INS Aided GPS Integer Ambiguity Resolution

Anning Chen, Dongfang Zheng, Arvind Ramanandan and Jay A. Farrell

Abstract— Real-time high precision GPS positioning is basedon carrier phase measurements, which requires fast and preciseon-the-fly integer ambiguity resolution. In some navigationapplications, external sensors are available that provide aux-iliary measurements. For example, in GPS/INS navigationsystems, the inertial sensors allow computation of prior positionestimates when GPS signals become available. This informationcan be used to aid GPS integer ambiguity resolution, offeringa higher probability of obtaining correct integers, especially inchallenging GPS conditions (e.g., few satellites, high measure-ment noise).

This paper describes a fast and efficient technique for integerambiguity resolution when auxiliary information from INS isavailable. The theoretical derivation will be presented andsimulation result will show the effectiveness of the proposedmethod.

I. INTRODUCTION

Real-time high precision positioning from the global po-

sitioning system (GPS) is based on the carrier phase mea-

surements, which requires fast and reliable on-the-fly integer

ambiguity resolution. In many navigation applications, exter-

nal sensors are available that provide auxiliary measurements

which can improve integer ambiguity resolution, especially

in GPS challenging conditions (e.g., few satellites available,

high multipath). For example, in land vehicle control and

guidance, the altitude of the roadway as a function of

arclength might be available [2].

Integrated GPS/INS (Inertial Navigation System) is a

popular tool for localization [4], [5]. Localization accura-

cies of a few centimeters can be achieved using carrier

phase processing assuming rapid and accurate on-the-fly

integer ambiguity resolution. One of the main advantages

of GPS/INS integration over a stand alone GPS system is

the capability of the former to maintain position estimate

accuracy during short GPS denied periods. This INS state

estimate can be used to facilitate GPS integer ambiguity

resolution which is the topic of this paper. This would

be helpful especially in GPS challenged areas (e.g. Urban

canyons, tunnels, thick canopy etc.) where the GPS receiver

may not be able to track a sufficient number of satellites to

resolve the integer ambiguities.

Incorporating INS measurements in GPS integer ambigu-

ity resolution has been studied in [9], in which the INS data

were used to reduce integer searching space. However, in [9],

the closed form of the searching space was not derived, and

the weighting factor between GPS and INS measurements are

Anning Chen, Dongfang Zheng, Arvind Ramanandan and JayA. Farrell are with Department of Electrical Engineering, Univer-sity of California Riverside, Riverside, CA, 92521, U.S.A. Email:{achen;dzheng;aramanandan;farrell}@ee.ucr.edu.

based on experiences, both of which reduce the computation

efficiency and estimation performance of the algorithm.

In this paper, we extend the approach in [1] with auxiliary

position estimate measurements from an INS. We present the

theoretical derivation and introduc a fast and efficient method

for GPS integer ambiguity resolution. Two sets of simulation

result shows the effectiveness of the proposed approach.

II. MEASUREMENTS MODEL

A. DGPS Measurements

Through this paper, we consider single difference GPS

(DGPS) measurements. For simplicity of notation, we as-

sume that the DGPS approach completely removes all

common-mode errors (e.g., ionosphere, troposphere, satellite

clock and ephemeris). The DGPS code and carrier phase

measurements for satellite i can be modeled as

ρ(i) = R(i) + cδ tr + ε(i) (1)

λφ (i) = R(i) + cδ tr +λN(i) +η(i) (2)

where R(i) = ‖X (i)−Xa‖ is the geometric distance between

the position X (i) of satellite i and position Xa of the receiver

antenna . The symbol cδ tr is the receiver clock bias. The

symbol λ represents the signal wavelength. The symbols

ε(i) and η(i) represent the measurement noises in code

and phase measurements. The symbol N(i) represents the

unknown integer ambiguity that is to be determined. The

index i = 1, · · · ,K, where K is the number of satellites in

view.

B. Residual Measurements

The approach works with residual measurements com-

puted relative to a position X0. It is assumed that xa0 = (Xa−X0) is sufficiently small such that the h.o.t are neglectable

after linearization. The residual measurements are

δρ(i) = ρ(i)−‖X (i)−X0‖ (3)

λδφ (i) = λφ (i)−‖X (i)−X0‖. (4)

The linearized residual measurements are modeled as

δρ(i) = h(i)xa0 + cδ tr + ε(i) (5)

λδφ (i) = h(i)xa0 + cδ tr +λN(i) +η(i) (6)

where and h(i) ∈ R3. We assume that ε(i) ∼ N (0,σ2ρ ) and

η(i)j ∼ N (0,σ2

Φ). In typical GPS applications, σρ is at the

meter level and σΦ is at the centimeter level. All the noise

terms are uncorrelated with each other.

The phase residual measurements can be put in matrix

form as

λδφφφ = Hx+λN+ηηη (7)

2011 IEEE International Conference on Control Applications (CCA)Part of 2011 IEEE Multi-Conference on Systems and ControlDenver, CO, USA. September 28-30, 2011

978-1-4577-1063-6/11/$26.00 ©2011 IEEE 567

where δφφφ =[

δφ (1), · · · , δφ (K)]�

, x =[x�a0 cδ tr

]�,

H =

⎡⎢⎣

h(1) 1...

...

h(K) 1

⎤⎥⎦, ηηη =

[η(1) · · · η(K)

]�, and N =

[N(1) · · · N(K)

]� ∈ZK is the integer ambiguity vector

that is to be determined.

For the simplicity of notation in the integer ambiguity

problem, we rewrite (7) as:

y = Gx+N+v (8)

where y = δφφφ ∈ RK represents the DGPS phase measure-

ments, x∈Rn and N∈ZK are the parameters to be estimated,

and n = 4. G = λ−1H ∈ RK×n is the observation matrix

characterizing the satellite-user-reference station geometry,

the noise term v=ηηη/λ ∈RK and v∼N (0,ΣΣΣvv), ΣΣΣvv =σ2

ΦIλ 2 .

C. INS Measurements

In GPS/INS systems, the INS propagates the navigation

state vector and the error covariance matrix during GPS

outage. Assume at some time, the position estimate from

the INS is xa0 with covariance P. This information from the

INS can be represented as

xa0 = J x+n, (9)

where J ∈ Rp×n, and cov(n) = P. In typical GPS/INS sys-

tems, p = 3 and J =[

I 0], as the INS state keeps track

of the 3 dimensional position, but not the receiver clock bias.

This prior knowledge of xa0 from the INS can facilitate GPS

integer ambiguity resolution.

III. INTEGER AMBIGUITY PROBLEM

A. Problem Statement

INS aided GPS integer ambiguity problem can be modeled

as a Bayesian problem, which is, given the prior of GPS

measurements y = Gx+N+ ηηη and INS measurements of

J x ∼ N (xa0,P), we would like to find the estimate of Nand x that

(N, x

)= arg max

N∈Zm,x∈Rnf (N,x|y).

According to Bayesian Rule,

(N, x

)= arg max

N∈Zm,x∈Rn

f (y|N,x) f (N|x) f (x)f (y)

As there is no uncertainty in N given x and f (y) is

independent of x and N,

(N, x

)= arg max

N∈Zm,x∈Rnf (y|N,x) f (x)

= arg maxN∈Zm,x∈Rn

ln( f (y|N,x) f (x)) . (10)

As

f (y|N,x) =e−

12 (y−Gx−N)T Σ−1

vv (y−Gx−N)

(2π)K/2 |ΣΣΣvv|1/2

and

f (x) = f (J x) =e−

12 (Jx−xa0)

T P−1(Jx−xa0)

(2π)K/2 |P|1/2.

Then,(N, x

)= arg max

N∈Zm,x∈Rn

((y−Gx−N)T Σ−1

vv

(y−Gx−N)+(Jx− xa0)T P−1 (Jx− xa0)

).(11)

Therefore, the INS aided GPS integer ambiguity problem

is an Integer Weighted Least-Square (IWLS) problem [10].

Our objective is to find N ∈ ZK , x ∈ Rn that minimize the

cost function

c(x,N) = (y−Gx−LN)T Σ−1vv (y−Gx−LN)

+(Jx− xa0)T P−1 (Jx− xa0) (12)

We reformulate the measurements by stacking the INS

prior with GPS only measurements as

y = Gx+LN+ n (13)

where the new measurement vector y� =[

y� x�a0

] ∈RK+p, the new measurement noise vector n� =[

v� n� ] ∈ RK+p, and G�=

[G� J�

] ∈ R(K+p)×n,

L� =[

I 0] ∈R(K+p)×K . For the simplicity of notation,

let m = K + p.

As INS state uncertainty n is not correlated with GPS

measurement noise v, the covariance matrix of the whole

measurement noise vector is

ΣΣΣ = cov([

vn

])=

[ΣΣΣvv 00 P

]. (14)

We can rewrite the cost function in (12) as:

c(x,N) =(y− Gx−LN

)� ΣΣΣ−1(y− Gx−LN

)= ‖y− Gx−LN‖2

ΣΣΣ. (15)

B. Generating Searching Candidates

One of the leading algorithms in integer ambiguity prob-

lem is LMS [8]. LMS is based on a very useful insight

[6] that although (13) contains (K +n) unknown variables,

there are only n degrees of freedom. Given x, all the integer

ambiguities can be resolved. 1 These remarks show that not

all combinations of integers are admissible and the challenge

is to reformulate (13) properly to find admissible integer

vectors efficiently. The basic idea of LMS is to search only

over the admissible combinations of integer candidates so

that the searching space can be decreased. The original LMS

procedure of LMS is in [8]. Alternative implementations are

presented in [5], [7], [13].

Divide the integer vector N into two subvectors NC and

ND, where ND contains n integers and NC contains the

remaining (K−n) integers. The integers in ND are searched

exhaustively over some range of d integers, the remaining

integers are computed as real value estimates and are rounded

to an optimally selected integer (described below) to get the

1Note that this is equivalent to the statement that precedes (10).

568

estimate of NC. This yields dn integer vectors in ZK . We can

evaluate each integer vector candidate to find the one with

least cost. As in LMS, this decreases the search dimension

from K to n, which decrease the integer vector candidates

from dK to dn.

Starting from (13), for a given integer ambiguity N, the

weighted least square estimate of x would be:

x =(

GT ΣΣΣ−1G)−1

GT ΣΣΣ−1 (y−LN) (16)

and the residual vector is

εεε = y− G · x−LN

=

(I− G

(GT G

)−1ΣΣΣ−1GT ΣΣΣ−1

)(y−LN)

= QΣΣΣ (y−LN) . (17)

where

PΣΣΣ = G(

GT ΣΣΣ−1G)−1

GT ΣΣΣ−1, (18)

QΣΣΣ = I− PΣΣΣ. (19)

Note that both PΣΣΣ and QΣΣΣ are idempotent and that Rank(P)=n and Rank(Q) = (m−n).

Because ΣΣΣ > 0 is a covariance matrix, ΣΣΣ−1 is symmetric

and positive definite. Thus it can be factored as

ΣΣΣ−1 = W�M�MW, (20)

where W∈Rm×m is a unitary matrix (i.e., WW� =W�W=I) and M� = M is a diagonal matrix with positive elements

on the diagonal. Substituting (20) into (18), we have

PΣΣΣ = G(G�ΣΣΣ−1G)−1G�ΣΣΣ−1

= G(G�W�M�MWG)−1G�W�M�MW= G(A�A)−1A�MW= W−1M−1A(A�A)−1A�MW= W−1M−1PMW, (21)

where P = A(A�A)−1A� and A = MWG. Eqn. (21) shows

that PΣΣΣ is similar to P where P is a projection matrix onto

the range of A; therefore, rank(P) = rank(A). Because Mand W are both nonsingular, rank(A) = rank(G) = n; hence,

rank(P) = n yields rank(PΣΣΣ) = n.

Let Q = I − P, then Q is a projection matrix onto the

subspace orthogonal to the range space of A, and has rank

(m−n). The following analysis shows that QΣΣΣ is similar to

Q:

QΣΣΣ = I− PΣΣΣ

= I−W�M−1PMW= W�M−1(I−P)MW= W−1M−1QMW. (22)

Therefore, QΣΣΣ also has rank (m−n).

From (15), the cost function evaluated from candidate Nis

c(N) = ‖y− Gx−LN‖2ΣΣΣ

= ‖QΣΣΣ(y−LN)‖2ΣΣΣ

= (y−LN)�Q�ΣΣΣ ΣΣΣ−1QΣΣΣ(y−LN)

= (y−LN)�Q0(y−LN) (23)

where

Q0 = Q�ΣΣΣ ΣΣΣ−1QΣΣΣ

=(I− PΣΣΣ

)� ΣΣΣ−1(I− PΣΣΣ

)= ΣΣΣ−1 −ΣΣΣ−1PΣΣΣ − P�

ΣΣΣ ΣΣΣ−1 + P�ΣΣΣ ΣΣΣ−1PΣΣΣ

= ΣΣΣ−1 −2ΣΣΣ−1G(G�ΣΣΣ−1G)−1G�ΣΣΣ−1

+ΣΣΣ−1G(G�ΣΣΣ−1G)−1G�ΣΣΣ−1G(G�ΣΣΣ−1vv G)−1G�ΣΣΣ−1

= ΣΣΣ−1 −ΣΣΣ−1G(G�ΣΣΣ−1G)−1G�ΣΣΣ−1

= ΣΣΣ−1(I− PΣΣΣ

)= ΣΣΣ−1QΣΣΣ (24)

Proposition 3.1: Rank(Q0) = m−n.

Proof: First we should notice that by use of (20) and

(22), Q0 can be written as:

Q0 = Q�ΣΣΣ ΣΣΣ−1QΣΣΣ

= (W−1M−1QMW)�

(W−1M�MW)(W−1M−1QMW)

= W−1M�QM−T WW−1M�MWW−1M−1QMW= W−1M�QMW. (25)

Following (19), we stated that Q is a projection matrix with

rank (m−n). Because M and W are all nonsingular, Q0 is

similar to Q; therefore, rank(Q0) = m−n.

Let the SVD (single value decomposition) of Q0 be

Q0 = US2U�,

where U is unitary and S is diagonal with diag(S) =[s1, . . . , sm−n,0, . . . ,0] with all si > 0 for i = 1, . . . ,m − n.

Define B = SU�such that

Q0 = B�B. (26)

where the last n rows of B are zero, and matrix B can be

represent as

B =

[A0

]=

[C D E0 0 0

],

where A ∈R(m−n)×m, C ∈R(m−n)×(K−n), D ∈R(m−n)×n and

E ∈R(m−n)×p.

Given the above analysis, the cost function of (23) can be

rewritten as

c(N) = (y−LN)�B�B(y−LN)

= ‖B(y−LN)‖2. (27)

569

Because B does not have full rank, the null space of B is

not empty. Therefore, there exists (non-unique) N ∈Rm such

that (y−LN) is in the null space of B:

B(y−LN) = 0. (28)

Let the last n elements of N to be integers. We denote this

subvector as ND. Our goal is to find N such that

By = BLN (29)

[A0

]y =

[A0

][I0

]N

[A0

]y =

[C D E0 0 0

][I0

][NCND

][

A0

]y =

[C D0 0

][NCND

]

Ay = CNC + DND,

Therefore, if we decompose y as y =[

y�C y�D]�

with

yC ∈ R(K−n) and yD ∈ Rn and given a hypothesized vector

ND ∈Zn, then the real-value estimate of NC is:

NC = (C�C)−1C� (Ay− DND

)= yC +(C�C)−1C� (

Exa0 + D(yD − ND

)), (30)

In (30), in addition to the information from GPS measure-

ments, the information from INS states is also involved in

calculation through the term Exa0. Note that the satellites

must be ordered such that C is invertible with good numeric

properties.

The integer candidates ND can be searched exhaustively

over some finite range of integers using n “for” loops.

C. Rounding NC

Having NC, to get the optimal integer estimate of NC, we

would like to find an integer vector NC which is close to

NC. As discussed in [1], [11], [12], as the integer estimation

error vector can be highly correlated. This is visualized as the

level curve of the cost function being a tilted and elongated

ellipse. In this situation, directly rounding NC to NC may

yield incorrect integer estimate and cause a significant cost

increase. In this section, we reformulate the approach of [1]

to get the integer estimate NC.

Proposition 3.2: Consider the cost function

J(NC) = ‖NC − NC‖2ΣΣΣNC

= (NC − NC)�ΣΣΣ−1

NC(NC − NC), (31)

where from (30)

ΣΣΣNC= ΣΣΣCC +

(C�C

)−1C

(EPE�

+ DΣΣΣDDD�) C�(C�C

)−�, (32)

ΣΣΣCC = cov(yC),

ΣΣΣDD = cov(yD).

Then the cost function c(N) defined in (15) will be mini-

mized by the same integer estimate that minimize J(NC).

Proof: From (27), for any N ∈Zm,

c(N) = (y−LN)�B�B(y−LN)

= (y−LN−LN+LN)�B�B(y−LN−LN+LN)

= ‖B(y−LN)‖2 +‖BL(N− N)‖2

−2(N− N)�L�B�B(y−LN) (33)

where N is the optimal real-valued estimate of N, which by

(28) satisfies B(y−LN)) = 0 and therefore

c(N) = ‖BL(N− N)‖2.

Denote

QΣΣΣ =

⎡⎣ QCC QCD QCP

QDC QDD QCPQPC QPD QPP

⎤⎦

where QCC ∈ R(K−n)×(K−n), QDD ∈ Rn×n, QDD ∈ Rp×p,

QCD,Q�DC ∈ R(K−n)×n, QCP,Q�

PC ∈ R(K−n)×p and

QDP,Q�PD ∈Rn×p.

As the covariance matrix ΣΣΣ is block diagonal, let

ΣΣΣ =

⎡⎣ ΣΣΣCC 0 0

0 ΣΣΣDD 00 0 P

⎤⎦ ,

where ΣΣΣCC ∈R(m−n)×(m−n),ΣΣΣDD ∈Rn×n and P ∈Rp×p. We

also have

ΣΣΣ−1 =

⎡⎣ ΣΣΣ−1

CC 0 00 ΣΣΣ−1

DD 00 0 P−1

⎤⎦ .

Then, from (26) and (24)

L�B�BL = L�Q0L= L�ΣΣΣ−1QΣΣΣL

=[

I 0]⎡⎣ ΣΣΣ−1

CC 0 00 ΣΣΣ−1

DD 00 0 P−1

⎤⎦

⎡⎣ QCC QCD QCP

QDC QDD QCPQPC QPD QPP

⎤⎦[

I0

]

=

[ΣΣΣ−1

CCQCC ΣΣΣ−1CCQCD

ΣΣΣ−1DDQDC ΣΣΣ−1

DDQDD

](34)

By the method that N is generated, ND = N is an integer

vector. Hence, the cost function c(N) can be written as

c(N) = ‖BL(N− N)‖2

= (N− N)�L�B�BL(N− N)

=

[NC − NC

0

][ΣΣΣ−1

CCQCC ΣΣΣ−1CCQCD

ΣΣΣ−1DDQDC ΣΣΣ−1

DDQDD

][

NC − NC0

]

= (NC − NC)� (

ΣΣΣ−1CCQCC

)NC − NC) (35)

Comparison of (31) and (35) shows that if we can prove

ΣΣΣ−1CCQCC = ΣΣΣ−1

NC, then these two cost functions are equivalent.

570

From (28), we know that BLN = By. Multiplying on

the left by B� yields B�BLN = B�By which provides the

following constraint on the covariance

B�BLΣΣΣNNL�B�B = B�BΣΣΣB�B. (36)

where

ΣΣΣNN =

[ΣΣΣNC

00 0

]

as there is no uncertainty in ND.

From (24) and (26), (36) can be written as

ΣΣΣ−1QΣΣΣLΣΣΣNNL�Q�ΣΣΣ ΣΣΣ−1 = ΣΣΣ−1QΣΣΣΣΣΣQ�

ΣΣΣ ΣΣΣ−1. (37)

Because ΣΣΣ is nonsingular, (37) reduces to

QΣΣΣLΣΣΣNNL�Q�ΣΣΣ = QΣΣΣΣΣΣQ�

ΣΣΣ (38)

= QΣΣΣΣΣΣ. (39)

Therefore,

QΣΣΣ

(LΣΣΣNNL�Q�

ΣΣΣ −ΣΣΣ)= 0, (40)

which can be written as⎡⎣ QCC QCD QCP

QDC QDD QCPQPC QPD QPP

⎤⎦

⎡⎣ ΣΣΣNC

Q�CC −ΣΣΣCC ΣΣΣNC

Q�DC 0

0 −ΣΣΣDD 00 0 −P

⎤⎦= 0.

From Sylvesters rank inequality “If A is a m-by-n matrix and

B n-by-k, then rank(A)+rank(B)−n≤ rank(AB)”. As QΣΣΣ ∈Rm×m,

(LΣΣΣNNL�Q�

ΣΣΣ −ΣΣΣ)∈ Rm×m and rank(QΣΣΣ) = m− n.

Therefore,

(m−n)+ rank((

LΣΣΣNNL�Q�ΣΣΣ −ΣΣΣ

))−m ≤ 0,

rank((

LΣΣΣNNL�Q�ΣΣΣ −ΣΣΣ

))≤ n.

As the block −ΣΣΣDD has rank n, therefore,

ΣΣΣNCQ�

CC −ΣΣΣCC = 0,

ΣΣΣNCQ�

CC = ΣΣΣCC,

ΣΣΣNC= ΣΣΣCCQ−�

CC ,

ΣΣΣ−1

NC= Q�

CCΣΣΣ−1CC.

As ΣΣΣNCis symmetric,

ΣΣΣ−1

NC= ΣΣΣ−�

NC= ΣΣΣ−1

CCQCC

c(N) = (NC − NC)�ΣΣΣ−1

NCNC − NC),

which completes the proof.

To find the integer vector that minimizes (31), we follow

the idea of LAMBDA to find a matrix ZZZ ∈ Z(m−n)×(m−n),

such that ZZZ−1 ∈ Z(m−n)×(m−n), and (ZZZΣΣΣNCZZZ�)−1 is nearly

diagonal. The procedure to find the Z-transformation was

described in detail in [3].

Let MC = ZNC, then the cost function written in terms of

MC is

J(MC) = (MC −MC)�ΣΣΣ−1

MC(MC −MC), (41)

where ΣΣΣMC= ZZZΣΣΣNC

ZZZ�. Because ΣΣΣ−1MC

is nearly diagonal,

J(MC) can be minimized by rounding MC to the nearest

integer; therefore, the integer-valued estimate of NC can be

computed as:

MC = ZZZNC (42)

MC = [MC]roundo f f (43)

NC = ZZZ−1MC (44)

At this point we have an integer vector candidate [N�C N�

D ]�.

One such candidate will be generated for each iteration of

the “for” loop. We can compare each integer vector candidate

using (23). Selecting the candidate vector with the lowest

value (subject to validity tests) as the best. By rounding off

the float estimate NC in the decorrelated domain of MC, we

have a better chance to achieve optimal integer estimate NC.

From (30), the integer candidates ND can be searched ex-

haustively over some finite range of integers. The algorithm

is summarize as Algorithm 1.

Algorithm 1: Triple “for” loop to compute N for thecase where n = 4.

A = C−1Dfor i =−d : d

for j =−d : dfor k =−d : d

ND = [i, j,k,0]�

NC = yC +(

C�C)−1

C� (Exa0 + D

(yD − ND

)). . . use NC to compute NC minimizing J(NC)N =

[NC ND

]if c(N)< current minimum

Save Ncurrent minimum = c(N)

...

IV. TEST RESULT

Two sets of tests for the proposed method have been

performed both in MATLAB simulation.

A. Test over different noise level

In this MATLAB simulation, the test is epoch-by-epoch

with a set of 8 single difference GPS L1 (λ ≈ 0.19m)

carrier phase measurements at different noise levels. For

each noise level, 1000 measurement epochs with randomly

picked satellite elevation and azimuth angles were generated.

We compared the success rate of getting all the integers

correctly using GPS only and with INS aiding. The success

rates of GPS only and GPS aided INS are plot versus

different noise levels in Fig. 1. As the wavelength of GPS

phase measurement is about 0.19cm, here we only simulate

the situations where the standard deviation of INS error

covariance is less than 0.5m. INS measurement with standard

deviation bigger than 0.5m would not be used to aid GPS

integer ambiguity resolution, in which case we will use the

algorithm proposed in [1].

From the figure, we can see that the success rate with INS

aiding is significantly improved. Even when the standard

571

10−3 10−2 10−10

10

20

30

40

50

60

70

80

90

100

phase noise, σ, m

perc

enta

ge c

orre

ct, %

GPS onlyINS Aieded, σ=0.05mINS Aieded, σ=0.1mINS Aieded, σ=0.5m

Fig. 1. Rate of Single Frequency Correct Integer Resolution

Fig. 2. Rate of Single Frequency Correct Integer Resolution

deviation of the INS position estimate is as big as 0.5m(which is more than 2.5 times the wavelength), the success

rate is still better than the GPS only resolution.

B. Test over different number of satellites

The second test is performed to show the performance of

the proposed approach versus the number of satellites. For

each number of satellites, 1000 measurement epochs with

randomly picked satellite elevation and azimuth angles were

generated with sσφ = 0.01m and standard deviation of INS

position estimate set to 0.1m. We compared the success rate

of getting all the integers correct in each single epoch. The

success rate of each method is plot versus different noise

level in Fig. 2.

From the figure, we can see that INS aiding improved

the probability of correctly estimating the GPS integers,

especially when the number of satellites is low. For example,

with 5 satellites in view, it’s unlikely to get an correct integer

ambiguity with GPS only method while the success rate has

been improved to 67% with INS aiding.

V. CONCLUSIONS

One of the typical applications for GPS/INS system is

navigation and guidance of the vehicles. When the vehicle

passed a structure, such as a viaduct, it is desirable to rapidly

reacquire the integer ambiguities.

In this paper, we extend the approach in [1] with auxiliary

position estimate from INS. By presenting the theoretical

derivation, we introduced a fast and computationally efficient

method for GPS integer ambiguity resolution. Two sets of

simulation results show the effectiveness of the proposed

approach. On-vehicle tests with real-world data will be

performed in near future.

VI. ACKNOWLEDGMENTS

This article was prepared using support from the State

of California, Business, Transportation and Housing Agency,

Department of Transportation under Award 65A0261 and the

DOT Federal Highway Administration Agency Award No.

DTFH61-09-C-00018. The authors gratefully acknowledge

this support.

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