multiple imu integration for vehicular navigationintegration to lay the foundation for the multiple...

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,


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


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).


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).


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


-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.


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


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 )




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


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


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


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


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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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%


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

multiple imu integration for vehicular navigationintegration to lay the foundation for the multiple...

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