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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
2
OUTLINES
Introduction
Problem statement
Research objectives
Mathematical models
Results
Conclusion
3
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).
4
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.
5
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.
6
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.
7
(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
8
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)
9
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
10
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
11
(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
12
(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
13
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
14
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
15
latitude Longitude
Altitude
RESULTS (GPS AVAILABILTY)
16
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)
17
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.
18
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.
19
Thank sQ ue s t i ons