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Title goes hereDate | Presented by

Tightly Coupled Integration of GPS-PPP and MEMS-Based Inertial System Using EKF and

UKF

Mahmoud Abd Rabbou and Ahmed El-Rabbany

Department of Civil Engineering,

Ryerson University

The XXV FIG International Congress 2014

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OUTLINES

Introduction

Problem statement

Research objectives

Mathematical models

Results

Conclusion

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INTRODUCTION

The most common navigation systems are the Global Navigation Satellite System

(GNSS) and the Inertial Navigation Systems (INS).

Currently, GPS system is the most widespread GNSS system and the only fully

operational system.

GPS mainly provides worldwide positioning, velocity and time synchronization.

Unfortunately, accurate GPS solution may not always be available as a result of

multi-path, GPS partial or complete outages.

To overcome these limitations, GPS can be integrated with a relatively environment-

independent system, the inertial navigation system (INS).

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INTRODUCTION

Unlike GPS system, the INS performance is not affected in environments as urban

canyons; it is independent of external electro-magnetic signals.

In contrast with GPS, which need complicated system to give the attitude solution,

INS system gives the attitude solution directly.

However, the main drawback of an INS is the degradation of its performance withtime. In order to control the errors to an acceptable level continues updates from, forexample, GPS are necessary.

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PROBLEM STATEMENT

Currently, most of the integrated GPS/INS systems for precise positioning

applications are based on high-end INS grades and differential GPS (DGPS).

This involves the deployment of a base station, which controls the rangeof navigation area.

The need of the surplus equipment (minimum two receivers) increases thecomplexity and cost.

The high grades INS are very expensive which add other limitation.

Most of the integrated system research employed Extended Kalman filter

(EKF) as an estimation filter may cause divergence in positioning

estimation during GPS outages.

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RESEARCH OBJECTIVES

To overcome the limitation mentioned before this research aims to develop a new

integrated navigation system for precise positioning and attitude applications

based on;

Precise Point Positioning (PPP) technique is used insteadDifferential mode to reduce the complexity and the cost.

Both pseudorange and carrier phase GPS measurements areconsidered.

Low-cost micro-electro-mechanical sensors (MEMS) accelerometersis used reducing the cost of the system with fiber optic gyros forprecise attitude detecting.

Both EKF and UKF estimation filters are employed to develop theoptimal filter for the proposed integrated system.

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(GPS-PPP)

Both pseudorange and carrier phase GPS measurements are

processed.

1s s1 r 1 r p 1 p 1P = ( t , t - ) + c [ d t ( t ) - d t ( t - ) ] + T + I + c [ d ( t ) - d ( t - ) ] + m + e

22 2

s sr r p 2 p 2P = ( t , t - ) + c [ d t ( t ) - d t ( t - ) ] + T + I + c [ d ( t ) - d ( t - ) ] + m + e

1

1 1 1 0 1 0

s s1 r 1 r 1 1

sr

= ( t , t - ) + c [ d t ( t ) - d t ( t - ) ] + T - I + c [ ( t ) - ( t - ) ] + m + e

+ [ N + ( t ) - ( t ) ]

22 2 2 2

2 2 2 0 2 0

s sr r

sr

= ( t , t - ) + c [ d t ( t ) - d t ( t - ) ] + T - I + c [ ( t ) - ( t - ) ] + m + e

+ [ N + ( t ) - ( t ) ]

MATHEMATICAL MODELS

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Un-differenced ionosphere-free linear combinations of pseudorange andcarrier phase measurements is employed.

2 21 1 2 2

3 22 21 2

1 2

2 21 1 2 2

3 22 21 2

1

sr r 1 r

s s

sr r 1 r

s

f P f PP = = + c d t - c d t + T + c ( 2 . 5 4 6 d - 1 . 5 4 6 d )

f f

c ( 2 . 5 4 6 d - 1 . 5 4 6 d ) + e

f f= = + c d t - c d t + T + c ( 2 . 5 4 6 - 1 . 5 4 6 )

f f

c ( 2 . 5 4 6 - 1 . 5 4

2s6 ) + ( N ) +

3 3

3 3

3

3 3

s r sr p p p 3

s r sr

P = + c [ d t - d t ] + T + B B + e

= + c [ d t - d t ] + T + B B + N + e

MATHEMATICAL MODELS (GPS-PPP)

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A system of non-linear first-order differential equations

(mechanization equations) is used

The inertial measurements (specific force and angular rate) are

used to solve it to provide positions, velocities and attitudes

&r n

&V n

&Rbn

=D V n

Rbn f b ( 2ni e +

nen ) V

n + gn

Rbn( ni b

bi n )

10 0

10 0

0 0 1

M h

D( N h ) c o s

+ = +

[ ]0 Tnie cos sin =T

n n E Een

V V V tan

M h N h N h

= + + +

b b n nin n ie enR ( ) = +

(INS)MATHEMATICAL MODELS

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Loosely coupled

Both the GPS and the INS mathematical models are

processed separately and the integration is applied in the level

of the navigation outputs (positioning, velocities and

attitudes)

Tightly coupled

The GPS raw measurements are not processed in a separate

filter as in the Loosely coupled case, but are directly combined

in a unique filter.

The tight integration strategy is preferred to use during the

periods of partial GPS availability.

As a result, in this research Tightly coupled is employed

(Integration mechanism)MATHEMATICAL MODELS

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(Integrated system processing)

1. Process model

x F ( t ) x G( t ) w( t ) = +&

3 3 3 3

3 3

3 3

3 3 3 3 3 3

3 3 3 3 3 3

3 3 3 3 3 3

3 3 3 3 3 3

0 0 0 0 0 00 0 0 0

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

nn

n r r r v rn n b n

n v r v v v b bnn n b

r v b ba

b ag b g

a s a

s gg

o f f

d r i

rF F F rvF F F R R F vF F F R R Wb

b

SS

t

t

=

&

&

&

&

&

&

&

&

&

3 3 3 3 3 3

3 3 3 3 3

3 3 3 3 3

3 3 3 3 3 3

3 3 3 3 3 3

3 3 3 3 3 3

3 3 3 3 3 3

0 0 0 0 0 0 0 00 0 0 0 0 0 0

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

an gb

n b aba b gg

s aa s gg

o f fo f f

d r id r i

wwRwRb wIb wIS wIS wI

t wt

+

2. Measurement modelG P S I N S

i

G P S I N Si

G P S I N Si

P P

zP P

=& &

M

0 0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0 0

i

i

i

A

BH

C

=

LL

M M M

LL

M M M O

LL

M M M

The complete state vector consists of 23+n

states describing the basic state vector (the

nine navigation parameter errors), the inertial

sensors errors (bias drift and scale factor),

errors unique to the GPS measurements such

as clock offset and drift and finally the

ambiguity parameters (n)

Where A

, B

, C

are the partial derivatives of the pseudorange, phase and

Doppler, respectively, with respect to the receiver position X and velocity V;

MATHEMATICAL MODELS

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(Estimation filter)

1 1 1

1 1 1 1

k ,k k ,k k

Tk ,k k ,k k k ,k

x x

P P

=

=

) )

11 1

1 1

1

T Tk k ,k k k k ,k k k

k k ,k k k k k ,k

k k k ) k ,k

K P H ( H P H R )

x x K ( Z H x )

P ( I K H P

= += +

=

) ) )

1. Extended Kalman filter (EKF)Apply linearization to the nonlinear models using first order Tylor

expansion and neglecting higher order terms. Restrict the probability distribution of the motion and measurement

models to Gaussian distribution.

Prediction step

Update step

MATHEMATICAL MODELS

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Unscented Kalman filter

Nonlinear models are used no Tylor expansion used.

Restrict the probability distribution of the motion andmeasurement models to Gaussian distribution.

2. Sampling step1. Predict step 3. Update step

&r n

&V n

&Rbn

=D V n

Rbn f b ( 2ni e +

nen ) V

n + gn

Rbn( ni b

bi n )

0 0

2 2

0 0

2

0

2

0

1

2

1

2

i x i

i n x i n

n n

i i i i i ii i

nT

z i i ii

nT

x i i ii

X x , W( n )

X x ( n )P , W( n )

X x ( n )P , W( n )

z h( X ) , z W z , x W x

P W ( z z ) ( z z )

P W ( x x ) ( x x )

+ +

= =

=

=

= =+

= + + =+

= + =+

= = =

=

=

11 1

1 1

1

T Tk k,k k k k,k k k

k k,k k k k k,k

k k k ) k,k

K P H (H P H R )

x x K ( Z H x )

P (I K H P

= += +

=

) ) )

(Estimation filter)MATHEMATICAL MODELS

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A vehicular test was carried out through the core of downtown Kingston (Canada)and shows difficult scenarios for satellite navigation, with frequent partial GPSoutages of several seconds.

DGPS solution is developed to be the reference for the PPP positioning.

VEHICLE TEST

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latitude Longitude

Altitude

RESULTS (GPS AVAILABILTY)

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Estimation

filterEKF UKF

Positioning RMSE (m)Maximum error

(m)RMSE (m)

Maximum error

(m)

latitude 0.15756 0.20749 0.15441 0.20334

longitude 0.10866 0.27348 0.10740 0.26527

altitude 0.14151 0.46528 0.13727 0.45133

RESULTS (GPS AVAILABILTY)

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RESULTS (GPS OUTAGES)

Estimation filter EKF UKF

GPS outages 60s 30s 10s 60s 30s 10s

latitude (m) 0.51703 0.33407 0.20108 0.50668 0.32739 0.19705

longitude(m) 0.71550 0.42918 0.21442 0.70119 0.42060 0.21013

altitude (m) 0.40188 0.31040 0.15904 0.39384 0.30420 0.15585

To simulate the challenging conditions of the trajectory trip including

high and slow speeds, twelve simulated GPS outages of 60s, 30s and

10s are introduced.

Both EKF and UKF have similar accuracy-level during the outages.

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CONCLUSION

This research develops new algorithms for the integration of GPS PPP

and MEMS-based IMU.

un-differenced ionosphere-free linear combinations of pseudorange and

carrier phase measurements are processed.

Both EKF and UKF are used as estimation filters for the integrated

system.

The positioning results of integrated system show decimeter level

accuracy.

Both EKF and UKF results show the same level of accuracy for

positioning .

As a results, considering the computation cost of UKF compared with

EKF, EKF is sufficient as an estimated filter for the proposed GPS-

PPP/MEMS based IMU integrated system.

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Thank sQ ue s t i ons