positioning data all over the world. Although the data rate is only 1 Hz, the characteristic performance of which is long time stable [9]. Thus it is a good ideal to integrate INS with GPS, the resulting system is not only reliable but also cost-effective, therefore, it is applied in this paper as the central part of the vehicle navigation system.

The purpose of this paper is not only to study the performance of the vehicle navigation system of INS aided with GPS, but also to monitor the vehicle by GSM (Global System for Mobile communication) modem linking with Short Message (SM mode) communication, thus the Differential GPS (DGPS) operation [11]-[12] of the GPS receiver can also be obtained. The positioning outputs of the resulting DGPS receiver are taken as the measurements to calibrate the INS error parameters in the Kalman filter, such as the navigation error covariance matrices of positions, velocities, heading angle and angular speed, as well as the bias, drift and scale factor error parameters of the rate gyros and accelerometers. The hardware of the system includes one notebook PC, one rate gyro, two accelerometers, three A/D converters, one GSM modem, and one GPS receiver. On the other hand, the software program is based on TUBRO.C in order to speed up the real time signal processing as well as easy for later-on maintaining. For the practical consideration, not only the normal operating but also the lost tracking conditions of DGPS are performed and analyzed. This is scarcely discussed in the other papers [5]-[8]. The results show that the longer the DGPS locks, the better the error parameters of INS are calibrated, thus the integrated system can preserve the advantages and avoid the disadvantages of both systems, i.e., not only the high positioning rate of INS, but the long time stability of DGPS are preserved, while the degraded performance as times go by of INS, and the lower positioning rate of DGPS are avoided.

The other advantage of the proposed system is that it is modularized, and not sensitive to the environment effect of the vehicle. For example, the heading of the vehicle is obtained by using the integrated velocity results of DGPS / INS (which is derived in (29) later), there is no need of gyrocompass. Thus it is not sensitive to the magnetic field effects due to the earth or the environments. By the way the wearing and/or slipping effects of the tire, for the Dead-Reckoning (DR) navigation systems by using the odometer [10] can also be avoided .In addition, the vehicle monitoring system is also realized by adding the GSM modems to the vehicle as well as the base station, respectively. Thus the base station can not only monitor the vehicle but make the differential GPS (DGPS) correction via GSM modem linking.

International Conference on Engineering Education July 21-25, 2003, Valencia, Spain. 2

The other parts of this paper are organized as follows: The problem is formulated in Section 2. The experiment results with GPS in normal operating or lost tracking conditions are given in Section 3. Finally, some conclusions are drawn in Section 4. PROBLEM FORMULATION The major error sources of INS are rate gyro and accelerometer. Let the sensing axes of the rate gyro and two other

accelerometers are respectively along the vehicle’s z, x and y axes, respectively, and ϕ is the heading angle between the

vehicle’s x axis (center line) and the north direction as shown in Figure 1. In general, there are four types of rate gyro errors [1], such as random bias B g

, scale factor S F g, random drift ε g

and random noise ω g , i.e.,

ωωεϕ *gggg SFB +++=•

(1)

Where ω is the angular rate of heading. The errors of B g

and S F gcan be modeled by the following equations:

0=•

gB (2)

0=•

gSF (3)

The random drift ε g is a white noise with zero mean and standard deviation

}{ )()()( 2 τδστεε ε −=Ε tt ggg (4)

The random noise ω g is modeled as

gnggg ωτωω +−=•

/ (5)

where τ g

is the correlation time of the random noise ω g , and ω gn

is a white noise with zero mean and standard

deviation as :

}{ )()()( 2 τδστωω −=Ε tt gngngn (6)

The power spectrum of the white noise ω gn is [13]

ggngnQ τσ 22= (7)

Similarly, the output of the accelerometers ( ia ) ( i x y= , ) can be expressed in terms of the errors such as random drift

i∆ , random noise iaB , and scale factor ( )SF i x yi = , , i.e.,

International Conference on Engineering Education July 21-25, 2003, Valencia, Spain. 3

iiiaiii fSFBVa ∗++∆==•

( )i x y= , (8)

Where f i is the acceleration along the vehicle’s i axis ( )i x y= , , and the errors of scale factor ( )SF i x yi = , and

random noise iaB can be modeled as follows:

0=•

iFS ( )i x y= , (9)

iaiaiaia BB ωτ +−=•

/ ( )i x y= , (10)

Where iaτ is the correlation time of the random noise

iaB , and iaω is a white noise with zero mean and standard

deviation as:

( ) ( ){ } ( )τδστωω −=Ε tt iaiaia2 ( )i x y= , (11)

The power spectrum of the white noise ω ai

is

iaiaiaQ τσ 22= ( )i x y= , (12)

The random drift ∆ i is a white noise with zero mean and standard deviation:

( ) ( ){ } ( )Ε ∆ ∆ ∆i it ti

τ σ δ τ= −2 ( )i x y= , (13)

It should be noted that for the sake of speeding up the Kalman filtering processes, the random biases of the accelerometers

are neglected.

Since the positioning results should be expressed in the navigation coordinates, thus the acceleration components of

vehicle in the north f N and the east f E are respectively obtained as:

−−

=

y

x

E

N

aa

cssc

ff

ϕϕϕϕ (14)

Where cϕ ϕ= c o s (15)

International Conference on Engineering Education July 21-25, 2003, Valencia, Spain. 4

sϕ ϕ= sin (16)

In general, the positioning outputs of GPS receives are in terms of the latitude L and the longitude λ , one can transform

them into the coordinates along the navigation axis, i.e., the positions in the north PN , and the east PE are obtained

respectively as follows:

eN RLLP )( 0−= (17)

00 )( CLRP eE λλ −= (18)

Where R e is the radius of the earth, L 0 0( )λ is the initial latitude (longitude) of the vehicle.

By the above derivation, one can formulate the state equation of the vehicle in the navigation system as [14]:

x t Ax t t•

= +( ) ( ) ( )ω (19)

Where

][)( gengagaxgENEN SFSFSFBBBVVPPtx ϕω= (20)

The system matrix A is [14]

−

−−

−

=

−

−

−

•−

•−

•

•

000000000000

000000000000

000000000000

000000000000

00000000000

00000000000

00000000000

00100100000

000000

000000

00000000100

00000000010

1

1

1

ay

ax

g

e

e

ENIRg

NEIRg

I

I

fcsfsL

fscfsL

sL

sL

A

τ

τ

τ

ω

ϕϕλ

ϕϕλ

λ

λ

Where

cLRV

e

EI =λ (22)

g m= 9 8 2. sec (23)

and the input noise vectorω ( )t in (19) is defined as:

[ ]TayaxgngENt 000000)( ωωωεω ∆∆= (24)

(21)

International Conference on Engineering Education July 21-25, 2003, Valencia, Spain. 5

Which is zero mean and with covariance as

}000000{ 222222ayxggENdiagQ ωωωε σσσσσσ ∆∆= (25)

The measurement equation is

z t H t x t v t( ) ( ) ( ) ( )= + (26)

Where

H t( ) =

1 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0

(27)

and v t( ) is the measurement noise of GPS with zero mean and covariance

{ }22222 ,,,,)( ϕσσσσσENEN VVPPdiagtR = (28)

where σ σ σ σP P V VN E N E

2 2 2 2( ), ( ) and σ ϕ2 are respectively the GPS standard deviation errors of the north (east) position ,

north (east) velocity and heading angle , and

ϕ = −t a n ( )1 vv

EN

(29)

The GPS/INS integrating method uses the loosely-coupled closed-loop structure, as shown in Figure 2, by the trade-offs of

system implementation, accuracy and reliability.

Let the discrete state and measurement equations are respectively as [6]

x xk k k k k= +− − −Φ Γ1 1 1ω (30)

and

kkkk vxHz += (31)

Then the navigation equations, for the integrated-system with INS and GPS , can be solved by the Kalman filtering

algorithm as follows:

x x tk k k

∧

−

∧

−− =( ) ( )Φ 1 1 (32)

International Conference on Engineering Education July 21-25, 2003, Valencia, Spain. 6

Tkkk

Tkkkk QtPP 111111 )()( −−−−−− ΓΓ+ΦΦ=− (33)

[ ]K P H H P H Rk k kT

k k kT

k= − − + −( ) ( ) 1 (34)

x t x K z H xk k k k k k

∧ ∧ ∧= − + − −

( ) ( ) ( ) (35)

[ ]P t I K H Pk k k k( ) ( )= − − (36)

Where x Pk k

∧− −( ) ( ( ) ) and x Pk k

∧+ +( ) ( ( ) ) are the estimated state (covariance matrix) before and after taking the

measurements into consideration, and K k is the Kalman filter gain matrix. By the way the sampling period of INS is 40 ms

in this study. EXPERIMENT RESULTS The block diagram of the vehicle navigation and monitoring system is shown in Figure 3, in which the Wavecom WMOD2B dual-band 900/1800 MH GSM modems are added to the vehicle as well as the base station, respectively, such that the differential mode operation of the GPS receiver can be obtained, and the positioning information of the vehicle, e.g., the navigation results obtained by DGPS/INS can be received by the base station via GSM short message (SM mode) communication.

The test site for evaluating the performance of the integrated system is in a trapezoid region in Hsin- Chu, Taiwan as shown in Figure 4. The sequence of the path is along the points s s1 2, , s s3 4, , s5 , and returning back to S 1

finally. The average speed is 40 km/hr. The total distance is about 3.4 km, and the testing period is 320 seconds. The parts number and the error parameters of the accelerometer, rate gyro as well as the GPS receiver are listed in Table I. Figure 5 shows the positioning result obtained by the system with only INS, as it is anticipated that the accuracy of which is degraded with time. Figure 6 shows the positioning output of DGPS receiver in which the differential GPS correction is accomplished by GSM linking from the base station, thus the standard deviation of the positioning result is reduced to 5 m. Although the results are not degraded with time, but the data rate is only one output per second. On the other hand, the result obtained by the integrating method is shown in Figure 8. It can be seen that the characteristic performances of both DGPS long-time stability and INS high- speed data rate are preserved. In addition, the standard deviations of the position, velocity and heading errors are shown in Figures 7-10. It can be seen that all the errors are converged as long as the DGPS locks.

If the DGPS loses track during two periods of time, i.e., from S 2 to ( )S 3 6 0 1 4 0− s e c and S 4 to

S 5 2 2 0 2 8 0( s e c )− as shown in Figure 11, then the result of the integrated system is shown in Figure 12, which shows that the overall performance is acceptable. The standard deviation of the position, velocity and heading errors are shown in Figs.13-14. It can be seen that as long as the DGPS locks, then the errors are converged. On the other hand, if it loses track, then the errors would grow up. Thus, the integrated navigation system of INS aided with DGPS is better than those systems with only INS or DGPS. CONCLUSIONS By the results of experiments one can draw some conclusions: • It is a good ideal to integrate INS with DGPS. This method is not only reliable but also cost-effective. • The integrated system can preserve the advantages while avoiding the disadvantages • of both systems, i.e., the high positioning rate of INS and the long time stability of DGPS are preserved , while the

degraded performance as times go by of INS , and the lower positioning rate of DGPS are avoided . • The longer the DGPS locks, the better the errors of INS are calibrated, thus the • navigation system of INS aided with DGPS is better then those systems with only DGPS or INS . This performance

analysis is scarcely discussed in the other papers. • The other advantage of the proposed system is that it is modularized, e.g., the DGPS/INS navigation module, and GSM

communication module, which can be easily applied for the vehicle navigation and monitoring system. In addition, it is neither sensitive to the magnetic field of the earth or the environments, nor the wearing and/or slipping effects of the tire, which are the disadvantages of the vehicle navigation systems developed by using the gyrocompass and odometer

International Conference on Engineering Education July 21-25, 2003, Valencia, Spain. 7

of the Dead-Reckoning (DR) system. ACKNOWLEDGEMENT

This work was supported by the National Science Council (grant no. NSC-89-2212-E-216-002), and the Ministry of

Education (grant no. 2002 Avionics Education Improvement Program ) of the Republic of China. REFERENCES

[1] Slater, J.M., et. al., “Inertial Navigation Analysis and Design”, McGraw-Hill Book Co.,1964.

[2] Cunningham, J.R., “Performance of GPS-Aided INS During High-Dynamics Maneuvers”, Master Thesis, Dept. of the Air Force Institute of Technology Air University, AFIT/ GE/ENG/87D-12 1987.

[3] Bancroft, S., “An Algebraic Solution of the GPS Equations”, IEEE Trans. on Aerospace and Electronic Systems, Vol. AES21, No,7, 1985, pp.56-59.

[4] Krause, L.O., “A Direct Solution to GPS-Type Navigation Equations”, IEEE Trans. Aerospace and Electronic Systems, Vol.AES-23, No. 2, 1987,pp.225-232.

[5] Lin, H.C., “An Integrated Study of INS with GPS Aiding,” Master Thesis, School of Aeronautics and Astronautics, Chung-Hua University 1995.

[6] Lin, J. M, Lin, H. C, and P. K. Chang, “A study of INS Aided by GPS”, Trans. of Aeronautics and Astronautics Society of the Republic of China, Vol. 26, No, 3,1995, pp.213-221.

[7] Lin, J. M. and Lin, H.C,“GPS Navigation Data Signal Processing ”, Trans. of Aeronautics and Astronautics Society of the Republic of China, Vol.28, No.2, 1996, pp. 153-161.

[8] Lin, J. M. and Lin, H. C, ”Performance Analyses On Some of GPS Navigation Equation and Data Smoothing Method”, J. Control, Systems and Technology, Vol.5, No,2, 1998, pp. 35-144,

[9] Spilker, J. J.“GPS Signal Structure and Performance Characteristics”, Navigation, J. Institute of Navigation, Vol.25, No.2, 1978, pp. 121-146.

[10] Wang, K. B,“Computer Architecture Study of Integrated GPS/INS”, Master Thesis, School of Aeronautics and Astronautics, Chung-Hua University, 1995.

[11] Blackwell. E G., ”Overview of Differential GPS”, Navigation, Journal of the Institute of Navigation, Vol.35, 1988, pp. 89-100.

[12] Hunter, J, Kosmalski, W., and P. Truong, “Vehicle Navigation Using Differential GPS”, IEEE Conference of Position Location and Navigation System, 1990, pp. 392-398.

[13] Stallard, D.V., ”Classical and Modern Guidance of Homing Interceptor Missiles”, Seminar of Department of Aeronautics and Astronautics, Massachusetts Institute Technology, Cambridge, MA, 1968,pp. 60-62.

[14] Bar-Itzhack, I.Y., and N. Berman, “ Control Theoretic Approach to Inertial Navigation Systems”, J. of Guidance, Control, and Dynamics, Vol.11, 1988, pp.237-245.

FIGURES AND TABLES

TABLE 1 THE PART NUMBERS AND THE ERROR PARAMETERS OF THE ACCELEROMETER, RATE GYRO AS WELL AS THE GPS RECEIVER.

Accelerometer (Lucas LSBC-2) 0.04g/secBa = 0.04g/sec∆ a = 0.08gσ a = 0.01secτ a =

Rate Gyro (Murata ENV-0.5A)

0.1deg/secBg =

0.1deg/secσεg =

c0.44deg/seσgn =

0.01secτg =

GPS Receivers (Astech-G12) 20mσ

NgP = 20mσEgP = 0.5m/secσ

NgV = 0.5m/secσEgV =

International Conference on Engineering Education July 21-25, 2003, Valencia, Spain. 8

FIGURE. 1

THE RELATIONSHIP BETWEEN THE COORDINATES OF VEHICLE AND THE INERTIAL NAVIGATION SYSTEM .

FIGURE. 2

THE LOOSELY-COUPLED CLOSED-LOOP STRUCTURE OF DGPS/INS.

INERTIALMEASUREMENT

UNIT

NAVIGATIONEQUATION

GPSRECEIVER

GPSKALMANFILTER

INERTIALKALMANFILTER

CORRECTEDPOSITION,VELOCITY,

AND ATTITUDE

INSTRUMENT ERRORS ANDPOSITION, VELOCITY,ANDATTITUDE CORRECTIONS

GPS POSITIONAND VELOCITY

N(North)

E(East)

x

yϕ ( )0

ϕa x

a y

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FIGURE. 3

THE BLOCK DIAGRAM OF THE DGPS/INS/GSM NAVIGATION SYSTEM.

FIGURE. 4

THE TEST SITE FOR EVALUATING THE POSITIONING PERFORMANCE OF THE INTEGRATED SYSTEM.

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FIGURE. 5

THE POSITIONING RESULT OBTAINED BY THE SYSTEM WITH ONLY INS.

FIGURE. 6

THE POSITIONING OUTPUT OF DGPS RECEIVER.

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FIGURE. 7

THE RESULT OBTAINED BY THE INTEGRATED SYSTEM WITH DGPS RECEIVER LOCKED ALL THE WAY.

FIGURE. 8

THE STANDARD DEVIATIONS OF THE POSITION ERRORS FOR THE INTEGRATED-SYSTEM.

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FIGURE. 9

THE STANDARD DEVIATIONS OF THE VELOCITY ERRORS FOR THE INTEGRATED-SYSTEM.

FIGURE. 10

THE STANDARD DEVIATIONS OF THE HEADING ANGLE AND TURNING RATE ERRORS WITH THE INTEGRATED SYSTEM.

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FIGURE. 11

THE POSITIONING RESULT OF DGPS RECEIVER FOR LOST TRACK DURING 60~140 SEC. AND 220~280 SEC.

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FIGURE. 12

THE POSITIONING RESULT OBTAINED BY THE INTEGRATED-SYSTEM WITH DGPS LOST TRACKING FOR TWO PERIODS

OF TIME.

FIGURE. 13

THE STANDARD DEVIATIONS OF THE POSITION ERRORS FOR THE INTEGRATED SYSTEM WITH DGPS RECEIVER LOST

TRACKING FOR TWO PERIODS OF TIME.

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FIGURE. 14

THE STANDARD DEVIATIONS OF THE VELOCITY ERRORS FOR THE INTEGRATED SYSTEM WITH DGPS RECEIVER LOST

TRACKS FOR TWO PERIODS OF TIME.