# GPS/Reduced IMU with a Local Terrain Predictor in  Land Vehicle Navigation

Authors: Debo Sun; Mark G. Petovello; M. Elizabeth Cannon
Journal: International Journal of Navigation and Observation
 (2008)

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## Abstract

In order to reduce the cost and volume of land vehicle navigation (LVN) systems, a “reduced” inertial measurement unit (IMU) consisting of only one vertical gyro and two or three accelerometers is generally used and is often integrated with other sensors. Since there are no horizontal gyros in a reduced IMU, the pitch and roll cannot be calculated or observed directly from the inertial data, and the navigation performance is thus affected by local terrain variations. In this work, a reduced IMU is integrated with global positioning system (GPS) data and a novel local terrain predictor (LTP) algorithm. The latter is used primarily to help estimate the pitch and roll of the reduced IMU system and thus to improve the navigation performance. In this paper, two reduced IMU configurations and two grades of IMUs are investigated using field data. Test results show that the LTP is valid. Specifically, inclusion of the LTP provides more than an 80% horizontal velocity improvement relative to the case when the LTP is not used in a GPS/reduced IMU configuration.


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## Body

## 1. Introduction

With recent  automobile technology development,  vehicle safety
and control have been given more attention both in civil and military
applications. To this end, land vehicle navigation (LVN) plays a critical role
and this is evident from the large number of publications on the topic in
recent years.  Specifically, LVN studies
have focused on low-cost
micro-electro-mechanical system
(MEMS) inertial measurement units (IMUs) [1–4], reduced
IMUs [5–11], and on GPS and inertial navigation
system (INS) integration [4, 12]
to GPS/IMU plus more vehicle sensors integration [2, 3].In inertial navigation, a full six degree of freedom IMU consists
of three orthogonal accelerometers and three orthogonal gyros. The three
gyro measurements are used to calculate attitude, and the three accelerometer measurements
are used to calculate velocity and position. Gyros are used to measure
rotations in space, so the INS always knows the direction in which the
accelerometers are pointing [13, 14].  By contrast, in a reduced IMU, where there may
be only one vertical gyro, the pitch
and roll cannot be measured. Similarly, if fewer than three accelerometers are used,
full knowledge of the vehicle’s acceleration is unavailable. Both situations introduce
errors in the navigation system. This paper focuses on reduced IMUs comprised of two or three accelerometers
and one vertical gyro which are
typical in LVNs to
reduce cost. These configurations are based on the supposition that roads are
relatively flat such that horizontal gyros and (in some cases) one vertical
accelerometer provide relatively little information [6]. However, in reality,
these sensors do provide critical information, and their omission inevitably degrades the performance of the
navigation system.In order to overcome the disadvantages of
reduced IMUs, integrating reduced IMUs with other navigation systems or
constraints is a means of improving performance; for example, GPS/reduced IMU
[6, 7] and GPS/reduced IMU with vehicular constraints [5]. To this end, since many new model vehicles are already equipped
with GPS receivers and some newer vehicles also contain reduced IMUs, integrating
the two systems provides the opportunity for a minimal cost land vehicle navigation
system. Such a system can also provide a base for other (augmented) integration
strategies. For example, integrating a GPS/reduced IMU with other vehicular
sensors such as a wheel speed sensor can be considered as a hardware-based augmentation
method. Conversely, vehicle dynamic constraints such as nonholonomic constraints
provide a software-based augmentation option (e.g., [5]). Although hardware
augmentations can be used to improve navigation performance, they can increase the cost of an LVNS if additional
vehicular sensors need to be installed in the vehicle. Software methods are popular
because they do not require hardware to be installed, thus reducing cost.The local terrain predictor method
presented in this paper is a form of software augmentation. It can be envisaged
as an error modeling method for reduced IMUs that compensates for
terrain-induced roll and pitch variations. It does not increase the measurement variables of GPS/reduced IMU, but
only changes the navigation equation and error model of the reduced IMU.For GPS/reduced IMU
systems with an LTP, if GPS is available, the local terrain (or equivalently,
the pitch and roll) can be estimated. Once the local terrain is modeled, the terrain
model can be used to predict the pitch and roll in an attempt to improve performance
relative to the case where the pitch and roll are assumed zero.  This prediction applies to both the case when
GPS is continually available and when GPS is unavailable.  In the latter case, the terrain effects are
assumed to be constant over time.The proposed
algorithm is evaluated using real data collected with both a tactical-grade and
MEMS-grade IMU and different reduced IMU sensor configurations. In particular, two
kinds of reduced IMU configurations are considered; a three accelerometer and
one vertical gyro (3A1G) configuration; and a two horizontal accelerometers and
one vertical gyro (2A1G) configuration. System performance in the absence of
GPS data is also evaluated.The paper begins
with a derivation of the navigation equations and the corresponding error model
of the reduced IMU. During the derivation, the local terrain model is established
and integrated into the error model. Then the field test and data analysis
procedures are described followed by an analysis of the results.  Finally, conclusions are drawn based on the
results presented.


## 2. Mechanization of a Reduced IMU

There are many different reduced IMU configurations [5],
but for land vehicle navigation, there are only a few which are practical such
as 3A1G and 2A1G as previously described. In the following, first, the relevant
coordinate systems are described, then the derivation of the mechanization and
corresponding error model of the reduced IMU is given based on a 3A1G
configuration, and finally the mechanization and corresponding error model of a
2A1G configuration are obtained.


### 2.1. Coordinate Systems

The coordinate systems used in this
paper are as follows:(i)
inertial frame (i-frame),(ii)
earth-fixed frame (e-frame),(iii)
local level frame  (ℓ-frame),(iv)
body frame (b-frame), and(v)
alternative level frame  (ℓ1-frame).
The i-frame is a fixed coordinate with the center of the Earth
as its origin. Its z axis is parallel to the spin axis of the Earth, x points
to the mean vernal equinox, and y is determined by x and z in a right-handed
system. The e-frame coincides with the i-frame at the origin
but rotates with the Earth rate. Its z axis is parallel to the z axis of the
i-frame, x points to the mean meridian of Greenwich,
and y is determined by x and z in a right-handed system. The ℓ-frame is an east, north, up rectangular
coordinate system, which is called ENU. The b-frame is rigidly attached to the
vehicle of interest, usually at a fixed point such as the center of gravity. The
x, y, and z axes of the b-frame, respectively, point in the right, forward, and
up directions. The ℓ1-frame is another local level frame which has
the same origin and vertical axis with those of ℓ-frame, but the directions of other two axes
(x and y) are determined by the directions of the corresponding two axes (x and
y) of the b-frame.


### 2.2. Reduced IMU Mechanization    Equations  and Error Model

In the following, the derivation of mechanization equations
and error model is based on the 3A1G configuration in which the acceleration
information is complete, but the rotation information is incomplete.


#### 2.2.1. Mechanization Equations

The navigation
equations for the 3A1G can be derived from the equations of motion for a full
IMU as shown in (A.1) in the Appendix.
Furthermore, only the attitude equation needs to be derived, as below. For
details on notation, please refer to the Appendix.First, from the direction cosine
matrix (from b-frame toℓ-frame):(1)Rbℓ=Az(ψ)Ax(−p)Ay(−r), where ψ is azimuth, p is pitch, r is roll, Aj(ϖ),j=x,y,z,ϖ=p,r,ψ,
is a rotation matrix about j-axis by angle ϖ.
Taking the derivative of (1), the
following equation can be obtained:(2)R˙bℓ=ψ˙A˙z(ψ)Ax(−p)Ay(−r)+p˙Az(ψ)A˙x(−p)Ay(−r)+r˙Az(ψ)Ax(−p)A˙y(−r), where A˙z(ψ)=dAz(ψ)/dψ,  A˙x(−p)=dAx(−p)/dp,
and A˙y(−r)=dAy(−r)/dr.Premultiplying  both sides of (2) by Az(−ψ) yields(3)Az(−ψ)R˙bℓ=ψ˙Az(−ψ)A˙z(ψ)Ax(−p)Ay(−r)+p˙A˙x(−p)Ay(−r)+r˙Ax(−p)A˙y(−r). From the rotation matrices and their derivatives, the following equations are
obtained:(4)A˙x(−p)Ax(p)=[00000−1010],A˙y(−r)Ay(r)=[001000−100],Az(−ψ)A˙z(ψ)=[010−100000]. Postmultiplying (3) with Ay(r)Ax(p) and substituting (4), the
following is obtained:(5)Az(−ψ)R˙bℓAy(r)Ax(p)=[0ψ˙−r˙sinpr˙cosp−ψ˙+r˙sinp0−p˙−r˙cospp˙0]=[Θ×],  where(6)Θ=[p˙r˙cosp−ψ˙+r˙sinp]. From (A.1), R˙bℓ=Rbℓ(Ωibb−Ωiℓb) yielding(7)Az(−ψ)R˙bℓAy(r)Ax(p)=Az(−ψ)Rbℓ(Ωibb−Ωiℓb)Ay(r)Ax(p). Substituting (1) into
(7) gives(8)Az(−ψ)R˙bℓAy(r)Ax(p)=Ax(−p)Ay(−r)(Ωibb−Ωiℓb)Ay(r)Ax(p). According to the definition
of ℓ1-frame, the following is obtained:(9)Rbℓ1=Ax(−p)Ay(−r)=Az(−ψ)Rbℓ. With (9), (8) is
changed as follows:(10)Az(−ψ)R˙bℓAy(r)Ax(p)=Rbℓ1(Ωibb−Ωiℓb)Rℓ1b=Ωibℓ1−Ωiℓℓ1=[(ωibℓ1−ωiℓℓ1)]. Comparing (5) and (10),
we obtain(11)Θ=(ωibℓ1−ωiℓℓ1).With (9), the following can be obtained:(12)ωibℓ1=Rbℓ1ωibb=Ax(−p)Ay(−r)[ωibxbωibybωibzb]=[ωibxbcosr+ωibzbsinrωibxbsinpsinr+ωibybcosp−ωibzbsinpcosr−ωibxbcospsinr+ωibybsinp+ωibzbcospcosr],ωiℓℓ1=Az(−ψ)ωiℓℓ=Az(−ψ)[ωiℓxℓωiℓyℓωiℓzℓ]=[ωiℓxℓcosψ−ωiℓyℓsinψωiℓxℓsinψ+ωiℓyℓcosψωiℓzℓ]. Substituting (12), and the standard expression for the
turn rate ωiℓℓ with velocity components ve, vn,  meridian radius M,
prime vertical radius N,  height h,
latitude φ,
and rotation rate of the earth ωe,
into (11), we get the following:(13)p˙=ωibxbcosr+ωibzbsinr−((−vnM+h)cosψ−(veN+h+ωecosφ)sinψ),r˙cosp=ωibxbsinpsinr+ωibybcosp−ωibzbsinpcosr−((−vnM+h)sinψ+(veN+h+ωecosφ)cosψ),−ψ˙+r˙sinp=−ωibxbcospsinr+ωibybsinp+ωibzbcospcosr−(vetanφN+h+ωesinφ). Therefore, the
azimuth rate equation is obtained from (13) as(14)ψ˙−r˙sinp=ωibxbcospsinr−ωibybsinp−ωibzbcospcosr+(vetanφN+h+ωesinφ). From (13), it can be seen that since ωibxb and ωibyb are unknown, pitch and roll cannot be
calculated. Because pitch and roll are generally small in land vehicle
applications, they can be considered as error terms and will, therefore, be modelled
accordingly.  However, for the
mechanization equations, they are assumed to be zero p=0,r=0.  With this assumption, the azimuth rate
equation and attitude direction cosine matrix can be written as(15)ψ˙=−ωibzb+(vetanφN+h+ωesinφ),Rbℓ=Az(ψ). Substituting (15)
into (A.1), the navigation equations of reduced IMU (3A1G), see (31), are obtained.


#### 2.2.2. Error Model

The corresponding error model of the reduced IMU (3A1G) needs
to be derived from (A.2) (see the Appendix).
Its position error model and accelerometer bias model are the same
as those of (A.2), but the other error state models (i.e., velocity, attitude,
and gyro biases) are derived in the following.First, the attitude error model is derived.
Recall that in (15), the pitch and roll are assumed to be zero and considered as
error terms. However, the pitch and roll are determined by the local terrain
and since this terrain can be expressed as a first-order Gaussian Markov
process [15], the pitch and roll can also be expressed as first-order Gaussian
Markov processes:(16)p˙=−αpp+wp,r˙=−αrr+wr, where αp,αr are the inverse of correlation time of the
process, and wp,wr are driving noise terms.The definition of the
azimuth error is(17)δψ˙=ψ˜˙−ψ˙, where ψ˙ is the “true” azimuth rate which can be
obtained from (14), ψ˜˙ is azimuth
rate with error and can be obtained from (15).
Accordingly,(18)ψ˜˙=−ω˜ibzb+(v˜etanφ˜N˜+h˜+ωesinφ˜), where ω˜ibzb is the vertical gyro measurement, v˜e,φ˜,h˜ are the calculated east velocity, latitude,
and height, and N˜ is the calculated prime vertical radius
of the reference ellipsoid.Let(19)ω˜ibzb=ωibzb−dz, where dz is the drift of vertical gyro. Since the second term on the right-hand side of
(18) is very small, especially relative to the first term, the following
approximation is made:(20)(vetanφN+h+ωesinφ)−(v˜etanφ˜N˜+h˜+ωesinφ˜)≈0.Substituting (14) and (18) into (17)  and applying (20), we obtain(21)δψ˙≈−r˙sinp−ωibxbcospsinr+ωibybsinp+dz+(cospcosr−1)ωibzb. Since ωibxb, ωibyb, and r˙ are unknown and since pitch and roll are small, so the terms r˙sinp, ωibxbcospsinr, and ωibybsinp are small values and considered as noise. For the
term (cospcosr−1)ωibzb,
since pitch and roll are only a few
degrees, cospcosr−1≈−(p2+r2)/2 is smaller and, in most cases, ωibzb is a few degrees per second, so (cospcosr−1)ωibzb is also small value and considered as noise in
order to simplify calculation. Therefore, the sum of the above small terms is approximated as white noise, and the
azimuth error model can be approximately expressed as(22)δψ˙≈dz+wψ, where wψ is the equivalent white noise of approximation
error of (22) and in general cases, dz can be expressed as a first-order Gaussian
Markov process (e.g., [12]):(23)d˙z=−αddz+wd, where αd is the inverse of the correlation time, wd is the driving noise, and (23) is determined
by the expression of vertical gyro drift.In order to express the attitude errorεℓ1 with respect to the pitch, roll, and azimuth
errors, the relationship between εℓ1 and these errors first needs to be
determined. The attitude error can be
written as follows [16]:(24)ε˙b=δωℓbb=[δωℓbxbδωℓbybδωℓbzb]T. Premultiplying both sides of (24) by Rbℓ1 yields(25)ε˙ℓ1=δωℓbℓ1=[δωℓbxℓ1δωℓbyℓ1δωℓbzℓ1]T. Perturbing (11), we obtain(26)δΘ=δωibℓ1−δωiℓℓ1=δωℓbℓ1. Comparing (25) and (26) gives(27)ε˙ℓ1=δΘ. Since p and r are small angles of only a few degrees, and in
the navigation equations, p and r are chosen as zero, the following is obtained:(28)ε˙ℓ1=δΘ≈[δp˙δr˙−δψ˙]=[−p˙−r˙−δψ˙]. From (16), (22), and (28), the attitude error model can be
written as(29)[ε˙xℓ1ε˙yℓ1ε˙zℓ1]=[−αp000−αr0000][εxℓ1εyℓ1εzℓ1]+[00−1]dz+[wpwrwψ]. The velocity error model
can be obtained from (A.2) as follows:(30)δv˙ℓ=Bδvℓ−Fℓεℓ+Rbℓb=Bδvℓ−Rℓ1ℓFℓ1εℓ1+Rbℓb. Finally, substituting (23), (29), and (30) into (A.2), the error
model of reduced IMU (3A1G)    (see (32))  is obtained.


### 2.3. Equation Summary for 3A1G Configuration

The navigation equations are as
follows:(31)r˙ℓ=D−1vℓ,v˙ℓ=Rbℓfb−(2Ωieℓ+Ωeℓℓ)vℓ+gℓ,ψ˙=−ωibzb+(vetanφN+h+ωesinφ),Rbℓ=Az(ψ), where ψ is azimuth, Az(ψ) is a rotation matrix about the z-axis by angle ψ,
and the other variables and parameters have the same definitions as those of a
full six degree of freedom IMU; see (A.1).The corresponding error model is given by(32)[δr˙ℓδv˙ℓε˙ℓ1d˙zb˙]=[03×3D−103×303×103×303×3B−Rℓ1ℓFℓ103×1Rbℓ03×303×3ΓεMd03×301×301×301×3−αd01×303×303×303×303×1−Λ][δrℓδvℓεℓ1dzb]+[03×103×1WεwdWb], where Rℓ1ℓ=Az(ψ), Fℓ1=Rbℓ1[fb×] is a skew symmetric matrix. Γε=diag(−αp−αr0)−αp−αr0, αp,αr are the inverse of the correlation time of the
local terrain (or pitch and roll). Md=[00−1]T. Wε=[wpwrwψ]T, wp,wr are driving noise of local terrain, wψ is the equivalent white noise of approximation
error of azimuth error equation. dz is vertical gyro’s drift, αd is the inverse of correlation time of dz,
and wd is the driving noise of dz.
Other variables and parameters have the same definition with those in (A.2).


### 2.4. Mechanization Equations and Error Model of 2A1G Configuration

In this configuration, there are two horizontal
accelerometers and one vertical gyro. The acceleration information and rotation
information are, therefore, both incomplete, and the difference of 2A1G from
3A1G is that in the former the vertical specific force needs to be calculated.Suppose the
specific force can be expressed asfℓ=[fefng+δfu]T in the local level frame (ℓ-frame), where g is acceleration due to gravity and δfu is the vehicle’s true vertical acceleration, fe and fn are specific forces in east and north,
respectively. So in the body frame (b-frame), specific force can be expressed
as
(33)fb=[fxbfybfzb]T=Rℓbfℓ=[𝒜𝒞−cospsinrsinψcospcosψcospsinpB𝒟cospcosr]fℓ, where 𝒜 denotes  cosψcosr+sinψsinpsinr, B denotes cosψsinr−sinψsinpcosr, 𝒞 denotes −sinψcosr+cosψsinpsinr, and 𝒟 denotes  −sinψsinr−cosψsinpcosr, so(34)fzb=fe(cosψsinr−sinψsinpcosr)+fn(−sinψsinr−cosψsinpcosr)+gcospcosr+δfucospcosr. Since pitch,  roll,
and δfu are generally
small (most of δfu is less than g/10 and behaves like noise, and δfu is unknown), and in most vehicle navigation applications, fe and fn are less than g,
(34) can be simplified as(35)fzb≈gcospcosr. Since the approximation
error of (35) is related to the local terrain, which is expressed as a first-order
Gauss Markov process, the approximation error of (35) is, therefore, expressed
as a first-order Gauss Markov process:(36)b˙z=−βzbz+wfz, where βz is the inverse of the correlation time of bz,
and wfz is the driving noise of bz.Substituting (35)
into (31), the navigation equations of 2A1G are obtained, which have the same
form as (31) except that the vertical acceleration in body frame needs to be
calculated with (35) and the best
available estimates of the pitch p and roll r.Substituting (36)
into (32), the
corresponding error model of 2A1G is obtained, which has the same form as (32)
except for the vertical accelerometer bias term. Although the vertical accelerometer bias has the same
expression in both 3A1G and 2A1G, it has different parameters or meanings in
the two error models. In 3A1G, the
bias model comes from the actual vertical
accelerometer, but in 2A1G, it comes from the vertical acceleration calculation
error, which is related to the local terrain.


## 2.1. Coordinate Systems

The coordinate systems used in this
paper are as follows:(i)
inertial frame (i-frame),(ii)
earth-fixed frame (e-frame),(iii)
local level frame  (ℓ-frame),(iv)
body frame (b-frame), and(v)
alternative level frame  (ℓ1-frame).
The i-frame is a fixed coordinate with the center of the Earth
as its origin. Its z axis is parallel to the spin axis of the Earth, x points
to the mean vernal equinox, and y is determined by x and z in a right-handed
system. The e-frame coincides with the i-frame at the origin
but rotates with the Earth rate. Its z axis is parallel to the z axis of the
i-frame, x points to the mean meridian of Greenwich,
and y is determined by x and z in a right-handed system. The ℓ-frame is an east, north, up rectangular
coordinate system, which is called ENU. The b-frame is rigidly attached to the
vehicle of interest, usually at a fixed point such as the center of gravity. The
x, y, and z axes of the b-frame, respectively, point in the right, forward, and
up directions. The ℓ1-frame is another local level frame which has
the same origin and vertical axis with those of ℓ-frame, but the directions of other two axes
(x and y) are determined by the directions of the corresponding two axes (x and
y) of the b-frame.


## 2.2. Reduced IMU Mechanization    Equations  and Error Model

In the following, the derivation of mechanization equations
and error model is based on the 3A1G configuration in which the acceleration
information is complete, but the rotation information is incomplete.


### 2.2.1. Mechanization Equations

The navigation
equations for the 3A1G can be derived from the equations of motion for a full
IMU as shown in (A.1) in the Appendix.
Furthermore, only the attitude equation needs to be derived, as below. For
details on notation, please refer to the Appendix.First, from the direction cosine
matrix (from b-frame toℓ-frame):(1)Rbℓ=Az(ψ)Ax(−p)Ay(−r), where ψ is azimuth, p is pitch, r is roll, Aj(ϖ),j=x,y,z,ϖ=p,r,ψ,
is a rotation matrix about j-axis by angle ϖ.
Taking the derivative of (1), the
following equation can be obtained:(2)R˙bℓ=ψ˙A˙z(ψ)Ax(−p)Ay(−r)+p˙Az(ψ)A˙x(−p)Ay(−r)+r˙Az(ψ)Ax(−p)A˙y(−r), where A˙z(ψ)=dAz(ψ)/dψ,  A˙x(−p)=dAx(−p)/dp,
and A˙y(−r)=dAy(−r)/dr.Premultiplying  both sides of (2) by Az(−ψ) yields(3)Az(−ψ)R˙bℓ=ψ˙Az(−ψ)A˙z(ψ)Ax(−p)Ay(−r)+p˙A˙x(−p)Ay(−r)+r˙Ax(−p)A˙y(−r). From the rotation matrices and their derivatives, the following equations are
obtained:(4)A˙x(−p)Ax(p)=[00000−1010],A˙y(−r)Ay(r)=[001000−100],Az(−ψ)A˙z(ψ)=[010−100000]. Postmultiplying (3) with Ay(r)Ax(p) and substituting (4), the
following is obtained:(5)Az(−ψ)R˙bℓAy(r)Ax(p)=[0ψ˙−r˙sinpr˙cosp−ψ˙+r˙sinp0−p˙−r˙cospp˙0]=[Θ×],  where(6)Θ=[p˙r˙cosp−ψ˙+r˙sinp]. From (A.1), R˙bℓ=Rbℓ(Ωibb−Ωiℓb) yielding(7)Az(−ψ)R˙bℓAy(r)Ax(p)=Az(−ψ)Rbℓ(Ωibb−Ωiℓb)Ay(r)Ax(p). Substituting (1) into
(7) gives(8)Az(−ψ)R˙bℓAy(r)Ax(p)=Ax(−p)Ay(−r)(Ωibb−Ωiℓb)Ay(r)Ax(p). According to the definition
of ℓ1-frame, the following is obtained:(9)Rbℓ1=Ax(−p)Ay(−r)=Az(−ψ)Rbℓ. With (9), (8) is
changed as follows:(10)Az(−ψ)R˙bℓAy(r)Ax(p)=Rbℓ1(Ωibb−Ωiℓb)Rℓ1b=Ωibℓ1−Ωiℓℓ1=[(ωibℓ1−ωiℓℓ1)]. Comparing (5) and (10),
we obtain(11)Θ=(ωibℓ1−ωiℓℓ1).With (9), the following can be obtained:(12)ωibℓ1=Rbℓ1ωibb=Ax(−p)Ay(−r)[ωibxbωibybωibzb]=[ωibxbcosr+ωibzbsinrωibxbsinpsinr+ωibybcosp−ωibzbsinpcosr−ωibxbcospsinr+ωibybsinp+ωibzbcospcosr],ωiℓℓ1=Az(−ψ)ωiℓℓ=Az(−ψ)[ωiℓxℓωiℓyℓωiℓzℓ]=[ωiℓxℓcosψ−ωiℓyℓsinψωiℓxℓsinψ+ωiℓyℓcosψωiℓzℓ]. Substituting (12), and the standard expression for the
turn rate ωiℓℓ with velocity components ve, vn,  meridian radius M,
prime vertical radius N,  height h,
latitude φ,
and rotation rate of the earth ωe,
into (11), we get the following:(13)p˙=ωibxbcosr+ωibzbsinr−((−vnM+h)cosψ−(veN+h+ωecosφ)sinψ),r˙cosp=ωibxbsinpsinr+ωibybcosp−ωibzbsinpcosr−((−vnM+h)sinψ+(veN+h+ωecosφ)cosψ),−ψ˙+r˙sinp=−ωibxbcospsinr+ωibybsinp+ωibzbcospcosr−(vetanφN+h+ωesinφ). Therefore, the
azimuth rate equation is obtained from (13) as(14)ψ˙−r˙sinp=ωibxbcospsinr−ωibybsinp−ωibzbcospcosr+(vetanφN+h+ωesinφ). From (13), it can be seen that since ωibxb and ωibyb are unknown, pitch and roll cannot be
calculated. Because pitch and roll are generally small in land vehicle
applications, they can be considered as error terms and will, therefore, be modelled
accordingly.  However, for the
mechanization equations, they are assumed to be zero p=0,r=0.  With this assumption, the azimuth rate
equation and attitude direction cosine matrix can be written as(15)ψ˙=−ωibzb+(vetanφN+h+ωesinφ),Rbℓ=Az(ψ). Substituting (15)
into (A.1), the navigation equations of reduced IMU (3A1G), see (31), are obtained.


### 2.2.2. Error Model

The corresponding error model of the reduced IMU (3A1G) needs
to be derived from (A.2) (see the Appendix).
Its position error model and accelerometer bias model are the same
as those of (A.2), but the other error state models (i.e., velocity, attitude,
and gyro biases) are derived in the following.First, the attitude error model is derived.
Recall that in (15), the pitch and roll are assumed to be zero and considered as
error terms. However, the pitch and roll are determined by the local terrain
and since this terrain can be expressed as a first-order Gaussian Markov
process [15], the pitch and roll can also be expressed as first-order Gaussian
Markov processes:(16)p˙=−αpp+wp,r˙=−αrr+wr, where αp,αr are the inverse of correlation time of the
process, and wp,wr are driving noise terms.The definition of the
azimuth error is(17)δψ˙=ψ˜˙−ψ˙, where ψ˙ is the “true” azimuth rate which can be
obtained from (14), ψ˜˙ is azimuth
rate with error and can be obtained from (15).
Accordingly,(18)ψ˜˙=−ω˜ibzb+(v˜etanφ˜N˜+h˜+ωesinφ˜), where ω˜ibzb is the vertical gyro measurement, v˜e,φ˜,h˜ are the calculated east velocity, latitude,
and height, and N˜ is the calculated prime vertical radius
of the reference ellipsoid.Let(19)ω˜ibzb=ωibzb−dz, where dz is the drift of vertical gyro. Since the second term on the right-hand side of
(18) is very small, especially relative to the first term, the following
approximation is made:(20)(vetanφN+h+ωesinφ)−(v˜etanφ˜N˜+h˜+ωesinφ˜)≈0.Substituting (14) and (18) into (17)  and applying (20), we obtain(21)δψ˙≈−r˙sinp−ωibxbcospsinr+ωibybsinp+dz+(cospcosr−1)ωibzb. Since ωibxb, ωibyb, and r˙ are unknown and since pitch and roll are small, so the terms r˙sinp, ωibxbcospsinr, and ωibybsinp are small values and considered as noise. For the
term (cospcosr−1)ωibzb,
since pitch and roll are only a few
degrees, cospcosr−1≈−(p2+r2)/2 is smaller and, in most cases, ωibzb is a few degrees per second, so (cospcosr−1)ωibzb is also small value and considered as noise in
order to simplify calculation. Therefore, the sum of the above small terms is approximated as white noise, and the
azimuth error model can be approximately expressed as(22)δψ˙≈dz+wψ, where wψ is the equivalent white noise of approximation
error of (22) and in general cases, dz can be expressed as a first-order Gaussian
Markov process (e.g., [12]):(23)d˙z=−αddz+wd, where αd is the inverse of the correlation time, wd is the driving noise, and (23) is determined
by the expression of vertical gyro drift.In order to express the attitude errorεℓ1 with respect to the pitch, roll, and azimuth
errors, the relationship between εℓ1 and these errors first needs to be
determined. The attitude error can be
written as follows [16]:(24)ε˙b=δωℓbb=[δωℓbxbδωℓbybδωℓbzb]T. Premultiplying both sides of (24) by Rbℓ1 yields(25)ε˙ℓ1=δωℓbℓ1=[δωℓbxℓ1δωℓbyℓ1δωℓbzℓ1]T. Perturbing (11), we obtain(26)δΘ=δωibℓ1−δωiℓℓ1=δωℓbℓ1. Comparing (25) and (26) gives(27)ε˙ℓ1=δΘ. Since p and r are small angles of only a few degrees, and in
the navigation equations, p and r are chosen as zero, the following is obtained:(28)ε˙ℓ1=δΘ≈[δp˙δr˙−δψ˙]=[−p˙−r˙−δψ˙]. From (16), (22), and (28), the attitude error model can be
written as(29)[ε˙xℓ1ε˙yℓ1ε˙zℓ1]=[−αp000−αr0000][εxℓ1εyℓ1εzℓ1]+[00−1]dz+[wpwrwψ]. The velocity error model
can be obtained from (A.2) as follows:(30)δv˙ℓ=Bδvℓ−Fℓεℓ+Rbℓb=Bδvℓ−Rℓ1ℓFℓ1εℓ1+Rbℓb. Finally, substituting (23), (29), and (30) into (A.2), the error
model of reduced IMU (3A1G)    (see (32))  is obtained.


## 2.2.1. Mechanization Equations

The navigation
equations for the 3A1G can be derived from the equations of motion for a full
IMU as shown in (A.1) in the Appendix.
Furthermore, only the attitude equation needs to be derived, as below. For
details on notation, please refer to the Appendix.First, from the direction cosine
matrix (from b-frame toℓ-frame):(1)Rbℓ=Az(ψ)Ax(−p)Ay(−r), where ψ is azimuth, p is pitch, r is roll, Aj(ϖ),j=x,y,z,ϖ=p,r,ψ,
is a rotation matrix about j-axis by angle ϖ.
Taking the derivative of (1), the
following equation can be obtained:(2)R˙bℓ=ψ˙A˙z(ψ)Ax(−p)Ay(−r)+p˙Az(ψ)A˙x(−p)Ay(−r)+r˙Az(ψ)Ax(−p)A˙y(−r), where A˙z(ψ)=dAz(ψ)/dψ,  A˙x(−p)=dAx(−p)/dp,
and A˙y(−r)=dAy(−r)/dr.Premultiplying  both sides of (2) by Az(−ψ) yields(3)Az(−ψ)R˙bℓ=ψ˙Az(−ψ)A˙z(ψ)Ax(−p)Ay(−r)+p˙A˙x(−p)Ay(−r)+r˙Ax(−p)A˙y(−r). From the rotation matrices and their derivatives, the following equations are
obtained:(4)A˙x(−p)Ax(p)=[00000−1010],A˙y(−r)Ay(r)=[001000−100],Az(−ψ)A˙z(ψ)=[010−100000]. Postmultiplying (3) with Ay(r)Ax(p) and substituting (4), the
following is obtained:(5)Az(−ψ)R˙bℓAy(r)Ax(p)=[0ψ˙−r˙sinpr˙cosp−ψ˙+r˙sinp0−p˙−r˙cospp˙0]=[Θ×],  where(6)Θ=[p˙r˙cosp−ψ˙+r˙sinp]. From (A.1), R˙bℓ=Rbℓ(Ωibb−Ωiℓb) yielding(7)Az(−ψ)R˙bℓAy(r)Ax(p)=Az(−ψ)Rbℓ(Ωibb−Ωiℓb)Ay(r)Ax(p). Substituting (1) into
(7) gives(8)Az(−ψ)R˙bℓAy(r)Ax(p)=Ax(−p)Ay(−r)(Ωibb−Ωiℓb)Ay(r)Ax(p). According to the definition
of ℓ1-frame, the following is obtained:(9)Rbℓ1=Ax(−p)Ay(−r)=Az(−ψ)Rbℓ. With (9), (8) is
changed as follows:(10)Az(−ψ)R˙bℓAy(r)Ax(p)=Rbℓ1(Ωibb−Ωiℓb)Rℓ1b=Ωibℓ1−Ωiℓℓ1=[(ωibℓ1−ωiℓℓ1)]. Comparing (5) and (10),
we obtain(11)Θ=(ωibℓ1−ωiℓℓ1).With (9), the following can be obtained:(12)ωibℓ1=Rbℓ1ωibb=Ax(−p)Ay(−r)[ωibxbωibybωibzb]=[ωibxbcosr+ωibzbsinrωibxbsinpsinr+ωibybcosp−ωibzbsinpcosr−ωibxbcospsinr+ωibybsinp+ωibzbcospcosr],ωiℓℓ1=Az(−ψ)ωiℓℓ=Az(−ψ)[ωiℓxℓωiℓyℓωiℓzℓ]=[ωiℓxℓcosψ−ωiℓyℓsinψωiℓxℓsinψ+ωiℓyℓcosψωiℓzℓ]. Substituting (12), and the standard expression for the
turn rate ωiℓℓ with velocity components ve, vn,  meridian radius M,
prime vertical radius N,  height h,
latitude φ,
and rotation rate of the earth ωe,
into (11), we get the following:(13)p˙=ωibxbcosr+ωibzbsinr−((−vnM+h)cosψ−(veN+h+ωecosφ)sinψ),r˙cosp=ωibxbsinpsinr+ωibybcosp−ωibzbsinpcosr−((−vnM+h)sinψ+(veN+h+ωecosφ)cosψ),−ψ˙+r˙sinp=−ωibxbcospsinr+ωibybsinp+ωibzbcospcosr−(vetanφN+h+ωesinφ). Therefore, the
azimuth rate equation is obtained from (13) as(14)ψ˙−r˙sinp=ωibxbcospsinr−ωibybsinp−ωibzbcospcosr+(vetanφN+h+ωesinφ). From (13), it can be seen that since ωibxb and ωibyb are unknown, pitch and roll cannot be
calculated. Because pitch and roll are generally small in land vehicle
applications, they can be considered as error terms and will, therefore, be modelled
accordingly.  However, for the
mechanization equations, they are assumed to be zero p=0,r=0.  With this assumption, the azimuth rate
equation and attitude direction cosine matrix can be written as(15)ψ˙=−ωibzb+(vetanφN+h+ωesinφ),Rbℓ=Az(ψ). Substituting (15)
into (A.1), the navigation equations of reduced IMU (3A1G), see (31), are obtained.


## 2.2.2. Error Model

The corresponding error model of the reduced IMU (3A1G) needs
to be derived from (A.2) (see the Appendix).
Its position error model and accelerometer bias model are the same
as those of (A.2), but the other error state models (i.e., velocity, attitude,
and gyro biases) are derived in the following.First, the attitude error model is derived.
Recall that in (15), the pitch and roll are assumed to be zero and considered as
error terms. However, the pitch and roll are determined by the local terrain
and since this terrain can be expressed as a first-order Gaussian Markov
process [15], the pitch and roll can also be expressed as first-order Gaussian
Markov processes:(16)p˙=−αpp+wp,r˙=−αrr+wr, where αp,αr are the inverse of correlation time of the
process, and wp,wr are driving noise terms.The definition of the
azimuth error is(17)δψ˙=ψ˜˙−ψ˙, where ψ˙ is the “true” azimuth rate which can be
obtained from (14), ψ˜˙ is azimuth
rate with error and can be obtained from (15).
Accordingly,(18)ψ˜˙=−ω˜ibzb+(v˜etanφ˜N˜+h˜+ωesinφ˜), where ω˜ibzb is the vertical gyro measurement, v˜e,φ˜,h˜ are the calculated east velocity, latitude,
and height, and N˜ is the calculated prime vertical radius
of the reference ellipsoid.Let(19)ω˜ibzb=ωibzb−dz, where dz is the drift of vertical gyro. Since the second term on the right-hand side of
(18) is very small, especially relative to the first term, the following
approximation is made:(20)(vetanφN+h+ωesinφ)−(v˜etanφ˜N˜+h˜+ωesinφ˜)≈0.Substituting (14) and (18) into (17)  and applying (20), we obtain(21)δψ˙≈−r˙sinp−ωibxbcospsinr+ωibybsinp+dz+(cospcosr−1)ωibzb. Since ωibxb, ωibyb, and r˙ are unknown and since pitch and roll are small, so the terms r˙sinp, ωibxbcospsinr, and ωibybsinp are small values and considered as noise. For the
term (cospcosr−1)ωibzb,
since pitch and roll are only a few
degrees, cospcosr−1≈−(p2+r2)/2 is smaller and, in most cases, ωibzb is a few degrees per second, so (cospcosr−1)ωibzb is also small value and considered as noise in
order to simplify calculation. Therefore, the sum of the above small terms is approximated as white noise, and the
azimuth error model can be approximately expressed as(22)δψ˙≈dz+wψ, where wψ is the equivalent white noise of approximation
error of (22) and in general cases, dz can be expressed as a first-order Gaussian
Markov process (e.g., [12]):(23)d˙z=−αddz+wd, where αd is the inverse of the correlation time, wd is the driving noise, and (23) is determined
by the expression of vertical gyro drift.In order to express the attitude errorεℓ1 with respect to the pitch, roll, and azimuth
errors, the relationship between εℓ1 and these errors first needs to be
determined. The attitude error can be
written as follows [16]:(24)ε˙b=δωℓbb=[δωℓbxbδωℓbybδωℓbzb]T. Premultiplying both sides of (24) by Rbℓ1 yields(25)ε˙ℓ1=δωℓbℓ1=[δωℓbxℓ1δωℓbyℓ1δωℓbzℓ1]T. Perturbing (11), we obtain(26)δΘ=δωibℓ1−δωiℓℓ1=δωℓbℓ1. Comparing (25) and (26) gives(27)ε˙ℓ1=δΘ. Since p and r are small angles of only a few degrees, and in
the navigation equations, p and r are chosen as zero, the following is obtained:(28)ε˙ℓ1=δΘ≈[δp˙δr˙−δψ˙]=[−p˙−r˙−δψ˙]. From (16), (22), and (28), the attitude error model can be
written as(29)[ε˙xℓ1ε˙yℓ1ε˙zℓ1]=[−αp000−αr0000][εxℓ1εyℓ1εzℓ1]+[00−1]dz+[wpwrwψ]. The velocity error model
can be obtained from (A.2) as follows:(30)δv˙ℓ=Bδvℓ−Fℓεℓ+Rbℓb=Bδvℓ−Rℓ1ℓFℓ1εℓ1+Rbℓb. Finally, substituting (23), (29), and (30) into (A.2), the error
model of reduced IMU (3A1G)    (see (32))  is obtained.


## 2.3. Equation Summary for 3A1G Configuration

The navigation equations are as
follows:(31)r˙ℓ=D−1vℓ,v˙ℓ=Rbℓfb−(2Ωieℓ+Ωeℓℓ)vℓ+gℓ,ψ˙=−ωibzb+(vetanφN+h+ωesinφ),Rbℓ=Az(ψ), where ψ is azimuth, Az(ψ) is a rotation matrix about the z-axis by angle ψ,
and the other variables and parameters have the same definitions as those of a
full six degree of freedom IMU; see (A.1).The corresponding error model is given by(32)[δr˙ℓδv˙ℓε˙ℓ1d˙zb˙]=[03×3D−103×303×103×303×3B−Rℓ1ℓFℓ103×1Rbℓ03×303×3ΓεMd03×301×301×301×3−αd01×303×303×303×303×1−Λ][δrℓδvℓεℓ1dzb]+[03×103×1WεwdWb], where Rℓ1ℓ=Az(ψ), Fℓ1=Rbℓ1[fb×] is a skew symmetric matrix. Γε=diag(−αp−αr0)−αp−αr0, αp,αr are the inverse of the correlation time of the
local terrain (or pitch and roll). Md=[00−1]T. Wε=[wpwrwψ]T, wp,wr are driving noise of local terrain, wψ is the equivalent white noise of approximation
error of azimuth error equation. dz is vertical gyro’s drift, αd is the inverse of correlation time of dz,
and wd is the driving noise of dz.
Other variables and parameters have the same definition with those in (A.2).


## 2.4. Mechanization Equations and Error Model of 2A1G Configuration

In this configuration, there are two horizontal
accelerometers and one vertical gyro. The acceleration information and rotation
information are, therefore, both incomplete, and the difference of 2A1G from
3A1G is that in the former the vertical specific force needs to be calculated.Suppose the
specific force can be expressed asfℓ=[fefng+δfu]T in the local level frame (ℓ-frame), where g is acceleration due to gravity and δfu is the vehicle’s true vertical acceleration, fe and fn are specific forces in east and north,
respectively. So in the body frame (b-frame), specific force can be expressed
as
(33)fb=[fxbfybfzb]T=Rℓbfℓ=[𝒜𝒞−cospsinrsinψcospcosψcospsinpB𝒟cospcosr]fℓ, where 𝒜 denotes  cosψcosr+sinψsinpsinr, B denotes cosψsinr−sinψsinpcosr, 𝒞 denotes −sinψcosr+cosψsinpsinr, and 𝒟 denotes  −sinψsinr−cosψsinpcosr, so(34)fzb=fe(cosψsinr−sinψsinpcosr)+fn(−sinψsinr−cosψsinpcosr)+gcospcosr+δfucospcosr. Since pitch,  roll,
and δfu are generally
small (most of δfu is less than g/10 and behaves like noise, and δfu is unknown), and in most vehicle navigation applications, fe and fn are less than g,
(34) can be simplified as(35)fzb≈gcospcosr. Since the approximation
error of (35) is related to the local terrain, which is expressed as a first-order
Gauss Markov process, the approximation error of (35) is, therefore, expressed
as a first-order Gauss Markov process:(36)b˙z=−βzbz+wfz, where βz is the inverse of the correlation time of bz,
and wfz is the driving noise of bz.Substituting (35)
into (31), the navigation equations of 2A1G are obtained, which have the same
form as (31) except that the vertical acceleration in body frame needs to be
calculated with (35) and the best
available estimates of the pitch p and roll r.Substituting (36)
into (32), the
corresponding error model of 2A1G is obtained, which has the same form as (32)
except for the vertical accelerometer bias term. Although the vertical accelerometer bias has the same
expression in both 3A1G and 2A1G, it has different parameters or meanings in
the two error models. In 3A1G, the
bias model comes from the actual vertical
accelerometer, but in 2A1G, it comes from the vertical acceleration calculation
error, which is related to the local terrain.


## 3. Field Test Description

In order to investigate the validity of the above method, a field
vehicle test was conducted in a suburban area of Calgary
in October 2007. Data from the field test
was collected and stored for postmission processing.


### 3.1. Field Test Setup

Two grades of
IMUs were used during the field test: a tactical-grade IMU (Honeywell HG1700)
and an MEMS-grade IMU (Crista IMU). For the HG1700 IMU, the gyro drift is 1 deg/h,
and the accelerometer bias is 1 milli-g [12]. For the Crista IMU, the gyro turn
on drift is 2000–5000 deg/h, noise
is 200–300 deg/h/√Hz, and
the accelerometer turn on bias and noise are 0.3-0.5 m/s2 and 0.003-0.004 g/√Hz, respectively
[17, 18]. In the field test, a NovAtel SPAN system, which contains a NovAtel OEM4
dual-frequency GPS receiver and a Honeywell HG1700 AG11 IMU, was used.
The GPS receiver can
provide high-quality code, Doppler, and carrier
phase measurements, and the IMU data is time tagged with GPS time thus ensuring  synchronicity of the data to better than 1 millisecond. The GPS antennas (one
for the SPAN system and another for an unrelated experiment), HG1700 IMU, and
Crista IMU were installed on the roof of the test vehicle. Data collection system
and power were put inside the vehicle. Figure 1 is the picture of the vehicle setup.Figure 1
Field test setup.


### 3.2. Data Collection

The data
collection system is shown in Figure2.  The
NovAtel SPAN system provides GPS data and IMU data. For the Crista IMU, the pulse-per-second
(PPS) signal from the SPAN system’s GPS receiver is used for time tagging
purposes. The GPS sampling frequency is 1 Hz, whereas the IMU sampling
frequency (for both the HG1700 and Crista) is 100 Hz.Figure 2
Data collection block diagram.


### 3.3. GPS Availability

GPS availability is shown in Figure3. It can be seen that,
in most cases, the number of tracked GPS satellites is seven to nine. Only in a
few very short periods, the number of tracked satellites is four to six. As
such, it is concluded that there are no naturally occurring GPS outages (<4 SVs) during the field test.Figure 3
Number of tracked satellites.


### 3.4. Trajectory, Velocity, and Attitude

The field test trajectory is shown in Figure4, velocity in
Figure 5, and attitude in Figure 6. The total vehicle test duration is more than
12 minutes. The south-north distance change is about 3 km, and is 500 m in the east-west
direction. The vertical variation is more than 100 m. Figure 5 shows the velocity profile  with the maximum horizontal velocity being
about 25 m/s and the maximum vertical velocity being less than 2 m/s. The
reference attitude solution is shown in Figure 6. As expected, the azimuth solution
roughly shows the orientation of the road on which the vehicle moves. In
contrast, pitch and roll generally show the slope of the road (some vehicle specific
attitude variations are also included, but these are expected to be small
compared to the terrain variations and have relatively short duration). In the field
test, the pitch and roll mostly range between −3 and 3 degrees. The maximum
pitch and roll is 4 degrees in each case. The root mean square (RMS) of the pitch
and roll is 2.1 degrees. In the following discussion, the term “local terrain”
will mean pitch and roll variations only (not azimuth).Field test trajectory.
(a)(b)Field test velocity.
(a)(b)Reference pitch, roll, and azimuth.
(a)(b)The reference
solution was obtained using a differential GPS solution integrated with the HG1700
IMU. It is assumed that the reference solution has a similar accuracy level
with that in [18], which is also
generated using a DGPS/HG1700 IMU system:
the RMS of the position error of the reference solution in each
direction is about 0.23 m, the RMS of velocity accuracy 0.015 m/s in each
direction, and the RMS of attitude accuracy is 0.03 degrees in pitch and roll
and 0.17 degrees in azimuth.


## 3.1. Field Test Setup

Two grades of
IMUs were used during the field test: a tactical-grade IMU (Honeywell HG1700)
and an MEMS-grade IMU (Crista IMU). For the HG1700 IMU, the gyro drift is 1 deg/h,
and the accelerometer bias is 1 milli-g [12]. For the Crista IMU, the gyro turn
on drift is 2000–5000 deg/h, noise
is 200–300 deg/h/√Hz, and
the accelerometer turn on bias and noise are 0.3-0.5 m/s2 and 0.003-0.004 g/√Hz, respectively
[17, 18]. In the field test, a NovAtel SPAN system, which contains a NovAtel OEM4
dual-frequency GPS receiver and a Honeywell HG1700 AG11 IMU, was used.
The GPS receiver can
provide high-quality code, Doppler, and carrier
phase measurements, and the IMU data is time tagged with GPS time thus ensuring  synchronicity of the data to better than 1 millisecond. The GPS antennas (one
for the SPAN system and another for an unrelated experiment), HG1700 IMU, and
Crista IMU were installed on the roof of the test vehicle. Data collection system
and power were put inside the vehicle. Figure 1 is the picture of the vehicle setup.Figure 1
Field test setup.


## 3.2. Data Collection

The data
collection system is shown in Figure2.  The
NovAtel SPAN system provides GPS data and IMU data. For the Crista IMU, the pulse-per-second
(PPS) signal from the SPAN system’s GPS receiver is used for time tagging
purposes. The GPS sampling frequency is 1 Hz, whereas the IMU sampling
frequency (for both the HG1700 and Crista) is 100 Hz.Figure 2
Data collection block diagram.


## 3.3. GPS Availability

GPS availability is shown in Figure3. It can be seen that,
in most cases, the number of tracked GPS satellites is seven to nine. Only in a
few very short periods, the number of tracked satellites is four to six. As
such, it is concluded that there are no naturally occurring GPS outages (<4 SVs) during the field test.Figure 3
Number of tracked satellites.


## 3.4. Trajectory, Velocity, and Attitude

The field test trajectory is shown in Figure4, velocity in
Figure 5, and attitude in Figure 6. The total vehicle test duration is more than
12 minutes. The south-north distance change is about 3 km, and is 500 m in the east-west
direction. The vertical variation is more than 100 m. Figure 5 shows the velocity profile  with the maximum horizontal velocity being
about 25 m/s and the maximum vertical velocity being less than 2 m/s. The
reference attitude solution is shown in Figure 6. As expected, the azimuth solution
roughly shows the orientation of the road on which the vehicle moves. In
contrast, pitch and roll generally show the slope of the road (some vehicle specific
attitude variations are also included, but these are expected to be small
compared to the terrain variations and have relatively short duration). In the field
test, the pitch and roll mostly range between −3 and 3 degrees. The maximum
pitch and roll is 4 degrees in each case. The root mean square (RMS) of the pitch
and roll is 2.1 degrees. In the following discussion, the term “local terrain”
will mean pitch and roll variations only (not azimuth).Field test trajectory.
(a)(b)Field test velocity.
(a)(b)Reference pitch, roll, and azimuth.
(a)(b)The reference
solution was obtained using a differential GPS solution integrated with the HG1700
IMU. It is assumed that the reference solution has a similar accuracy level
with that in [18], which is also
generated using a DGPS/HG1700 IMU system:
the RMS of the position error of the reference solution in each
direction is about 0.23 m, the RMS of velocity accuracy 0.015 m/s in each
direction, and the RMS of attitude accuracy is 0.03 degrees in pitch and roll
and 0.17 degrees in azimuth.


## 4. Field Test Data Processing

The data processing strategy is shown in Figure7.  A loose integration strategy was adopted to
simplify development and to assess algorithm performance. In each reduced IMU configuration
(details below), both the tactical and MEMS-grade IMUs were used. The GPS-only
solution was obtained with GPS solution software (C3NAVG2) developed in
the PLAN group at the University of Calgary. The GPS measurement
update rate of the Kalman filter was 1 Hz, and the integrated solution output rate was 10 Hz.Figure 7
Data
processing block diagram.


## 5. Gps/Reduced IMU Evaluation

In order to verify
the LTP method and to evaluate the performance of the GPS/reduced IMU with a
local terrain predictor, a series of tests were performed using different IMU
configurations.  In particular, both
reduced IMU configurations (3A1G and 2A1G) were tested using each grade of IMU
(tactical and MEMS), for a total of four combinations. All reduced IMUs are integrated
with GPS using a loose coupling strategy.


### 5.1. GPS/3A1G Integration

Figure8 shows the
velocity and attitude errors of GPS/3A1G (HG1700) with LTP and Figure 9 is the
velocity and attitude errors of GPS/3A1G (Crista) with LTP.  Statistics from the two figures are shown in
Table 1 and it can be seen that the RMS of the pitch and roll errors for both the
HG1700 and Crista IMU are reduced to less than 0.8 degrees from the actual local
terrain value of 2.1 degrees. This means that the LTP can help to estimate the
pitch and roll, suggesting the model is valid in the GPS/3A1G case. Furthermore,
comparing the performance of the HG1700 and Crista IMUs from Table 1, it can be
seen that the lower grade IMU (Crista) has poorer performance. The azimuth
error using the Crista is near twice that when using the HG1700. It is affected
by the grade of the IMU since the azimuth is calculated from the vertical gyro
measurement. But the difference in the pitch and roll errors between the two
systems is small. With the HG1700 IMU, the pitch and roll errors are only
reduced about 20% compared to the Crista IMU, which is not as much as in the azimuth
direction. They are affected less by the grade of the IMU since the pitch and
roll of the two systems come from the same terrain model and the accuracy of the
accelerometers have less effect on the pitch and roll estimation. The east and
north velocity errors of the two systems are the same. The reason for this
result will be given in the following paragraph. In contrast, the vertical
velocity error is affected by the grade of the IMU since the vertical velocity is
calculated from the vertical accelerometer measurement. Finally, from Figures 8
and 9, it can be seen that the velocity error is related to the attitude error,
especially to the pitch and roll errors. When pitch and/or roll have large
errors, the velocity error will increase.Table 1
Attitude and velocity
error statistics of GPS/3A1G with LTP.
Reduced IMURMS of attitudeRMS of velocityerror (Deg.)error (m/s)PitchRollAzimuthEastNorthUp3A1G (HG1700)0.570.581.380.080.080.053A1G (Crista)0.720.702.380.080.080.08Velocity and attitude errors of GPS/3A1G (HG1700)
with LTP.
(a)(b)Velocity and attitude errors of GPS/3A1G (Crista)
with LTP.
(a)(b)In the above paragraph, it is stated that with
a higher grade IMU (HG1700), although its pitch and roll estimation errors are
reduced about 20% compared to the Crista IMU, its east and north velocity errors
are not reduced. This may seem contradictory because reducing pitch and
roll estimation errors generally reduces velocity error. The reason for this
result is that although the Crista IMU has lower pitch and roll estimation
accuracy, some of its pitch and roll information is “estimated out” in its horizontal
accelerometer biases. So the total effect from the Crista IMU on horizontal
velocity is the same with that from the HG1700. The further explanation
for this result should be based on the separability of pitch, roll, and horizontal accelerometer biases. To this
end, from (32), it can be seen that the cross product of the specific
force and attitude errors (Fℓ1εℓ1) and
accelerometer bias (b) have a similar effect on velocity
error and are thus coupled since Rℓ1ℓ≈Rbℓ when pitch and roll are small (only a few
degrees). Since the specific force (Fℓ1) is known, it can help to separate the
attitude error (εℓ1), especially for pitch and roll, and the accelerometer
bias (b), especially for horizontal
accelerometer biases. But the attitude error and the accelerometer bias cannot
be separated completely, and the horizontal accelerometer biases and specific
force affect the pitch and roll estimation accuracy.In fact, reducing pitch
and roll estimation errors can reduce velocity error if the pitch and roll are
not estimated in other variables. This is why the LTP method is presented in this
paper. If only the velocity error
caused by pitch, roll, and gravity acceleration is considered, the theoretical horizontal
velocity errors for GPS/reduced IMU during GPS sampling period can be obtained
with the following approximate equation [19]:(37)δvx=δrgt,δvy=−δpgt, where δvx and δvy are velocity errors in the right and forward
directions in b-frame, respectively, δp and δr are pitch and roll estimation errors,
respectively, g is gravity acceleration, and t is time. For local terrain, if pitch and roll
are not estimated, suppose δp=δr=2.1 degrees, which are the original RMSs of pitch
and roll, and suppose t=1 s, the GPS sampling period, the horizontal
velocity errors calculated from (37) are |δvx|=|δvy|≈0.36 m/s. In contrast, if the pitch and roll are
estimated, suppose δp=δr=0.7 degrees, which are the RMSs of pitch and roll
errors obtained above, the horizontal velocity errors from (37) are |δvx|=|δvy|≈0.12 m/s. From the above calculation results, it
can be seen that the pitch and roll estimation in a GPS/reduced IMU (3A1G) with
an LTP can reduce velocity estimation error. This assertion is supported by the
results obtained when the pitch and roll values are fixed to zero (i.e., no
LTP).  The corresponding results obtained
using the Crista IMU are summarized in Table 2. Comparing Tables 1 and 2, it
can be seen that without the LTP, the velocity error increases greatly,
especially for horizontal velocities, and so does the azimuth error. This larger
velocity error is caused by pitch and roll errors, and also by the larger
azimuth error.Table 2
Attitude and velocity error statistics of GPS/3A1G
(Crista) without LTP.
Reduced IMURMS of attitudeRMS of velocityerror (Deg.)error (m/s)PitchRollAzimuthEastNorthUp3A1G (Crista)2.171.9915.900.690.740.13Returning to the original analysis, from Figures8(a) and 9(a) it can be seen that there are some larger velocity errors (more than
0.5 m/s) in the GPS/reduced IMU (3A1G) with an LTP. These errors are caused by
pitch and roll estimation errors. As shown in Figure 8(b), these pitch and roll estimation
errors can reach more than 3 degrees. Suppose δp=δr=3 degrees, and t=1 s, from (37), the horizontal velocity error is
calculated as δvx2+δvy2≈0.73 m/s, which is almost the same value observed
in Figure 8(a). These larger pitch and roll estimation errors are caused by the
local terrain variations. When pitch or/and roll changed during test, the
GPS/reduced IMU with an LTP will try to track this change or to estimate the
new pitch and roll value. But it needs time to track the new value or converge at
the new value in estimation. Before the estimation converges to the new value,
it will produce larger estimation error, which can be explained with estimation
theory (e.g., [20]).Finally, it can be concluded that the velocity error of GPS/reduced IMU with an LTP is related to local terrain variations,
especially to pitch and roll variations. In order to reduce the effect of local
terrain variation on the
velocity error, the GPS sampling
rate should be increased.


### 5.2. GPS/2A1G Integration

In the test, as
before, both HG1700 and Crista IMU were used in this configuration. The test
results for both IMUs are shown in Figures10 and 11, respectively. Their
statistics are listed in Table 3.Table 3
Attitude
and velocity error statistics of GPS/2A1G with LTP.
Reduced IMURMS of attitudeRMS of velocityerror (Deg.)error (m/s)PitchRollAzimuthEastNorthUp2A1G (HG1700)0.570.591.350.080.080.092A1G (Crista)0.710.702.380.080.080.09Velocity and attitude errors of GPS/2A1G
(HG1700) with LTP.
(a)(b)Velocity and attitude errors of GPS/2A1G
(Crista) with LTP.
(a)(b)From the statistics, it can be seen that the pitch and roll RMS errors
for both the HG1700 and Crista IMUs are reduced greatly from the local terrain
original value, just as in GPS/3A1G case. This suggests that the LTP is valid
in the GPS/2A1G case as well. Further, from Table3, it can be seen that the Crista reduced
IMU has larger attitude errors than the HG1700 reduced IMU, as expected. As
with the 3A1G configuration, the azimuth error of the GPS/2A1G (Crista) is almost
two times that of the GPS/2A1G (HG1700) and the pitch and roll errors of the HG1700
IMU are only reduced about 20% compared to the Crista IMU for the same reasons
as before. From Table 3, it also can be seen that the two grades of IMUs have
the same velocity error. For the east and north velocity errors, the reason for
this result is the same as with the 3A1G configuration. But for the vertical
velocity error, the reason is that the vertical accelerations for both grades
of IMUs are calculated from the same formula as in the 2A1G configuration, which
has no vertical accelerometer (and thus is not a function of IMU quality). Finally,
from Figures 10 and 11, it can be seen that velocity error is related to attitude
error, as in the 3A1G configuration.As above, the GPS/2A1G (Crista) configuration
with the pitch and roll values fixed to zero was also tested. The results are summarized
in Table4. Comparing with Table 3, it can be seen that without the estimation
of pitch and roll, the velocity error and azimuth error increase greatly, just as
in the GPS/3A1G case.Table 4
Attitude and velocity error statistics of GPS/2A1G (Crista) without
LTP.
Reduced IMURMS of attitudeRMS of velocityerror (Deg.)error (m/s)PitchRollAzimuthEastNorthUp2A1G (Crista)2.171.9915.900.690.740.09


### 5.3. Comparison of GPS/3A1G and GPS/2A1G

In order to facilitate
the performance comparison for the two configurations, the above results are summarized
in Table5.Table 5
Attitude and velocity error statistics of GPS/reduced IMU with
LTP.
Reduced IMURMS of attitudeRMS of velocityerror (Deg.)error (m/s)PitchRollAzimuthEastNorthUp3A1G (HG1700)0.570.581.380.080.080.052A1G (HG1700)0.570.591.350.080.080.093A1G (Crista)0.720.702.380.080.080.082A1G (Crista)0.710.702.380.080.080.09From Table5, it can be seen that the two
configurations (3A1G and 2A1G) have almost the same attitude result for a
certain grade of IMU. It is concluded, therefore, that, for the datasets
considered here, the reduced IMU configuration has little effect on the attitude
performance of the GPS/reduced IMU. This is because the attitude estimation in the
GPS/reduced IMU is mainly determined by GPS, horizontal accelerometer biases, local
terrain, and the vertical gyro measurement. Vertical accelerometer has no
direct effect on the attitude estimation. For velocity error, Table 5 shows
that two IMU configurations have the same east and north velocity errors, but the
3A1G configuration has a smaller vertical velocity error. This means that the
horizontal velocity errors are not affected by the configuration or the vertical
accelerometer, but the vertical velocity error is affected by the configuration
because the vertical velocity is calculated directly from vertical acceleration.
From the above analysis, it can be concluded that the vertical accelerometer in
a reduced IMU only can mitigate some vertical velocity error, but cannot reduce
other velocity and attitude errors. Therefore, the 2A1G configuration is a
reasonable choice for vehicular applications when the tradeoff between cost and
performance is considered.


## 5.1. GPS/3A1G Integration

Figure8 shows the
velocity and attitude errors of GPS/3A1G (HG1700) with LTP and Figure 9 is the
velocity and attitude errors of GPS/3A1G (Crista) with LTP.  Statistics from the two figures are shown in
Table 1 and it can be seen that the RMS of the pitch and roll errors for both the
HG1700 and Crista IMU are reduced to less than 0.8 degrees from the actual local
terrain value of 2.1 degrees. This means that the LTP can help to estimate the
pitch and roll, suggesting the model is valid in the GPS/3A1G case. Furthermore,
comparing the performance of the HG1700 and Crista IMUs from Table 1, it can be
seen that the lower grade IMU (Crista) has poorer performance. The azimuth
error using the Crista is near twice that when using the HG1700. It is affected
by the grade of the IMU since the azimuth is calculated from the vertical gyro
measurement. But the difference in the pitch and roll errors between the two
systems is small. With the HG1700 IMU, the pitch and roll errors are only
reduced about 20% compared to the Crista IMU, which is not as much as in the azimuth
direction. They are affected less by the grade of the IMU since the pitch and
roll of the two systems come from the same terrain model and the accuracy of the
accelerometers have less effect on the pitch and roll estimation. The east and
north velocity errors of the two systems are the same. The reason for this
result will be given in the following paragraph. In contrast, the vertical
velocity error is affected by the grade of the IMU since the vertical velocity is
calculated from the vertical accelerometer measurement. Finally, from Figures 8
and 9, it can be seen that the velocity error is related to the attitude error,
especially to the pitch and roll errors. When pitch and/or roll have large
errors, the velocity error will increase.Table 1
Attitude and velocity
error statistics of GPS/3A1G with LTP.
Reduced IMURMS of attitudeRMS of velocityerror (Deg.)error (m/s)PitchRollAzimuthEastNorthUp3A1G (HG1700)0.570.581.380.080.080.053A1G (Crista)0.720.702.380.080.080.08Velocity and attitude errors of GPS/3A1G (HG1700)
with LTP.
(a)(b)Velocity and attitude errors of GPS/3A1G (Crista)
with LTP.
(a)(b)In the above paragraph, it is stated that with
a higher grade IMU (HG1700), although its pitch and roll estimation errors are
reduced about 20% compared to the Crista IMU, its east and north velocity errors
are not reduced. This may seem contradictory because reducing pitch and
roll estimation errors generally reduces velocity error. The reason for this
result is that although the Crista IMU has lower pitch and roll estimation
accuracy, some of its pitch and roll information is “estimated out” in its horizontal
accelerometer biases. So the total effect from the Crista IMU on horizontal
velocity is the same with that from the HG1700. The further explanation
for this result should be based on the separability of pitch, roll, and horizontal accelerometer biases. To this
end, from (32), it can be seen that the cross product of the specific
force and attitude errors (Fℓ1εℓ1) and
accelerometer bias (b) have a similar effect on velocity
error and are thus coupled since Rℓ1ℓ≈Rbℓ when pitch and roll are small (only a few
degrees). Since the specific force (Fℓ1) is known, it can help to separate the
attitude error (εℓ1), especially for pitch and roll, and the accelerometer
bias (b), especially for horizontal
accelerometer biases. But the attitude error and the accelerometer bias cannot
be separated completely, and the horizontal accelerometer biases and specific
force affect the pitch and roll estimation accuracy.In fact, reducing pitch
and roll estimation errors can reduce velocity error if the pitch and roll are
not estimated in other variables. This is why the LTP method is presented in this
paper. If only the velocity error
caused by pitch, roll, and gravity acceleration is considered, the theoretical horizontal
velocity errors for GPS/reduced IMU during GPS sampling period can be obtained
with the following approximate equation [19]:(37)δvx=δrgt,δvy=−δpgt, where δvx and δvy are velocity errors in the right and forward
directions in b-frame, respectively, δp and δr are pitch and roll estimation errors,
respectively, g is gravity acceleration, and t is time. For local terrain, if pitch and roll
are not estimated, suppose δp=δr=2.1 degrees, which are the original RMSs of pitch
and roll, and suppose t=1 s, the GPS sampling period, the horizontal
velocity errors calculated from (37) are |δvx|=|δvy|≈0.36 m/s. In contrast, if the pitch and roll are
estimated, suppose δp=δr=0.7 degrees, which are the RMSs of pitch and roll
errors obtained above, the horizontal velocity errors from (37) are |δvx|=|δvy|≈0.12 m/s. From the above calculation results, it
can be seen that the pitch and roll estimation in a GPS/reduced IMU (3A1G) with
an LTP can reduce velocity estimation error. This assertion is supported by the
results obtained when the pitch and roll values are fixed to zero (i.e., no
LTP).  The corresponding results obtained
using the Crista IMU are summarized in Table 2. Comparing Tables 1 and 2, it
can be seen that without the LTP, the velocity error increases greatly,
especially for horizontal velocities, and so does the azimuth error. This larger
velocity error is caused by pitch and roll errors, and also by the larger
azimuth error.Table 2
Attitude and velocity error statistics of GPS/3A1G
(Crista) without LTP.
Reduced IMURMS of attitudeRMS of velocityerror (Deg.)error (m/s)PitchRollAzimuthEastNorthUp3A1G (Crista)2.171.9915.900.690.740.13Returning to the original analysis, from Figures8(a) and 9(a) it can be seen that there are some larger velocity errors (more than
0.5 m/s) in the GPS/reduced IMU (3A1G) with an LTP. These errors are caused by
pitch and roll estimation errors. As shown in Figure 8(b), these pitch and roll estimation
errors can reach more than 3 degrees. Suppose δp=δr=3 degrees, and t=1 s, from (37), the horizontal velocity error is
calculated as δvx2+δvy2≈0.73 m/s, which is almost the same value observed
in Figure 8(a). These larger pitch and roll estimation errors are caused by the
local terrain variations. When pitch or/and roll changed during test, the
GPS/reduced IMU with an LTP will try to track this change or to estimate the
new pitch and roll value. But it needs time to track the new value or converge at
the new value in estimation. Before the estimation converges to the new value,
it will produce larger estimation error, which can be explained with estimation
theory (e.g., [20]).Finally, it can be concluded that the velocity error of GPS/reduced IMU with an LTP is related to local terrain variations,
especially to pitch and roll variations. In order to reduce the effect of local
terrain variation on the
velocity error, the GPS sampling
rate should be increased.


## 5.2. GPS/2A1G Integration

In the test, as
before, both HG1700 and Crista IMU were used in this configuration. The test
results for both IMUs are shown in Figures10 and 11, respectively. Their
statistics are listed in Table 3.Table 3
Attitude
and velocity error statistics of GPS/2A1G with LTP.
Reduced IMURMS of attitudeRMS of velocityerror (Deg.)error (m/s)PitchRollAzimuthEastNorthUp2A1G (HG1700)0.570.591.350.080.080.092A1G (Crista)0.710.702.380.080.080.09Velocity and attitude errors of GPS/2A1G
(HG1700) with LTP.
(a)(b)Velocity and attitude errors of GPS/2A1G
(Crista) with LTP.
(a)(b)From the statistics, it can be seen that the pitch and roll RMS errors
for both the HG1700 and Crista IMUs are reduced greatly from the local terrain
original value, just as in GPS/3A1G case. This suggests that the LTP is valid
in the GPS/2A1G case as well. Further, from Table3, it can be seen that the Crista reduced
IMU has larger attitude errors than the HG1700 reduced IMU, as expected. As
with the 3A1G configuration, the azimuth error of the GPS/2A1G (Crista) is almost
two times that of the GPS/2A1G (HG1700) and the pitch and roll errors of the HG1700
IMU are only reduced about 20% compared to the Crista IMU for the same reasons
as before. From Table 3, it also can be seen that the two grades of IMUs have
the same velocity error. For the east and north velocity errors, the reason for
this result is the same as with the 3A1G configuration. But for the vertical
velocity error, the reason is that the vertical accelerations for both grades
of IMUs are calculated from the same formula as in the 2A1G configuration, which
has no vertical accelerometer (and thus is not a function of IMU quality). Finally,
from Figures 10 and 11, it can be seen that velocity error is related to attitude
error, as in the 3A1G configuration.As above, the GPS/2A1G (Crista) configuration
with the pitch and roll values fixed to zero was also tested. The results are summarized
in Table4. Comparing with Table 3, it can be seen that without the estimation
of pitch and roll, the velocity error and azimuth error increase greatly, just as
in the GPS/3A1G case.Table 4
Attitude and velocity error statistics of GPS/2A1G (Crista) without
LTP.
Reduced IMURMS of attitudeRMS of velocityerror (Deg.)error (m/s)PitchRollAzimuthEastNorthUp2A1G (Crista)2.171.9915.900.690.740.09


## 5.3. Comparison of GPS/3A1G and GPS/2A1G

In order to facilitate
the performance comparison for the two configurations, the above results are summarized
in Table5.Table 5
Attitude and velocity error statistics of GPS/reduced IMU with
LTP.
Reduced IMURMS of attitudeRMS of velocityerror (Deg.)error (m/s)PitchRollAzimuthEastNorthUp3A1G (HG1700)0.570.581.380.080.080.052A1G (HG1700)0.570.591.350.080.080.093A1G (Crista)0.720.702.380.080.080.082A1G (Crista)0.710.702.380.080.080.09From Table5, it can be seen that the two
configurations (3A1G and 2A1G) have almost the same attitude result for a
certain grade of IMU. It is concluded, therefore, that, for the datasets
considered here, the reduced IMU configuration has little effect on the attitude
performance of the GPS/reduced IMU. This is because the attitude estimation in the
GPS/reduced IMU is mainly determined by GPS, horizontal accelerometer biases, local
terrain, and the vertical gyro measurement. Vertical accelerometer has no
direct effect on the attitude estimation. For velocity error, Table 5 shows
that two IMU configurations have the same east and north velocity errors, but the
3A1G configuration has a smaller vertical velocity error. This means that the
horizontal velocity errors are not affected by the configuration or the vertical
accelerometer, but the vertical velocity error is affected by the configuration
because the vertical velocity is calculated directly from vertical acceleration.
From the above analysis, it can be concluded that the vertical accelerometer in
a reduced IMU only can mitigate some vertical velocity error, but cannot reduce
other velocity and attitude errors. Therefore, the 2A1G configuration is a
reasonable choice for vehicular applications when the tradeoff between cost and
performance is considered.


## 6. GPS Outage Test

To assess the
performance of the GPS/reduced IMU system with an LTP during GPS outage, a
series of GPS outage tests were conducted.


### 6.1. Test Description

In the outage tests, both
grades of IMU and both reduced IMU configurations were used. Ten 30 seconds
long GPS outages were simulated in the data by artificially omitting the
satellites during postmission processing. These outages are carefully selected
to represent varying vehicle dynamics, as shown in Figure12. In the outage
tests, GPS/reduced
IMU with an LTP means that the pitch and roll estimates are almost constant value
during GPS outage (they vary slightly because of the first-order Gauss-Markov
process model with long correlation time (500 seconds) used); GPS/reduced IMU without
an LTP means that the pitch and roll estimates are chosen as zero during GPS
outage. In the tests, the RMS of position or velocity
error is calculated from the following equation:(38)RMS(ti)=110∑j=110(xj(ti)−xr(ti))2,
where ti is GPS outage duration (from 0 to 30 seconds), xj(ti) is the output of reduced IMU in the jth GPS outage at ti, and xr(ti) is the reference solution at ti.Figure 12
Ten 30 seconds
GPS outage gaps.


### 6.2. Results

The position and
velocity RMS errors as a function of time since the last GPS measurement are
shown in Figures13, 14, and 15, and the results at the end of GPS outage are
summarized in Table 6.  For the vertical position,
since the GPS/reduced IMU has a larger bias (about 4 m) from GPS compared to
the reference solution, in order to facilitate the analysis, this bias is
removed when calculating the RMS of the vertical position with (38). If this
bias is not removed in the RMS calculation, the result for the Crista IMU is
shown in Figure 16.Table 6
Horizontal and vertical position error
comparison between GPS/reduced IMU with and without LTP at the end
of GPS outage.
Reduced IMURMS of position error (m)HorizontalVerticalLTPNo LTPLTPNo LTP3A1G
(HG1700)10421910152A1G
(HG1700)10422312123A1G
(Crista)10322111122A1G
(Crista)1032201112Horizontal and vertical position error comparison
between GPS/reduced
IMU (HG1700) with and without LTP.
(a)(b)Horizontal position and velocity error comparison
between GPS/reduced IMU (Crista) with and
without LTP.
(a)(b)Vertical position and velocity error comparison
between GPS/reduced
IMU (Crista) with and without LTP.
(a)(b)Figure 16
Vertical position
error comparison between GPS/reduced IMU (Crista) with and without LTP When the vertical
position bias of GPS/reduced IMU is not removed in RMS calculation.


### 6.3. Analysis

From Table6, it can be seen that, for the GPS/reduced IMU without
an LTP, the horizontal position errors for different grades of IMU (HG1700 and
Crista) and different reduced IMU configurations (3A1G and 2A1G) are almost the
same, from 219 to 223 m, since the horizontal position error is mainly determined
by the local terrain when there is no LTP used during GPS outage. During a GPS outage, if only
the position error caused by pitch, roll, and gravity acceleration is
considered, the theoretical horizontal position errors for the GPS/reduced IMU
without an LTP can be obtained with the following approximate equation derived by
integrating (37):(39)δx=12δrgt2,δy=−12δpgt2, where δx and δy are position errors in the right and forward directions,
respectively, in the b-frame, δp and δr are pitch and roll estimation errors,
respectively, g is gravity acceleration, and t is time. For local terrain, the RMS of the
actual pitch and roll are all about 2.1 degrees. The GPS outage duration is 30
seconds. Suppose δp=δr=2.1 degrees, and t=30 seconds, the position errors in the right and forward
directions can be calculated from (39) to be |δx|=|δy|≈162 m. Then the RMS of the horizontal position
error is 1622≈229 m. So the outage test results and theoretical
results are very close.With an LTP, the horizontal position errors
for different IMU grades and different configurations are reduced greatly, as
shown in Figures13(a) and 14(a) and Table 6. From Table 6, it can be seen that with an
LTP, even though the grade and configuration of the reduced IMUs are different,
they have similar results; the RMSs of the horizontal position errors for all IMUs
and configurations are almost the same, from 103 to 104 m, which is less than half
the error without an LPT.The explanation for this result is that the
position error of the GPS/reduced IMU with an LTP is also affected by the local
terrain during GPS outages because it is impossible to predict the local
terrain accurately when there is no other external information. When GPS is
unavailable, the only thing that can be used for pitch and roll estimation is
the previous estimates of pitch and roll. In order to use the previous estimates
of pitch and roll in the GPS outage, the correlation times of the pitch and roll models are chosen to
be large. If the pitch and roll changed during the GPS outage, it will
introduce more position errors. In fact, during GPS outages, the horizontal
position error comes from two parts: the first is produced by the initial pitch
and roll errors; the second is produced by the pitch and roll change during a GPS
outage.Figure14(b) shows that for the Crista IMU under
both configurations, the horizontal velocity error of the GPS/reduced IMU with an
LTP is about half that of the GPS/reduced IMU without an LTP. It further
demonstrates that the LTP is valid in a GPS/reduced IMU during GPS outage.Figure13(b) shows the vertical position
errors of two configurations of the GPS/reduced IMU (HG1700) with and without an
LTP. Figure 15(a) shows the vertical position errors of two configurations of a GPS/reduced
IMU (Crista) with and without an LTP. Their statistics are listed in Table 6
and it can be seen that the vertical position error is reduced only slightly,
if at all, with an LTP. For the vertical position error, if only the position error
caused by pitch, roll, and gravity acceleration is considered, its approximate
formula can be derived from (34) and is as follows:(40)δz=12g(cospcosr−cosp^cosr^)t2≈−14g((p+p^)δp+(r+r^)δr)t2, where p is pitch, r is roll, δp=p−p^, δr=r−r^, g is gravity acceleration, and t is time. For the local terrain, the RMSs of its
original pitch and roll are about 2.1 degrees, suppose p=r=2.1 degrees, p^=r^=0,
so δp=δr=2.1 degrees, and suppose the outage duration t=30 seconds, from (40), the vertical position
error can be calculated as |δz|=5.9 m; when the RMSs of pitch and roll  errors are about 0.7 degrees, suppose p=r=2.1 degrees, p^=r^=1.4 degrees, and δp=δr=0.7 degrees, the vertical position error is calculated
as |δz|=3.3 m. Comparing the theoretical results and outage
test results (in Table 6), it can be seen that the test results are larger. The
reasons for this are as follows.  First, during
a GPS outage, the pitch and roll changes. Second, since the vertical position
error is small, other factors, which can also cause vertical position error,
need to be considered such as the vertical projection of the vehicle’s
acceleration and the cross product of the pitch angular velocity and the vehicle’s
velocity. All the above factors will accumulate the vertical position error. From
Table 6, it also can be seen that the vertical position errors are
almost the same as for different grades of IMUs and different configurations. This
is because the vertical position error depends largely on the local terrain,
and it is not significantly affected by the grade of IMU or the configuration.The vertical
velocity errors of the GPS/reduced
IMU (Crista) are shown in Figure15(b). From Figure 15(b), it can be seen that
the vertical velocity error is reduced
only marginally or remains the same with an LTP, just like the vertical
position error. The explanation for this result is the same as that for vertical
position error. However, if
the vertical position bias of the GPS/reduced IMU (Crista)
is not removed in the RMS calculation, the result shown in Figure 16 does not show
a consistent relationship with the vertical velocity error shown in Figure
15(b).


## 6.1. Test Description

In the outage tests, both
grades of IMU and both reduced IMU configurations were used. Ten 30 seconds
long GPS outages were simulated in the data by artificially omitting the
satellites during postmission processing. These outages are carefully selected
to represent varying vehicle dynamics, as shown in Figure12. In the outage
tests, GPS/reduced
IMU with an LTP means that the pitch and roll estimates are almost constant value
during GPS outage (they vary slightly because of the first-order Gauss-Markov
process model with long correlation time (500 seconds) used); GPS/reduced IMU without
an LTP means that the pitch and roll estimates are chosen as zero during GPS
outage. In the tests, the RMS of position or velocity
error is calculated from the following equation:(38)RMS(ti)=110∑j=110(xj(ti)−xr(ti))2,
where ti is GPS outage duration (from 0 to 30 seconds), xj(ti) is the output of reduced IMU in the jth GPS outage at ti, and xr(ti) is the reference solution at ti.Figure 12
Ten 30 seconds
GPS outage gaps.


## 6.2. Results

The position and
velocity RMS errors as a function of time since the last GPS measurement are
shown in Figures13, 14, and 15, and the results at the end of GPS outage are
summarized in Table 6.  For the vertical position,
since the GPS/reduced IMU has a larger bias (about 4 m) from GPS compared to
the reference solution, in order to facilitate the analysis, this bias is
removed when calculating the RMS of the vertical position with (38). If this
bias is not removed in the RMS calculation, the result for the Crista IMU is
shown in Figure 16.Table 6
Horizontal and vertical position error
comparison between GPS/reduced IMU with and without LTP at the end
of GPS outage.
Reduced IMURMS of position error (m)HorizontalVerticalLTPNo LTPLTPNo LTP3A1G
(HG1700)10421910152A1G
(HG1700)10422312123A1G
(Crista)10322111122A1G
(Crista)1032201112Horizontal and vertical position error comparison
between GPS/reduced
IMU (HG1700) with and without LTP.
(a)(b)Horizontal position and velocity error comparison
between GPS/reduced IMU (Crista) with and
without LTP.
(a)(b)Vertical position and velocity error comparison
between GPS/reduced
IMU (Crista) with and without LTP.
(a)(b)Figure 16
Vertical position
error comparison between GPS/reduced IMU (Crista) with and without LTP When the vertical
position bias of GPS/reduced IMU is not removed in RMS calculation.


## 6.3. Analysis

From Table6, it can be seen that, for the GPS/reduced IMU without
an LTP, the horizontal position errors for different grades of IMU (HG1700 and
Crista) and different reduced IMU configurations (3A1G and 2A1G) are almost the
same, from 219 to 223 m, since the horizontal position error is mainly determined
by the local terrain when there is no LTP used during GPS outage. During a GPS outage, if only
the position error caused by pitch, roll, and gravity acceleration is
considered, the theoretical horizontal position errors for the GPS/reduced IMU
without an LTP can be obtained with the following approximate equation derived by
integrating (37):(39)δx=12δrgt2,δy=−12δpgt2, where δx and δy are position errors in the right and forward directions,
respectively, in the b-frame, δp and δr are pitch and roll estimation errors,
respectively, g is gravity acceleration, and t is time. For local terrain, the RMS of the
actual pitch and roll are all about 2.1 degrees. The GPS outage duration is 30
seconds. Suppose δp=δr=2.1 degrees, and t=30 seconds, the position errors in the right and forward
directions can be calculated from (39) to be |δx|=|δy|≈162 m. Then the RMS of the horizontal position
error is 1622≈229 m. So the outage test results and theoretical
results are very close.With an LTP, the horizontal position errors
for different IMU grades and different configurations are reduced greatly, as
shown in Figures13(a) and 14(a) and Table 6. From Table 6, it can be seen that with an
LTP, even though the grade and configuration of the reduced IMUs are different,
they have similar results; the RMSs of the horizontal position errors for all IMUs
and configurations are almost the same, from 103 to 104 m, which is less than half
the error without an LPT.The explanation for this result is that the
position error of the GPS/reduced IMU with an LTP is also affected by the local
terrain during GPS outages because it is impossible to predict the local
terrain accurately when there is no other external information. When GPS is
unavailable, the only thing that can be used for pitch and roll estimation is
the previous estimates of pitch and roll. In order to use the previous estimates
of pitch and roll in the GPS outage, the correlation times of the pitch and roll models are chosen to
be large. If the pitch and roll changed during the GPS outage, it will
introduce more position errors. In fact, during GPS outages, the horizontal
position error comes from two parts: the first is produced by the initial pitch
and roll errors; the second is produced by the pitch and roll change during a GPS
outage.Figure14(b) shows that for the Crista IMU under
both configurations, the horizontal velocity error of the GPS/reduced IMU with an
LTP is about half that of the GPS/reduced IMU without an LTP. It further
demonstrates that the LTP is valid in a GPS/reduced IMU during GPS outage.Figure13(b) shows the vertical position
errors of two configurations of the GPS/reduced IMU (HG1700) with and without an
LTP. Figure 15(a) shows the vertical position errors of two configurations of a GPS/reduced
IMU (Crista) with and without an LTP. Their statistics are listed in Table 6
and it can be seen that the vertical position error is reduced only slightly,
if at all, with an LTP. For the vertical position error, if only the position error
caused by pitch, roll, and gravity acceleration is considered, its approximate
formula can be derived from (34) and is as follows:(40)δz=12g(cospcosr−cosp^cosr^)t2≈−14g((p+p^)δp+(r+r^)δr)t2, where p is pitch, r is roll, δp=p−p^, δr=r−r^, g is gravity acceleration, and t is time. For the local terrain, the RMSs of its
original pitch and roll are about 2.1 degrees, suppose p=r=2.1 degrees, p^=r^=0,
so δp=δr=2.1 degrees, and suppose the outage duration t=30 seconds, from (40), the vertical position
error can be calculated as |δz|=5.9 m; when the RMSs of pitch and roll  errors are about 0.7 degrees, suppose p=r=2.1 degrees, p^=r^=1.4 degrees, and δp=δr=0.7 degrees, the vertical position error is calculated
as |δz|=3.3 m. Comparing the theoretical results and outage
test results (in Table 6), it can be seen that the test results are larger. The
reasons for this are as follows.  First, during
a GPS outage, the pitch and roll changes. Second, since the vertical position
error is small, other factors, which can also cause vertical position error,
need to be considered such as the vertical projection of the vehicle’s
acceleration and the cross product of the pitch angular velocity and the vehicle’s
velocity. All the above factors will accumulate the vertical position error. From
Table 6, it also can be seen that the vertical position errors are
almost the same as for different grades of IMUs and different configurations. This
is because the vertical position error depends largely on the local terrain,
and it is not significantly affected by the grade of IMU or the configuration.The vertical
velocity errors of the GPS/reduced
IMU (Crista) are shown in Figure15(b). From Figure 15(b), it can be seen that
the vertical velocity error is reduced
only marginally or remains the same with an LTP, just like the vertical
position error. The explanation for this result is the same as that for vertical
position error. However, if
the vertical position bias of the GPS/reduced IMU (Crista)
is not removed in the RMS calculation, the result shown in Figure 16 does not show
a consistent relationship with the vertical velocity error shown in Figure
15(b).


## 7. Summary and Conclusions

This paper presents a local terrain predictor (LTP) for reduced
IMU in land vehicle navigation. Based on the LTP, the navigation equations and
error model of the reduced IMU are established. Then GPS/reduced IMU with an LTP
is introduced. To verify the LTP and investigate the performance of GPS/reduced
IMU with an LTP, a field vehicle test was conduced to collect GPS and IMU data
for integration tests. In the field test, two grades of IMUs were used: a
tactical-grade IMU (Honeywell HG1700), and an MEMS-grade IMU (Crista). After
the data collection, a series of system configuration tests were conducted to
verify the new method. In these tests, two kinds of reduced IMU configurations were
used: three accelerometers plus one vertical gyro (3A1G) and two horizontal
accelerometers plus one vertical gyro (2A1G). Test results show that higher grade
IMU has higher accuracy of attitude estimation, especially for azimuth; two
grade IMUs have the same horizontal velocity errors; two configurations have
almost the same attitude error, and have the same horizontal velocity errors,
but 3A1G configuration has smaller vertical velocity error and higher accuracy
vertical accelerometer has smaller vertical velocity error; 2A1G configuration
for different grade IMU has the same vertical velocity error; velocity error is
related to local terrain variations. Based on the above results, the following conclusions
are drawn:(1)
with an LTP, pitch and roll can be
estimated, and velocity error can be reduced, especially for horizontal velocity
error, it can be reduced by more than 80% compared to without LTP;(2)
2A1G is the better configuration
when cost and performance are considered.Following the
system configuration tests, some GPS outage tests were conducted. In this case, ten 30 seconds GPS gaps were
simulated. Two conclusions are drawn from the following results(1)
the LTP is valid, and can reduce position and velocity error during GPS
outage;(2)
with an LTP, the
horizontal position and velocity errors of different grade IMUs and different
configurations are reduced greatly, but the vertical position and velocity
errors are reduced only a little bit or kept the same value.In conclusion, LTP
is a valid attitude error model (for pitch and roll) for reduced IMU, and can
improve the navigation performance of GPS/reduced IMU. Tests confirm that the
LTP can provide better navigation performance in areas with limited terrain
variation. The algorithms are, however, applicable to more variable terrain but
the expected improvement may degrade. This will be investigated in later work.



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*Source: 813821-2008-11-03.xml*