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ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 1
Multiple IMU Integration for Vehicular
Navigation
Jared B. Bancroft
Position, Location And Navigation Group
Department of Geomatics Engineering
Schulich School of Engineering
University of Calgary
BIOGRAPHY
Jared B. Bancroft is a Ph.D. candidate in the PLAN
Group of the Department of Geomatics Engineering at the
University of Calgary. He received his B.Sc. in
Geomatics Engineering in 2007 and has worked with
navigation techniques since 2004. Jared's research
interests include pedestrian and vehicular navigation
through data fusion of multiple inertial units and satellite
navigation data.
ABSTRACT
One of the most widespread applications of the Global
Positioning System (GPS) is vehicular navigation.
Improving the navigation accuracy continues to be a focus
of research, commonly answered by the use of additional
sensors. A sensor commonly fused with GPS is the
inertial measurement unit (IMU). Due to the fact that the
requirements of commercial systems are low cost, small
size, and power conservative, micro-electro mechanical
sensors (MEMS) IMUs are used. They provide
navigation capability even in the absence of GPS signals
or in the presence of high multipath or jamming.
This paper addresses a centralized filter construction
whereby navigation solutions from multiple IMUs are
fused together to improve accuracy in GPS degraded
areas. The proposed filter is a collection of several single
IMU block filters. Each block filter is a 21 state IMU
filter. Because each block filter estimates position,
velocity and attitude, the system can utilize relative
updates between the IMUs. These relative updates
provide a method of reducing the position drift in the
absence of GPS observations.
The proposed filter’s performance is analyzed as a
function of the number of IMUs used and relative update
type, using a data set consisting of GPS outages, urban
canyons and residential open sky conditions. While the
use of additional IMUs (including a single IMU) provides
negligible improvement in open sky conditions (where
GPS alone is sufficient), the use of two, three, four and
five IMUs provided a horizontal position improvement of
25 %, 29 %, 32 %, and 34 %, respectively, when GPS
observations are removed for 30 seconds. Similarly, the
velocity RMS improved by 25 %, 31%, 33%, and 34% for
two, three, four and five IMUs, respectively. Attitude
estimation also improves significantly ranging from 30 %
– 76 %. Results also indicate that the use of more IMUs
provides the system with better multipath rejection and
performance in urban canyons.
INTRODUCTION
Since its inception, GPS has been widely used for
vehicular navigation. With the cost of OEM GPS
receivers declining and the increased user-friendliness of
commercial systems, GPS continues to propel the
advancement of vehicular navigation. While the position
remains at the core of any navigation system, the solution
can often be incorrect or unavailable. Thus research in
improving the accuracy and availability of the navigation
solution in vehicles using GPS is warranted.
Among the most detrimental factors affecting a GPS
based vehicular navigation system is the obstruction of
the lines of sight between the vehicle and the satellites.
As users travel in urban canyons, parkades and high
foliage areas, the ability for GPS to provide an acceptable
position is compromised. Although high sensitivity GPS
(HSGPS) receivers can track weak signals through fading
effects, this renders them susceptible to multipath. Thus
manufacturers have examined other sensors to integrate
with GPS to provide continuity when GPS signals are
unavailable.
Inertial measurement units (IMUs) are among the most
common source of integration to provide a continuous
navigation solution and thereby form an inertial
navigation system (INS). As the competitive consumer
market drives the price of vehicular navigation devices
down, an increasingly common choice for IMUs are
micro electro-mechanical sensors (MEMS). The size,
ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 2
cost, weight, and low power consumption make this an
attractive IMU type; however the in-run biases, scale
factors and high noise require the integration scheme to
be more robust.
While ongoing research typically involves one IMU, the
purpose of this study is to investigate the use of multiple
IMUs in a navigation solution. In particular, this study
investigates one approach to integrating multiple IMUs in
a centralized filter and the constraints that can be used to
further improve the accuracy in GPS signal degraded
areas.
The objectives of this paper are to:
1. Present a Kalman filter capable of incorporating
multiple IMUs and GPS observations.
2. Develop a series of relative updates to link the
inertial units together to improve performance.
3. Test the performance of the system developed in
software and analyze the results as a function of
the number of IMUs used.
Godha (2006) thoroughly discusses the use of MEMS
IMUs for vehicular navigation, including the use of height
and non-holonomic velocity constraints using both loose
and tight integration schemes. Other research in this area
includes: Salychev et al (2000), Mathur & Grass (2000),
Kealy et al (2001), Cao et al (2002), Hide (2003), Park
(2004), Shin (2005) and Godha (2005a & 2005b).
McMillan et al (1993) incorporated two IMUs in a
Kalman filter for marine applications. The system, called
Dual Inertial Integrated Navigation Systems (DIINS), was
used as a reference system to test other navigation
systems, e.g. Scherzinger et al (1996 & 1997). While it
provided fault detection on the IMU measurements, its
main focus was to provide redundancy in case of a single
IMU failure.
Brand and Philips (2003) introduced the use of two IMUs
for pedestrian navigation, using MEMS IMUs. While
their filter is similar to the one proposed herein, their
method used additional RF observables to directly
observe the distance between the IMUs (which were
placed on the fore foot). Bancroft et al (2008) also
applied two inertial units placed on the feet in a
centralized filter utilizing relative updates from the
predictable nature of the gait cycle.
Petovello et al (2005) used a dual-GPS/INS methodology
to quantify ship flexure in aircraft carriers. Two sets of
INS were used to determine the relative position of each
inertial system, and each INS was provided GPS
observables through the use of a GPS antenna and
receiver.
Waegli et al (2008), Osman et al (2006), Colomina et al
(2004), and Sukkarieh et al (2000) all propose multiple
IMU integration methods whereby the set of raw IMU
measurements are combined to form a synthetic
measurement. This approach has several benefits
including: (1) reduced input noise from the IMUs, (2)
fault detection on individual IMUs and (3) improved
navigation and state error estimation. Navigation
improvement for this approach typically ranges from 30-
45 % (Waegli et al 2008). Waegli also proposed a similar
multi-IMU model as presented herein utilizing relative
updates.
Because this paper addresses the use of multiple inertial
units, a brief description of the single IMU configuration
in an extended Kalman Filter (KF) is provided. The multi-
IMU filter is then discussed, followed by a description of
the relative updates used within the multi-IMU filter. The
filter performance is analyzed through data collected
under operational conditions (i.e. urban canyons and GPS
outages).
SINGLE IMU INERTIAL NAVIGATION SYSTEMS
The proposed multi-IMU filter is composed of single
IMU filters. Therefore this section will review the basic
integration to lay the foundation for the multiple IMU
filter.
The linear model of the KF appear as:
1 , 1k k k k kx x w
1 1 1k k kz H x
(1)
(2)
where,
1,kk is the transition matrix,
kx
is the corrections to the expansion point
(state vector),
H
is the design matrix,
kw
is the process noise,
1k is the observation noise, and
1kz
is the observation misclosure vector.
The single IMU Kalman filter estimates accelerometer
and gyro biases (6), scale factor errors (6) and corrections
to the linearization points of the position (3), velocity (3),
attitude (3) components. This combination is known as
the 21-state filter (Godha 2006). As this study utilizes
double differenced GPS observations, no additional states
are required to estimate the GPS receiver clock errors
(although if the filter operated in single point mode two
additional states would be required for the GPS receiver
clock bias and drift). Observations into the filter include
ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 3
double differenced pseudoranges and Doppler
observations.
The filter operates in a tight integration mode, allowing
the system to be updated with less than four satellites, a
key ability if poor GPS observability is expected (Knight
1999). Inertial measurements are corrected by the
estimated errors, mechanized into the navigation frame
(Earth Centered Earth Fixed, or ECEF) and used to
predict the filter forward. The system is updated as GPS
observations become available. Zero-velocity updates
may be applied while the vehicle is at rest. Figure 1 is a
flow chart describing the centralized integration method.
Figure 1 - Single IMU Integration
The details of the design, transition, process noise, and
observation covariance matrix are described in Godha
(2006).
Common constraints applied to the IMU filter are
non-holonomic velocity and height constraints. The
former constrains the across track and vertical velocity in
the vehicle’s body frame to zero, while the height
constraint fixes the absolute height of the system for short
durations. The combination of these constraints can
improve the accuracy of the solution by 73-85 % (Godha
2006).
MULTIPLE IMU INERTIAL NAVIGATION
SYSTEM
The multi-IMU filter is a combination of several single
IMU filters. Each “block” filter (i.e. each single IMU
filter) contains the same filter components, although are
related to a specific IMU. The block filters can be
updated at the same time, or updated individually. The
filter is generic enough that any number of IMUs can be
accommodated. It is noteworthy that each block filter
could contain different error states or state models,
allowing for varying IMU qualities, however for the
purposes of this study all block filters have the same
structure.
The multiple IMU filter can be expressed in terms of
equation (1) and (2) as:
11 1 1
, 11
22 2 2, 11
, 11
0 0 0
0 0 0
0 0 0
0 0 0
k kk k k
k kk k k
nn n nk kk k k
x x w
x x w
x x w
1 1 1 1
1 1 1 1
2 2 2 2
1 1 1 1
1 1 1 1
0 0 0
0 0 0
0 0 0
0 0 0
k k k k
k k k k
n n n n
k k k k
z H x n
z H x n
z H x n
(3)
(4)
where,
, 1
n
k k is the n
th block filter transition matrix,
1
n
kx is the nth
block filter states (21 state model),
1
n
kz is the misclosure vector from the nth
block
filter of the observations, n
kw is the process driving noise of the nth
block
filter, and
1
n
k is the measurement noise of the nth
block
filter.
Given that the block filters will contain the same states
(although related to different IMUs), an adaptation to the
filter must be discussed. Consider a GPS update is to be
performed with three available double differenced
pseudoranges. Because the misclosure is formed by the
difference between the actual observation and the
prediction of the observation, there will be a set of
misclosures for each block filter. Thus there will be 3 × n
observations (where n is the number of IMUs), although
only three are provided by the receiver. This allows for a
more practical approach to determining 21 × n states,
when in fact there are many more unknown parameters
than observations (as in the case of 5 IMUs being used
where there are 105 states).
The output of the filter will be n navigation solutions; the
weighted mean thereof providing the ultimate multi-IMU
navigation solution corresponding to the location of the
antenna. The weighted mean uses the estimated standard
deviation of each of the calculated solutions.
Figure 2 is a flow chart describing the centralized
integration method. The chart shows the observations
into the filter are measured GPS pseudorange and Doppler
observations and relative updates (as discussed next).
ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 4
Figure 2 - Multiple INS Configuration
RELATING THE NAVIGATION SOLUTIONS
Because the filter contains repeated sets of navigation
solutions (i.e. position, velocity and attitude), it is possible
to constrain the system so that all IMUs have
corresponding navigation solutions. In this section the
relative position, velocity and attitude update are
described. The position and distance update were
described in Bancroft et al (2008) in the context of
pedestrian navigation and the position update is reviewed
briefly below for context.
Relative Position Updates (RPUPT)
Because the IMUs are rigidly mounted, and the vector
between them (in the body frame) remains constant, it is
possible to use this information as an update to the
system. This update does not provide an absolute position
correction (as a GPS update does), but provides a method
of constraining the rate of divergence between two IMUs.
The inter-IMU vector in the body frame is computed by
differencing the lever arms (i.e. the vector from the GPS
antenna to the IMU in the body frame). When expressed
according to the form of equation (2), the update is given
by:
1 2 2,1
1 18 1 18
1 2 2,1
1 18 1 18
1 2 2,1 1 18 1 18
ˆ ˆ1 0 1
ˆ ˆ 1 0 1
1 0 1ˆ ˆ
x x xx x
y y y x x
x xz z z
r r r
r r r x
r r r
where,
1
x̂r is the estimated X coordinate of the 1st block filter
and 2,1
xr is the a priori known X axis distance between the
IMUs.
The misclosure is formed by differencing the two IMU
solutions and the known vector between the IMUs. It is
important to note that by differencing the lever arms, a
vector is formed in the body frame. Since the navigation
frame for this research is the ECEF frame, the vector must
be rotated into that frame. Consequently, there is an
inherent relationship between the efficacy of the RPUPT
and the error in the knowledge of the orientation of the
body frame relative to the ECEF frame
The estimated standard deviation of the RPUPT was set to
1 cm (a diagonal matrix) which was the precision of the
measured lever arms.
Relative Velocity Updates (RVUPT)
The relative velocity of a point on a moving rigid body is
given by Marion & Thornton (1995) and in the context of
this paper is:
2,1 2,1
1r r (5)
where, 2,1r
is the relative velocity between the IMUs 2
and 1,
1 is the rotation rate vector measured by IMU
1, and 2,1r is the relative vector between IMUs 1 and 2
(as determined from the lever arms).
It is with this observation model that the relative velocity
can be observed and used to update the system. The
relative vector between the IMUs is assumed to be known
a priori and the rotation vector is directly observed by the
inertial unit. This update therefore derives its input from
the observation of the IMU, whose accuracy is ultimately
controlled by the filter. When expressed according to the
form of equation (2), the update is given by:
1 2 2,1
1 3 1 18 1 15
1 2 2,1
1 3 1 18 1 15
1 2 2,1 1 3 1 18 1 15
ˆ ˆ0 1 0 1 0
ˆ ˆ 0 1 0 1 0
0 1 0 1 0ˆ ˆ
x x xx x x
y y y x x x
x x xz z z
v v r
v v r x
v v r
where,
1ˆxv is the X velocity of the 1
st block filter,
2ˆxv is the X velocity of the 2
nd block filter, and
2,1r
is computed as per equation (5).
As with the relative position update, the relative velocity
is “measured” in the body frame and must be rotated into
the navigation frame, thus creating a similar relationship
between the error of the rotation and the RVUPT.
The standard deviation used for RVUPTs was 2 cm/s.
This was derived using the propagation of variances of
equation (5).
ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 5
Relative Attitude Updates (RAUPT)
The relative attitude update follows a similar pattern to
the relative position update. The misclosures are formed
using the rotation between the individual estimated IMUs
attitude. In this research the IMUs are fixed on the same
platform ensuring a constant relative orientation. When
expressed according to the form of equation (2), the
update is given by:
2
1 1 6 1 18 1 12
2
1 1 6 1 18 1 12
2
1 1 6 1 18 1 12
0 1 0 1 0
0 1 0 1 0
0 1 0 1 0
B
B x x x
B
B x x x
B
B x x x
x
where,
2
1
B
B is the misclosure of the roll between the first and
second IMU body frames, 2
1
B
B is the misclosure of the pitch between the first and
second IMU body frames, and 2
1
B
B is the misclosure of the yaw between the first and
second IMU body frames.
In this research the relative attitude (rotation from IMU A
to IMU B) was assumed to be zero. This assumption
proved to be accurate to within a few degrees, when
compared with a GPS ambiguity resolved navigation
solution. Accordingly, the standard deviation of this
observation is 0.1 rad (5.7°).
DATA COLLECTION AND PERFORMANCE
EVALUTION
The 37 minute data set used to evaluate the filter contains
residential open sky conditions (25 %), urban canyons
(45 %) and seven 30-second simulated GPS outages
(30 %) (see Figure 3). The test consisted of speeds
ranging from zero to 70 km/h. The antenna and all IMUs
were mounted on the roof of a minivan.
Figure 3 - Map of Test Trajectory (Calgary, AB)
Data was collected from five functioning Cloud Cap
Technology MEMS IMUs (logging at 100 Hz) and a
NovAtel OEMV2 GNSS receiver (logging at 20 Hz) with
a NovAtel 702 antenna. The sensor assembly on the
vehicle is seen in Figure 4. The base station was a
NovAtel OEM4 receiver with a NovAtel 702 antenna
(logging at 20 Hz), located atop a campus building with
excellent satellite visibility. Data from each of the IMUs
was time tagged, but not synchronized. Data was then
aligned in a pre-processing step using linear interpolation.
Figure 4 - Sensor Assembly on Vehicle
A NovAtel OEM4 receiver and HG1700 tactical-grade
IMU (1 deg/hr) were used to provide the reference
solution. The data was processed with SAINTTM
(Petovello 2003) with carrier phase ambiguity resolution
enabled. The estimated position standard deviations of
the reference solution are shown in Figure 5. The
durations with larger variances are associated with the
time spent in the urban canyons. The largest horizontal
error (1 σ) is approximately 1 m, and a vertical error of
1.7 m.
0 500 1000 1500 2000 25000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Test Duration (s)
Std
(1
) (m
)
Reference Trajectory Quality
Horizontal Reference Std
Vertical Reference Std
Figure 5 - Reference Trajectory Estimated Position
Error Standard Deviations
GPS observability while on the residential streets usually
ranged from five to eight satellites, with the occasional
reduction to three satellites. During the urban canyons,
the number of satellites observed was erratic, and ranged
between zero and eight satellites. Figure 6 shows the
number of satellites observations over the course of the
data collection.
Urban Canyons
Simulated 30 s
Outages
ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 6
-2000 -1000 0 1000 2000 3000 4000 5000 6000-4500
-4000
-3500
-3000
-2500
-2000
-1500
-1000
-500
0
500
Easting (m)
No
rth
ing
(m
)NovAtel V1 Rx - Satellites Obsevered
Reference Trajectory
1-2 Satellites
3-4 Satellites
5-6 Satellites
7-8 Satellites
>=9 Satellites
Figure 6 - Satellite Observability during Test
A GPS only solution was derived from an extended
Kalman filter. In the urban canyons, the position solution
error increased to hundreds of metres, but provided
acceptable levels of accuracy during the residential areas
as seen in Figure 7. For comparison purposes, the HDOP
is also shown, indicating the areas of high position errors
are fundamentally linked to poor satellite geometry.
0 500 1000 1500 2000 25000
10
20
30
40
50
60
70
80
90
100
Test Duration (s)
Ho
rizo
nta
l E
rro
r (m
) -
Re
d
0 500 1000 1500 2000 25000
5
10
15
20
25
HD
OP
- B
lue
Figure 7 - GPS Only Solution Error
Because the combinations of IMUs are the focus of the
paper, the following notation is adopted to ease
explanations. Each IMU is designated A, B, C, D, or E.
Processing IMUs A and B in the multi-IMU filter is
denoted as AB. Four IMUs processed together would be
ABCD. Averaged solutions are denoted as
(ABC+ABD)/2, the result of averaging the combinations
ABC and ABD. Relative updates between two IMUs are
shown with an arrow, A→C indicates an update between
A and C.
The five IMUs were processed independently in single
IMU mode and then averaged to provide a benchmark,
(A+B+C+D+E)/5, to compare the use of additional IMUs.
This is the equivalent of an end user using five INS
solutions and performing a weighted average.
The level of performance of each IMU, as quantified by
biases and scale-factor errors are different. Therefore, to
obtain a single metric to show the improvement in the
navigation solution as a function of the number of IMUs,
all possible pairs of IMUs were processed and then
averaged to get (AB+AC+AD+AE+BC
+BD+BE+CD+CE+DE)/10. The process was also
followed for three and four IMU combinations. In this
manner, comparisons between different numbers of IMUs
is possible without being selective of the different
qualities and performance.
In the presence of strong satellite geometry (e.g. HDOP <
1.5) the system’s performance becomes independent of
the number of IMUs used; the RMS position accuracy is
typically better than 1 m. While redundant solutions
provide a more robust solution, the absolute accuracy of
the system does not improve with the use of more IMUs.
However, in the absence of GPS observations, or in the
presence of degraded GPS observations, additional IMUs
provide superior accuracy, as is shown in the next section.
RESULTS DURING SIMULATED OUTAGES
Simulated outages provide the ability to compare the
performances of the various systems in the absence of
GPS data. Seven (7) thirty second outages were
simulated every 100 s leaving 70 s for the filter to re-
converge. During the 30 s outages no GPS observables
were used in the filter.
Maximum errors during the 30 second outages ranged
between 25.7 m and 94.7 m for the single IMU solutions
(A, B, C, D). While the solution was inconsistent
between IMUs, the estimated variances corresponded to
the solution accuracy; this indicates that the required filter
tuning was sufficient for each IMU. Filter parameters for
each IMU remained the same, whether processed in single
or multiple IMU modes. Figure 8 shows the time-
matched RMS errors and the estimated standard
deviations for all seven outages.
ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 7
0 5 10 15 20 25 300
10
20
30
40
50
60
70
80
90
100
GPS Outage Duration (s)
Ho
rizo
nta
l P
os
itio
n E
rro
r (m
)RMS Error of All(7) 30 s GPS outages
IMU A Error
IMU B Error
IMU C Error
IMU D Error
IMU E Error
IMU A Std (1 )
IMU B Std (1 )
IMU C Std (1 )
IMU D Std (1 )
IMU E Std (1 )
Figure 8 - RMS of Position Errors during Seven 30 s
Outages (Single IMU Configuration)
It should be noted that no other constraints, e.g. velocity
or height constraints, have been applied in any of the
solutions. This is to showcase the direct impact of
additional IMUs and their updates relative to the single
IMU case. Using additional constraints would improve
the accuracy of each block INS, but reduce the ability to
provide insight into the multiple IMU filter performance.
When a “relative update” is said to be applied, it means
that for each IMU combination being employed, a single
update for that combination is applied. For example,
there are 6 updates applied between 4 inertial units (i.e.
A→B, A→C, A→D, B→C, B→D, and C→D). When
applying all three relative updates (i.e. RPUPTs,
RVUPTs, and RAUPTs) there would be a total of 18
observations (relative updates) to the system.
Applying the RPUPT shows a 23 %, 29 %, 32 %, and
34 % improvement in horizontal position error for two,
three, four, and five IMUs respectively in comparison to
the benchmark solution. Applying the RVUPT shows a
23%, 24%, 31% and 33% improvement in the horizontal
position error for two, three, four, and five IMUs
respectively in comparison to the benchmark solution.
Figure 9 show the results when using the RAUPT.
Applying the RAUPT shows a 22 %, 23 %, 27 %, 28 %
improvement for two, three, four, and five IMUs
respectively in comparison to the benchmark solution.
Also noteworthy is the increased estimated standard
deviations, which in comparison shows a more
pessimistic solution as seen in Figure 9.
0 5 10 15 20 25 300
10
20
30
40
50
60
GPS Outage Duration (s)
Ho
rizo
nta
l P
os
itio
n E
rro
r (m
)
RMS Error of All(7) 30 s GPS outages
Benchmark Error
2 IMU RAUPT Error
3 IMU RAUPT Error
4 IMU RAUPT Error
5 IMU RAUPT Error
Benchmark Std (1 )
2 IMU RAUPT Std (1 )
3 IMU RAUPT Std (1 )
4 IMU RAUPT Std (1 )
5 IMU RAUPT Std (1 )
Figure 9 – RMS of Position Errors during Seven 30 s
Outages with RAUPT Applied (Multiple IMU
Configurations)
Figure 10 shows the results of the system as a function of
time when applying RPUPTs, RVUPTs, and RPUPTs.
Applying both updates shows a 25%, 29 %, 32 %, 34 %
improvement for two, three, four, and five respectively,
when compared with the benchmark solution. It is clear
that the combination of the three update types provides
superior accuracy as opposed to any single update (in the
position domain). It is also clear that the position update
performs better than the relative velocity and attitude
update. See APPENDIX A, which shows the statistical
values of the horizontal error during all seven outages
using the different combinations of update types.
0 5 10 15 20 25 300
10
20
30
40
50
60
GPS Outage Duration (s)
Horizonta
l P
ositio
n E
rror
(m)
RMS Error of All(7) 30 s GPS outages
Benchmark Error
2 IMU Error
3 IMU Error
4 IMU Error
5 IMU Error
Benchmark Std (1 )
2 IMU Std (1 )
3 IMU Std (1 )
4 IMU Std (1 )
5 IMU Std (1 )
Figure 10 – RMS of Position Errors during Seven 30 s
Outages with RPUPT, RVUPT and RAUPT Applied
(Multiple IMU Configurations)
RESULTS DURING URBAN CANYONS
Urban canyons present a challenging environment,
coupling poor satellite visibility and high multipath on
ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 8
those satellites which are observable. Figure 11 shows
the horizontal errors during the time spent in the urban
canyon. It also contains the HDOP (Horizontal Dilution
of Precision) of the satellite observations used in the GPS
updates. From Figure 11 it is apparent that there is a
direct relation to the availability of satellite observations,
and the error of the solution. Thus the system is still
dependent on GPS for IMU error mitigation, but
performance does improve with additional IMUs. The
most relevant portion of the data consists from 100 - 500
seconds where the satellite observability fluctuates and is
particularly poor. During this period there is a 51 %
improvement in the RMS (over the 400 s period) with the
use of two IMUs and 54 % for the use of three IMUs and
55 % for the use of four and five IMUs. The
improvement is less substantial when looking at the
maximum error. The reduction of the maximum
horizontal error is 26 %, 34 %, 37 %, and 40 %
improvement for two, three, four and five IMUs.
0 100 200 300 400 500 600 700 800 900 10000
10
20
30
40
50
60
70
80
90
100
Test Duration (s)
Ho
rizo
nta
l E
rro
r (m
)
Benchmark Error
2 IMU Error
3 IMU Error
4 IMU Error
5 IMU Error
0 100 200 300 400 500 600 700 800 900 10000
5
10
15
20
25
HD
OP
- B
lue
Figure 11 HDOP and Horizontal Errors during Urban
Canyons with RPUTP, RVUPT and RAUPT Applied
(Multiple IMU Configurations)
An important part of any integrated system with GPS is
the detection and rejection of faulty GPS measurements.
With the unique characteristic of the proposed filter, the
fault detection system is now discussed, although it is not
different than that of a typical single IMU INS.
Fault detection is the process of analyzing the innovation
sequences in the filter update cycle for values lying
outside a specific region of the distribution. Given that an
innovation sequence is the difference between actual and
predicted observations (derived from the prediction stage
of the filter), the distribution of innovation sequences is
normal, assuming the predicted observations and actual
observations are normally distributed, an assumption
made in all Kalman filter applications. Thus the
individual values in the sequence can be tested to see if
they in fact lie within a specified area under the normal
distribution curve (Petovello 2003).
The null and alternative hypothesis in the fault detection
test is given as:
0| ~ (0, )H vv N C (6)
| ~ ( , )AH vv N M C (7)
where, v is the innovation sequence
vC is the innovation sequences variance is the
covariance matrix,
M is the fault mapping matrix (also referred to
as the blunder design matrix),
is the vector of blunders, and
( , )N p q indicates a normal distribution of mean p and
variance q.
The test statistic (Petovello 2003) to determine the
validity of the alternative hypothesis is computed as:
1
1 1 1T T T
v v vT v C M M C M M C v . (8)
As shown in (8), the only input into the test statistic is the
innovation sequence, the fault mapping matrix and the
innovation covariance matrix. Therefore, in considering
the difference between a single set of GPS observations in
a typical INS update and the proposed multi-IMU filter
update test statistic, the innovation and its variance are
analyzed next. The innovation sequence using two IMUs
is:
1 1( )1
2 2 2( )
ˆ
ˆ
z H xv
v z H x. (9)
The innovations sequence is segmented according to the
block states and block design matrices, which show that
(9) is in fact still normally distributed (pending the
observations and states are normal). The covariance of
the innovation sequence is defined as: T
x zvC H C H C and in the context of the two IMUs is
given as:
1 12 1 12
21 2 21 2
1 1
2 2
0 0 0
0 0 0
T
zv v x x
zv v x x
C C C CH H C
C C C CH H C
(10)
1 1
1 1T
zv xC H C H C (11)
2 2
2 2T
zv xC H C H C (12)
12 21 12
1 2T T
v v xC C H C H (13)
The diagonal nature of the design matrix for the multi-
IMU filter ensures the decoupling of the covariance of the
ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 9
innovation sequence (i.e. equations (11) and (12) are not
functions of each other). Therefore the innovation test is
still valid and performed on all values in the innovation
sequence. This research assumed a single fault in the
entire innovation sequence and congruently multiple
faults are recursively detected. Future work will include a
more correct assumption whereby the GPS observations
are assumed to be the fault. The distinguishing factor in
these two assumptions forms the definition of a fault,
whether the fault exists in the element of the innovation
sequence or the GPS observation, the latter being
preferred.
The urban canyon data experienced approximately 0.5 %
pseudorange rejection and 1.9 % Doppler rejection (which
fluctuates slightly with the number of IMUs and type of
relative updates applied). The number of GPS
observations can be seen in Figure 12; red points
indicating the number of rejected observations for the five
IMU RPUPT, RVUPT and RAUPT solutions.
0 100 200 300 400 500 600 700 800 900 1000
2
4
6
1
3
5
GP
S P
su
ed
ora
ng
es
0 100 200 300 400 500 600 700 800 900 1000
2
4
6
1
3
5
Test Duration (s)
GP
S D
op
ple
r
Ob
se
rva
tio
ns
GPS Observations Used Rejected GPS Observation
Figure 12 - GPS Observations Rejected during Urban
Canyons with RPUPT, RVUPT and RAUPT Applied
In the urban canyon data set there was an increase in
horizontal position errors when the RAUPTs were
applied, however this was not observed in the 30 s outage
periods (see Table 2 in APPENDIX A). It was shown in
Figure 9 that the relative attitude updates do not decrease
the position variance as effectively as the relative position
updates. Thus, the estimated standard deviations of the
position while using RAUPTs are pessimistic during
times of GPS outages. In this case, the system is more
willing to accept GPS observations, thereby reducing the
ability to provide optimal fault detection. This can be
seen as an increase in misdetections of GPS blunders
(type II errors) as a result of pessimistic position
variances. Therefore the system accepts blunders of a
higher magnitude and incorporates them into the solution
rather than rejecting them as done when RPUPTS are
applied. This effect is removed when the combination of
RPUPT and RAUPT is applied.
VELOCITY AND ATTITUDE ESTIMATION
The application of RVUPTs and RAUPTS increases the
accuracy of the velocity and attitude estimation. During
the entire data set (with RPUPTs, RVUPTS and RAUPTS
applied), the velocity improvement relative to the
benchmark solution was 10 %, 18 %, 20 %, and 30 %
with two, three, four and five IMUs, respectively.
Pertaining to the entire data set the roll improved 57 %,
45 %, 39 % and 34 %, with two, three, four and five
IMUs respectively. This presents a peculiarity in that as
more IMUs are added, the corresponding improvement
decreases. The absolute roll error however remained less
than a few degrees, and is likely a result of the direct
comparison between the reference solution and the noisy
output of the INS solution's. The pitch conversely
improved with the number of IMUs, to 43 %, 57 %, 57 %,
and 63 % for two, three, four and five IMUs, respectively.
However, the most improved parameter in the entire
multi-IMU filter is heading. For two, three, four and five
IMUs, the improvement was 30 %, 48 %, 59 % and 76 %,
respectively. This proves to be very valuable in
recognizing that the heading error is typically the limiting
factor in GPS/MEMS integration.
Figure 13 shows the percent improvement of all
navigation parameters as a function of IMUs when
RPUPT, RVUPT and RAUPT are applied. It is important
to note that the benchmark solution from which the
improvement is derived is an average of five
independently processed solutions. Figure 13 therefore
shows that using two IMUs utilizing the relative updates
is superior to the use of five units independently
processed and averaged together.
2 3 4 520
30
40
50
60
70
80
IMUs Used in Solution
Pe
rce
nt
Imp
rov
em
en
t
Horizontal Position
Horizontal Velocity
Roll
Pitch
Heading
Figure 13 - Summary of Percentage Improvement as a
Function of IMUs Used with RPUPT, RVUPT, and
RAUPT Applied
ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 10
INTER-BLOCK FILTER CORRELATION
Because the IMU filters are stacked and the relative
updates are applied to the position, velocity, and attitude,
correlation between the block filters is inevitable.
Bancroft et al (2008) showed that in the pedestrian case,
the filter correlation was negligible between states (other
than position) in the case of a twin IMU filter. However,
given the increased size of the state matrix and the
associated numerical instability of larger matrices (and
inversion), a similar analysis is repeated here with three
IMUs, using RPUPTs, RVUPTs and RAUPTs. Also in
Bancroft et al (2008), the IMUs were placed on the feet,
resulting in larger dynamics and the ability to provide
zero velocity updates which yield other filter effects.
APPENDIX B shows the correlation coefficient matrix as
a color map. Warmer colors represent higher correlations.
The figure is subdivided into nine blocks, the diagonal
block representing the inner correlation between states of
a block filter.
The position estimates all have high inner and
inter-correlation (i.e. > 80 %). This is expected as a result
of the relative position updates, and the fact that they are
almost co-located. High correlations (~ 65%) also exist
between the inter-IMU velocity and attitude states, but are
not as substantial as the position states, nor as repeatable.
The majority of the IMU error states are estimated as
uncorrelated (i.e. < 15%) with exception of the gyro
biases. Sporadic correlations exist on the Z
accelerometers (20 - 40 %), but do not appear to follow a
particular trend or pattern. Another noteworthy exception
is the high correlation between the gyro biases and the
navigation states of the other IMUs. In any case, the fact
that inter-correlation exists between blocks is unfortunate,
but again appears not to be significant.
CONCLUSIONS
A centralized integrated navigation system whereby any
number of inertial units can be fused together was
proposed. The filter combines n IMUs and relates the
common states through relative updates between IMUs.
It was shown that relative attitude updates provide
pessimistic position variances, while the relative position
and velocity updates provide more realistic position
variances. The combination of position, velocity, and
attitude updates provides the most significant
improvement. In seven 30 s outages, the position
improvement using five inertial units was 34 %, while in
urban canyons there was an improvement of 55 % on the
RMS error during 400 s of data. The improvement
percentages are given relative to five inertial solutions
determined independently and finally averaged. Inter-
IMU correlation was observed in the multi-IMU filter, but
the correlation between IMU error states is minor.
ACKNOWLEDGMENTS
The author would like to thank his supervisors Professors
Gérard Lachapelle and Elizabeth Cannon for their
continuous encouragement and support. The author
would also like thank the Natural Science and
Engineering Council of Canada (NSERC) and the
Informatics Circle Of Excellence (iCORE) for financial
support. The author also acknowledges Dr. Thomas
Williams for assistance in this research. Also
acknowledged is Aiden Morrison, PhD candidate in the
PLAN Group, for the design and construction of the
NavBox™ used to collect the data.
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ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 12
APPENDIX A
Table 1 – Seven 30 Second Outage Position Accuracy Statistics
Benchmark IMUs
Used
RPUPT RAUPT RVUPT
RPUPT & RVUPT &
RAUPT
RMS
(m)
Max
(m)
RMS
(m)
Max
(m)
Max
Improvement
RMS
(m)
Max
(m)
Max
Improvement
RMS
(m)
Max
(m)
Max
Improvement
RMS
(m)
Max
(m)
Max
Improvement
17.8 46.2
2 13.6 35.5 23.2% 14.1 36.1 21.8% 13.6 35.5 23.1% 13.3 34.5 25.4%
3 12.6 32.9 28.8% 14.0 35.6 23.0% 13.5 35.3 23.7% 12.9 32.8 29.0%
4 12.1 31.5 31.8% 13.3 33.8 26.8% 12.2 31.8 31.2% 12.3 31.4 32.1%
5 11.8 30.7 33.6% 13.0 33.3 28.0% 11.9 31.1 32.7% 12.0 30.5 34.0%
Table 2 - Urban Canyon Position Accuracy Statistics
Benchmark IMUs
Used
RPUPT RAUPT RVUPT
RPUPT & RVUPT &
RAUPT
RMS
(m)
Max
(m)
RMS
(m)
Max
(m)
RMS
Improvement
RMS
(m)
Max
(m)
RMS
Improvement
RMS
(m)
Max
(m)
RMS
Improvement
RMS
(m)
Max
(m)
RMS
Improvement
11.0 80.7
2 9.2 56.0 16.4% 12.3 85.5 -12.2% 9.3 55.4 15.7% 9.1 58.0 17.4%
3 9.1 51.7 17.0% 11.1 72.9 -1.3% 9.4 58.7 14.0% 8.6 51.9 21.3%
4 9.0 51.6 18.1% 11.3 76.4 -3.1% 9.1 56.1 17.2% 8.6 49.0 21.9%
5 9.2 53.5 16.3% 11.1 74.6 -1.1% 9.1 57.6 16.7% 8.5 46.8 22.3%
Table 3 - Entire Data Set Position Accuracy Results (includes outages)
Benchmark IMUs
Used
RPUPT RAUPT RVUPT
RPUPT & RVUPT &
RAUPT
RMS
(m)
Max
(m)
RMS
(m)
Max
(m)
RMS
Improvement
RMS
(m)
Max
(m)
RMS
Improvement
RMS
(m)
Max
(m)
RMS
Improvement
RMS
(m)
Max
(m)
RMS
Improvement
9.0 80.6
2 7.8 59.6 13.3% 9.7 85.5 -7.2% 7.9 59.9 12.9% 7.7 59.4 14.9%
3 7.6 55.1 16.3% 9.0 72.9 0.9% 7.9 59.2 12.3% 7.3 53.6 18.6%
4 7.4 52.6 18.4% 8.9 76.4 1.2% 7.4 56.1 17.5% 7.2 50.6 20.4%
5 7.4 53.5 17.8% 8.8 74.6 3.0% 7.4 57.6 17.7% 7.1 48.7 21.7%
ION GNSS 2009, Session D2, Savannah, GA, 22-25 September, 2009 13
APPENDIX B
Correlation Coeffecient of Covariance Matrix
GPS Time: 331859.90 s
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
State Vector Order
Position (X,Y,Z)
Velocity (Vx,Vy,Vz)
Attitude (Pitch,Roll,Yaw)
Gyro Bias (X,Y,Z)
Accel Bias (X,Y,Z)
Accel Scale Factor (X,Y,Z)
Gyro Scale Factor (X,Y,Z) C
orr
ela
tio
n C
oe
ffic
ien
t M
ag
nit
ud
e IMU A
IMU B
IMU C