LEVER ARM CORRECTIONS DURING INS TRANSFER ALIGNMENT
WITH WIDE ANGLE INITIAL HEADING ERROR
Paul G. Savage
Strapdown Associates, Inc.
SAI-WBN-14008
www.strapdownassociates.com
April 17, 2015 (Updated 09/15/16 - See last page for changes)
ABSTRACT
A recent article developed the methodology for Kalman filter alignment of an inertial
navigation system (INS) with unknown initial heading under dynamic trajectory conditions.
Reference data provided to the Kalman alignment process was GPS velocity with no heading
input. To simplify analytics, the article assumed collocation of the INS and GPS antenna. This
article expands on the wide heading alignment concept to account for lever arm displacement
between the INS and GPS antenna using two alternative Kalman input measurement approaches;
velocity matching (as in the original article), and integrated velocity matching. Included is a
detailed discussion of transitioning from Kalman wide-heading-angle alignment into free-inertial
or aided-inertial navigation.
INTRODUCTION
To assure accuracy, traditional linearized Kalman filters generally require operation under
small system error conditions to minimize second order errors effects. When applied to INS
initial alignment, this has required that initial three-axis attitude errors be small, necessitating a
“Coarse Alignment” attitude initialization process before Kalman filter aided alignment can be
initiated. A recent article [6] has shown how the input measurement to an INS Kalman
alignment filter can be structured for minimum linearization error under large initial heading
angle conditions. Using this approach, attitude initialization only requires “Coarse Leveling” to
a small horizontal attitude error prior to Kalman alignment engagement. Reference [7] shows
how rapid Coarse Leveling can be achieved in an INS using velocity data provided by a GPS
receiver (or equivalent).
The [6] Kalman alignment article was based on forming the measurement with velocity data
provided by a separate navigation reference device (e.g., a GPS receiver). To simplify the
analytical development, [6] assumed that the INS and navigation reference were collocated in the
user vehicle. In practice, physical separation between the INS and navigation reference device
must be included in the Kalman alignment filter formulation to account for differences in INS
and reference device velocity under vehicle angular rate (the so-called “lever arm” effect). (Note
- Lever arm compensation is included in the [7] Coarse Leveling method.) This article expands
on the [6] Kalman alignment approach to include lever arm compensation for two different
measurement configurations: “velocity matching” based on a velocity comparison measurement
(as in [6]), and “integrated velocity matching” based on a velocity integral comparison for the
measurement. A comparison between the alternative measurement approaches is provided,
1
demonstrating the advantages of integrated velocity matching in modeling lever arm effects
under dynamic motion, including random flexure. This article also shows how uncertainty in
GPS reference data can be incorporated within the Kalman alignment filter structure.
MATHEMATICAL NOTATION
The mathematical notation used in this article is the same as in references [1] - [7]:
V = Vector without specific coordinate frame designation. A vector is a parameter that
has length and direction. Vectors used in this article are classified as “free vectors”,
hence, have no preferred location in coordinate frames in which they are
analytically described.
| V | = Magnitude of vector V.
V A = Column matrix with elements equal to the projection of V on coordinate frame A
axes. The projection of V on each frame A axis equals the dot product of V with a
unit vector parallel to that coordinate axis.
(V A ×) = Skew symmetric (or cross-product) form of V A
⎡ 0
⎢
matrix ⎢ V ZA
⎢− V
⎣ YA
− VZA
0
V XA
represented by the square
V YA ⎤
⎥
− VXA ⎥ in which V XA , V YA, V ZA are the components of
0 ⎥⎦
(
)
V A . The matrix product of V A × with another A frame vector equals the cross-
(
)
product of V A with the vector in the A frame, i.e.: V A × W A = V A × W A .
A
CA 1 = Direction cosine matrix that transforms a vector from its coordinate frame A2
2
projection form to its coordinate frame A1 projection form, i.e.: V A1 = C A1 V A 2 .
A2
A
The columns of C 1 are projections on A1 axes of unit vectors parallel to A2 axes.
A2
Conversely, the rows of C A1 are projections on A2 axes of unit vectors parallel to
A2
A1 axes. An important property of C A1 is that its inverse equals its transpose.
A2
.
( )=
d( )
= Time derivative of ( ).
dt
ωA
= Angular rate vector of coordinate frame A2 relative to coordinate frame A1 .
1 A2
2
COORDINATE FRAMES
N = Locally level navigation coordinate frame (with Z axis up) used for attitude
referencing and velocity/position integration operations in the INS. By definition,
in this article, the initial heading of the N Frame is assumed to be nominal, i.e.,
error-free. Initial heading alignment of the N Frame relative to another known
reference frame (N*) is accounted for by defining the N frame to be nominally
misaligned in heading from the N* frame. It is further assumed that the N and N *
frames are rotated at the same angular rate so that the heading angle (around the Z
up axis) between the two remains constant. Appendix A shows how this stipulation
can be removed using a heading angle correction.
N* = Locally level navigation coordinate frame (with Z axis up) used by the reference
navigation device to deliver position/velocity data to the INS being aligned. The
potentially large heading angle misalignment between the N and N* frames is the
means to account for initial heading error in the INS attitude data at the start of
alignment. As defined, the Z axis of the N* frame is parallel to the Z axis of the N
Frame.
B = Strapdown inertial sensor coordinates (“body frame”) with axes parallel to nominal
right handed orthogonal INS sensor input axes.
I = Non-rotating inertial coordinate frame used as a symbolic reference for INS gyro
angular rotation rate measurements as defined in [4].
E = Coordinate (earth) frame aligned with axes fixed to the earth (e.g., one axis parallel
to earth’s rotation axis, the other two axes parallel to earth’s equatorial plane).
GENERAL KALMAN ALIGNMENT FILTER STRUCTURE
Kalman alignment is a specialized application of Kalman filter inertial aiding, a dynamic
process in which INS computed navigation data is periodically compared with equivalent
reference data (at cycle rate n), and used in feedback fashion to update (i.e., correct) INS
navigation parameters. Equations (1) summarize the general updating process:
(
m = f ξ
,ξ
M
INS n Ref n
n
l x (−)
z = H
n n
n
(
)
m − z
z Res = M
n
n
n
x (+ ) = x (−) + K n z
Res
n
n
u c = f x n(+ )
n
)
(1)
n
Applied To ξ INS For Correction
.
t
x n+1 (−) = x n(+) + ∫ n+1 x dt + u c
tn
n
3
.
x = Al x
Parameters shown with a (l) designation in (1) identify computed estimates within the INS and
reference navigation device of actual equivalent ( ) parameters. The (-) designation in (1) refers
to the parameter value at the current n cycle, before it is updated (i.e., corrected) by the Kalman
filter; the (+) designation refers to the parameter value at the current n cycle after Kalman
updating. A summary description of the (1) operations is provided next. The analytic details are
provided in [1 - pp. 415 - 486], [2 - Chapt. 15], [5], and [6].
In (1), data computed from INS navigation parameters ξ INS are compared against equivalent
m . Observation M
m is
data from the navigation reference ξ
to form “observation” vector M
Ref
input to the Kalman filter where at each Kalman updating cycle n, it is compared against
m . The equation for z is based on linearized
“measurement” z , a linearized estimate of M
estimates of expected system errors (embodied in the error state vector column matrix x ), and
l . The error state
how they couple into measurement z through "measurement matrix" H
dynamic matrix Al in (1) defines the dynamics of how x “propagates” from the last n cycle to
the current n cycle.
m and estimated measurement z (the "measurement
The difference between observation M
residual" z Res ), is multiplied by Kalman gain matrix K to generate corrections to the Kalman
filter error estimates. The control vector u formed from INS error estimates x (including
c
provisions for x computation delay) is used to correct the INS data by subtraction from the
equivalent INS parameters. To account for u corrections applied to the INS, the u vector is
c
c
also used to update the Kalman filter x error model for the applied INS error correction.
Kalman gain matrix K in (1) is computed at each n cycle with a statistical model of the
expected uncertainty in the (1) linearized updating process, a function of the error state
covariance matrix P:
(
l
lT
l Tn H
l n P n ( −) H
l Tn + G
K n = P n ( −) H
M n RM GM n
(
)
(
l n P n ( −) I − K n H
ln
P n (+ ) = I − K n H
.
t
P n+1 (−) = P n(+ ) + ∫ n+1 P dt
tn
.
)
T
)
-1
T
l
lT
+ Kn G
M n RM GM n K n
(2)
l P QP G
l TP
P = Al P + P Al T + G
where I is the identity matrix. The P covariance is analytically defined as
(
E χ χT
) where E
is the expected value operator and χ is the uncertainty in the error state estimate x compared
with the x true value. The covariance matrix measures how the initial uncertainty in x (at the
start of Kalman alignment) is progressively reduced by the (1) dynamic estimation/updating
4
process, and how unaccounted noise effects (in x propagation between updates and z Res
measurement updating) delay and limit the convergence process. Noise parameters incorporated
in the (2) gain determination operations are the Q P process noise matrix that accounts for
l P matrix that couples the
random INS error buildup in ξ
and ξ
between n cycles, the G
INS
INS
process noise into error state uncertainty components, the measurement noise matrix R M that
l M matrix
accounts for random errors in the observation and measurement residual, and the G
that couples measurement noise into the measurement residual components, [2 - Sect. 15.1] and
[1 - pp. 428].
The remainder of this article will refer to equations (1) and (2), showing how derived
velocity and integrated velocity alignment process equations fit into the general Kalman
alignment structure.
ANALYTICS OF LEVER ARM COMPENSATION
The analytical basis for lever arm compensation derives from the positioning equation:
R INS = R Ref + l
(3)
where
R = Position vector from earth’s center to a designated location.
INS = Subscript identifying the parameter value at the INS location.
Ref = Subscript identifying the parameter value at the reference navigation device
location.
l = Distance vector (“lever arm”) from the reference navigation device to the INS.
The projection of (3) on E frame axes is:
E
E
RE
INS = R Ref + l
(4)
The time rate of change of (4) defines the relationship between velocities relative to the earth at
the INS and reference navigation device locations:
.
E
E
vE
INS = v Ref + l
.
E
vE
INS ≡ R INS
.
E
E
≡ R Ref
v Ref
where
v = Velocity relative to the earth.
Transformed to N frame axes, (5) becomes
5
(5)
.
N
N
= v Ref
+ C EN l E
v INS
(6)
N
in
The velocity output from the reference device is provided in N* coordinates, hence, v Ref
(6) can be expressed as:
N
N*
= CN
v Ref
N * v Ref
(7)
where
CN
N * = Direction cosine matrix between the N* and N frames which, by virtue of the
frame definitions, represents a transformation operation around the local vertical,
and is constant because both N* and N are rotated at the same angular rate around
the local vertical.
.
The C EN l E term in (6) can be expressed in terms of B frame components through the following
development:
B
lE = CE
Bl
.
.B
.
l E = C BE l B + C E
l
B
.E
C EN l
(8)
.B
.E
= C EN C B l B + C BN l
But,
.
C BN =
d
E = .N E + N .E
C EN C B
CE CB CE CB
dt
(
)
(9)
hence:
.
.
.
C EN C BE = C BN − C EN C E
B
(10)
With (10), the last equation in (8) becomes
.
.
.
.
B
N B
+
l
C EN l E = C BN − C EN C E
C
l
B
B
(
)
(11)
Further development of (11) depends on whether the measurement approach is velocity matching
or integrated velocity matching.
Lever Arm Effects For Velocity Matching Alignment
.
.
For a velocity matching Kalman alignment approach, the C BN and C EN terms in (11) are
from [2 - Eq. (3.3.2-13)] and the transpose of [2 - Eq. (3.3.2-6)]:
6
.
(
) (
)
B
N
C BN = C BN ω IB × − ω IN × C BN
.
(
)
N
C EN = − ω EN × C EN
(12)
in which it has been recognized that the transpose of C EN equals C EN , the transpose of the
(
)
general skew symmetric form V A × equals its negative, and where
ω BIB = Angular rate of the B frame relative to non-rotating inertial space (I) projected on
B frame axes; the angular rate measured by the INS strapdown gyros.
N
ω EN
= Angular rate of the N frame relative to the E frame projected on N frame axes.
N
ω IN
= Angular rate of the N frame relative to non-rotating inertial space (I) projected on
N
N
N frame axes; equal to the sum of ω EN
plus ω IE
, earth’s angular rate relative to
inertial space.
N
N
N
Combining equations (12) into the (11) bracketed term with ω IN
finds:
= ω IE
+ ω EN
.
.
(
) ( ) ( )
( ) ( )
N
B
N
N T
N
N
N ⎧⎡ N T N ⎤ ⎫
= C BN (ω B
IB × ) − C B ( C B ) (ω IE × ) C B = C B (ω IB × ) − C B ⎨ ⎢( C B ) ω IE ⎥ ×⎬
⎦ ⎭
⎩⎣
B
N
N
B
N
N
N
N
N
N
C BN − C EN C E
B = C B ω IB × − ω IN × C B + ω EN × C B = C B ω IB × − ω IE × C B
(13)
⎧⎡
N T N⎤ ⎫
= C BN ⎨ ⎢ω B
IB − C B ω IE ⎥ ×⎬
⎦ ⎭
⎩⎣
( )
Substituting (13) in (11) yields:
.E
.B ⎫
T N⎤
⎧⎡ B
B
C EN l = C BN ⎨ ⎢ω IB − C BN ω IE ⎥ × l + l ⎬
⎦
⎩⎣
⎭
( )
(14)
With (7) and (14), (6) then becomes:
.
N
N*
N ⎧ ⎡ω B −
N Tω N ⎤ × l B + l B ⎫
= CN
+
v INS
v
C
C
⎨
⎬
B ⎢ IB
B
IE ⎥
N * Ref
⎦
⎩⎣
⎭
( )
(15)
Kalman alignment filter design using a velocity matching measurement is based on (15).
Lever Arm Effects For Integrated Velocity Matching Alignment
For an integrated velocity matching Kalman alignment approach, the equivalent to (15) is
.
found from (11) with C EN from (12):
7
.E
.
.
.
.
.
.
B
N l B = N lB + N l B − N E lB
+
l
C EN l = C BN − C EN C E
C
C
C
C
B
B
B
B
E CB
. E B d N B
d
B
N
B
=
l =
C BN l − C EN C B
C B l + ω EN × C BN l
dt
dt
(
(
)
)
(
)
(
)
(16)
Substituting (16) in (6) with (7) then obtains:
N
N*
= CN
v INS
N * v Ref +
(
)
(
d
B
N
B
C N l + ω EN × C BN l
dt B
)
(17)
Kalman alignment filter design using an integrated velocity matching measurement is based on
(17).
Lever Arm Modeling
N
equations (15) and (17), the l B lever arm term can be represented as
For v INS
B
l B = l 0B + l Flex
(18)
where
l 0B = Value of l B at the start of Kalman alignment.
B
lB
Flex = Small variation in l since the start of Kalman alignment due to structural
bending (flexure) between the INS and reference navigation device.
.
The derivative of (18) shows that for l B in (15),
.
.
B
l B = l Flex
(19)
INERTIAL NAVIGATION OPERATIONS DURING KALMAN ALIGNMENT
Kalman alignment operations in the INS computer are configured as a Kalman filter aided
inertial navigation operation in which ξ INS inertial navigation updating operations in (1)
between Kalman n cycles are from [6 - Eqs. (28) - (34)] and [2 - Sects. 8.1.1.1 - 8.1.1.2]:
8
(
) (
)(
)(
N
N
n N
n
lN
C
N * = I + sinβ u ZN × + 1 − cosβ u ZN × u ZN ×
d n
sinβ = 0
dt
d n
cosβ = 0
dt
N*
N*
l =ω
l
ω
IE
IE
N*
Ref
N
N
N
N*
N
N
Ref
N*
l
= lN l
ω
EN C N * ω EN
B
)(
(
N*
B
B
i −K
i −K
l =ω
l
l Scal / Mis ω
ω
IB
IB
Bias
IB
l =ω
l +ω
l
ω
IN
IE
EN
.
N*
l
=l
=l
ω
EN ω EN * ω EN *
l =C
l
l N* ω
ω
IE
IE
N
)
)
lB × − ω
lN × C
l BN = C
l BN ω
l BN
C
IB
IN
.
l BN dt
+ ∫tn C
l BN = C
l BN
C
n
n −1 t n −1
(20)
(
)
B
N ⎡l N l B
× l ⎤
ω
v ZN n = v ZN *Ref / n + u ZN . ⎢C
B
⎣ n IB n 0 n ⎦⎥
N
N*
lN
gl = C
gl
N
*
P
P
Ref
.
N
v INS
H
B
B
Scal / Mis a B − L
a SF = a SF − L
SF
Bias
)(
(
N
N
l N +2 ω
l N × v N
l BN a B + gl − ω
= ⎡⎢C
+ v ZN u ZN
SF
EN
IE
INS
P
H
⎣
)⎤⎥⎦
H
.
N
tn
N
N
v INS H / n = v INS H / n −1 + ∫t n −1 v INS H dt
where
(l) = Designation for a computed or measured parameter containing error compared to
the same but idealized error-free ( ) parameter.
(i) = Designation for inertial sensor (gyro or accelerometer) output parameter containing
error compared to the sensor input error-free parameter (i.e., gyro sensed angular
B
rate ω B
IB or accelerometer sensed specific force acceleration a SF ).
β = Constant heading angle between the N and N* frames measured positive around the
upward defined N and N* frame Z axes.
I = Identity matrix.
9
N*
l
ω
IE
Ref
l N*
,ω
EN *Ref
l N*
,ω
EN Ref
= Angular rates ω IE , ω EN calculated in N* coordinates
using standard INS computation techniques, e.g., [2 - Sects. 4.1.1 & 5.3], but based
on N* frame navigation data provided to the Kalman alignment process by the
reference navigation device (and that N and N* rotate at the same angular rate).
Scal / Mis = Gyro, accelerometer scale-factor/misalignment correction
l Scal / Mis, L
K
matrices.
l
,
K
Bias L Bias = Gyro, accelerometer bias correction vectors.
v ZN = Component of v N along the N frame vertical Z axis.
v ZN *Ref = Component of
N*
v Ref along the N* frame vertical Z axis.
a SF = Acceleration relative to gravitational inertial space [4], measured by strapdown
B
INS accelerometers as a SF
, generating the accelerometer output vector
containing errors.
B
a SF
N*
gl
= Plumb-bob gravity calculated in N* coordinates using standard INS computation
P
Ref
techniques, e.g., [2 - Sects. 5.4 & 5.4.1], but based on N* frame navigation data
provided to the Kalman alignment process by the reference navigation device.
H = Subscript indicating horizontal components of the designated vector.
n
lN
Note in (20) that C
N * is represented by the two scalar sine and cosine parameters sinβ and
n . Errors in these parameters are part of the x error state vector estimated by the Kalman
cosβ
alignment filter in (1), and are corrected (updated) at each n cycle by the control vector u c . At
n
lN
Kalman alignment completion, the final estimate for C
N * determines the heading of the N frame
relative the known reference device N* heading, and uses it to initialize heading for subsequent
navigation mode operations [6].
Note that the INS vertical velocity component v ZN n in (20) is equated to vertical reference
B
velocity v ZN *Ref / n as in [6], but with a correction for the estimated lever arm l 0 n between the
reference device and INS locations (based on an approximate form of (15) with (18)). The error
B
B
in l 0 n is updated in (1) by the u c control vector based on estimates for δ l 0 error included in
n
the x error state vector. Note also that the l B term in (18) is not present in the (20) v
Flex
10
ZN n
equation. This is because of its potential for rapid change (compared to the Kalman update cycle
time), and because of the difficulty in representing l B
Flex with an accurate analytical dynamic
model.
INERTIAL NAVIGATION ERROR STATES DURING KALMAN ALIGNMENT
The x error state vector components estimated in (1) during Kalman alignment consist of
B
inertial navigation (20) parameter errors as in [6], plus inaccuracy in lever arm estimate l 0 :
Kalman INS Alignment Estimated Error States
N
N
B
δ K Scal / Mis, δ K Bias, δ L Scal / Mis, δ L Bias, Δsinβ , Δcosβ , γ , Δv INS H , δ l 0
(21)
where
δ ( ) = Designation for errors that are small compared with ( ) .
Δ ( ) = Designation for errors that can be as large as ( ) .
l BN matrix (considering the N
γ N = Small rotation angle error vector associated with the C
frame to be misaligned), as projected on N frame axes.
The Δ ( ) designation is assigned to Δsinβ and Δcosβ in (21) because of the initial wide angle
heading uncertainty. The Δ ( ) designation for Δ v N
INS H
in (21) is used because the horizontal
N
N*
components of v INS are initialized with v Ref
, i.e., assuming no heading misalignment between
N
l BN is defined to have
the N and N* frames [6]. The γ error is small because the N frame in C
l BN
zero heading error at the start of Kalman alignment, and because the initial leveling error in C
is small by virtue of the Coarse Leveling process.
The error state dynamic equations for the (21) error states are derived in [6] (exclusive of l 0B
lever arm effects) and summarized by [6 - Eqs. (37) - (44)]. Appendix C lists the [6] estimated
error state dynamic rate equations, including lever arm error δ l 0B in the (20) v ZN equation, and
.
n
equating δ l B to zero (based on the definition of l 0B being the lever arm value at the start of
0
alignment, hence, constant as well as its error). The Appendix C equations comprise the
.
elements of x in (1) for estimated error state vector propagation between Kalman update cycles.
Because of the difficulty in accurately modeling input reference data error, Appendix C and
(21) as in [6] contain no allowances for reference data error estimation. However, statistically
11
estimated input data error uncertainties can still be provisioned in the Kalman estimation
process, not as errors to be estimated, but as part of the K n Kalman gain calculations in (2) (to
be discussed subsequently).
VELOCITY MATCHING OBSERVATION AND ESTIMATED MEASUREMENT VECTORS
m in (1) for a velocity matching Kalman alignment filter is based
The observation vector M
n
N
on [6]; by comparing the horizontal components of v INS with their equivalent as formulated
from (15) with (18) for l B :
m = v N
M
n
INS
H /n
lN
−C
N* v
N*
−
Ref H / n
( )
l BN
C
N
N*
lN
≈ v
−C
−
v
N
*
Ref H / n
INS H / n
T N⎤
⎧⎡ B
⎫
l − C
l ⎥ × l B ⎬
l BN ω
⎨ ⎢ω
0
IB
IE
H / n⎩⎣
⎦
⎭n
( )
( )
(22)
l B × l B
l BN
ω
C
H / n IB n 0 n
where
n = Subscript designation identifying a parameter value a Kalman update cycle n.
.
An estimate for the l B term in (15) is not included in (22) because from (19), it is generated by
flexure rate, a parameter of notorious high frequency and random nature (compared to the
Kalman alignment update frequency), with elusive representation by an accurate analytic model.
The estimated measurement vector z n in (1) for a velocity matching Kalman filter is derived
N
from the equivalent error-free form of (22) obtained as v INS
minus the equivalent from (15),
.B
with (18) for l B and (19) for l
H /n
:
.B ⎫
⎧⎡ B
N Tω N ⎤ × l B + l B
− C BN
−
+
ω
l Flex ⎬
C
⎨
B
IE ⎥
0
Flex
H /n
H /n
H / n ⎩ ⎢⎣ IB
⎦
⎭n (23)
.
N
N*
B
B
B ⎤
N
⎡ B
≈ v INS
− CN
N * v Ref H / n − C B H / n ⎢⎣ω IB × l 0 + l Flex + l Flex ⎥⎦
H /n
n
( )
N vN*
−CN
* Ref
N
M n = 0 = v INS
(
( )
(
( )
)
)
The (23) error-free parameters can be defined as the (22) parameters (containing errors), minus
the parameter errors:
N
v INS
H
= v
N
INS H
− Δv N
l BN − δ C BN
C BN = C
INS H
N*
v Ref
H
= v
N*
Ref H
B
lB −δωB
=ω
ω IB
IB
IB
N
lN
CN
N * = C N * − ΔC N *
12
N*
− δ v Ref
H
B
l 0B = l 0 − δ l 0B
(24)
lN
with, based on C
N * in (20),
(
)
(
)(
N
N
N
N
CN
* = I + sinβ u ZN × + (1 − cosβ ) u ZN × u ZN ×
n
n
Δsinβ ≡ sin
β − sinβ
Δcosβ ≡ cos
β − cosβ
(
)
(
)
(25)
)(
N
N
N
N
lN
ΔC N
N * = C N * − C N * = Δsinβ u ZN × − Δcosβ u ZN × u ZN ×
)
Substituting (24) into (23) and subtracting the result from (22) then finds:
(
(
)
N N*
N
l B × l B
m = Δv N
−
Δ
−
δ
C
C
v
ω
M
B
N
*
INS
n
Ref H / n
H /n
IB n 0 n
H /n
) ( ) (
( ) (
)
( ) (
)
B
B
B
B
lB ×δlB + CN
l BN
l BN
− C
× l − C
× l Flex
δ ω IB
ω IB
ω
B
0
n
n
n
n
0n
IB n
H /n
H /n
H /n
.
( )
)
(26)
N*
N*
+ C BN
+ ΔC N
l B − l N δ v Ref
N * δ v Ref H / n + other higher order terms
H / n Flex n C N *
H /n
Reference [6 - Appendix A] shows that the δ C BN error in (25) can be defined in terms of a
small angle error vector which, to first order, approximates as
( )
N
l BN
δ C BN ≈ − γ × C
(27)
Substituting (27) into (26) obtains:
(
)
(
N*
l B × l B
m = Δv N
lN ω
γN
− ΔC N
M
C
N * v Ref H / n + Lin×
INS H / n
n
Bn
IB n 0 n
H /n
) ( ) (
( ) (
)
( ) (
)
B
lB ×δlB + CN
l BN
l BN
δ ω BIB × l − C
− C
ω BIB × l BFlex
ω
B
0
n
n
n
n
0
IB
H /n
n
n
H /n
H /n
.
( )
)
(28)
B
N*
N*
N
lN
+ C BN
−C
l Flex
N * δ v Ref H / n + ΔC N * δ v Ref H / n + other higher order terms
n
H /n
m is the identical equivalent to (22), but expressed as a function of
Equation (28) for M
n
m is obtained from (28) by deleting higher order terms,
system errors. A linearized version of M
n
.N
.N
N
N
and using linearized error rates Δ v
, γ for integration into Δ v INS
,γ :
INS H
N
z n = Δv INS
(
)
(
H /n
N*
− ΔC N
v
N
*
Ref H / n
Lin / H / n
) ( ) (
l )
+ (C
(ωl × l ) + (Cl )
) ( ) (
B
N ×
l B × l B − C
lB ×δlB
lN ω
l BN
l BN
δ ω BIB × l − C
+ γ Lin
ω
C
0n
Bn
n
0
0
IB
IB n
n
n
n
H /n
H /n
H /n
N
B H /n
B
IB n
B
Flex n
.B
N
N*
lN
B H / n l Flex n − C N * δ v Ref H / n
13
)
(29)
where
Lin = Subscript designating an error parameter obtained by integrating a corresponding
linearized error dynamic rate equation provided in Appendix C
m .
z n = Linearized version of M
n
N*
Note that the ΔC N
N * δ v Ref H / n second order term in (28) has been dropped in (29), even though
its magnitude is initially first order by virtue of the large value of ΔC N
N * at the start of
N*
is generally very small, hence, neglecting
alignment. The rationale is that δ v Ref
H /n
N*
ΔC N
N * δ v Ref
H /n
will have negligible impact on early Kalman alignment convergence when
ΔC N
N * is large and has high visibility on the measurement. As Kalman alignment convergence
N*
proceeds and the small δ v Ref
error becomes important for estimating small system error
H /n
effects, the ΔC N
N * estimation error will have converged to a small value, making
N*
ΔC N
N * δ v Ref
H /n
second order and safely neglected. A more sophisticated treatment of
N*
neglecting ΔC N
N * δ v Ref H / n in (29) accounts for its presence as adaptive second order
measurement noise [3].
The estimated measurement vector z n used for measurement residual z Res n determination
m ), is (29) with estimates for the error parameters, and neglecting
in (1) (by comparison with M
n
the flexure terms because they are random, hence, unknown:
n N*
n
z n = Δv N
− ΔC N
N * v Ref
INS
Lin / H / n
−
( ) (
l BN
C
H /n
(
⎛n
N ⎞
l B × l B
lN ω
+ ⎜ γ Lin× ⎟
C
B
n
IB n 0 n
H /n ⎝
⎠H / n
) ( ) (
B
n
l B × δn
l BN
ω
δ ω BIB n × l0 n − C
l 0B n
IB n
H /n
)
)
(30)
l n for calculating z from x in (1) would be formed from the error
The measurement matrix H
n
n
coefficients in (30). The x elements for (30) are calculated as part of the (1)
n
.
integration/estimation/control process using Appendix C for x error state dynamic rates in (1).
N*
Note that as in (21), that velocity reference error δ v Ref
has not been included in (30) for
H
Kalman alignment estimation. For velocity reference data provided within a “tightly coupled”
GPS/INS Kalman aiding configuration, the GPS receiver input to the INS computer would be
pseudo-range measurements to GPS satellites, containing GPS receiver clock phase/frequency
error. The GPS/INS Kalman filter would be structured to include estimation/correction of the
14
N*
GPS clock error during INS alignment. Thus, an estimatable equivalent to δ v Ref
would in
H
principle, be included in the GPS/INS tightly-coupled Kalman measurement model as a function
of estimated GPS clock phase/frequency error. As of this writing, however, a GPS/INS tightlycoupled Kalman alignment filter has yet to formulated from [6], for wide initial heading angle
error application.
For a traditional “loosely coupled” GPS/INS
is corrected within the GPS receiver system, and
configuration, GPS clock frequency/phase error
N*
would be the output provided to the INS.
v Ref
H
N*
However, for lack of an accurate GPS receiver processing model, estimating δ v Ref
in this
H
case is usually not possible, and would typically be neglected as in (30). A random presence of
N*
δ v Ref
in the measurement can still be accounted for, however, as part of Kalman gain
H
calculation in (2) (to be discussed subsequently).
INTEGRATED VELOCITY MATCHING OBSERVATION AND ESTIMATED
MEASUREMENT VECTORS
m and z ) in (1) for an integrated
The observation and estimated measurement vectors ( M
n
n
velocity matching Kalman alignment approach are formulated from the integrated difference
between the INS horizontal velocity and its equivalent using (17) with (18) for l B . For the
idealized error-free form of the observation vector, the integrated velocity comparison M n is:
(
)
(
N
N*
N
⎡ N B B ⎤
= ∫ t n {v INS
− CN
N * v Ref − ω EN × ⎢⎣C B ( l 0 +l Flex ) ⎥⎦} dt
0
H
B
B
B
− C BN
(l0 + l Flex ) + C BN l0
d
⎧ N
N*
⎤ − ω N × ⎡C N l B + l B
− CN
− ⎡⎢C BN l 0B +l B
v Ref
M n = 0 = ∫ t n ⎨v INS
Flex
EN ⎣⎢ B 0 Flex
N
*
⎣
⎦⎥
0 ⎩
dt
H /n
)⎤⎦⎥ ⎫⎬⎭H dt
(31)
H /0
n
in which it has been recognized from the definition of l 0B , that l B
Flex is zero at the start of
alignment, and where
t = Time from the start of Kalman alignment.
Equation (31) simplifies by defining
.
t .
S ≡ ∫ S dt
0
{
(
)}
N
N*
N
⎡ N B B ⎤
− CN
S = 0 = v INS
N * v Ref − ω EN × ⎣⎢C B l 0 +l Flex ⎦⎥
H
N
N*
N
N B ⎤
≈ ⎡⎢v INS
− CN
N * v Ref − ω EN × C B l 0 ⎦⎥
⎣
H
(
so that (31) becomes
15
)
(32)
(
)
− C BN
l B − C BN
lB
M n = 0 = S n − C BN
H /n
H /0 0
H / n Flex n
(33)
Recognizing from the definition of the N and N* frames that C N
N * is constant, an equivalent
recursive form of S n can be structured for (33) based on (32) as:
S n = 0 = S n −1 + ∫ t n
t n −1
(
)
t n v N * dt
N
⎡v N
− ω EN
× C BN l 0B ⎤ dt − C N
∫
Ref H
N
*
⎢⎣ INS H
⎥
t
⎦
H
n −1
(34)
The (34) form is preferred as the basis for deriving an equation for the observation vector in (1)
because, as will be discussed in the next section, it allows the option for an accurate evaluation
N*
dt term using n cycle positioning data provided by the reference navigation
of the ∫ t n v Ref
t n −1
device.
Observation Vector
m for (1) is obtained from (33) and (34) by
The INS calculated observation vector M
n
substituting actual values (containing errors) for the idealized error-free parameters, deleting the
lB
Flex term in M n as non-estimatable (because of its random nature with high frequency
components compared to the Kalman filter updating frequency), and deleting the zero equalities
in (33) and (34) because, unlike the idealized version, the actual observation contains errors,
hence, is generally non-zero:
t
S n = S n −1 + ∫ n
t n −1
(
(
⎡ N
lN
⎢⎣ v INS H − ω EN ×
)
H
l BN
C
)
B
m =S − C
lN
− lN
M
n
n
B H / n C B H /0 l0 n
l B ⎤ dt − C
l N ∫ t n v N * dt
0 ⎥⎦
N *n t n −1 Ref H
(35)
(36)
N*
Equation (35) for S n allows an accurate approximation for the ∫ t n v Ref dt term in
t n −1
H
N*
is not available at a high enough rate to effect an accurate digital
applications when v
Ref H
N*
integration process (i.e., when v Ref contains high frequency components compared to the
H
Kalman alignment filter update frequency). Under such conditions, the positioning output from
N*
the reference device can be used to derive v
based on the development in Appendix B:
Ref H
16
N*
⎞ ⎤ dt
lE − R
lE
l EN * ⎛⎜ R
×) C
⎟
)n ⎝⎜⎛ Rl ERef − Rl ERef ⎠⎟⎞ + ∫ttnn−1 (ωl EN
Ref
Ref
⎝
⎠ ⎦⎥ H
1 ⎡ l N*
⎧ l N*
⎞
lN
l E −R
lE
l EN *) + (ω
l EN *) ⎤ T n ⎫⎬ ⎛⎜ R
×) ( C
×) ( C
⎨( C E ) + ⎢(ω
⎟
Ref
Ref
⎥
EN n-1
n 4 ⎣ EN n
n
n −1⎦
n
n-1 ⎠ H
⎩
⎭⎝
∫t nn−1 v Ref H dt = ⎣⎢( Cl E
t
≈
⎡
N*
N*
n-1
n
n-1
(37)
where
T n = Kalman cycle time interval (from t n −1 to t n ).
l E terms in (37) are functions of standard position outputs provided by the
l EN * and R
The C
Ref
navigation reference device (e.g., latitude, longitude, and altitude) and are readily calculated for
N*
t
N*
∫t nn−1 v Ref H dt determination; e.g., the transpose of [2 - Eq. (4.4.2.1-2)] for Cl E , and [2 - Eq.
l E using [2 - Eq. (4.4.2.2-4)] for u E = u E and [2 - Eq. (5.1-10)] for R ' .
(5.2.2-1)] for R
S
Ref
Up
ZN
Estimated Measurement Vector
The estimated measurement vector z n in (1) for an integrated velocity matching Kalman
alignment filter is derived from the (33) error-free observation form by first setting the (33)
parameters equal to the difference between their computed values (containing errors) minus the
m in (36), and noting that M in (33) is zero:
associated errors, subtracting the result from M
n
n
S n = S n − Δ S n
lN −δCN
C BNH = C
BH
BH
(
)
(
B
l 0B = l − δ l 0B
n
n
0n
(38)
)
N
m = ΔS − δ C N
B − C
lN
−
− lN
δ l 0B + C BN
+ h.o.t.
lB
M
n
n
B H / n δ C B H /0 l0 n
B H / n C B H /0
H / n Flex n
n
(39)
where
h.o.t. = Higher order terms.
The δ C BN terms in (39) can be approximated [6 - Appendix A] in terms of the small angle error
H
l BN :
γ N in C
( )
N
δ C BNH / n ≈ − γ n ×
H
lN
C
Bn
( )
N
δ C BNH / 0 ≈ − γ 0 ×
Substitution in (39) then finds:
17
H
lN
C
B0
(40)
( )
( )
m = Δ S + ⎡ γ N × l N − γ N × l N ⎤ l B
M
C
n
n ⎢ n
0 H C B H / 0 ⎥⎦ 0 n
⎣
H Bn
(
lN
− C
− lN
B H / n C B H /0
)
(41)
+ h.o.t.
δ l 0B + C BN
lB
H / n Flex n
n
Equation (41) is the exact equivalent to observation equation (36), but expressed as a function of
parameter errors that make the computed observation non-zero (as opposed to the idealized
error-free form in (33) for M n that was constructed to be identically zero).
The “measurement vector” z n is the linearized form of (41):
(
)
( )
N
zn = ΔS n + ⎡ γ N × C
lN
l N ⎤ B
⎢⎣ Lin n H B n − γ 0 × H C B H / 0 ⎥⎦ l 0 n
(
lN
− C
− lN
B H / n C B H /0
)
(42)
B
δ l 0B + C BN l Flex
H /n
n
n
The estimated measurement vector z n in (1) is obtained from (42) using estimated values for the
error parameters, with the l B
random term deleted as not estimatable:
Flex
n
(
)
⎡⎛ n ⎞
⎤ B
n
N ×⎞ l N
m
l N − ⎛⎜ γn
lN
lN
−C
z n = Δ
S n + ⎢⎜ γ N × ⎟ C
⎟
δ l 0B n
C
⎥ l0n − C
B
B
B
B
n
H
/
0
H
/
n
H
/
0
Lin
Lin
n ⎠H
0 ⎠H
⎝
⎣⎝
⎦
(43)
l in (1). The
The error coefficients in (43) form the elements of measurement matrix H
n
estimated errors in (43) are elements of estimated error state vector x used in (1) to calculate z n .
n
n
N
mS , γn
Equation (43) shows that Δ
, γ N , and δ l 0B are required to form the estimated
n
Lin
Lin
n
n
0
N
mS , γn
to the (21) estimated error states in x for
measurement, necessitating the addition of Δ
n
Lin
0
the velocity matching Kalman alignment filter. The corresponding additional estimated error
.
state dynamic rate components for x in (1) (additions to Appendix C) derive from the estimated
.
N
value of S in (32) with γ Lin constant:
0
.
(
)
N*
⎡ N
lN × C
lN
l BN l B ⎤
S = ⎢ v INS − C
v Ref − ω
N
*
0 ⎥⎦
EN
⎣
H
.
n
N
γ Lin = 0
0
(44)
Substituting from (24) and (38) for the parameters in (32), and subtracting the (32) result from
(44) obtains:
18
.
N
Δ S = Δ v INS
N*
N − ΔC N
* v Ref
H
)
(
H
(
(
)
)
N
lN × δCN
l BN l B − ω
− Δω EN
×
C
B
0
EN H
H
l B
0
N*
N*
N
lN × C
l N δ l 0B − C
lN
− ω
N * δ v Ref H + ΔC N * δ v Ref H + other higher order terms
EN H B
N
≈ Δ v INS
N*
H
− ΔC N
N * v Ref
H
N*
lN
−C
N * δ v Ref
H
N*
+ ΔC N
N * δ v Ref
H
(45)
+ other higher order terms
or with linearization:
.
N
Δ S Lin = Δv INS
H / Lin
N*
− ΔC N
N* v
Ref H
N
N*
lN
−C
* δ v Ref
H
(46)
N*
The comments following (29) also apply to (46) regarding eliminating ΔC N
N * δ v Ref
H
N*
linearization process from (45) to (46). In this case, the presence of ΔC N
N * δ v Ref
in (46) can
H
in the
be represented as adaptive second order process noise [3], rather than adaptive measurement
noise for (29).
.
.
The estimated version of Δ S for x in (1) is (46) using estimates for the error parameters:
.
n
n N*
N
n
Δ
S Lin = Δ v INS Lin / H − ΔC N
N * v Ref
(47)
H
N*
In (47) as in (30), velocity reference error δ v Ref
has not been included in (47) for Kalman
H
alignment estimation. The rationale is the same as in the paragraphs following (30). As in (30),
N*
in (47) can be accounted for, as part of the Kalman gain
a random presence of δ v Ref
H
calculation in (2) (to be discussed subsequently).
The coefficients for the error terms in (47) constitute elements of the error state dynamic
n
matrix Al in (1) associated with Δ
S Lin propagation between Kalman cycles. From (44) for
.
n
N
N
γ Lin , the rows of the Al matrix in (1) for γ Lin
are zero.
0
0
N
Because dynamic changes in the γ Lin error are much smaller than the initial value over the
N
N
alignment period, the γ Lin error in (43) can be safely approximated by γ Lin , thereby
0
N
eliminating the need to include γ Lin as an error state to be estimated. If this approximation is
0
l BN updating
used, however, it should be recognized that u c control vector components in (1) for C
l N updating in measurement equation (43) and observation equation
should also be applied to C
B0
19
N
l N updating) . Using
(36) (as they would have if γ Lin was estimated separately and used for C
B0
0
the γ N
N
Lin0
≈ γ Lin approach, (42) and (43) then simplify to
(
N
zn = ΔS n + γ
×
Lin n
) (Cl
H
)
(
) δ l0B + C BN
)
(
)
N
B − C
lN
− lN
− lN
B n C B H /0 l 0 n
B H / n C B H /0
(
n
B
⎛ N ×⎞ l N − l N
n
m
lN
lN
−C
z n = Δ
S n + ⎜ γ
⎟ C B n C B H / 0 l 0 − C
δ l 0B n
B
B
H
/
n
H
/
0
Lin
n
n ⎠H
⎝
lB
H / n Flex n
(48)
(49)
MEASUREMENT NOISE
The measurement noise matrix R M used in (2) to calculate the K n Kalman gain matrix,
represents the presence of noise on measurement residual z Res in (1) used for estimated error
n
state vector x n updating. The relationship between measurement noise and R M is based on
m in (1) by the linearized measurement vector z , as in
approximating the observation vector M
n
n
(29) and (42):
m
zn ≡ M
Lin n
l
l n xn + G
=H
n
Mn Mn
(50)
where
n M = Vector of independent random components that are uncorrelated between Kalman
n
update cycles.
Using the z n equation in (1) with (50) to calculate the equivalent linearized version of z Res n in
(1) finds:
z Res
Lin / n
χ ≈ x − x
l
l n χ ( −) + G
=−H
n
n
Mn Mn
(51)
where
χ = Uncertainty in x .
Using z Res
for z Res in the (1) equation for x n(+) , and subtracting the true value identity
Lin / n
n
x n+1 ( + ) = x n(−) from the result, then obtains for the change in χ from the Kalman update:
l
l n ) χ ( −) + K n G
χ n(+ ) = ( I − K n H
n
n
Mn Mn
(52)
The P n (+) covariance updating operation in (2) is derived from (52) by substitution into the P
definition equation and expansion [2 - Sect. 15.1.2.1]:
20
(
T
P ≡E χ χ
)
(
T
RM ≡ E nM nM
)
(53)
The K n Kalman “optimal” gain equation in (2) is derived as being the K n that minimizes
P n (+) in (2) for a given P n (−) [2 - Sect. 15.1.2.1].
Velocity Matching Kalman Alignment Measurement Noise
For a velocity matching Kalman alignment filter, (50) is provided by (29). In (29), the
l
n M measurement noise term in (50) would comprise Δ v N
accelerometer
G
M
n
n
Lin / INS H / n
.
, l B , vN*
in the last line of (29):
quantization noise (as in [6]), and l B
Flex n Flex n δ Ref H / n
.
B
(Cl BN )H / n (ωl BIB × l BFlex ) + (Cl BN )H / n l Flex
n
n
n
N*
lN
−C
N * δ v Ref
H /n
l
. The G
M n measurement noise
dynamic coupling matrix in (50) would have unity for the quantization noise rows, and would be
.B
,
formed from the coefficients of n M in the previous expression for the l B
l Flex ,
Flex
n
N*
δ v Ref
H /n
n
n
rows. Because the definition for n M is a column of independent random
n
.B
and
components, a difficulty arises for the l B
l Flex terms in n M because of their
Flex n
n
n
.B
is
the
integral
of
correlation (i.e., l B
l Flex ). A simple expedient has been to estimate
Flex
independent values for each in (52) and assume no correlation. Additionally, to assure that the
flexure measurement noise model is uncorrelated from Kalman cycle to cycle (the analytical
basis for n M ), the Kalman cycle time should be set sufficiently higher (e.g., two times) than
n
the lowest expected flexure mode frequency.
N*
A more interesting problem arises for the δ v Ref
term because as discussed earlier, it may
H /n
not be modelable as part of the estimated error states, but may be correlated between n cycles,
thereby violating the defining requirement for n M n . One method for dealing with a random
N*
nature for δ v Ref
, is to assume it is correlated with itself over some time period, but that the
H /n
correlation time is short compared with the Kalman update n cycle period. This would require
N*
an assumed δ v Ref
correlation time, with the Kalman update period then set, for example, to
H /n
N*
from n cycle
twice the assumed correlation time value, thus effectively un-correlating δ v Ref
H /n
N*
(e.g., a first
to n cycle. A more sophisticated method assumes a correlation model for δ v Ref
H /n
N*
as part of the error state
order Markov process, e.g., [2 - Eq. (12.5.6-3)], and treating δ v Ref
H /n
vector, but not to be estimated. This can easily be achieved using a “considered variable”
approach based on the (2) covariance update implementation.
21
With the considered variable method, the Kalman gain matrix K n is calculated as if
N*
δ v Ref
H /n
was part of the x error states (with a corresponding error state dynamic rate equation).
N*
are then set to zero, thus nullifying
Following the K n computation, the K n rows for δ v Ref
H /n
N*
N*
estimation, and eliminating the need to include δ v Ref
in (1) as part of
potential δ v Ref
H /n
H /n
random presence then only registers as part of the covariance
x or z . The δ v N *
n
n
Ref H / n
N*
propagation process (third equation in (2)) and by its impact on the non- δ v Ref
H /n
rows in K n
N*
considered variable modeling is provided in
for x n updating. Further discussion of δ v Ref
H /n
the Process Noise section to follow.
The considered variable approach is only valid if the P n (+) update equation is implemented
as shown in (2) (the “Joseph’s” form) whose derivation is general for any K n matrix. In some
applications, a simplified form of the P n (+) equation is used that is derived from the Joseph’s
form by analytically substituting the optimum (2) equation for K n (without allowance for
zeroing rows) [2 - Eq. (15.1.2.1.1-4)]:
P n ( + ) = ( I − K n H n ) P n ( −)
(54)
Using (54) to determine P n (+) with a K n adjusted for zeroed rows would generate an invalid
P n (+) update, producing error in subsequent calculations, and potential Kalman-loop instability.
A similar problem would arise if P n (+) was calculated based on “UD factorization” [8 - Sect.
9.5], another approach used by some analysts, but also formulated assuming an optimum K n ,
without allowances for zeroing rows.
Integrated Velocity Matching Kalman Alignment Measurement Noise
For an integrated velocity matching Kalman alignment approach, (50) is provided by (42),
l
n M measurement noise term in (50). Then
lB
with C BN
in (42) representing the G
M
H / n Flex n
n
n
B
N
l
the G
M n measurement noise dynamic coupling matrix would be C B H / n and l Flex n would be
n M . The components of l B
would be modeled as independent random variables that are
n
Flex n
uncorrelated from n cycle to n cycle. To assure n M model validity, the Kalman cycle time
n
should be set higher (e.g., two times) than the lowest expected flexure mode frequency.
22
PROCESS NOISE
The process noise matrix Q P used in (2) for P covariance propagation between Kalman
updates, represents the effect of noise integrated into parameters forming the observation vector
m in (1). The relationship between the integrated noise and Q is based on approximating
M
P
n
m by the linearized measurement vector z in (50). Integrated process noise is generated in
M
n
n
.
z n from its buildup in x n between Kalman cycles according to the equivalent of x estimation
equation in (1) that includes non-estimatable random components:
.
lP n
x = Al x + G
P
(55)
where
n P = Vector of independent white noise components.
l P = Estimated process noise dynamic coupling matrix.
G
.
The difference between (55) and x provides an equation for the build-up rate in x uncertainty
between Kalman cycles:
.
l P nP
χ = Al χ − G
(56)
.
The covariance rate P in (2) is obtained by substituting (56) into the derivative (53) for P, and
expanding:
.
(
.
.
)
T
T
l TP
P = E χ χ + χ χ = Al P + P Al T + G P Q P G
where
(57)
Q P = Diagonal matrix with diagonal elements equal to the n white noise densities.
P
For the Kalman alignment filter, integrated process noise is created by random components
B
i B output vectors in (20), generating error build-up in
in the gyro/accelerometer a , ω
SF
IB
N
m observation vector
l BN , and ultimately entering the M
calculated velocity v INS and attitude C
n
n
H
l P and Q matrices in (2) are formed
input to the Kalman filter. Reference [6] shows how the G
P
from gyro/accelerometer process noise elements in a velocity matching Kalman alignment
application. For an integrated velocity matching approach, the [6] gyro/accelerometer process
N*
lN
noise results also apply, but with additional random components on Δv N
and C
N * δ v Ref in
INS H
23
.
.
the (46) Δ S expression. (Recall that the integral of Δ S in (45) generates Δ S n in the (41) form
m .)
of observation M
n
N
Accelerometer quantization δυ Quant is a random element in Δv INS
that directly impacts
H
.
N
N
Δ S as white process noise. It is modeled within Δv INS
as C B H δ υ Quant [2 - Eq. (12.5-13)].
H
(In [6] and discussed in the previous section, the same term is treated as measurement noise for
.
.
.
velocity matching Kalman alignment.) For the Δ S portion of x in (55) and the Δ S covariance
.
l P for δυ
equivalent of P in (57) and (2), the elements in G
Quant quantization noise coupling
.
.
N
into x (and P ) would be C B H . Elements would then be included in Q P for (57) and (2) equal
to the δυ Quant quantization noise densities, a function of the digital integration algorithm update
)
(
⎡ N
lN × C
l BN l B ⎤ dt in S n equation (35) - See [2 frequency for implementing ∫ t n ⎢v
− ω
0 ⎥⎦
EN
t n −1 ⎣ INS H
H
Sect. 16.2.3.1] for the equivalent effect when calculating position by velocity integration.
.
N*
lN
Random reference velocity components in Δ S from C
N * δ v Ref are easily modeled in the
integrated velocity Kalman alignment filter using the considered variable approach discussed
.
N*
earlier for the velocity matching type measurement. The δ v Ref
error in Δ S would be treated as
part of error state vector x, without estimation in Mx of (1), but with inclusion in the covariance
l N coupling of the δ v N *
matrix P of (2). The Al matrix in (2) would be formatted to include C
.
N*
.
Ref
covariance into the Δ S covariance rate elements of P . The Kalman gain matrix K n would be
N*
N*
calculated as in (2), with the computed δ v Ref
rows then set to zero, thereby nullifying δ v Ref
.
N*
error would be modeled in the generic (55) x format as an easily defined
estimation. The δ v Ref
.
N*
N*
dynamic process, e.g., first order Markov for each δ v Ref
component i:
correlated δ v Ref
.
N* = l
N* + G
l
δ v Ref
AvRef δ v Ref
P Ref / i n P vRef / i
i
i
i
(58)
with a selected correlation time embedded in (58) as the negative reciprocal of l
AvRef i for each
.
N*
N* from (58)
as an un-estimated part of error state vector x (with δ v Ref
element i. Treating δ v Ref
i
.
.
in x ), the Q P process noise matrix in (2) for P would be structured to include n P vRef white
N*
variance [2 noise densities, with the densities set to achieve a specified steady δ v Ref
.
l P process noise coupling matrix for P in (2) would be structured to
Sect.16.2.3.1]. The G
24
.
l
l
include G
P vRef coupling of n P vRef into P , and the A error state dynamic coupling matrix
.
N*
would include l
AvRef coupling of δ v Ref into P .
VELOCITY MATCHING VERSUS INTEGRATED VELOCIY MATCHING
Integrated velocity matching has some advantages compared with the velocity matching
measurement approach [2 - Sect. 15.2.2.3] that can best be ascertained by their observation
equations in a pure estimation Kalman alignment configuration. For a Kalman estimator, the
control vector in (1) is not utilized, and the Kalman filter would estimate system errors as in (1),
but without estimated error corrections. To further simplify the measurement approach
N
comparison, it is expedient to approximate the uncontrolled horizontal velocity error Δv INS
as
H
having little change from its initial value
derives from the initial estimated version
N
v INS
H /0
(
N
. An analytical
Δ v INS
H /0
N
in (15):
of v
INS H
expression for
)
l B × l B ⎤
l N v N * + C
lN ω
= ⎡⎢C
N
*
B
Ref
IB0 0 ⎥⎦ H
0
0
0
⎣
N
Δ v INS
H /0
(59)
N
N
The Δ v INS
error is derived as (59) minus the equivalent v INS
from (15) using error
H /0
H /0
definitions in (24):
N
Δv INS
H /0
= ΔC N
N* v
N*
+"
Ref H / 0
(60)
N
Approximating Δv INS
during Kalman alignment by (60), velocity matching observation
H
equation (26) becomes in terms of system errors:
.B
⎞+ N
B
B
N
m ≈ −ΔC N ⎛⎜ v N *
N*
−
×
+
+"
ω
l
l
M
C
C
v
⎟
B
B
N*
n
Flex n
IB n
H /n
H / n Flex n
⎝ Ref H / n Ref H /0 ⎠
( ) (
) ( )
(61)
Similarly, (41) with the integral of (45) gives for the integrated velocity matching observation
equation:
⎞ dt + N
B
m ≈ − ΔC N t n ⎛⎜ v N * − v N *
M
C B H / n l Flex n + "
N * ∫0 ⎝ Ref H
n
Ref H /0 ⎟⎠
(62)
Equations (61) and (62) illustrate how lever-arm flexing impacts the observation for the two
alignment approaches, both representing measurement noise on the Kalman filter observation
input. The form of (61) versus (62) benefits the integrated velocity measurement approach in
estimating lever-arm flexing measurement noise for R M in (2), and in establishing maneuver
requirements to achieve a specified heading determination accuracy.
25
Estimating lever-arm flexing measurement noise for integrated velocity matching entails
estimating the magnitude of l B
Flex in (62) for R M in (2) under dynamic flight conditions. (In
some cases this might be by intuitive judgment of the Kalman filter system design engineer
based on a general understanding of the user vehicle in its operational environment,) In contrast,
.B
for a velocity matching measurement, estimates of both l B
and
l Flex magnitude in (61) are
Flex
required for R M , the latter typically having a larger impact on Kalman alignment performance.
.B
Flexing rate l Flex
is a function of both l B
Flex magnitude and corresponding excited structural
.B
bending mode shapes/frequencies that generate l Flex
proportional to the l B
Flex mode amplitudes.
In general, flexing mode frequency responses are more difficult to estimate than l B
Flex amplitude,
.B
adding a higher degree of uncertainty in estimating l Flex
magnitude for measurement noise
matrix R M .
Maneuver requirements for determining heading to a specified ΔC N
accuracy can
N *Required
to measurement-noise ratio in (61) and (62). For a
be ascertained from the ΔC N
N *Required
velocity matching measurement, the measurement-noise/heading-accuracy ratio in (61) for the
.
B
dominant flexing rate noise term is symbolically l Flex
/ ΔC N
, which sets requirements
N *Required
n
N*
N*
⎞ maneuver in (61) to achieve Δ N
− v
on the ⎛⎜ v
C N *Required . For example, for a 0.02
⎟
Ref
Ref
H /n
H /0 ⎠
⎝
ft estimated flexure amplitude at an oscillation frequency of 2 Hz = 12.6 rad/sec, the
.B
amplitude would be 0.02 × 12.6 = 0.25 fps. For a 1 milli-rad
corresponding flexure rate l Flex
.
B
/ ΔC N
ratio would be 0.25 / 0.001
heading estimation accuracy requirement, the l Flex
N *Required
n
N*
N*
⎞ single maneuver of 250 fps to
− v
= 250 fps. From (61), this requires a ⎛⎜ v
⎟
Ref
Ref
H /n
H /0 ⎠
⎝
achieve the required heading accuracy in a single Kalman estimation cycle. To reduce the
maneuver amplitude, multiple measurements would be processed to average out the flexure noise
(e.g., for factor of 10 reduction, 10 2 = 100 measurements (and Kalman cycles) would be
required). An added complication is that to maintain the same average trajectory direction
N*
N*
during the alignment period, the average of v
must approximate v
, necessitating a
Ref H / n
Ref H /0
N*
trajectory
change in horizontal velocity magnitude, or executing a lateral oscillatory v Ref
H /n
N*
(the so-called dynamic “S” maneuver).
change pattern around v Ref
H /0
In contrast, for an integrated velocity matching measurement, (62) shows that the
measurement-noise/heading-accuracy ratio is symbolically l B
/ ΔC N
. For the same
N *Required
Flex n
26
0.02 ft lever arm flexure and 1 milli-rad heading error accuracy, l B
/ ΔC N
would
N *Required
Flex n
N*
N*
⎞ horizontal maneuver
−
be 20 ft. From (62), this corresponds to a ∫ t n ⎛⎜ v
0 ⎝ Ref H v Ref H /0 ⎟⎠
requirement of 20 ft to achieve the required heading accuracy from a single Kalman estimate.
Processing 10 measurements would reduce this by √10 to 6.3 ft. Note, however, that
N*
⎞
t ⎛ N*
∫0 n ⎜⎝ v Ref H − v Ref H /0 ⎟⎠ represents a horizontal position displacement from the nominal
trajectory. This maneuver can be achieved with a small lateral turn followed by a turn back to
the original trajectory direction after the required position displacement has been achieved (i.e., a
single half-cycle of an S maneuver). Alternatively, the horizontal velocity magnitude can be
slightly altered until the position change has been achieved, the velocity magnitude then returned
to its original value. Continued Kalman estimates can then proceed as desired along the
modified trajectory with no additional maneuvering.
The previous discussion demonstrates how measurement noise modeling is simplified, and
both maneuver and alignment time requirements are reduced for integrated velocity matching
compared with velocity matching Kalman alignment. It should also be noted, that due to other
system errors (other than heading) being estimated during Kalman alignment, the previous
numerical examples would require additional alignment time and/or maneuver increases to
achieve the required accuracy.
KALMAN ALIGNMENT INITIALIZATION
Initialization of navigation parameters at the start of Kalman alignment would be as in [6] for
the equation (20) parameters, but including a lever arm correction for initial horizontal velocity
N
l N initialized at identity as in [6]. The C
l N attitude matrix would be
v
as in (59) with C
N*
INS H
B
initialized to the value at Coarse Leveling completion. The best estimate available would be
B
used for l lever arm initialization (zero would be acceptable if completely unknown), and the
0
integrated velocity matching S parameter would be initialized at zero. All estimated error state
parameters would be initialized at zero as in [6].
Initialization of the covariance matrix P would be as in [6] for the (21) navigation error states
with a best estimate for the estimated δ l 0B lever arm error covariances, and zero for the
integrated velocity matching error state Δ S covariances. If first order Markov processes are used
N*
as in (58) to model the δ v Ref
error components, the associated covariances in P would be
initialized at their steady state values [2 - Sect. 16.2.3.1].
27
TRANSITIONING TO NAVIGATION AT KALMAN ALIGNMENT COMPLETION
Kalman alignment completion is declared when the covariance of the (21) heading error
parameter Δsinβ falls below a specified level. Then the basic navigation parameters would be
initialized for the next phase of either free-inertial or Kalman-aided-inertial navigation. Two
methods can be considered in this regard; initialization based on maintaining the N frame at its
l BN matrix, or initialization based on N frame
estimated attitude relative to the B frame in the C
alignment with the reference N* frame (or with the reference N** frame if using an Appendix A
augmentation approach). For either case, navigation mode processing following initialization
would then proceed using traditional inertial navigation integration operations, e.g., [2 - Tables
5.6-1, 7.5-1 & 8.4-1] with (1) and (2) as appropriate.
Initialization For Subsequent Free-Inertial Navigation
When the desired navigation mode N frame is defined to be at its alignment completion
l N and v N would be initialized for the navigation mode at
orientation relative to the B frame, C
B
INS
their alignment completion values, altitude would be initialized at the referenced device input
B
altitude h
with a l lever arm correction, and N frame attitude relative to the E frame (i.e.,
Ref
0
E
l N ) would be initialized at the reference input derived C
l EN * transformed to the N frame:
C
(
N
l BN
h INS = h Ref + u ZN . C
l B
0
)
( )
N
l EN = C
l EN * C
lN
C
*
T
(63)
B
lN
The l 0 term in (63) would be set to its alignment completion value, and C
N * in (63) would be
n and cosβ
n heading parameters at alignment
calculated as in (20) using the estimated sinβ
E
l N * matrix in (63) would be computed from reference device input position
completion. The C
data using appropriate equivalency formulas, e.g., as [2 - Eq. (4.4.2.1-2)] for latitude, longitude,
wander angle reference inputs. (Note - The reference wander angle would be zero for a
l EN matrix is also required in the navigation
geographic type N* frame.) (Another note - The C
mode as a means for converting N frame data to a known coordinate frame for INS output, e.g.,
[2 - Eqs. (4.4.2.1-3), (4.1.2-1), (4.1.2-2), & (4.3.1-4)] for latitude, longitude, north/east velocity,
l EN would be continuously updated during the navigation mode following
roll/pitch/heading. C
.
l NE , e.g., [2 - Eq. (4.4.1.1-1)].
alignment completion by integrating C
When the desired navigation mode N frame is defined to be initially aligned with the N*
N
l N and v
frame attitude, C
would be initialized for navigation mode entry at
B
INS
28
( )
( )
T
N
lN
v INS (+) = C
N*
l BN (+) = C
lN
lN
C
N * C B (−)
where
T
N
v INS (−)
(64)
(+), (-) = Values at align completion (-) and at navigation mode entry (+).
l EN matrix would be initialized at C
l EN * , and h INS altitude would be initialized as in (63).
The C
When the desired navigation mode N frame is defined to be initially aligned with the N**
N
l N and v
frame of Appendix A, C
would be initialized at navigation mode entry as
B
(
INS
)
T
l BN (+) = C
lN
l N* l N
C
N * C N ** C B (−)
(
N
lN
l N*
v INS (+) = C
N * C N **
)
T
N
v INS (−)
(65)
l EN matrix would be initialized at C
l EN ** , and h INS altitude would be initialized as in (63).
The C
Initialization For Subsequent Kalman Aided Inertial Navigation
l BN ,
If Kalman alignment is to be followed by a Kalman aided inertial navigation mode, C
N
l EN , and h INS would be initialized as in the previous section for free inertial navigation.
v INS , C
In addition, the covariance matrix and initial error states must also be initialized for the aided
inertial navigation mode Kalman filter. Two options can be considered: 1. Where the initial
navigation mode N frame is aligned with the end-of-alignment N frame, and 2. When the initial
navigation mode N frame is aligned with the end-of-alignment N* (or N**) frame.
n
sinβ estimated
For the former case (maintaining N at the end-of-alignment orientation), the Δ
m
N
to form an updated
error state would be merged with the estimated γn
ZN component of γ
m
N
version of γ . This would include summing the end-of-alignment Δsinβ elements in
covariance matrix P with those of γ ZN (including summing covariances with other error states).
Then, the Δsinβ , Δcosβ error states and corresponding rows/columns of P would be deleted.
Lastly, a vertical velocity error state would be added with, based on vn in (20), an estimated
ZN
initial navigation mode uncertainty of
⎤
N ⎡l N ⎛ l B χ
+ u ZN
. ⎢C B ⎜ ω IB × δ l B ⎞⎟ ⎥
0 ⎠⎦
⎝
⎣
χ δ v ZN
INS
= χδ v
χ δ v ZN
INS
, χ δ v ZN *Ref , χ δ l B = δ v ZN INS , δ v ZN *Ref , δ l 0B error uncertainties.
0
ZN * Ref
(66)
where
29
v ZN INS error state estimate would be initialized at zero in the navigation mode error
The new δn
state vector, and the corresponding elements of the covariance matrix P would be initialized
based on (66) as
⎛
⎞
E ⎛⎜ χ δ v ZN INS 2 ⎞⎟ = E ⎜ χ δ v ZN *Ref 2 ⎟
⎝
⎠
⎝
⎠
(
( )
N T lN lB
+ u ZN
C B ω IB ×
)
T
⎛
T ⎞ ⎡ N T N lB ⎤
E ⎜ χ δ l 0B χ δ l B ⎟ ⎢ u ZN Cl B ω IB× ⎥
0 ⎠⎣
⎝
⎦
( )
( )
(
N T lN lB
E ⎛⎜ χ δ v ZN INS χ δ l BT ⎞⎟ = u ZN
C B ω IB ×
0 ⎠
⎝
)
( )
⎛
T⎞
E ⎜ χ δ l 0B χ δ l B ⎟
0 ⎠
⎝
E ⎛⎜ χ δ l 0B χ δ v ZN INS ⎞⎟ = ⎡⎢E ⎛⎜ χ δ l B χ δ v ZN INS ⎞⎟ ⎤⎥
⎝
The
⎛
E ⎜ χ δ l 0B χ δ l B
⎠
⎣ ⎝
(67)
T
⎠⎦
0
T⎞
⎟ term in (67) would be set to the equivalent in the end-of-alignment
⎠
⎛
⎞
covariance matrix P. The E ⎜ χ δ v ZN *Ref 2 ⎟ term would be set to an estimate based on
⎝
⎠
knowledge of reference navigation device accuracy.
⎝
0
For the case when the navigation mode N frame is aligned with the end-of-alignment N* ( or
N**) frame, the previous initialization process would also apply in addition to a transformation
of the estimated error state and covariance matrix elements from N to N* (or N**). The former
is straightforward but the latter can be complicated regarding correlations with other error states.
To avoid this complexity, maintaining the N frame orientation at its end-of-alignment value
might be preferred.
Another covariance initialization approach that avoids much of the complexity with the
previous methods is to treat the wide angle Kalman alignment process as a Coarse Heading
determination operation that follows Coarse Leveling, both then comprising a combined Coarse
Alignment function. Course Alignment would be terminated when the Δsinβ variance on the P
diagonal became small enough to accurately approximate γ ZN with a linearized model (i.e., a
2
variance of 0.017 corresponding to an uncertainty of 1 deg = 0.017 rad), hence, compatible with
a traditional aided INS linearized Kalman filter. Then initial attitude for the navigation mode
l BN at Coarse Alignment completion (with appropriate (64) - (65) N* or N**
would be set to C
conversion if desired), the initial γ ZN variance in P would be set to the specified Coarse
N
Heading Kalman alignment completion value, and the initial γ H variances would be set to the
expected Coarse Leveling accuracies, i.e., the same as when initializing the Coarse Heading
alignment Kalman filter. All other parameters for the Kalman aided navigation mode would be
initialized as they were at Coarse Heading Kalman alignment initiation, the exception being that
30
the Δsinβ , Δcosβ error states would not be included, and the γ ZN variance in the covariance
matrix would be initialized at its expected Coarse Alignment completion value.
The penalty for the latter approach is the additional time for the navigation mode Kalman
N
filter to converge the γ H horizontal attitude errors to their alignment completion values, a minor
penalty since these errors would be very visible on the Kalman filter observation vector, hence,
attacked immediately at navigation mode initiation (with or without maneuvering). The benefit
for the latter approach is avoidance of the analytic development required for transitioning the
end-of- alignment Kalman filter into the navigation mode version, the associated time for
software development/validation, and the possibility for incurring design errors during the
development process, particularly if an N to N* conversion is required. An additional advantage
is that the combined dual Coarse-Leveling/Coarse-Heading Alignment function is easily
interfaced with previously validated Kalman aided inertial navigation software configurations
with their individual error modeling approaches.
APPENDIX A - COMPENSATING FOR DIFFERENCES
IN N AND N* FRAME HEADING RATES
This article has assumed that both the N and N* frames are rotated about the vertical at the
same rate. For cases when reference data is being supplied in a coordinate frame N** having a
different vertical rotation rate than N and N*, a simple transformation operation can be used to
convert the N** vector data into the equivalent N* frame format assumed in the article. The
transformation matrix would represent a rotation around the vertical equal to the integrated
difference between the N** and N* vertical rotation rates. For example, consider that the N and
N* frames are of the locally level wander azimuth type [2 - Sect. 4.5] whose angular rate around
the vertical relative to the earth fixed E frame is zero, i.e.,
ω EN ZN = 0
(A-1)
where
ω EN ZN = Component of ω EN along the N and N* frame upward Z axes.
Assume that the N** frame used for reference vector data transfer to the INS is of the locally
level geographic type [2 - Sect. 4.5] in which one of the horizontal axes (e.g., Y) is controlled by
ω EN ZN to point north. Reference [2 - Eq.( 4.4.3-5)] shows that for this situation:
ω EN ZN ** = ω EN YN ** tan l
(A-2)
where
ω EN ZN ** = Component of ω EN ** along the N** frame upward Z axis.
31
ω EN YN ** = North (Y axis) component of ω
EN ** .
l = Latitude
The transformation matrix for converting vector data from N** to N* coordinates would be
about the vertical through an angle equal to the integrated difference between (A-1) and (A-2) [2
- Eq. (3.3.2-6)]:
t
.
.
(
N* = N* ω N
CN
** C N ** N * N ** ×
N*
*
CN
N ** = ∫0 C N ** dt
)
N
N
ωN
* N ** = ω EN YN ** tan l u ZN
(A-3)
Then any vector V provided by the reference navigation device in N** coordinates would be
converted to the N* frame, e.g., for (20), using:
N **
*
V N* = C N
N ** V
(A-4)
Computations using the transformed data would then proceed assuming N and N* are rotating at
the same rate as described in the article.
N*
APPENDIX B - EVALUATING ∫ t
v Ref dt FROM REFERENCE POSITION DATA
t n −1
From the v E
Ref definition in (5):
.
.
t
E
vE
Ref ≡ R Ref
t
E
E
E
ΔRE
dt = ∫
vE
R Ref
Ref ≡ R Ref − R Ref n-1 = ∫t
Ref dt
t
n −1
n −1
(B-1)
in which Δ in this appendix signifies change, not error (as in the main article). Defining:
N*
Δ R Ref
= C EN *Δ R E
Ref
(B-2)
It follows that:
.E
.
.
d
d
N*
N*
E
E
E
Δ R Ref
=
C EN *Δ R Ref = C EN * Δ R Ref + C EN *R Ref = C EN * Δ R Ref + v Ref
dt
dt
(
)
(
)
(B-3)
.
The C EN * term in (B-3) is by similarity with (12) and recognizing that N and N* rotate at the
same angular rate:
.
(
)
(
E
E
C EN * = − C EN * ω EN * × = − C EN * ω EN ×
)
Then (B-3) becomes
32
(B-4)
(
(
)
)
d
E
N*
N*
N*
N*
E
N*
Δ R Ref
= − C EN * ω EN
× Δ R Ref
+ v Ref
= − ω EN
× Δ R Ref
+ v Ref
dt
(B-5)
Integrating (B-5) finds at time t:
N*
N*
N*
N*
dt + ∫ t
dt
Δ R Ref
= − ∫ t ω EN
× Δ R Ref
v Ref
t n −1
t n −1
(B-6)
N*
Rearranging (B-6) and substituting Δ R E
Ref from (B-1) with the Δ R Ref definition from (B-2)
N*
then obtains for ∫ t
v Ref dt at t = t n
t n −1
t
N*
∫t nn−1 v Ref
(
)
(
)
(
)
N* ⎡ N* E
⎤ dt
dt = C EN * R E
− RE
+ ∫ t n ω EN
× C E R Ref − R E
Ref
Ref
Ref
⎢
n
n-1
n-1 ⎥
n
t n −1
⎣
⎦
(B-7)
APPENDIX C - ESTIMATED ERROR STATE DYNAMIC EQUATIONS FOR VELOCITY
MATCHING ALIGNMENT KALMAN FILTER
d n
d n
Δsinβ = 0
Δcosβ = 0
dt
dt
n n N
N
n N
ΔC N
N * = Δsinβ u ZN × − Δcosβ u ZN × u ZN ×
(
)
(
n
n l N*
N
Δω IE
= ΔC N
N * ω IE Ref
H
)(
EN Ref
H
H
B
n n
N
(C-2)
H
d n
d n
δ K Bias = 0
δ K Scal / Mis = 0
dt
dt
n
n
B
δ ω BIB = δn
K Bias
K Scal / Mis ω IB + δn
.
)
n
n l N*
N
Δω EN
= ΔC N
N* ω
n
n
n
N
N
N
Δω IN
= Δω IE
+ Δω EN
n
N
(C-1)
(C-3)
n
N
l − ωN ×γ
l BN δ ω
+ Δω IN
γ Lin = − C
IN
Lin
IB
H
(C-4)
d n
d n
δ L Bias = 0
δ L Scal / Mis = 0
dt
dt
n
B
B
= δn
δ a SF
L Bias
L Scal / Mis a SF + δn
(C-5)
33
.
N ⎡l N ⎛ l B n
B ⎞⎤
δn
v ZN = u ZN . ⎢C
B ⎜ ω IB × δ l 0 ⎟ ⎥
⎝
⎠⎦
⎣
.N
Δv
INS Lin / H
⎡
⎢v ZN
N
− u ZN
× ⎢
H ⎢
⎢⎣
(
)
n
N
n
δ l 0B = 0
n N*
δ g P = ΔC N
N* g
n
⎛n
⎞
B
B
l N δ a SF
=C
+
⎜ a SF × ⎟
BH
⎝
⎠H
P Ref
n
N
n
γ Lin
+ δ gN
PH
(
n
n
⎛Δ
N
N ⎞ n lN
lN
+
2
ω
⎜ ω EN H + 2 Δω IE H ⎟ + δ v ZN ω
EN
IE
⎝
⎠
n n n
N
+ ω IN + ω IE
Δ v INS
(
ZN
ZN
(C-6)
)
Lin / H
) ⎤⎥⎥
(C-7)
⎥
⎥⎦
REFERENCES
[1] Savage, P.G., Introduction To Strapdown Inertial Navigation Systems, Strapdown Associates,
Inc., 1981 - 2010, available for purchase at www.strapdownassociates.com.
[2] Savage, P.G., Strapdown Analytics, Second Edition, Strapdown Associates, Inc., 2000,
available for purchase at www.strapdownassociates.com.
[3] Savage, P.G., "Mitigating Second Order Error Effects In Linear Kalman Filters Using
Adaptive Process And Measurement Noise", SAI-WBN-14001, Strapdown Associates, Inc.,
May 16, 2014, free access available at www.strapdownassociates.com.
[4] Savage, P.G., "Redefining Gravity And Newtonian Natural Motion", SAI-WBN-14002,
Strapdown Associates, Inc., May 21, 2014, free access available at
www.strapdownassociates.com.
[5] Savage, P.G., "Modifying The Kalman Filter Measurement To Mitigate Second Order Error
Amplification In INS Velocity Matching Alignment Applications", SAI-WBN-14005,
Strapdown Associates, Inc., July, 15, 2014, free access available at
www.strapdownassociates.com.
[6] Savage, P.G., "Moving Base Alignment With Large Initial Heading Error", SAI-WBN14006, Strapdown Associates, Inc., October 3, 2014, free access available at
www.strapdownassociates.com.
[7] Savage, P.G., "Coarse Leveling Of INS Attitude Under Dynamic Trajectory Conditions",
SAI-WBN-14007, Strapdown Associates, Inc., January 28, 2015, free access available at
www.strapdownassociates.com.
[8] Brown, G.B. and Hwang, P.Y.C., Introduction To Random Signals And Applied Kalman
Filtering, John Wiley & Sons, Inc., 1997.
34
CHANGES SINCE ORIGINAL APRIL 17, 2015 PUBLICATION
l N*
Eq. (20) - Clarified ω EN definition equation.
Pg. 10 - Clarified definitions at top of page for Eq. (20) change.
Eq. (37) - Changed based on revised Appendix B Eq. (B-7).
Appendix B - Clarified and updated text/equations following Eq. (B-3) for N and N* rates being
the same, to correct a transpose error in Eq. (B-7), and to have the Eq. (B-7) result a function
of N* frame parameters provided by the reference navigation device.
35