The Effect of Carouseling on MEMS IMU
Performance for Gyrocompassing Applications
by
Benjamin Matthew Renkoski
Submitted to the Department of Aeronautics & Astronautics
in partial fulfillment of the requirements for the degree of
Master of Science in Aeronautics & Astronautics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2008
@ Massachusetts Institute of Technology 2008 . All rights reserved.
Author ....................
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Department of Aeronautics & Astronautics
May 23, 2008
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Matthew Bottkol
Charles Stark Draper Laboratory
Thesis Supervisor
Certified by..................
Emilio Frazzoli
Associate Professor of Aeronautics & Astronautics
Thesis Supervisor
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Accepted by ..............
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Chair, Committee on Graduate Studies
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MASSACHUSETTS IN
OF TECHNOLOGY
JUN 2 4 2009
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The Effect of Carouseling on MEMS IMU Performance for
Gyrocompassing Applications
by
Benjamin Matthew Renkoski
Submitted to the Department of Aeronautics & Astronautics
on May 23, 2008, in partial fulfillment of the
requirements for the degree of
Master of Science in Aeronautics & Astronautics
Abstract
The concept of carouseling an IMU is simulated in order to improve the accuracy
of MEMS IMUs. Carouseling consists of slewing the IMU through a pre-determined
trajectory that is selected based on inherent properties that lead to improved performance. MEMS devices typically have far more uncertainty than standard inertial
measurement devices, yet are considerably less massive and require less power, so implementing this carouseling scheme could make the use of these lightweight systems
possible even in high-accuracy situations, such as gyrocompassing. In gyrocompassing, the most significant benefit provided by the carouseling scheme is the reduction
in the error contribution of gyroscope bias, as this error is almost completely eliminated. Additionally, it was found that although implementing the carouseling scheme
required the addition of error states to account for the size effect, in many cases these
error states may not be necessary. Overall, the carouseling of the MEMS IMUs was
shown to be effective in reducing azimuth error covariance significantly.
Thesis Supervisor: Matthew Bottkol
Title: Charles Stark Draper Laboratory
Thesis Supervisor: Emilio Frazzoli
Title: Associate Professor of Aeronautics & Astronautics
Acknowledgments
First I would like to thank the Charles Stark Draper Laboratory and the Massachusetts Institute of Technology for giving me such an incredible opportunity to
broaden my academic horizons. I am profoundly grateful that I had the privilege to
attend the most prestigious engineering school in the world and work for one of the
most respected labs in aerospace. I would like to thank my thesis advisor Matthew
Bottkol for his dedicated support and invaluable guidance during the creation of this
thesis. In addition, I would like to thank Olivier de Weck and Emilio Frazzoli, my
advisors at M.I.T. who provided me with wisdom and encouragement along the way.
Numerous professors and personell at M.I.T., the University of Missouri, and Draper
provided me with innumerable amounts of help that assisted me on my journey, and
a special thanks goes to the late Michael Ash, who I never had the pleasure to meet,
but who did me the great service of bringing me in to the Draper Laboratory.
I thank my parents, Angela and Matthew Renkoski for almost 25 years of unwavering support and love. They taught me the benefits of hard work and instilled in
me the drive to succeed and do my best in all my endeavors. There is no doubt I
never would have accomplished the completion of this thesis if not for their positive
and nurturing influence.
A great amount of thanks goes to all my family and friends for the encouragement
and kindness they have constantly shown me. I can not imagine a more caring or
thoughtful group of people to help me through my thesis or my life. In particular I
would like to thank Julie Haesemeier for her unconditional support and dedication.
Throughout this thesis writing process she kept my spirits high and was a constant
source of inspiration.
This thesis was prepared at The Charles Stark Draper Laboratory under IR&D
and Contract GC 009256, sponsored by the United States Army Night Vision and
Electronic Sensors Directorate at Ft. Belvoir, VA. Publication of this thesis does not
constitute approval by Draper or the sponsoring agency of the findings or conclusions
contained herein. It is published for the exchange and stimulation of ideas.
THIS PAGE INTENTIONALLY LEFT BLANK
Contents
1
2
Introduction
1.1
Thesis Objective
1.2
Carouseling Background ...................
1.3
Background on MEMS Instruments . ..................
...................
.........
.
9
10
......
11
Carouseling Description
15
2.1
Motivation ...................
2.2
The Baseball Stitch Slew ...................
2.3
Analytical basis for Carouseling ...................
...........
..
15
......
16
..
19
2.3.1
Gyroscope Bias Elimination ...................
19
2.3.2
Accelerometer Bias Elimination .................
23
2.4
Expanding on the Baseball Stitch Slew .................
25
2.5
Comparing Variations of the Baseball Stitch Slew . ..........
28
2.6
Modeling Carouseling ...................
2.7
Error State Navigation Filter
2.8
3
9
......
...................
2.7.1
Covariance and State Propagation . ...............
2.7.2
Kalman Filter Update
Size Effect Compensation
...
42
44
48
...................
...................
.
...
49
.....
51
Gyrocompassing Analysis
57
3.1
Basic description of gyrocompassing ...................
57
3.2
Kalman Filter Update
3.3
Simulation Details
......
...................
..........
.......
.......
58
.
59
3.4
Azimuth Covariance Determination Results . ..........
3.5
Sensitivity Analysis .
3.6
4
.
.....
.....
3.5.1
Error Budget Formulation . ........
3.5.2
Sensitivity Analysis Results
.....
.
.
60
. .
.
69
...........
70
................
...
Size Effect Sensitivity ...................
......
71
..
Conclusions
77
85
4.1
Carouseling Effect on Azimuth Accuracy ....
4.2
Size Effect ...................
4.3
Other Considerations .................
4.4
Future Work ........
............
85
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..
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...
86
87
. .. . .
87
A MEMS IMU Error Specifications
89
Bibliography
89
List of Figures
93
List of Tables
96
Chapter 1
Introduction
1.1
Thesis Objective
The objective for this thesis is to demonstrate the utility of implementing a carouseling technique to improve Inertial Measurement Unit (IMU) navigation performance.
The main goal is to see the effect of carouseling in conjunction with Micro-ElectroMechanical System (MEMS) instruments to see if the improvements in accuracy can
lead to the utilization of these instruments in higher-accuracy applications such as gyrocompassing. This is highly desirable as MEMS instruments are lighter and cheaper
than standard IMUs, yet have significantly less accuracy. First the carouseling concept will be investigated analytically in order to explore the theoretical benefits associated with it. Then, to ascertain if carouseling improves performance, the carouseling
maneuver will be simulated and navigated with the results compared to those of a
standard navigation system.
The remainder of the first chapter of this thesis will provide explanatory background information regarding the carouseling scheme in general and MEMS instruments, particularly those developed by Draper. In the second chapter, a more thorough investigation of the carouseling maneuver will be undertaken including an in
depth analysis of its theoretical benefits. The process for simulating the maneuver
and the navigation scheme including the error filter shall be described in detail. The
third chapter focuses on the results of the carouseling maneuver when applied to gy-
CHAPTER 1. INTRODUCTION
rocompassing. This includes covariance analysis and sensitivity analysis, including
error budget comparisons. Specifically, the error contributions of the size effect involved with the addition of lever arm states is investigated. Conclusions and ideas
for future work are included in chapter four.
1.2
Carouseling Background
The concept of carouseling, also known as "slewing" or "maytagging", is a concept that
has been developed for use in high-precision scenarios using high accuracy IMUs. The
basic operation is to put gimballed IMUs through an inertially referenced slew. The
main design of the slewing maneuver is to direct the instruments so that they are
pointed in different directions so that the total directional effect is averaged out over
time to prevent the build up of biases (the analytical basis and equations of motion
regarding this will be examined more thoroughly in Chapter 2). Theoretically the bias
errors of both the gyroscopes and accelerometers will vanish at the end of each period
since they have not been allowed to build up in any one direction. However, since
the IMUs now have an added angular velocity and are not collocated, this could add
errors that previously were negligible that could cancel out the improvements made
in the biases. Therefore it would be advantageous to fully simulate and investigate
the effects of the carouseling maneuver. There is growing interest at Draper and in
the navigation community for use of this estimation scheme with current and future
MEMS applications. If it is effective in reducing error in MEMS IMUs, it could be
a substantial enough improvement so that MEMS devices could be used in place of
standard IMUs with little or no loss in quality.
1.3. BACKGROUND ON MEMS INSTRUMENTS
1.3
Background on MEMS Instruments
Micromachined silicon inertial sensors are much smaller and less expensive to produce
than standard IMUs. These characteristics make them extremely attractive for reallife systems; especially certain aerospace and military applications where mass is at a
premium. MEMS IMUs can be batch produced using semiconductor manufacturing
processes which reduces the cost significantly. Additionally, silicon is inexpensive
itself and each silicon wafer can contain a large magnitude of inertial measurement
devices.
The smaller size of the instruments (only a few square millimeters) not
only decreases total mass and volume taken up by the instruments, but also leads
to reduced power usage; a highly desirable attribute in many applications. All these
advantages makes MEMS IMUs an area of much interest, but their lower accuracy
has limited widespread use so far.
Draper Laboratory has been developing MEMS sensors for over 15 years, and
has become a leader in their development. Draper MEMS instruments have been
implemented in various military and space scenarios, including being tested in the use
of guiding gun-launched munitions and being designed for use on the Space Shuttle.
The MEMS gyroscope Draper has developed a silicon planar tuning fork gyroscope,
photographed by a scanning electron microscope (SEM) in Fig. 1-1.
This model MEMS gyroscope consists of an etched silicon piece anodically bonded
to a glass substrate [9]. Like all tuning fork gyroscopes, it contains two vibrating proof
masses suspended by beam springs. In this MEMS case, the two symmetrical proof
masses are flexurally suspended over capacitor electrodes in the glass substrate. When
the device experiences a rotation about its input axis, parallel to the proof mass plane,
the Coriolis effect causes the masses to oscillate up and down out of the plane 16].
The amplitude of the resultant motion, which is proportional to the angular rate, is
measured by the capacitive plates underneath the proof masses in the glass substrate.
This signal is amplified and demodulated by the integrated electronics to provide an
output voltage proportional to the angular rate.
CHAPTER 1. INTRODUCTION
Figure 1-1: SEM photo of MEMS tuning fork gyroscope
Draper has also developed MEMS accelerometers which are pendulous mass displacement accelerometers 16]. A MEMS 100-g accelerometer is shown in Fig. 1-2.
Figure 1-2: SEM photo of MEMS accelerometer
The accelerometer is also manufactured using a dissolved wafer silicon on insulator process that results in an unbalanced proof mass suspended by torsional spring
flexures. The proof mass is suspended in a see-saw type configuration so that when
there is an acceleration felt by the accelerometer, the proof mass rotates around the
1.3. BACKGROUND ON AMEMS INSTRUMENTS
hinged axis. This motion is measured by capacitive plates in the glass substrate and
is proportional to the acceleration, similar to the MEMS gyroscope. This output also
provides the input for a closed loop control operation that is used to drive torquer
plates to rebalance the proof mass.
These MEMS devices are becoming accurate enough for use in many systems,
with improvement continuing. They have been used in guiding projectiles, avionics,
and numerous automotive applications. New high aspect ratio etch technology and
highly integrated fabrication techniques are increasing the overall performance, theoretically for use in more high-performance tactical applications in military and space
systems. If found to be effective, the carouseling concept could be used to further
this performance improvement.
CHAPTER 1. INTRODUCTION
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Chapter 2
Carouseling Description
2.1
Motivation
As previously discussed, MEMS instruments provide many desirable attributes for
implementation into high-precision applications. However, the desired performance
has not yet been achieved in the instruments themselves, so alternative ways to improve performance have been developed. One of these concepts is the carouseling
scheme which would theoretically improve accuracy and retain the high weight and
cost saving associated with MEMS devices. The main goal of implementing an estimation scheme using carouseling is to eliminate errors due to accelerometer and
gyroscope biases. As described earlier, the basic theory is that if the IMU was spun
around in a certain trajectory which will average out the pointing of the IMU so that
no direction is favored, the build-up of gyroscope biases would be eliminated. If this
process can improve accuracy enough, MEMS devices could then be implemented in
higher accuracy applications. There would be additional mass associated with the
slewing device, but this could be minimized so that the total system would still be
significantly smaller and require less power than a conventional IMU.
This chapter will investigate the analytical basis for the performance improvement
and describe the kinematics of the carouseling scheme. First the slewing maneuver
CHAPTER 2. CAROUSELING DESCRIPTION
itself will be introduced and its effect on the contributions from gyroscope and accelerometer error shall be analyzed and simulated. Variations of the original slew will
be developed and compared to select the optimum slewing motion to eliminate the
most error. Then the equations used to simulate the slewing maneuver with error
contributions will be developed as well as navigation equations, including filter updates. Finally, the complications of the addition of lever arms and the size effect will
be investigated more fully.
2.2
The Baseball Stitch Slew
Carouseling is meant to reduce gyroscope and accelerometer biases by averaging each
axis to be pointing in all directions for an equal amount of time. The concept is that
since biases are directional errors that build up along an axis, if the overall direction
is averaged out by the slewing trajectory, the net effect of the bias errors will go to
zero. If the IMU can then be made to follow a trajectory that properly averages out
this effect, then the bias contribution to the total error would vanish at the end of
each period. Therefore, the first task is to select a trajectory that accomplishes the
desired effect.
There are a number of slews that achieve this characteristic, the most common
being what is called the Baseball Stitch Slew, so named for its resemblance to the
path of the stitching along a baseball. The baseball stitch slewing maneuver can be
mathematically described as the combination of two orthogonal rotations with the
angular velocities of the two rotations related by an integer ratio. For example, in the
commonly used 2:1 baseball stitch slew, the second gimbal rotation has double the
angular velocity of the first. To create this motion, each IMU will be dual-gimballed
so it is allowed to follow the specified path controlled by servo-mechanisms operating
each rotation. This rotation is performed in the inertial reference franme, which is an
important property of the baseball stitch slew. The slew is essentially a rotation from
2.2. THE BASEBALL STITCH SLEW
body coordinates to inertial coordinates. Figure 2-1 shows the baseball stitch slew in
a three dimensional representation, demonstrating the trace of the endpoint of one
axis for the duration of the slew. Now that the concept of the baseball stitch slew is
.......
... ......]...
.....
...
.
.........
Si.....
..........
..
........
i
Figure 2-1: 3-D Trace of a 2:1 Baseball Stitch Slew Depiction of the three dimensional trace
of the endpoint of one of the rotating axes, closely resembling the stitching of a baseball
established, its properties of importance can be analyzed. The equations governing
its motion will be produced and summarized in this section to provide the basis for
this analysis. In matrix form, the 2:1 baseball stitch rotation, Rbs, can be represented
as the combination of the two orthogonal rotations comprising the slew:
Rbs(t)
cos(wt)
sin(wt) 0
0
0
- sin(wt)
cos(wt) 0
cos(2wt)
sin(2wt)
0
cos(wt)
- sin(wt)
0
0
1
sin(wt) cos(2wt)
(
- sin(2wt) cos(2wt)
sin(wt)
cos(wt) cos(2wt) cos(wt) sin(2Lot)
- sin(2wt)
cos(2wt)
(2.1)
CHAPTER 2. CAROUSELING DESCRIPTION
where w is the angular velocity of the first rotation matrix and t is time. In this case,
it can be seen the first rotation is about the z-axis and the second rotations is at twice
the angular rate and performed about the x-axis. It is important to remember this
slew is inertially referenced, not Earth-fixed. The baseball stitch slew Rbs(t), has a
valuable property which will be addressed later in this section that demonstrates the
benefit of its implementation.
To find the angular velocity, use the following equation to define the time rate of
change of any three-axis rotation:
R = R[ x]
(2.2)
where [ x] is the skew symmetric form of the angular velocity vector,
= (w, W2 , W3).
The skew symmetric matrix representation of the angular velocity takes this form:
[Ux] =
0
-w3
L2
W)3
0
-L)1
--W)2
L1
(2.3)
0
Again, this rotation R is the transformation from the body axis to the inertial axis
and the angular rates are in the inertial frame. This is important as gyroscopes
measure inertial rates, since they are inertial instruments. Using the rotation Rbs(t)
from Eq. 2.1 as the rotation matrix in Eq. 2.2, the vector 0 can be solved for to
yield
2w
Qbs
=
csin(2ct)
.
(2.4)
w cos(2wt)
This angular rate vector Obs is the output of an ideal gyroscope which is fixed to
a body that executes the baseball stitch slew with respect to inertial space.
2.3. ANALYTICAL BASIS FOR CAROUSELING
2.3
Analytical basis for Carouseling
In this section the knowledge gained from Section 2.2 will be used to show how
gyroscope and accelerometer biases can be eliminated using the baseball stitch slew.
Basic equations of motion will be introduced that will be implemented later into the
navigation scheme, but the main focus of this section will be solely on the effect of
the baseball stitch on these equations. The overall navigation scheme will be fully
described in Section 2.7.
2.3.1
Gyroscope Bias Elimination
In order to better understand the effect of slewing on gyroscope bias error, it is
important to see what makes up the gyroscope error contribution to angular velocity,
defined here as 6Q.
To find this error, first linearize Eq.
2.2, where R(t) is the body to inertial
coordinate transformation, to result in
b
= 6R[ x] + R[6 x]
(2.5)
where the gyroscope error 6Q is the error in sensed angular rate. It can be assumed
that the error in the angular rotation can be represented as an error rotation matrix
in body coordinates, called T. This can be defined by Eq.2.6:
6R = R[ix]
(2.6)
This can be substituted into Eq. 2.5 to yield
R[,x] + R[Qx][*x] = R[*x][x] + R[6x]
(2.7)
CHAPTER 2. CAROUSELING DESCRIPTION
which can be be converted into a vector equation in body coordinates as
I = -Q x q + mO.
(2.8)
To express the error in inertial coordinates, differentiate the equation
I = R(t)F
(2.9)
where I' is the error rotation in inertial coordinates. After differentiating Eq. 2.9
and substituting the result of Eq. 2.8, this results in
d
dt
= RW+
MI
4R
= R(-Q x x, + 6Q) + R[ x]q,
= R(-Q x xI + t
x * + 6Q)
= R(t)6Q2
(2.10)
Therefore the total contribution from the gyroscope error to the error rotation in the
inertial frame is simply the gyroscope error multiplied by the rotation matrix. Now
the gyroscope error 6Q, can be broken down into its components to further examine
the contribution to the error. The gyroscope error can be assumed to be consisting of
four main error sources: gyroscope bias error bg, gyroscope scale factor sfg, gyroscope
misalignment m,, and gyroscope non-orthogonality ng. Each of these quantities are
vectors consisting of 3 scalar values (one in each direction of the body frame). The
errors are inherent to the gyroscope itself and are a function of the error characteristics
of that gyroscope. They can be combined to form the overall gyroscope error in the
following equation:
S1 = bg + ([sfg.] + [mgx] + [ng(]) Q
(2.11)
2.3. ANALYTICAL BASIS FOR CAROUSELING
where [sfg,] represents a 3 x 3 diagonal matrix formed from the 3-vector containing
the gyroscope scale factor errors, sfg = (sfl, sf 2 , sf3), in the following way:
0
0
0
s f2
0f3
0
0
sf3
sfi
[sfg] =
(2.12)
[m, x] is a 3 x 3 skew-symmetric matrix formed from the three misalignment error
terms of the 3-vector, m 9 = (ml, m2, m3):
0
m2
-m3
0
-m2
-m
1
(2.13)
0
M1
and [ng® ] is a symmetric 3x3 matrix created from the non-orthogonality terms of
the 3-vector n, = (ni, n2, n 3 ), such that
0
[ng®] =
n2
(2.14)
n3
n2
1
7i
0J
Now that these matrices have been defined, the change in the rotation error, AI',
can be calculated by integrating Eq. 2.10 from time zero to some discrete time T,
and combining it with Eq. 2.11:
A
I
-
=
jI/R(t)60dt
+
TR(t) (b([sfg.]
+ [mgx] + [ng0]) Q) dt
(2.15)
Equation 2.15 can be rewritten by distributing the diagonal, skew-symmetric, and
symmetric matrix representations to the angular velocity vector Q, instead of the
CHAPTER 2. CAROUSELING DESCRIPTION
scale factor, misalignment, and non-orthogonality error vectors, respectively.
The
results of this re-distribution create Eq. 2.16:
,A*
()(bg + [.]sfg - [lx]m 9 + [
=
]ng)
(2.16)
This puts the equation into a form that can be more easily used to demonstrate the
effect the slewing motion has on the rotation error. Notice that the diagonal and
symmetric operations can switch to apply to the vector 0 and cause no change, while
changing the skew-symmetric operation from the misalignment vector to Q results in
a change in sign.
Now the gyroscope error source vectors can be completely separated from the
integral, since they are assumed to be constant, and Eq. 2.16 can be rewritten as
Ak I = SB(T)bg + SSF(T)sfg + SA,(T)mg + SN(T)ng
(2.17)
where the matrices of the form Sx are known as "sensitivity matrices" and are integrals
used to quantify the effect of the four error sources on the rotation error matrix. The
sensitives are all integrals over the time interval 0 - T that result in 3 x 3 matrices.
The bias, scale factor, misalignment, and non-orthogonality sensitivity matrices can
be respectively described as:
SB(T) =
S(,(T) =
SA,(T) = -
S (T)
J
T
/T
(2.18)
[0(
)(t)dt,
(2.19)
R(t) [Q(t) x]dt,
(2.20)
R(t) [ (t)(] (t)dt.
(2.21)
The baseball stitch slew, with rotation matrix
Rb, attained from Eq. 2.1, is particu-
2.3. ANALYTICAL BASIS FOR CAROUSELING
larly interesting and useful because it has the property,
SB(T) =
Rbs (t)dt = 0.
(2.22)
This mathematically expresses the fact that the overall accumulation of the rotation
Rbs is zero, meaning over time T no one direction is favored. To prove this, it can be
seen that integral of each element in the baseball stitch rotation matrix from Eq. 2.1
equals zero when taken over the time of one period. The integral of the trigonometric
functions sin and cosine over one period is always equal to zero and using the property
of orthogonality of the trigonometric functions, the integral of the other elements of
the rotation matrix will also be equal to zero. Since this entire integral is zero, the
sensitivity matrix in Eq. 2.18 is equal to zero, and it eliminates the contribution of
gyroscope bias to the overall rotation error A*, in Eq. 2.15. Since the bias sensitivity
matrix goes to zero, no matter how large the gyroscope bias error, its contribution
will always vanish at time T when the baseball stitch slew is performed. In this way
the baseball stitch slew is shown to achieve the goal of eliminating the contribution
from the gyroscope bias error, but no other error source. The elimination of these
other gyroscope errors will be investigated in Section 2.4.
2.3.2
Accelerometer Bias Elimination
While the previous subsection detailed how gyroscope bias errors can be eliminated
by carouseling using the baseball stitch slew, contributions from accelerometer bias to
the error in acceleration can also be reduced. In fact, it was for the express purpose of
eliminating accelerometer bias error that the slew was developed, and was later found
to also eliminate the gyroscope bias. However, in the gyrocompassing application, the
attitude error is more important than the velocity error so the gyroscope errors are
more crucial.
As was seen in Eq. 2.15, there is no contribution from accelerometer error to the
CHAPTER 2. CAROUSELING DESCRIPTION
rotation error. However, accelerometer error plays a large part in the error in velocity
felt by the IMU. In the following equation, derived from equations of motion, can be
seen the components of error in acceleration:
= G(t)6x - 2[QE x]5v + R[
×x]f + 6a
(2.23)
where 6a is the accelerometer error. The derivation of this equation and the
components of the first three terms will be described in detail in Section 2.7, but for
now the only term of concern is the accelerometer error. It can be assumed that the
accelerometer error is made up of the contributions from four sources, much like the
gyroscope error: accelerometer bias, scale factor, misalignment, and nonorthogonality.
They combine to contribute to the total accelerometer error as follows:
aa = R(6ba + [f-]6sfa - [fx]6ma + [f0fn)
(2.24)
Where R is the body to inertial coordinate rotation used in the gyroscope equations as well. Ignoring the first three terms and integrating Eq. 2.23 with Eq. 2.24
inserted for Sa, the contribution of accelerometer error to the velocity error can be
seen:
T
6adt
v
=
v
= SB(T)ba + / R([f.]sf
/T0
-Lfxlma
+ [f®]n)dt
(2.25)
where SB(T) is the bias sensitivity matrix from Eq. 2.18 in the previous section,
and has been proven to vanish over the time interval 0 - T when using the baseball
stitch slew. Therefore, the contribution from accelerometer bias error to the velocity
error vanishes as well, no matter what the accelerometer bias value is. Even though
the area of research for this thesis is the gyrocompassing application, which is far
2.4. EXPANDING ON THE BASEBALL STITCH SLEW
25
more concerned with the angle error, this increased performance in the velocity error
is still a valuable attribute of the carouseling scheme. And although this does not
directly increase accuracy of the azimuth angle, when filtering is used and all errors are
intermixed, better accuracy in any measurement can contribute to overall accuracy.
2.4
Expanding on the Baseball Stitch Slew
As was shown in Section 2.3, the baseball stitch slew is effective in eliminating the
contribution of gyroscope bias to the total error rotation I.
However, it does not
eliminate contribution from other error sources, namely the integrals SsF(T), SM,(T),
and SN(T) do not reduce to zero at the end of each period. So while the bias term
returns to zero, these other errors continue to accumulate. This is undesirable, but
it has been found that by modifying the baseball stitch slew, these errors can also
be eliminated. In this section, modifications to the baseball stitch slew will be made
that will make it more effective at improving accuracy. In order to provide a goal
for the improved slew, the sensitivity matrix format will again be used to determine
another possible way for carouseling to provide benefits. To investigate this problem,
first create the matrix expression
M = [sfg,] + [m,x] + [n9 ®]
(2.26)
which can be seen in Eq. 2.11 as the 3 x 3 matrix multiplied by the angular velocity
to contribute to the gyroscope error. Also notice that this matrix M is given by nine
free parameters, so it can be said to represent an arbitrary 3 x 3 matrix with nine free
entries. From this knowledge it can be proven that the three sensitivity matrices it is
desired to eliminate (SsF(T),SnA(T), and SN(T)), will be equal to zero if and only if
=-R(t)MA(t)dt
0.
(2.27)
CHAPTER 2. CAROUSELING DESCRIPTION
for any 3 x 3 matrix Al. That is, no matter the magnitude of Al (in this case the scale
factor, misalignment and nonorthogonality errors), this will hold true. So a rotation
R(t) must be found that satisfies Eq. 2.27. This equation holds true no matter the
value of M, which means no matter how poor the accuracy, with the right rotation,
the error contribution for all three of these sources can be eliminated at time T.
Now a slew will be created that satisfies the above equation while still satisfying
Eq. 2.22 as well. The equations governing this slew will be developed and then it will
be proven that it does satisfy equation 2.27. This new slew can be created from the
concept of reversing the baseball stitch slew. The IMU will travel through the slew
completely, and then travel over the same trajectory in the opposite direction. This
was created to "unwind" the effects of scale factor, misalignment, and nonorthogonality, while still eliminating the bias contribution. To investigate the effects of slew
reversal, define slew R+(t) to be any slew, and create a piecewise slew R(t) defined
as
R(t)
R±(t) O<t<T
R(t) T<t<2T
(2.28)
where R_(t) is the reverse in time of R+(t) and can be mathematically defined as:
R_(t) = R(2T - t)
T < t < 2T
(2.29)
From Eq. 2.2, the time rate of change for the rotations can be defined as
R+ (t) = R+(t)[Q(t)x]
R_(t) = R (t)[_(t)x]
To find
0
t <T
T < t < 2T
(2.30)
(2.31)
_(t) in terms of R+ and Q+, differentiate Eq. 2.29 and substitute Eq. 2.30
2.4. EXPANDING ON THE BASEBALL STITCH SLEW
to obtain
R_(t) = -R±(2T-t)
= -R+(2T-
(2.32)
t)[,+(2T - t)x]
Setting Eq. 2.32 equal to Eq. 2.31 and using Eq. 2.29 produces
I_ (t)
= -R+(2T-t)[,+(2T-t)x]
t)[Q+(2T-t)x]
R_(t)[_ (t)x] = -R+(2T[Q_(t)x]
= -[a+(2T-t)x]
=
-(t)
(2.33)
2+(2T- t)
over the interval T < t < 2T. Using this relation, it can be shown that Eq. 2.27 is
true for any trajectory that undergoes a slew reversal as described in Eq. 2.28. Take
Eq. 2.27 and use the definitions of the rotation matrix and angular velocity with slew
reversal to show
R(t)M
2T
R(t)M
R+(t)M +(t)dt +
(t)dt
R+(t)MQ+(t)dt -
=
R+(t)M +(t)dt +
=
1
2T
(t)dt
(2.34)
R+(2T - t)MA+(2T - t)dt
R+(t)()MOa(t)dt
0
Notice that this is true for any 3x3 matrix M and by Eq. 2.27, SF(2T),SI(2T), and
SN(2T) will all be equal to zero when there is a slew reversal. Notice also that this
holds for any rotation R(t), not necessarily the baseball stitch slew. However, since
it is based on the baseball stitch slew, Eq. 2.22 still holds true and the contribution
due to both accelerometer and gyroscope biases will continue to vanish at the end of
CHAPTER 2. CAROUSELING DESCRIPTION
28
each period. Combined with the reversal, the contribution of all four gyroscope error
sources is now seen to vanish.
Comparing Variations of the Baseball Stitch Slew
2.5
To better compare the effects of the baseball stitch slew and its derivatives in terms of
their contribution to T, this section will focus on exploring the error contributions
in Eq. 2.17 graphically for each slew.
For the regular baseball stitch slew, first observe the motion of the gimbal angles
over a period of 60 seconds (implementing Eqs. 2.1 and 2.4).
Carouseling Gimbal Angles
~%U
'4
<2
0
10
20
30
40
50
Angular rate vector in Body Frame
-50 1
0
10
20
30
Time (s)
40
50
Figure 2-2: Plots of gimbal angles and angular rates of a 60 second Baseball Stitch
Slew
It can be seen in the first subplot that the gimbal angles continue to increase
2.5. COMPARING VARIATIONS OF THE BASEBALL STITCH SLEW
29
as the slew progresses an the gimbals continue to rotate constantly in one direction,
with the outer gimbal increasing at twice the rate as its rotation has twice the angular
velocity. The second subplot shows the angular velocity components for the entire
slew. Notice two of the angular velocity components follow sinusoidal behavior while
the third element remains a constant equal to the angular rate of the second slew, as
was shown in Eq. 2.4.
Now performing the integrations of the sensitivity matrices in Eq. 2.17, the total
contribution to the rotation error can be plotted for the baseball stitch slew. Values
for the gyroscope bias, scale factor, misalignment, and nonorthogonality errors are
assumed to be those of the current Draper MEMS IMU model, which will be discussed
later in Section 3.3 (the specifications can be found in Table A.1 in Appendix A).
The exact values of the errors are not important at this point, but rather the pattern
of behavior of the error contributions over time for one period. In Fig. 2-3 is plotted
the contributions of each error term to the 9 elements of the AI matrix in Eq. 2.17
for the baseball stitch slew.
Notice the bias terms vanish at the end of the period as was shown from Eq. 2.22.
Also the misalignment and non-orthogonality contributions do vanish, but two elements of the scale factor error continue to build up. Note that the bias, misalignment,
and non-orthogonality errors not only vanish at the end of the period, but also every
15 seconds which is equal to one fourth of a period. As can be seen in Fig. 2-2, this
corresponds to the times when both gimbals have performed a whole number amount
of revolutions. Also observe that the scale for the plot of the error contribution due to
gyroscope scale factor is an order of magnitude larger than the scale use for the other
three error sources. These errors combine to form the total rotation error shown here
in Fig. 2-4.
In Fig. 2-4 it can be seen that most of the error elements return to zero at the end
of each period, but two elements attributable to the scale factor error still continue
to build up. The errors that do return to zero oscillate in the manner observed before
CHAPTER 2. CAROUSELING DESCRIPTION
T error due to Gyroscope Bias
x 10 .4
10
30
.20
40
50
T'error due to Gyroscope Scale Factor
x 10.3
0
!I
I
|
I
20
30
40
50
-2
-41
0
5
10
E
Y error due to Gyroscope Misalignment
"
x 10 5
0
-5
-10
10
20
30
40
50
'1 error due to Gyroscope Non-orthoganality
x 10.4
-EJ
-21
0
10
20
30
Time (s)
40
50
60
Figure 2-3: Contributions of different error sources to the overall rotation error for the
Baseball Stitch Slew
2.5. COMPARING VARIATIONS OF THE BASEBALL STITCH SLEW
0.5
x 103
1
1
10
20
Total Error in y
,
0
-0.5
-1.5
-2
-2.5
30
Time (s)
40
50
Figure 2-4: The total contribution to rotation error from gyroscope errors for the
Baseball Stitch Slew
in which they also return to zero ever one fourth of a period. As a point of reference,
compare the above plot to the total error for a non-rotating MEMS IMU (i.e. the
rotation matrix is equal to identity in Eq. 2.17), which yields the total error plot seen
in Fig. 2-5.
First, notice that only one element changes from zero, and that is known to be due
to the bias. The other errors contribute no error as the body is not moving. So even
though the baseball stitch slew does an excellent job of eliminating the gyroscope
bias, by introducing rotation, the other errors become a factor. And even though
misalignment and non-orthogonality are also cyclically eliminated, the scale factor
error builds up which effectively eliminates the improvements made in reducing the
bias. In fact, Fig. 2-4 shows that the additional scale factor error grows slightly
CHAPTER 2. CAROUSELING DESCRIPTION
1
x 10.3
Total Error in T
0.9
0.8
0.7
0.6
"
0.5
0.4
0.3
0.2
0.1
10
20
30
Time (s)
40
50
Figure 2-5: The total contribution to rotation error from gyroscope errors for a nonrotating body
larger than the original bias error seen in Fig. 2-5. So the normal baseball stitch
slew performs well at eliminating bias, misalignment and nonorthogonality, but the
addition of the scale factor makes it unattractive for implementation. However, as
discussed in Section 2.4, if the baseball stitch slew performs a reversal as defined in
Eq. 2.28, the scale factor error will also reduce to zero at the end of each period.
For the baseball stitch slew with reversal, the gimbal angles and angular velocity now
behave as seen in Fig. 2-6.
As can be seen in the figure, both gimbals are reversed halfway through the 60
second slew. Instead of the gimbal angles continuing to increase, they now reverse
direction and are in fact both equal to zero at the end of the period. Notice the
2.5. COMPARING VARIATIONS OF THE BASEBALL STITCH SLEW
Carouseling Gimbal Angles
---
Outer Gimbal
I-nner Gimbal
.... .... ...........
..
..
.... .....
...... ....
Angular rate vector in Body Frame
40
°
°
20
cc
0
-20
-40
-.
...
...
........
..........
........
.....
.......
....
....
..... .......
...
..........
'
|
0
10
|
20
30
Time (s)
I
!
40
50
E
Figure 2-6: Plots of gimbal angles and angular velocities for Baseball Stitch Slew with
Reversal
elements of angular velocity reverse sign at the time of slew reversal as well. Once
again using Eq. 2.17, but this time integrating the baseball stitch slew with reversal
rotation, the contributions to the 9 elements of the A'I matrix can be seen for the
baseball stitch slew with reversal in Fig. 2-7.
Now all nine elements of the four error source sensitivity matrices return to zero
at the end of the slew. This is the exact result desired. It would appear that the
baseball stitch slew with reversal would be the best slew to use for carouseling, but
there is another aspect that has not been taken into account.
Although the focus of gyrocompassing is on the azimuth angle so the velocity
error is not the main concern, in the real navigator using a Kalman filter, it is still
highly desired to have accurate velocity measurements. So to better ascertain if the
CHAPTER 2. CAROUSELING DESCRIPTION
T error due to Gyroscope Bias
x 10-4
5
3
4
0
2
20
25
30
25
30
-
x 10,
-10
15
T error due to Gyroscope Scale Factor
x 10
10
10
y error due to Gyroscope Misalignment
x
5
10
15
20
'1 error due to Gyroscope Non-orthoganality
x 10,4
0
-2
0
5
10
15
2n
Time (s)
Figure 2-7: Contributions of different error sources to the overall rotation error for the
Baseball Stitch Slew with Reversal
2.5. COMPARING VARIATIONS OF THE BASEBALL STITCH SLEW
35
baseball stitch with reversal is the optimum slew, it would be valuable to investigate
the gyroscope error contribution to the velocity error as well.
Recall from Eq. 2.23, that the I error actually does contribute to the velocity
error in the third term, which is R['IFx]f. If the result for AkF from Eq. 2.17 is
inserted for * ,this can be integrated to determine the contribution of the gyroscope
errors to the velocity error (represented as Av in this analysis):
Av(T) =
=
Aq
x f(t)dt
jT(SB(T)b + SsF(T)sf + SM(T)mg + S(T)ng) x f(t)dt(2.35)
In this equation it can be seen that instead of the error contributions merely
being the sensitivity matrices, it is now the integral of the sensitivity matrices that
determines the contribution to the velocity error. Performing the integrations in Eq.
2.35 for the baseball stitch slew with reversal yields the results shown in Fig. 2-8.
Notice the contribution due to gyroscope bias continues to vanish, however the
other three sources continue to build up. Particularly the error contribution due to
scale factor is a problem, as it is an order of magnitude greater than the misalignment
and non-orthogonality errors.
So the slew reversal fails to eliminate these errors.
However, it has previously been found that this reversal concept can be extended to
provide better performance. As there are two separate gimbals, both gimbals do not
have to be reversed simultaneously. It was discovered that a concept involving only
reversing the outer gimbal every other time would improve upon the performance of
the baseball stitch slew with reversal. The plots of the gimbal angles and angular
velocities for this scheme, called the baseball stitch slew with reversal extended can
be seen in Fig. 2-9.
This figure demonstrates how the outer gimbal is not reversed at 30 seconds when
the inner gimbal is reversed. Also notice that when the next period starts the inner
CHAPTER 2. CAROUSELING DESCRIPTION
Error in velocity due to Gyroscope Bias
0.02
-" '--" ' -----"..
_.
.
:
-
_ _--- -
0
-0.02
-
I
I
0
Error in velocity due to Gyroscope Scale Factor
U.
-0.5
0
10
5
15
20
25
31
25
30
Error in velocity due to Gyroscope Misalignment
0.02
0
-0.02
-0.04
3
10
5
15
20
Error in velocity due to Gyroscope Non-orthoganality
0.05
I
•
0
I
I
-0.05
Time (s)
Figure 2-8: Contributions of different gyroscope error sources to the change in velocity
error for the Baseball Stitch Slew with Reversal
2.5. COMPARING VARIATIONS OF THE BASEBALL STITCH SLEW
Carouseling Gimbal Angles
-
10
D
Outer Gimbal
Inner Gimbal
40
30
20
50
61
Angular rate vector in Body Frame
i
t~I
I
-50
0
t
lt
iVI'
I~
30
Time (s)
Figure 2-9: Plots of gimbal angles and angular velocities for Baseball Stitch Slew with
Reversal Extended
gimbal will be reversed but the outer gimbal continues to rotate in the same direction.
To verify that this slew continues to provide the nulling capabilities of gyroscope error
contribution to A* of the baseball stitch with slew reversal, Eq. 2.17 can again be
used to plot out these results for the baseball stitch with slew reversal extended,
shown here in Fig. 2-10.
Notice that all the error contributions continue to vanish at the end of each period,
and also the scale factor contribution vanishes at the 30 second mark. The other three
sources show the familiar behavior of all elements returning to zero at each one fourth
of a period. It is interesting to note that the error contribution plots for each source
can be divided at the 30 second point and the two halves are reverse images of each
other.
CHAPTER 2. CAROUSELING DESCRIPTION
Y error due to Gyroscope Bias
x 10-4
1,I~
I
0
I
I
i
10
20
30
x 10 3
T error due to Gyroscope Scale Factor
10
20
30
40
50
T error due to Gyroscope Misalignment
x 10-5
10
2
I
40
20
30
40
50
T error due to Gyroscope Non-orthoganality
x 10.4
SI
I
I
tr(0
-203
SI
10
20
i
I
30
Time (s)
40
50
E
Figure 2-10: Contributions of different error sources to overall rotation error for Baseball Stitch with Reversal Extended
2.5. COMPARING VARIATIONS OF THE BASEBALL STITCH SLEW
39
Now to investigate if the baseball stitch with reversal extended provides any benefit
in the velocity error, the error contributions to the velocity error are presented in Fig.
2-11.
The scale factor error contribution is now reduced back to zero at the end of
the period, along with the contribution from the gyroscope bias.
Unfortunately,
the misalignment and non-orthogonality errors persist, but their error contribution
is significantly less than the previous contribution from scale factor for the simple
reversal in Fig. 2-8. These misalignment and non-orthogonality errors are therefore
not of critical importance to eliminate, and as will be seen in the simulations, their
contributions are minimal and do not greatly impact azimuth error.
The total error contribution from gyroscope errors to F for the baseball stitch
slew with reversal extended can be seen in Fig. 2-12:
Now the contributions from all error sources vanish at the end of the period. The
scale factor error does not build up like it did for the normal baseball slew as seen
in Fig. 2-4, and the bias terms do not continue to grow as seen when no rotation
at all is applied as seen in Fig. 2-5. All errors reduce back to zero and do not grow
larger than 8 x 10- 4 radians in magnitude at any point, with most elements remaining
significantly smaller than that, including all reducing to zero at the mid-point as well.
It is important to note that all of these results are for ideal cases where there
is no filtering and no updates whatsoever. In real applications there will be filter
updates that will affect the contributions of these error sources. Cross-correlations
will develop and other error sources such as random walk are incorporated into the
navigation as well. Therefore, the exact results found in this section should not be
expected to be repeated in the simulations. However, the belief is that the trends seen
here will be reflected in the results, as in, it is expected to see that the carouseling
will reduce the effects of gyroscope error sources on angle error. It is not expected
that these contributions will be exactly zero or that any of the other errors will show
identical behavior to what has been demonstrated in these idealized plots.
CHAPTER 2. CAROUSELING DESCRIPTION
Error in velocity due to Gyroscope Bias
0.02
Hi
0
-0.02
10
30
20
40
50
i
Error in velocity due to Gyroscope Scale Factor
0.5
--L
l
--
~---L
0
I
i
-
-0.5
Error in velocity due to Gyroscope Misalignment
0.1
0
-0.1
-
I
I
I
Error in velocity due to Gyroscope Non-orthoganality
0.1
--~
0
-0.1
I
i
i
1
I
C
Time (s)
Figure 2-11: Contributions of different gyroscope error sources to velocity error for
Baseball Stitch with Reversal Extended
2.5. COMPARING VARIATIONS OF THE BASEBALL STITCH SLEW
Total Error in Y
x 10- 4
0
10
20
30
Time (s)
40
50
60
Figure 2-12: Total Contribution to angle error from gyroscope errors for Baseball
Stitch Slew with Reversal Extended
Now that the baseball stitch slew with reversal extended as been shown to be
superior to the regular regular baseball stitch slew, it will be selected for use in the
navigation simulations. There would be no practical reason to implement the regular
baseball stitch slew instead of the slew with reversal extended as it performs either the
same of better in all areas. As the term "baseball stitch slew with reversal extended"
is a long and convoluted phrase, for the remainder of this thesis, the slew will be
referred to simply as the "baseball stitch slew", unless otherwise specified. So when
the baseball stitch slew's performance is referred to from now on, it is understood to
be referring to the baseball stitch slew with reversal extended.
CHAPTER 2. CAROUSELING DESCRIPTION
2.6
Modeling Carouseling
The theoretical benefits of carouseling have been shown, but the real goal of this
thesis is to show what these benefits equate to in a numerical simulation which models
the true performance of MEMS IMUs in real-life applications. The first step is to
simulate the output of the IMU experiencing the slew with ideal accelerometers and
gyroscopes at the center of the rotation. In reality, the IMUs will most likely not
be collocated at the center of rotation and this has potentially serious implications,
called the size effect, which will be investigated in Section 2.8. This section will be
concerned with establishing the equations used to model the baseball stitch slew and
generate the IMU outputs angular velocity and specific force. First the ideal outputs
will be created (i.e. no accelerometer or gyroscope errors added), then the errors will
be incorporated to create a simulated IMU output.
The ideal gyroscope output is the angular velocity vector Q, which has been
defined for the baseball stitch slew in Eq. 2.4 and for the reversal in Eq. 2.33. The
output of an ideal accelerometer is the specific force f felt by the accelerometer in the
body frame. This includes the acceleration of the navigation point in the body frame
and the gravitational acceleration:
f = R'
- GB
(2.36)
where RB is the rotation from the Earth Centered Inertial (ECI or simply, inertial)
frame to body coordinates. In this case RB is the inverse of the baseball stitch slew,
Rb8 = R . The acceleration of the navigation point, x' in inertial space is defined
as 3', which is determined by the trajectory of the overall body, i.e. if the navigated
point is standing still on the earth, the only movement will be due to the rotation
of the earth (and there will be no movement observed when using local level coordinates). If the IMU is being used for missile guidance, the missile's trajectory and
flight path will determine the motion felt by the body. The determination of the spe-
2.6. MODELING CAROUSELING
cific trajectory must also be modeled and will be discussed when applying to specific
situations. Also note that superscripts on variables indicate the coordinate frame of
reference, and for rotations, the subscript indicates the initial frame and the superscript indicates the frame into which the rotation is being performed.
The gravitational acceleration contains both the effects of gravity and the centripetal acceleration induced by the Earth's rotation:
GB = (g(x) - [QearthX]2XI)R B
(2.37)
where g(x') is the expression of gravitation depending on the inertial location of the
navigation point. This can be approximated by linearizing and ignoring perturbation
effects as:
g(x) =X
(I - 3uuT)
(2.38)
where p is the universal gravitational constant and u is the unit vector in the direction
of x I . The quantity [Kearth X ] is the skew symmetric formed from the earth's rotation
vector defined as
0
Qearth =
0
(2.39)
[Wearth
where Wearth is equal to the angular velocity of the Earth's rotation. As was stated
earlier, the specific force and angular velocity are the outputs of ideal inertial measurements, meaning there are no measurement errors. In application, there are accelerometer and gyroscope errors in the measurements. Therefore, these errors must
be simulated as well and their contributions taken into account. The simulated outputs, designated as a for accelerometer output and g for the gyroscope output vector
CHAPTER 2. CAROUSELING DESCRIPTION
are simulated as:
a =
f + R(ba + [f-]sfa - [fx]ma + [f @]na) + Wa
(2.40)
g =
+ b + [Qg.]sf - [x]m, + [W ]ng + wg
(2.41)
where the subscript a signifies an accelerometer error source and the subscript g
signifies gyroscope errors. The bias and scale factor terms are simulated as Markov
processes as well as constants and the misalignment and non-orthogonality terms as
random variables with covariances determined by the specific IMU. In addition to
the instrument errors, random walk effects, w, also affect both the accelerometer and
gyroscope output. The random walk can also be simulated as a random process with
specified covariances. Using Eqs. 2.40 and 2.41, the output of an IMU undergoing
carouseling can be simulated. An estimation scheme must be developed to navigate
this data so the performance can be evaluated.
2.7
Error State Navigation Filter
In order to navigate the slew (or any standard strap down system) an integration
scheme must be developed. In this section the basic concept of the navigator will be
described and the equations of motion will be presented and linearized. Subsection
2.7.1 presents the covariance propagation equations and Subsection 2.7.2 will describe
the Kalman filtering procedure.
The navigator will be designed to take the IMU outputs from the simulation and
integrate them back to attain the position, velocity and attitude of the body in the
Earth coordinate frame. This Earth-based navigator will use a Kalman filter and
compute the gains needed for the external updates. This navigator will specifically
keep track of the covariance of navigation errors to determine how accurately the
solution is being determined. The main errors desired to be known are in position
and velocity errors (Sx and 6v respectively) and attitude error represented by mis-
2.7. ERROR STATE NAVIGATION FILTER
alignment vector I. The full list of the 45 error states is shown in Table 2.1 below:
Error state
Position error
Velocity error
Attitude error
Accel. bias(turn-on)
Accel. bias(in-run)
Accel. scale factor(turn-on)
Accel. scale factor(in-run)
Accel. misalignment
Accel. nonorthogonality
Gyro. bias(turn-on)
Gyro. bias(in-run)
Gyro. scale factor(turn-on)
Gyro. scale factor(in-run)
Gyro. misalignment
Gyro. nonorthogonality
index
1-3
4-6
7-9
10-12
13-15
16-18
19-21
22-24
25-27
28-30
31-33
34-36
37-39
40-42
43-45
Table 2.1: Navigation Filter States
The linearized navigation states can be determined by first examining the differential navigation equations (in the Earth fixed frame):
=
v
v
RE =
(2.42)
g(x) - 2[L x]v + REf
(2.43)
RE[qx]
(2.44)
These can be linearized for navigation as:
6xC = 6v
(2.45)
6,
G(t)6x - 2[L x]6v + R E[*x]f + 6a
(2.46)
-[x]F + 5g.
(2.47)
S=
=
CHAPTER 2. CAROUSELING DESCRIPTION
where 6a is the accelerometer error and 6g is the gyroscope error. Notice Eqs.
2.46 and 2.47 were used earlier in Section 2.3 for determining the contribution from
instrument error sources. G(t) contains the effects of gravity and is equal to
(I - 3uuT) - [Qg x] 2 ,
G(t) = -
which is similar to Eq.
2.37, as seen before.
(2.48)
The three linear equations are
implemented in the navigator by integrating them at each time step, with initial
conditions applied.
The derivations of Eqs. 2.45 and 2.46 are fairly straight forward from Eqs. 2.42
and 2.43 respectively. To obtain the result found in Eq. 2.47, which was used earlier
in Eq. 2.8 as well, some derivation is required. First define R as the true rotation
(this holds true for any rotation from one frame to another, but in this case applies
to the body to earth coordinate transformation), and so R is the estimate for this
rotation defined as
iR =
R(I + [T x])
(2.49)
which can be rearranged as
R-'jR
I + ['x].
(2.50)
Taking the derivative of Eq. 2.50 yields
R)= [lx]
d(R
To find
(2.51)
(R-1 R), first examine derivatives of each part,
-
dt
d1
dt
1R
=
-[Qx]R
-
1
= R[(O + g)x]
(2.52)
(2.53)
2.7. ERROR STATE NAVIGATION FILTER
then take the overall derivative of R- 1R:
d(R-)
dt
= -[Qx]R- 1 + R-'R[(
+ 6g)x].
(2.54)
Substituting the value for R- 1R from Eq. 2.50, results in the following equation:
d
dt(R
) = - [x ](I + [
=
x
]) + (I + [ x ])[(Q + 6g) x]
[Tx][x] - [Qx][Wx] + [6gx]
(2.55)
Combined with Eq. 2.51, the expression becomes
[ x] = [*×x][x] - [Qx][9x] + [6gx]
(2.56)
which can be further simplified using the skew-symmetric property that for any two
matrices [ax] and [bx],
[ax][bx] - [bx][ax] = [(a x b)x].
(2.57)
Using this result, Eq. 2.56 can be re-written and then eliminating the skew symmetric
[9x] = [(
S=
v
W' =
x
x
-Q
)x]+[ gx]
+ 6g
x
+ 6g
(2.58)
which is equal to the result obtained in Eq. 2.47.
To determine the accelerometer and gyroscope errors used in Eqs. 2.46 and 2.47,
recall that the accelerometer and gyroscope outputs are equal to:
a = f + R(ba + [f']sfa - [fx]ma + [fl9]na) + Wa
(2.59)
CHAPTER 2. CAROUSELING DESCRIPTION
g =
0
+ bg + [Q.]sfg - [Q x]mg + [Q]n, + w,
(2.60)
The navigator compensates by subtracting the estimated errors:
6a =
Wa
(2.61)
+ [ -.]6sfg - [ x]6mg + [O®]6ng ) + wg
(2.62)
-R(6ba + [f -]sfa - [fx]6ma + [f®]6na) +
dg = -(b,
which are simulated as they were in the modeling of the trajectory, but with no
knowledge of the true values of the errors. This compensation is done prior to the
integration of Eqs. 2.45-2.47.
2.7.1
Covariance and State Propagation
Equations 2.45-2.47 describe the propagation of the estimate in the navigator, but the
estimate is bound by its covariance. The navigator begins to propagate the covariance
with initial estimates for the covariances of the error states in Table 2.1. The initial
estimates are formed from the specifications of the specific IMU. These comprise the
diagonal of the initial covariance matrix, P, which is a square matrix with dimensions
equal to the number of error states. The state and the covariance are updated at each
time step, At, by the following equations:
X+
= 4X-
P+ =
I)P-I)T +Q
(2.63)
(2.64)
where X- is the state immediately before the time step and X + is state immediately following the time step. P- is the covariance immediately before the time step
and P+ is the updated covariance directly following the time step. Q is the covariance
of the process noise and ) is the transition matrix which is estimated as
4 =
eFAt
(2.65)
2.7. ERROR STATE NAVIGATION FILTER
where F is the dynamics matrix and relates the error states to their derivatives:
6X = F6X
(2.66)
The linearized equations 2.45-2.47 are encompassed into F to propagate each
error state.
In this way the covariance and state are propagated each time step
into Earth coordinates. Notice the navigation is being performed in Earth coordinated even though it was stated earlier that the slew will be inertially referenced.
Earth-referenced navigation is easier and more common which is why it is being
implemented, while the properties of the baseball stitch slew are inherent to being
inertially referenced. The consequence of referencing the slew to the Earth will be
investigated later, however.
The described navigation filter will perform the integration of the estimate and
covariance, but provides no updates to maintain accuracy. To improve state and
covariance estimates by incorporating external sources of information, a Kalman filtering scheme can be developed as well.
2.7.2
Kalman Filter Update
To navigate using external updates, an extended Kalman filter algorithm can be
implemented in the integrated navigation system. The main concept is to combine
independent external sources of navigation information with the reference navigation
solution to obtain the optimal estimate of the navigation state.[131 An example would
be the Zero Velocity, or Zupt update Kalman filter, where at each time step the
velocity of the IMU is updated to be zero. This would be implemented when it is
known that the body is standing still, such as in the gyrocompassing application. The
Kalman filter works to better estimate the solution by updating the full state X at
regular intervals as:
CHAPTER 2. CAROUSELING DESCRIPTION
X + = X- + K(z - HX)
(2.67)
where X + is the state matrix after the update and X- is the state matrix before
the update. K is known as the Kalman Gain and is equal to
K = P-HT(HP-HT + R)-'
(2.68)
where H is the observation matrix and R is the covariance matrix of the observation, which is a diagonal matrix where the elements are equal to the measurement
noise squared. In Eq. 2.67, z is the measurement, equal to
z = HX- + v
where v is the measurement noise vector.
(2.69)
The covariance is updated by the
Kalman filter using Eq. 2.70:
P+ = (I - KH)P-
(2.70)
This update works only when the optimum Kalman gain is used. If a suboptimal
gain is used, the following update equation is implemented:
P+ = (I - KH)P-(I - KH)T + KRK'
(2.71)
This equation requires more computation, but it is accurate for any value for the
gain, not just the Kalman gain, which is beneficial for some of the sensitivity analysis
that will be performed later. In either case the gains are computed at each filtering
time step and used to update both the state and the covariance.
2.8. SIZE EFFECT COMPENSATION
2.8
Size Effect Compensation
As previously stated, the navigation scheme described navigates a point at the center
of the rotation. However, in practice, the IMUs will be mounted some distance away
from the center. By sheer logistics all 6 instruments (3 gyroscopes and 3 accelerometers) cannot be located at the exact center of the rotation, so some apparatus is
created that can provide the necessary amount of gimbal
= x0 + Rx
(2.72)
where R, is the rotation from body coordinates to inertial coordinates. If the accelerometers are assumed to be on fixed lever arms, then xB would be a constant
value. Taking the derivative of Eq. 2.72 results in:
xI = ix + RBC
+ R [~ x]xk
(2.73)
but the lever arm is assumed to be constant so 5xB is equal to zero and the second
term can be eliminated. Thus taking the second derivative of Eq. 2.72 to obtain
acceleration yields:
X =i + R L
x2
(2.74)
+ [ x] 2 xf)
+x]x
(2.75)
]xf + R
which, after combining rotations can be written as
I = R(fo + [
since the specific force of the point at the center of the rotation, fo, is the acceleration
of that point rotated into the body frame. Equation 2.75 can be changed into the
body frame, and fk can be defined as the specific force felt by the kth accelerometer
CHAPTER 2. CAROUSELING DESCRIPTION
(fk
=RI
). This results in Eq. 2.76:
fk = fo + [ x]xB + [X]2X
(2.76)
and rearranged to solve for fo, which is the specific force felt at the center of rotation
and the quantity desired for navigation:
fo = f -
×]x
- [
2
xk
(2.77)
The angular rate quantities can be determined from the gyroscope outputs and the
lever arms are assumed to have been measured, meaning that only the fk quantity is
left to be determined. This value can be determined from the scalar outputs of the
accelerometers. Each kth accelerometer outputs a scalar acceleration sensed in the
body frame, ak, felt along the accelerometer's sensitive axis, Sk, and can be expressed
as
ak = sf.fk
(2.78)
Substituting Eq. 2.78 into Eq. 2.77, the equations determining fo for all three accelerometers can be written as:
X]X B + [ x
]2X B )
(2.79)
sTfo = a, - sT([x]xB + [ x]2xB)
(2.80)
sTfo = a, -s T([2x]X
(2.81)
sTfo = a, - sT([
+ [Qx]2xB)
For convenience, Eqs. 2.79-2.81 can be written as one equation using matrices. The
3 x 3 matrix S is defined as
(2.82)
s=
S3
2.8. SIZE EFFECT COMPENSATION
From the three scalar accelerometer outputs the vector a can be formed,
a1
a=
a2
(2.83)
a3
And an additional vector named M is produced containing the remaining terms:
T([x ]Xl + [x ]2X )
M =
S([2X]x B + [ x]2x B )
.(2.84)
sT([2 x]xB + [ x]2XB)
In this manner, the Eqs. 2.79-2.81 can be expressed simply as
fo = S - a - S -1 M
(2.85)
where all quantities used to determine fo are either known beforehand or are outputs
of the accelerometers or gyroscopes. Notice also that if the IMU is designed so the
three sensitive axes are aligned with the lever arms, then the following equation is
true by definition:
s
k
(2.86)
If Eq. 2.86 holds, then the first term in each entry of the vector M will be equal to
zero and therefore fo is independent of 2. This can be advantageous as the angular
acceleration would not be required to be computed.
If the lever arms were known perfectly, Eq. 2.85 could be applied to the accelerometer outputs to yield fo, which would be identical to the specific force used
in modeling the slew with no size effect (Eq. 2.36). However, sometimes the lever
arms are not known, or even if they are, there is some limit to how well they can be
measured or how precisely they can be constructed. So there will be some error in the
lever arm values. As the lever arm is incorporated into the equations as xB and has
CHAPTER 2. CAROUSELING DESCRIPTION
three dimensions, there are three possible errors for each lever arm. In the same way
there are three possible sensitive axis errors for each lever arm. These errors must be
incorporated into the navigation as additional error states.
To incorporate the size effect into the navigator as described in Section 2.7, first
the result for fo found in Eq. 2.85 will be used to transform the outputs from the three
separate accelerometers into the theoretical output of one accelerometer that was at
the center of rotation. This specific force value can then be used in the estimator.
However, the possible errors in the lever arms must be accounted for as well. The
equations of motion in Eqs. 2.42-2.44 must be re-linearized, because no the specific
force error term is associated with it (due to the size effect). The linearized equations
now take the following form:
5x = 6v
6
(2.87)
= G(t)6x - 2[
=
-x
l
x]6v + Rfo + R[Lx]fo
(2.88)
+ 6g
(2.89)
where the only difference from the original linearized equations is the added Rf 0o
term in the velocity equation. The position and attitude equations remain completely
unchanged from the earlier navigation equations. It is necessary then to determine
an expression for 6fo so that it can be incorporated into the dynamics matrix for
propagation. The goal is to find a linear equation for 6fo in terms of the different
error states, that can be inserted into Eq. 2.88 and therefore the dynamics matrix.
To accomplish this, linearize Eqs. 2.79-2.81 to yield the resulting equations:
6s fo + s6fo
= 6a - s ([(x]
+ [Lx] 2)X - S([X +
s Tfo + sTfo
=
a2 - 6S([2x] + [{x] 2)x2 - s
s fo
=
a -
sfo
X 2 )6xB (2.90)
X([!x]
+ [x]
2
)6x
(2.91)
s ([{2x] + [ x] 2 )xB - s([X + LX]2)6x
(2.92)
2.8. SIZE EFFECTCOMPENSATION
where the 6 ak terms are the accelerometer errors, the 6sT terms are the sensitive axes
errors, and the 6x3 terms are the lever arm errors. Rearranging these equations to
isolate the 6fo terms yields:
sl6fo
=
6a, -
sT6fo
=
6a2 -
sT6fo
=
6a3 - 6sT (([ x] +
T(([2x] + [ x] 2 )X + fo) - S ([!x] + [QX]2)6x
sT(([x
] + [Q x] 2)XB + f0) - ST([!
B + fo) - ST(
[ x] 2 )X3
(2.93)
x ]
+ [QX]2)6x B (2.94)
X
+ [x2)6xB (2.95)
In order to combine Eqs. 2.93-2.95 to obtain a matrix expression for 5fo, each side of
the equations can be multiplied by S - 1 and a complicated looking expression can be
obtained in order to facilitate the direst correlation between the lever arm errors and
the specific force error for the implementation in the dynamics matrix. The equation
for 6fo then takes the form
3
fo = S-1'a -
E
3
S-
1
AkSk - E
S-Lkx~
(2.96)
k=1
k=1
where Ak is a 3x3 matrix of zeros except for the kth row defined for each sensitive
axis k as
Ak(k,:) = ([!ax] + [Qx] 2 )x B + fo
(2.97)
and similarly, the 3x3 matrix Lk has all elements equal to zero except for the kth
row which is defined by the following equation:
Lk(k,:) = s([2
x] + [2 x]2).
(2.98)
These equations can be used to express the specific force error precisely in terms of
the sensitive axis and lever arm error states for incorporation into Eq. 2.88 and the
dynamics matrix F of the navigation filter.
In a real system, it is highly desired to simplify these equations in order to reduce
CHAPTER 2. CAROUSELING DESCRIPTION
possible errors. If general procedures were to be followed, some assumptions can be
made, such as designing the IMU such that the sensitive axes are aligned with the
their respective lever arms so that Eq. 2.86 holds true and the effects of angular
acceleration can be greatly minimized, and possibly ignored completely. Also it is
standard practice to align the lever arms properly so that the three sensitive axes are
aligned with the body coordinate axes, meaning that S is equal to the identity. This
greatly simplifies calculations involving S or S - 1 as they will both be equal to the
identity.
In order to modify the original navigation filter, the extra error states are first
added and then Eqs. 2.87-2.89 can be implemented into the estimation scheme in
place of Eqs. 2.45-2.47 and integrated to find the navigated solution as earlier described. The propagation and Kalman updates are performed in the same manner as
before, except now there is the presence of extra lever arm states which will increase
computation time. Also recall that now the accelerometer input to the navigator is
the vector a which is really a collection of the three scalar non-collocated accelerometer outputs. Therefore a must be converted into fo using Eq. 2.85 before integration.
Accounting for the lever arm errors by incorporating extra error states allows the
filter to most accurately model the dynamics of the problem and correct for the size
effect. In the next chapter, the size effect encountered when not using these extra
error states will be analyzed in detail.
Chapter 3
Gyrocompassing Analysis
3.1
Basic description of gyrocompassing
Gyrocompassing is the process of finding North by measuring the direction of the
Earth's rotation vector and the rotation of the gravity plane. The rotation is sensed
by the gyroscopes and accelerometers in the IMU and this knowledge is used for
azimuth determination. Since gyrocompassing relies on the Earth's axis of rotation
and not it's magnetic field, it points to true north, as opposed to magnetic north like
standard compasses. Gyrocompassing has another advantage in that it will not be
interfered with by ferrous metals in the vicinity and is not subject to variations in
the Earth's magnetic field. The method was first used for marine navigation in the
late 19th/early 20th centuries and now its use has been expanded to UAVs, personal
navigation and military scenarios. For the purposes of this thesis the focus will be on
implementing carouseling into a gyrocompassing application for personal navigation
use that needs to be lightweight as it will be carried by a person, so the mass benefits
of MEMS instruments are highly desired.
The concept is a person can carry the
gyrocompass and set it down at intervals to take a bearing on the direction of north,
much like the utilization of a magnetic compass.
As gyrocompassing is typically a higher accuracy application, MEMS devices have
CHAPTER 3. GYROCOMPASSING ANALYSIS
generally not been selected for use in these scenarios. However, for certain gyrocompassing applications, there are severe mass and power constraints where it would be
highly beneficial to use the lighter, more energy efficient MEMS devices. Therefore,
this chapter will investigate the implementation of carouseling with MEMS IMUs
for use in gyrocompassing.
Kalman filtering techniques and MEMS specifications
used will be described and then the results of the simulation will be presented. The
carouseling scheme will be compared to the standard gyrocompassing method to see
if any benefits are produced. Numerous variables concerning the navigation will be
tested for their effect on performance. The influence on azimuth error of the size
effect in particular will be investigated, including the ramifications of not properly
compensating for the addition of lever arms.
3.2
Kalman Filter Update
In most gyrocompassing applications, the body is standing still in relation to the
Earth when a reading is taken. This extra amount of knowledge can be incorporated
into the Kalman filter to improve navigation. Since it is known for a fact that the
IMU is not moving in relation to the Earth, this knowledge can be implemented as
truth to the filter. One way to tell the filter the body is not moving is to implement
a zero-velocity update, known as a Zupt. This is a Kalman update where the earthrelative velocity is updated as zero with each update. In this case the observation
matrix H in Eq. 2.68 is a 3 x n matrix (where n is the size of the covariance matrix)
and all the elements are equal to zero except for a 3 x 3 identity matrix in the rows
corresponding to the velocity error.
This zupt update procedure indicates to the filter that the body has no earthrelative velocity at that point and implies there was no motion over the time step.
However, this does not provide the explicit information that the IMU has not moved
from its original spot. So another option is to use a delay state update instead, in
3.3. SIMULATION DETAILS
which the position is updated at each interval as being the same as the position at
the previous time step. This implies, but also does not explicitly state, that the
IMU did not move between time steps. In preliminary simulations, the delay state
method provided similar results to the zupting method in this case, but zupting is
more common in this application and will be used for all the simulations.
For this simulation, a measurement noise of 0.001 m/s was selected for the zupt
updates, which will occur with a frequency of 1 Hz. This seems to be a conservative
estimation for the noise that would be felt in this personal navigation setting. The
effect of changing the value of the measurement noise and the frequency of the zupts
on the results will be investigated in later sections.
3.3
Simulation Details
The Charles Stark Draper Laboratory has designed numerous MEMS instruments
ready for use in a variety of applications. The error specifications for the most accurate
MEMS accelerometer, were used for these simulations. For most of this analysis, the
simulations will be performed using the specifications of the current MEMS gyroscope
Draper has designed, but the improved model is in development and Draper has
estimated specifications for the improved accuracy it will be able to provide. Therefore
the current model IMU will be the main focus of this research, but the estimated
specifications of the improved IMU will also be studied as it will provide further
insight. Both IMUs implement the use of the same accelerometers, it is only in the
gyroscope performance that they differ. The complete error specifications of the two
IMUs can be seen in Table A.1 in the Appendix.
For this simulation, the baseball stitch slew will have a period of 60 seconds and
follow the angular rate profile shown in Fig. 2-9, which equates to a maximum 48 0 /s
angular rate for the outer gimbal (the outer gimbal performs 8 total revolutions of
CHAPTER 3. GYROCOMPASSING ANALYSIS
3600 for a total of 28800 in 60 seconds). Once again, the slew is inertially referenced,
as was shown in Section 2.2, although the effects of referencing with respect to the
Earth will be investigated as well. The size of the apparatus performing the slewing is
assumed to have a radius of approximately 3 cm, so this will be the assumed values for
the lever arms, with an error covariance of 1 mm 2 (therefore the standard deviation of
the error in all the lever arm measurements will be equal to 1 mm). The variations of
the lever arm parameters will have a great influence on the size effect experienced and
in turn the accuracy of the IMU, so they shall be investigated later in this chapter.
3.4
Azimuth Covariance Determination Results
The goal of this section is to show the effect of the carouseling scheme on the gyrocompassing accuracy. Simulations were run to compare the carouseling method to
the standard gyrocompassing method so the differences can be observed. The set-up
of the simulation is that of a typical gyrocompassing problem in which the body is
at rest on the earth's surface in order to take a bearing. Recall, that although the
body is at rest, the IMU will be going through the inertially-referenced carouseling
scheme in the body. The IMU navigates and essentially measures the Earth's rotation
vector in order to determine North, or the azimuth angle. The navigation scheme will
solve for the entire state, but the main focus of gyrocompassing is to determine the
azimuth angle. In any navigation problem there are initial error covariances in the
states, which have been set to be typical values shown in Table 3.1:
Error State
Position
Velocity
Attitude
Initial Covariance
10 mn
0.1 m/s
1o
Table 3.1: Initial Error States
Implementing these initial covariances along with the specifications from the IMU,
3.4. AZIMUTH COVARIANCE DETERMINATION RESULTS
the gyrocompassing problem can be navigated and the results will be analyzed. The
state estimate is not of much interest as it depends on the initial value and varies
according to random variables correlated by the covariance. Therefore, the focus will
be on the covariance analysis, as the covariance determines the "envelope" for the
error estimate.
The analysis begins by simulating the slewing trajectory using the equations described in Section 2.6 to obtain an IMU output of 6v and 60 along with the truth
values of position, velocity, and attitude. Then, using the navigator developed in
Sections 2.7 and 2.8, the IMU outputs can be integrated to obtain estimates for
the errors along with the covariances for all states. The estimates can be compared
against the true values for all states, however it is the azimuth error that is of most
importance in the gyrocompassing application. In Fig. 3-1 can be seen the azimuth
error when using the baseball stitch slew and the error specifications of the current
Draper MEMS IMU. In this plot, the covariance is depicted as the dotted red lines
and the error estimate is the solid blue line.
The navigation azimuth error estimate initially fluctuates but approximately approaches zero after starting a from a randomly distributed point with the covariance
range. The covariance decreases to slightly less than 0.3 degrees within 10 minutes
from an initial error of one degree. This drastic covariance decrease confirms the
results of Section 2.4, and reveals that the baseball stitch slew can provide accurate
results. The estimate is shown in this plot to demonstrate its behavior within the
covariance bounds, but as there are random value functions involved in the estimate
calculation, the focus will be on the covariance analysis.
As was seen in Fig. 3-1, the carouseling method does seem to perform well, but in
order to accurately gauge the performance of the baseball stitch slew, it is necessary
to have a standard for comparison. One standard would be to compare the results to
those of the scenario in which the body were stationary and made absolutely no movement over the time interval. However this is unrealistic, as gyrocompassing systems
CHAPTER 3. GYROCOMPASSING ANALYSIS
Azimuth Error and Covariance
ii
1
0.8
.....'C
-!
i..
. I........
.................
.......
........
l .
.
.
0.6
0.4
. .
.. .... ...
....
. ..
. .. ..
.. .. ... ... ..
•
....
. . . .
... . . ..
. . . . . . .. . . . . . . . .
.... ...
. .. . .. .. . . ...........
0.2
0
-0.2
' .. ............ .........
......... ..... ... . ..... .......
..
- . . . . . . . =.
-0.4
(~ - . .
~~~~~~
-0.6
..
•
i
~.-. ~
L I
..
-0.8
-1
0
1
2
3
4
6
5
Time (min)
7
8
9
10
Figure 3-1: Current MEMS IMU Azimuth Covariance for Baseball Stitch Slew The
estimation error remains inside the covariance bounds which indicate an accuracy of 0.2910 after 10
minutes.
implementing IMUs of lower quality take the extra measure of reversing the orientation of the body periodically in what is called 2-Position Gyrocompassing. Some
IMUs are accurate enough that this reorientaiton is unnecessary, but for the MEMS
devices selected it would certainly be employed. Therefore, this method shall be taken
as the standard to which the baseball stitch slew share be compared. 2-Position Gyrocompassing is essentially a less-intensive way to eliminate biases by simply reversing
the direction of the object and preventing the build up in one direction of biases. The
action can be physically picking up the gyrocompass and turning it around or could
be performed by some servo-mechanism. This rotation would be earth referenced in
the simulation, not inertially since it is just the physical reversing of direction on the
Earth's surface. The time between reversals can vary but for this simulation, the
3.4. AZIMUTH COVARIANCE DETERMINATION RESULTS
direction is reversed once every minute.
In the following plot shows the covariance of the baseball stitch compared with
the 2-position method over a period of 10 minutes:
Azimuth Error Covariance
I
I
I
I
I
0.8
....
...........
. .....
''I-,,-............
:
.........
-"' '' ...
0.6 . . ..
....... .
!.........
!.........i........
.. . ~........ !........
-...
-..
.....
.. ..
. . . ... .. . ...
..
"-_"-- .. . .. . .. . . . .. . . . :"
-. . ... .•. . ..
0.4
... . ;.........;............
•
0.2
:
........
..-
-.. ... • .........
- - ..,-. -.. .......
:
•
'
-,-
. ..........
._...
_ ...
_
...
.
...
. . . .
.
0
-0.2
-0.4
-- --........
4 ...... ;........
. . .. . ..;.................
• - ....................
i . .... ....
. ....
.. . .. .... ...
... ..... .... ...
.......
...........
.. ..... .. ;..........................
. .. :.
. ..- ...
... .. ...
.. .. .. ................
...
.. . .. .. i....
... . .........
-0.6
-0.8
...
.....
-1
.... ...... i......... ........ ......... ::.....
I I
--
1
2
Baseball Stitch Slew
-2 p s to
I
,
,
,
4
5
E
Time (minutes)
7
8
9
2-o
it o
=Ib
•
0
I
-
3
10
Figure 3-2: Current MEMS IMU Azimuth Covariance Comparison for Baseball Stitch
Slew and 2-Position Gyrocompassing The carouseling method provides a substantial decrease
in error of 50% over the conventional 2-postion method.
This shows that the baseball stitch slew does indeed provide a significant benefit.
After 10 minutes of operation the carouseling scheme has improved accuracy by over
50%, from 0.649 degrees to 0.291 degrees. Using carouseling, the current MEMS IMU
can achieve accuracy of less than half a degree in 4 minutes, where the 2-position
gyrocompassing method will never reach this level of accuracy. This certainly shows
64
CHAPTER 3. GYROCOMPASSING ANALYSIS
there is a substantial increase in accuracy when implementing carouseling. There are
some variable factors that affect these results which will need to be analyzed.
To further investigate these results, another comparison would be to determine
if the benefit gained from carouseling is due to the unique properties of the baseball
stitch slew as discussed in Section 2.3, or if it is due simply to the fact that rotating
the IMU in any slew will provide similar results. To test this, the baseball stitch slew
covariance can be compared to that when the body is simulated to simply rotate 3600
back and forth every minute. These results are shown in Fig. 3-3.
Azimuth Error Covariance
0.8 ......
- .-
....
.......... ...
...
...
.....
...
.....
.....
..
.
0.4
o)
=0.2
E
-0.8 ...... ..........
........................
-
-1
0
1
2
3
........
......
• . •Baseball Stitch Slew
- - 360* turns
5
6
4
Time (minutes)
7
8
9
10
Figure 3-3: Current MEMS IMU Azimuth Covariance Comparison for Baseball Stitch
Slew and ±3600 turns The carouseling technique provides the same benefit over the ±3600 turns
case as over the 2-position method.
Once again, the baseball stitch slew provides significantly more accuracy. In fact,
it can be seen that the ±360' turns provide no discernible benefit to the 2-position
3.4. AZIMUTH COVARIANCE DETERMINATION RESULTS
method (the final covariance for the ±3600 method is 0.648 degrees as compared to
0.649 degrees for the 2-position method). So it can be concluded that the special
properties of the baseball stitch slew as discussed in Chapter 2 are responsible for
this increased accuracy, not simply the rotating motion itself.
One facet of the baseball stitch slew that has been mentioned before is that the
slewing must be inertially referenced for the full benefits to be felt. However, in most
cases it is much easier to reference gimballing in the Earth coordinate instead. Since
both coordinate frames are centered at the center of the Earth, the only difference
is the small rotational motion of the Earth itself. So it would be valuable to know
how the IMU will perform if the slew actually is Earth referenced and not inertially
referenced.
Figure 3-4 shows the azimuth error covariance results comparing the
inertially referenced slew to the earth referenced slew.
This plot reveals that although the inertially referenced slew does perform better, as expected, the Earth referenced slew does not increase the error covariance
drastically. There is less than a .030 increase with the Earth referenced slew. This
would indicate it may be possible to Earth reference the slew to reduce complexity
while still attaining an almost 50% improvement in covariance performance over the
2-position method, depending on the required accuracy of the application. This is
a 10% increase in error over the inertially slewed case, so if the extra precision is
required for performance the slew will need to be intertially referenced to take full
advantage of the properties of the baseball stitch slew.
Another important aspect that affects both the 2-position and carouseling methods is the zupt update. It was stated earlier that the zupting was performed once
a second for these simulations. Although this frequency is not uncommon, it may
be desired or necessary to update less frequently, which could potentially have an
effect on the results. In Fig. 3-5, the azimuth covariance for both the 2-position
and carouseling schemes can be seen when the zupts are performed every 5 seconds
instead of every one second.
CHAPTER 3. GYROCOMPASSING ANALYSIS
66
Azimuth Error Covariance
0.2
0.4 --
ED
-0.2
.Referenced
.....
.............
i...
........
...
...
.....
-1.....
--
0
1
2
3
6
5
4
Time (minutes)
-
Earth Referenced
7
8
9
10
Figure 3-4: Current MEMS IMU Azimuth Covariance for Baseball Stitch Slew referenced in two different coordinate framesThe inertially referenced slew performs better than
the Earth referenced slew, but the difference is only marginal.
There is a slight increase in the covariance for both methods, but the increase is
consistent for both so the overall benefit of slewing is undiminished. The baseball
stitch slew still provides a 50% lower azimuth error covariance. The zupting interval
is therefore not a large factor in the comparison between the two methods.
The marked increase in azimuth performance when implementing carouseling for
use with the current MEMS IMU is now apparent, but if in the future the current
model can be improved upon, it will be helpful to see what kind of benefit the baseball
stitch slew will provide. The predicted specification of the improved IMU can be
integrated into the simulation to see how this change in IMU accuracy affects the
results. Using the same initial conditions, the carouseling and 2-position methods
3.4. AZIMUTH COVARIANCE DETERMINATION RESULTS
67
Azimuth Error Covariance
0
0
.8
0.2.....
0. ....
-0..
........................................................
. .........
Time (minutes)
Figure 3-5: Current MEMS IMU Azimuth Covariance Comparison with zupt update
performed every 5 seconds Even with zupting performed over longer intervals, the carouseling
method provides the approximate same amount of benefit.
can be simulated and their respective azimuth error covariance shown here in Fig.
3-6.
Although the slewing continues to provide increased performance over the 2position method, the final reduction in error is significantly lower than for the current
MEMS IMU. The carouseling method provides an azimuth accuracy of 0.1160 while
the 2-position method provides 0.1450. Both methods in fact perform better than
when the carouseling is used for the current MEMS IMU, and now the benefit of
slewing is reduced to about 0.040, which is still 20% improvement, but not a substantial magnitude. So if the specifications for the proposed improved design actually
can be met, if concerned with longer time applications it makes little sense to add
can be met, if concerned with longer time applications it makes little sense to add
68
CHAPTER 3. GYROCOMPASSING ANALYSIS
Azimuth Error Covariance
I
I
I
1*
00.2.2
. . .............................
........ ........ ....... ........
.............
...........
...
.........
........
.......
o ..........
.. ............................
........
................
•
-0.2 .... ...........
........
0 .........
-1
0
............... .......
.......
.......... ........
................. ........
...............................................
1
2
3
Baseball Stitch Slew
--
4
5
6
Time (minutes)
-2-position
7
8
9
10
Figure 3-6: Improved MEMS IMU Azimuth Covariance Comparison for Baseball Stitch
Slew and 2-Position Gyrocompassing With the improved gyroscopes, the carouseling method
still shows slightly better performance but it is not as significant as with the current model.
complexity by using the carouseling scheme when the accuracy is not improved drastically. However, notice that in the first 4 minutes of the simulation there is still a
sizable benefit of using the carouseling method. For example, the carouseling method
can reach an accuracy of one half of a degree in about 45 seconds while it takes 3
minutes for the 2-position method to reach that mark. At right before 2 minutes, the
accuracy of the carouseling method is about 0.25 degrees as compared to almost 0.8
degrees for the 2-position method. Therefore, if the given application is extremely
time-sensitive, there is still a large advantage to using the carouseling technique with
the improved MEMS IMU. This would be a likely scenario since it would be undesirable for a person using the gyrocompass to be required to wait over 5 minutes to
.0.4._
3.5. SENSITIVITY ANALYSIS
find which direction is North. Carouseling with the improved MEMS model would
provide that extra accuracy in a far more timely manner than the 2-position method.
These covariance plots have shown the overall benefit in using carouseling for this
particular application, but the reasons and sources of this improved accuracy can be
investigated further. By performing different types of sensitivity analysis, the contributions to the error can be broken down and compared to see in what way is the
carouseling providing benefit, and how changes to the problem will affect it. This is
the focus of the remaining of this section.
3.5
Sensitivity Analysis
The overall effect on azimuth covariance due to carouseling has been seen, but the
components of this error can be even further investigated. This section is devoted to
parsing out the total error covariance in order to see the exact benefits that carouseling
provides.
The goal is to discover in what ways the baseball stitch slew provides
improvement and how sensitive the results are to the changing of certain aspects of
the problem.
It would be desirable to see if the carouseling maneuver is eliminating the errors
predicted in the analysis of Chapter 2. Also, insight can gained form investigating
if there are any other errors unexpectedly building up in the process. It would be
beneficial to study the components of the total covariance error to see what contributions are the most important. A special type of sensitivity analysis known as an
error budget can be implemented to reveal this information and show the breakdown
of the entire covariance. An error budget determines the separate effects of the error
sources on the overall covariance errors 171. The process of creating an error budget
will be described in the next subsection.
CHAPTER 3. GYROCOMPASSING ANALYSIS
3.5.1
Error Budget Formulation
The main principle of the error budget is that when a filter is applied to the navigation solution, all of the error sources (even if they are not modeled in the filter)
contribute to the total error in the filter estimate. The error budget delineates these
separate contributions to the total covariance error [7]. To create this budget, first,
the time history of the filter gain matrix K must be acquired. This can be done by
simulating the system in the exact same way as the earlier covariance analysis with
initial covariance matrix P*(O), and recording the filter gain matrix at each time step.
This time history of the gains can be saved as K*.
To determine the contributions of each error source, the simulation is run again
with zero measurement noise and initial covariance P(O) where all error elements
not corresponding to the selected error source are set to zero. For example, if the
contribution due to accelerometer bias is to be measured, all entries into P(O) will
be equal to zero except for the turn-on and in-run accelerometer bias covariance
terms. This can be parsed out further if for example the contribution due solely to
the turn-on bias is desired to be tested. During these simulations, instead of the
filter generating its own gains, the gains from the time history K* are employed for
the duration of the simulation.
The final covariance values are recorded for each
simulation, which captures the effect due solely to that specific error source. The
final covariance for each simulation is the total contribution to the error solely due
to that specific error source.
The contribution for each error source can then be
recorded, including measurement error. To find the effect of measurement error (in
this case the zupt noise), all initial elements of the P matrix are set to zero and so
the measurement noise is the only error contribution.
After generating the final covariance for each separate error source, it can be seen
that squaring all the values and adding them together is equal to the covariance error
for the full state errors squared. In this way, the entirety of the error can be accounted
for, and the contribution from each error source can be compared. The larger the
3.5. SENSITIVITY ANALYSIS
contribution from one error source, the more influence this error source has on the
total error covariance and the more sensitive the covariance is to that error source.
By this logic, error sources with the largest contributions to the error budget are the
most desired to be eliminated or reduced as this will provide the most benefit to the
total results. In the next sections, the results from creating error budgets and other
sensitivity analyses will be used to determine the effects of changes in critical error
sources.
3.5.2
Sensitivity Analysis Results
Using the method described in Subsection 3.5.1, error budgets of azimuth error for
both the carouseling and 2-position method using the current MEMS IMU can be
generated. Table 3.2 compares the azimuth error results of both for 10 minute simulations with initial conditions as specified in Table 3.1.
Now it is possible to dissect the covariance errors for both methods and ascertain
what gains the slewing procedure is providing. First, notice that the largest reduction
in azimuth error comes in the gyroscope bias contribution. The baseball stitch slew
practically eliminates this entire error, leading to an RSS contribution of only 1% of
the gyroscope error in the 2-position method. This is an extremely attractive feature
of the carouseling scheme, and confirms the analytical results obtained in Chapter 2.
Using the 2-position method, the gyroscope bias is the largest error contributor, and
by using slewing this substantial error source is almost completely brought to zero. In
the 2-position method, the gyroscope bias contributes to 26% of the total covariance.
This is brought down to less than 1% by using carouseling. This also explains why
the slewing provides only marginal benefit over 2-position gyrocompassing when the
specificaitons for the improved MEMS model are used. The gyroscope bias of the
improved MEMS IMU is less than 10% of the bias for the current MEMS model.
Because the bias is already so low, there is less error for the slewing to eliminate.
It is also of interest that for the lever arm errors present only in the carouseling
CHAPTER 3. GYROCOMPASSING ANALYSIS
Error Source
Position
Velocity
Baseball Stitch
2-position
0.000
0.001
0.000
0.001
Angle
0.113
0.240
Gyro. Bias
Gyro Scale Factor
Gyro. Misalignment
Gyro. Non-orthogonality
Angle Random Walk
Accel. Bias
Accel. Scale Factor
Accel. Misalignment
Accel. Non-orthogonality
Velocity Random Walk
Lever Arm 1
Lever Arm 2
0.033
0.013
0.002
0.001
0.219
0.107
0.011
0.001
0.001
0.016
0.001
0.001
0.327
0.013
0.002
0.001
0.305
0.076
0.000
0.001
0.001
0.014
Lever Arm 3
0.001
Measurement Noise
Total Error
0.158
0.291
[
0.056
0.647
Table 3.2: Error Budget Comparison of Azimuth Error contributions in degrees for
Baseball Stitch Slew and 2-position Gyrocompassing
case, the total error contribution due to these errors is practically insignificant. The
sensitivity to these errors will be investigated in detail later in Section 3.6, but the
fact that these errors do not substantially decrease performance for the assumed scenario is a positive result. This indicates that for the lever arm specifications employed
(lover arm length of 3 cm and uncertainty of 1 mm), there is not a large penalty in
terms of overall accuracy.
Many of the other error sources contribute roughly equally in both the carouseling and 2-position case, for example, the accelerometer scale factor, misalignment,
and non-orthogonality errors contribute almost no error to the azimuth covariance.
This makes sense as the accelerometers measure change in velocity and do not factor into the attitude error directly. However there are a few anomalies noticeable
in Table 3.2. The carouseling scheme also provides significant improvement over the
3.5. SENSITIVITY ANALYSIS
2-position method in the contribution due to initial angle error and the angle random
walk. These gains are not as substantial as the gyroscope bias reduction, but they
still do provide a non-trivial amount of the total covariance improvement.
Conversely, the 2-position method, besides from not having any error due to the
lever arms, has a substantially lower amount of error due to measurement noise. This
is interesting, as the zupting measurement noise was equal to 0.001 m/s for both.
The error budget indicates that the 2-position method is in fact less sensitive to the
zupt noise than the baseball stitch slew. To further investigate this, another type of
sensitivity analysis can be done in order to determine the effect of measurement noise
on the azimuth performance.
Since the error budgets indicated that the measurement noise provides a substantial unforeseen disparity between the two methods, a sensitivity analysis of the
measurement noise shall be performed.
Now instead of generating an entire error
budget, only the measurement noise used for zupting will be changed and the total
azimuth covariance error will be recorded for both the 2-position and carouseling
schemes. The results can be seen in Fig. 3-7:
Notice the points which correspond to a measurement noise of 0.001 m/s reflect the
results found in the earlier simulations. However, the 2-position method is relatively
unaffected by any change in measurement noise, as the value of azimuth error scarcely
shows any change even with orders of magnitude changes in the measurement noise
(note the log scale of the measurement noise axis). This is expected from the error
budget results, as the contribution from the measurement noise was low for the 2position method. It was seen that the measurement noise contributed to less than 1%
of the RSS azimuth error, so any increase or decrease in the zupt noise will not have
much effect on the total error. However, when the baseball stitch slew is applied, the
azimuth covariance is much more sensitive to the measurement error. A full 26% of
the total error in azimuth is due to the measurement error in the error budget, and
as seen in Fig. 3-7, this contribution only increases with increasing noise. This is
CHAPTER 3. GYROCOMPASSING ANALYSIS
0.7
0.6
1 0.5
0
" 0.4
.E
0.2
2-position
I
0.1
---
0
0.0001
0.001
Baseball Stitch Slew
0.01
Measurement Noise (mis)
Figure 3-7: Azimuth Covariance Error after 10 minutes for Varying Measurement Noise
As the measurement noise increases, the contribution to azimuth error increases for the carouseling
case while remainly practically constant for the 2-position method.
valuable information, as it indicates that if the measurement noise can be reduced
in some way, the carouseling will provide even greater accuracy. For example, if the
measurement noise is reduced by a factor of 10, the overall accuracy improves by
40%. This would not be unreasonable as the measurement noise for zupt updates
can vary significantly with a given application. Conversely, if the environment of the
application is considerably noisy, much of the benefit of the carouseling sheme could
be negated.
For further analysis, the error budgets for both the 2-position and baseball stitch
trajectories were compiled for simulations lasting 5 minutes and 15 minutes in addition
to the error budget for 10 minutes in Table 3.2. In this way, any patterns that develop
over time can be investigated. Figure 3-8 shows the covariance contributions of four
3.5. SENSITIVITY ANALYSIS
75
selected error sources: gyroscope bias, zupt noise, accelerometer bias, and initial
attitude error. These four were selected as the other error sources either have very
minuscule contributions or did not exhibit particular interesting patterns over time.
So the columns in Fig. 3-8 are only partial error budgets and their total height is not
equal to the total error covariance in this case.
0.35
0.3
0.25
0.2
0 Gyro Bias
MZupt Noise
.
015
Accel Bias
. Initial Attitude
0.1
0.05
2-position
Carouseling
5 minutes
2-position
Carouseling
10 minutes
2-position
Carouseling
15 minutes
Figure 3-8: Selected Azimuth Covariance Error Contributions over time As time
in-
creases, the gyroscope bias contribution increases for the 2-position case while remaining negligible
when carouseling. Additionally, the carouseling maneuver eliminates the initial attitude error significantly more quickly.
Notice first that the contribution from gyroscope bias for the carouseling maneuver for all three times is so small it is practically not visible. However, the same
contribution for the 2-position case is not only substantial, but actually grows larger
over time. In fact, the gyroscope bias error completely dominates all other errors by
the time the simulation has been running for 15 minutes. So over time the benefit of
CHAPTER 3. GYROCOMPASSING ANALYSIS
carouseling will only become increasingly more substantial. Not only does carouseling
eliminate the gyroscope bias error almost completely, it keeps it from growing and
the covariance getting worse
As previously mentioned, the measurement noise contribution is seen to be larger
for the carouseling case than the 2-position case. This contribution does decrease as
time increases for both methods, but still remains a large contributor to the error for
carouseling. In fact, as time increases and the carouseling method continues to vastly
reduce all other errors, the portion of the total error due to the zupt noise increases.
At the same time, the measurement noise error for the 2-position case is hardly a
factor at 15 minutes. This again points to the possible improvements that could be
made for th carouseling case if the measurement noise could be reduced.
Also seen in Fig. 3-8, although the contribution from accelerometer bias is larger
for the 2-position case at 5 minutes, for the simulations of 10 and 15 minutes the
carouseling actually has a slightly larger accelerometer bias contribution.
This is
most likely due to the fact that because the carouseling maneuver involves constantly
slewing the IMU, whereas for the 2-position case the IMU was mainly stationary.
This added velocity creates a larger opportunity for velocity error in the carouseling
method. As for the 2-position case, velocity errors will not be as large as there is less
total velocity.
The last contribution seen in Fig. 3-8 is the that due to the initial attitude error
of one degree. It can be seen that the proportional amount of this error decreases
drastically for both methods over time, however the carouseling maneuver does a
substantially better job of eliminating this error quickly. For example, at 5 minutes,
the carouseling scheme has reduced this error to less than one eight of the same contribution for the 2-position method. The relative benefit in this error contribution
provided by the carouseling decreases over time, as can be seen for the 10 and 15
minute simulations, as both methods effectively eliminate this error source. But the
quicker response of the carouseling method gives it a distinct advantage for more
3.6. SIZE EFFECTSENSITIVITY
time-critical operations, especially if the initial error is ever larger than one degree,
in which case the benefit will be even more pronounced.
The analysis so far has been focused on general comparisons of the two methods in terms of error contributions. This has provided a picture of the benefits of the
carouseling scheme when implemented for the selected application. However, of possibly greater concern is the impact of the size effect on any gyrocompassing application
possibly employed. For this simulation, this size effect was seen to be negligible, but
it is desired to see how this can change in different situations. Investigating the sensitivity to size effect, an area of great concern regarding carouseling, is the aim of the
next section.
3.6
Size Effect Sensitivity
One of the biggest issues with implementing carouseling is the compensation of the
size effect. As seen from the error budget for this gyrocompassing application, the
increase in azimuth covariance due to the lever arm uncertainties is minimal (less
than 1% of the total error). This was based on assumed parameters for the lever
arms and their respective uncertainties as well as the slew rate, which may be true
for this scenario but could vary greatly from application to application. Recall from
Eq. 2.84 that the lever arm length and angular rate 0 are the main contributors to
the difference in the outputs of the uncollocated accelerometers from the specific force
at the center of rotation, fo. Since the lever arms are actual error states in the filter,
the navigator also attempts to estimate the lengths of these lever arms. As previously
stated, in the gyrocompassing simulation the lever arms are assumed to be 3 cm long
with an initial uncertainty of 1 mm. Also, the period of the baseball stitch slew was
60 seconds, which equates to a maximum gimbal speed of 480/s. To examine how
well the filter is accounting for the lever arm, the time history of the estimate for one
lever arm will be examined. In this situation, the lever arm was given a real error
CHAPTER 3. GYROCOMPASSING ANALYSIS
of 0.2 mm, and it is desired to see how well the filter does at estimating this error.
Figure 3-9 shows the time history of the estimate for this lever arm error.
The filter does an excellent job of estimating the error as the estimate approaches
Arm Error Estimate
-Lever
-
- Actual Lever Arm Error
1
- 0.8
Wm
0.6
S0.4
0.2
I
-------------------------------
SI
0
0
1
2
3
4
6
5
Time (min)
7
I
8
9
Figure 3-9: Lever Arm Error The estimate can be seen to approach the true error of 0.2 mm
the true value and is within 1% within 8 minutes. Note that although this is the
estimate for only one of the three lever arms, the other two show identical the identical behavior exhibited in Fig. 3-9. This shows the filter can accurately estimate
lever arm errors, which means this filter could also be implemented to determine the
true length of lever arms, possibly during calibration. In this scenario, since even the
true lever arm error is already on the scale of less than a milimeter, the total error
contribution to the covariance due to the lever arm uncertainty is not substantial.
However, in other applications, there could be longer lever arms and greater uncer-
3.6. SIZE EFFECT SENSITIVITY
tainty in the lever arm values as well as different slew rates. Since the implementation
of these extra lever arm states significantly increases the computation necessary and
the complexity of the filter, there is interest in studying if these error states are necessary or if they can be eliminated. If the additional errors for no compensating for
the additonal states are not significant, it could be desirable to not implement the
additional states to save on computaiton.
For example, the output of the three accelerometers with the 3cm lever arms can
be simulated as in previous analysis. But instead of compensating for the lever arms,
the navigation will be performed on the accelerometer outputs with a navigator that
does not account for the lever arms and treats this output as the specific force felt by
the body at the center of rotation. The azimuth error estimate when this procedure
is followed is compared the the normal covariance in Fig. 3-10.
It can be seen that the estimate does fluctuate more so than in the compensated
case (as seen previously in Fig.
3-1), but remains within the normal covariance
bounds. This would lead one to believe that it would be possible to neglect the
size effect in this case so the lever arm states would not need to be included in the
filter. As the lever arms are only 3 cm in length, the extra velocity added is not that
significant. The impact of the size effect is then not so great that it would require
the extra computation and complexity of incorporating the extra lever arm states.
If the lever arm length is longer than the 3 cm assumed for this application, the
size effect felt due to the added change in velocity will increase and it may no longer
be possible to ignore. For example, if the same simulation is run with 10 cm lever
arms where there is no compensation, the estimate will deviate significantly, as shown
in Fig. 3-11.
Clearly this is well out of the covariance bounds and attempting to implement
this navigation scheme would yield disastrous results in a real application. In fact,
the estimate in this case would be significantly worse than when using the 2-position
gyrocompassing scheme. In order to better quantify the value of the added lever arm
CHAPTER 3. GYROCOMPASSING ANALYSIS
Azimuth Error and Covariance
0.5
0
-0.5
-1.5
0
1
2
3
4
6
5
Time (min)
7
8
9
10
Figure 3-10: Navigated solution compared to Azimuth Covariance when 3 cm lever
arms are uncompensated The error estimate deviates but remains within covariance bounds
with a maximum outer gimbal rate of 480/s
states, a sensitivity analysis can be performed that accounts for filters with a reduced
number of states. [71
In this case the navigation filter using the lever arm states is the "optimal" or
"truth" filter as it accounts for all states of the true dynamics model, and the filter
not implementing the lever arms states is the "sub-optimal" filter. So the optimal filter
has more total error states and thus requires more computation than the sub-optimal
filter, while more accurately accounting for the dynamics of the system. Using the
sub-optimal filter to navigate will produce a time history of the sub-optimal gains used
for navigation. Much like with the error budget formulation, these sub-optimal gains
can be stored and then implemented into the optimal filter instead of the optimal
3.6. SIZE EFFECTSENSITIVITY
Azimuth Error
I
I
I
~
I
I
I
I
I
i
.......................................
•
I
•;
... .. .. ..:
• --..
...
..
.. . . ...
..
. . ...
.
...
.....
..
...
.
...
..
...
...
..
..
_...
. ,.
I. ... . . .. . ... . ... . . .. .
.
-2
. . .
. . . . . .
:
.
. . .. . .. ... .. . . .... . . .
. ..
.
......
•
. . ...
.
.
. ...
..
..
,..
.
. ..
..
. . ...
...
..
...
...
. ...
.. ...
. . .. ...
.. . . .. . . ...
..- -. ... - . ...
. . .. . .... . .. . . .. .. .
. .. . .. .. . .. . .
. . . .
....... .. .. .. .. .. .
. . . . . . .
-3
-4t
o
1
2
3
4
5
6
Time (min)
7
8
9
Figure 3-11: Navigated solution compared to Azimuth Covariance when 10 cm lever
arms are uncompensated In this case with a maximum outer gimbal rate of 480/s, the estimate
deviates significantly from the covariance bounds
gains, to obtain an azimuth covariance. This sub-optimal covariance result can be
compared to the result when the optimal filter performs updates implementing the
normal optimal gains. In this manner the penalty for not compensating for the size
effect is the difference between the navigation using the sub-optimal gains and the
optimal gains. This penalty can be observed for varying lever arm values with the
same 60 second baseball stitch slew with the maximum outer gimbal rate of 48 0 /s.
These results are compiled in Fig. 3-12.
The azimuth error can be seen to increase in an approximately linear relationship
with increases in lever arm length when angular rate is held constant. For lever arm
values less than 1 cm there is practically no appreciable difference in azimuth covari-
82
CHAPTER 3. GYROCOMPASSING ANALYSIS
140
I
I
---------
0----
0 +I
I,
I
0
0.05
0.1
0.15
02
0.25
Lever Arm Length (m)
Figure 3-12: Increase in Azimuth Covariance when implementing Sub-optimal gains
for varying lever arm lengths In this simulation the sub-optimal gains do not model the lever
arm states while the truth filter does include these additional states. The size effect contribution
appears to increase linearly with the increase in lever arm length while the maximum gimbal rate is
held constant at 480/s.
ance. And even up to a 4 cm lever arm equates to a 10% increase in error. Therefore
for smaller devices where the accelerometers will not be offset by more than 4 cm,
it may be advisable to use a filter that does not account for the lever arm states.
However, for lever arms longer than this, it becomes apparent that by not accounting
for the size effect, the overall accuracy of the IMU will suffer noticeably.
From the equations developed in Subsection 2.8, it can be seen that the size effect
is not only dependent on the length of the lever arm, but also the angular rate of
the rotating point. In this simulation, the baseball stitch slew was always assumed
to have a period of 1 minute, which meant the outer gimbal had a maximum angular
3.6. SIZE EFFECTSENSITIVITY
rate of 480/s. If the angular velocity were to be increased, the amount of change in
the velocity felt by the accelerometers due to the size effect would be increased. In
fact, as was seen in the equations developed in Subsection 2.8, the additional velocity
is directly proportional to the total angular rate felt by the accelerometers squared,
q2.
To see the effect this has on the navigation scheme implementing updates, a
sensitivity analysis can be performed similar to that seen in Fig. 3-12, where the
sub-optimal performance without accounting for the dynamics of the lever arms is
compared to that of the optimal filter that includes the additional lever arm states.
In this case the gimbal rate of the outer gimbal shall be varied instead of the length
of the lever arms (the lever arms are held constant at 3 cm as in the previous simulations). The results can be seen in Fig. 3-13.
This figure demonstrates that the increase in error does roughly increase proportionally to the gimbal rate squared. For the 3 cm lever arm, with angular rates less than
800/s, the increase in error is less than 10% and it would be probably be acceptable to
not implement the lever arm error states. With increasing gimbal rate, the increase
in errors due to the sub-optimal gains increases drastically and it becomes infeasible
to implement a filter that does not accurately account for the size effect.
As was shown in Figs. 3-12 and 3-13, the size effect can have a substantial impact
on the accuracy of the IMU and in many cases will need to be accounted for with
extra error states. But it is also very possible to eliminate the need for these extra
states by designing the IMU to be compact enough so the lever arms do not exceed
about 5 cm, and keeping the slew rate low. The rate of the slew in no way effects the
accuracy of the accelerometers, so keeping this low will add a great benefit. This is
because as was shown in the development of the sensitivity matrices in Section 2.5,
the sensitivity integrals are all of the special orthogonal group in three dimensions
SO(3), and therefore the speed of traversing the rotation does not effect the value
of the integrals. As long as the slew itself is fixed, the overall accuracy will remain
constant as well. So any scheme for carouseling using a low slew rate with small lever
CHAPTER 3. GYROCOMPASSING ANALYSIS
160%
-----------.--------------------
----
------------ ----
140%
---------- ---- --- - - ------
------------I------ ------- -- --- -- - - - -- -- --- ----
120%
I
I
100%
----------------------
80%
-------------. . .. . n. . . . .
. .
. . .
. . .
. .
. .
. . .-
. -.
. .
. .
------ -----------
---------. ..-
. .
,---
. .
. . ---- - - - - - - - - - -- - - - - - - - - - - -.
---------
%
60%
----
----
40%
20%
4--------- ------------
--
-,- ---
-
-
. .
0
20
40
60
,
. -.. -..
.
. . - -.
.
.
80
.
. -
. .
.
100
---
. . - . - .
. - . - .
120
.
,
140
. . ..-- - . . ..--
160
Outer Gimbal Angular Rate (degls)
Figure 3-13: Increase in Azimuth Covariance when implementing Sub-optimal gains
for varying outer gimbal angular rates The sub-optimal gains do not include the lever arm
states and do not as accurately model the system. The size effect increases roughly proportionally
to the gimbal rate squared
arms should safely be able to navigate without implementation of extra error states.
It should also be noted that although adding error states does increase computation,
depending on the application, there may be enough computing power where this is
not an issue and the savings in accuracy is of more critical importance. As with
all navigation problems, the trade offs between aspects such as mass and accuracy
requirements are specific to different situations and can vary greatly.
Chapter 4
Conclusions
4.1
Carouseling Effect on Azimuth Accuracy
As was seen in Fig. 3-2, implementing carouseling provides significantly better performance in azimuth error than the standard 2-position method for the gyrocompassing
problem when implementing the current Draper MEMS IMU. There is a 50% drop in
the covariance, a large portion of that reduction due to the almost total elimination
of gyroscope bias. With such marked improvements, this makes implementation of
carouseling with use of the current MEMS IMU a highly recommended option for
gyrcompassing.
However, there is an increased sensitivity to measurement noise when implementing carouseling that could be significant if implemented in a noisy environment. By
the same token, if the measurement noise can be reduced, the benefits of carouseling
become even more pronounced. Because of the decreased sensitivity to gyroscope
bias, if the gyroscope bias of the IMU being used is lowered, as with the improved
MEMS model, the comparative benefit of using the carouseling scheme over time is
reduced. However, for applications where performance within the first 5 minutes is
critical, the carouseling method provides considerable benefits when used in conjunction with the improved MEMS IMU.
CHAPTER 4. CONCLUSIONS
For the gyrocompassing application using current MEMS technology, implementation of the carouseling scheme would be highly beneficial to overall azimuth accuracy.
Using carouseling allows MEMS devices with significantly lower mass, cost and power
requirements, to be implemented in high-accuracy applications. If the measurenent
noise is not large and especially if the MEMS instrument selected has a large gyroscope bias or the initial attitude uncertainty is large, the carouseling will provide
even more benefit. Additionally, in situations where the accelerometers are offset by
less than 5 cm, it may not be necessary to add states to the filter as the size effect
will be negligible. This could be an added benefit, but even when adding the error
states to the filter, the benefits that carouseling provides in lower mass and power
requirements could greatly outweigh the burden of extra computation.
4.2
Size Effect
In the particular gyrocompassing problem laid out in this thesis, the size effect is not
a significant complication and the lever arm error states could possibly be eliminated.
However, this simulated scenario has small lever arms and small uncertainties which
might not be realistic to assume for other applications. The size effect was proved to
be related to the offset of the accelerometers (the lever arm) and the angular rate of
the slew squared. It was seen in this simulation, that for lever arms of 5 cm or more,
the size effect becomes substantial and for a navigator to be accurate, the lever arm
states must be implemented. Additionally, if the angular speed of the outer gimbal is
larger than 80 degrees per second, the added velocity from the uncompensated lever
arms will also cause a large error and it is necessary for the states to be incorporated
into the filter. If these lever arm states are accounted for, the same accuracy can be
attained, but more computation will be required as more states must be integrated
in the filter.
4.3. OTHER CONSIDERATIONS
4.3
Other Considerations
Other factors besides accuracy will determine the feasibility of implementing a carouscling scheme in the gyrocompassing application. It is important to remember that the
IMUs must be physically slewed and this rotation must be performed and controlled
by some mechanism. This mechanism will add some volume and mass (although still
have lower total mass than a typical high-accuracy IMU) but also must be accurate
enough to perform the slewing effectively. It was shown however, that the slew can
most likely be Earth referenced as this does not drastically change the results of
the inertially referenced slew, which can make the controller scheme less difficult to
implement. Overall, the added complications of the carouseling scheme are not so
severely intensive that they would preclude the implementation of the scheme for use
in real-life applications. With the expected benefit of a 50% increase in accuracy, the
added mass and designing of the gimballed control scheme can easily be justified.
4.4
Future Work
A major area of future work would be to simulate the carouseling maneuver for
trajectories and applications other than simply gyrocompassing.
There are many
other high-accuracy applications that could benefit greatly from the reduced mass and
cost of MEMS IMUs, both in space and for terrestrial navigation. Other applications
could implement other types of updates including GPS and might be concerned more
with velocity error more than azimuth error as was the case for the gyrocompassing
application. The specific challenges of different applications could affect the benefit
of carouseling and must be investigated.
Additionally, the effect of having more
uncertainty in the slewing trajectory itself could also indicate how sensitve the results
are to the preciseness of the slewing motion.
A next logical step in this research that would also be of importance is to design
actual hardware for use in real-world experiments to validate the results of the size
88
CHAPTER 4. CONCLUSIONS
effect simulations. With real-world data the true value of incorporating the lever
arm filter states can be ascertained and the ideal filter model for the gyrocompassing
application can possibly be determined. This thesis has been focused on simulations
for its results which must be confirmed by experimentation.
Appendix A
MEMS IMU Error Specifications
0.3
0.06
33
scale factor(in-run)
misalignment
nonorthogonality
random walk
3.3
0.3
170
170
6.25
6.25
0.05
units
dph
dph
ppm
8.25
6.25
6.25
0.01
ppm
arcsec
arcsec
deg/ hr
bias(turn-on)
bias(in-run)
scale factor(turn-on)
scale factor(in-run)
misalignment
nonorthogonality
random walk
300
240
50
50
6.25
6.25
0.009
300
240
50
50
6.25
6.25
0.009
p-g
P-g
ppm
ppm
arcsec
arcsec
mps/ /hr
900
900
sec
11 Current
Gyro.
Gyro.
Gyro.
Gyro.
Gyro.
Gyro.
Gyro.
Accel.
Accel.
Accel.
Accel.
Accel.
Accel.
Accel.
bias(turn-on)
bias(in-run)
scale factor(turn-on)
Time constant
Improved
Table A.I: Design Specifications for the Current and Improved Model MEMS IMUs
APPENDIX A. MEMS IMU ERROR SPECIFICATIONS
THIS PAGE INTENTIONALLY LEFT BLANK
Bibliography
[1] Theresia C. Becker. Approaches to optimal inertial instrument calibration using
slewing. Master's thesis, Massachusetts Institute of Technology, June 2005.
121 Esmat Bekir, editor. Introduction to Modern Navigation Systems. World Scientific Publishing, Singapore, 2007.
[31 Brown and Hwang. Introduction to Random Signals and Applied Kalman Filtering. John Wiley & Sons, New York, 3rd edition, 1997.
[4] Charles Broxmeyer. Inertial Navigation Systems. McGraw-Hill, New York, 1964.
[51 J. Connelly and A. Kourepenis. A micromechanical inertial measurement unit for
tactical applications. Technical Report AIAA-2002-5050, Charles Stark Draper
Laboratory, 2002.
[6] Charles S. Draper, Walter Wrigley, and John Hovorka, editors. Inertial Guidance.
International Series on Aeronautical Sciences and Space Flight. The Macmillan
Company, New York, 1960.
[71 Arthur Gelb, editor. Applied Optimal Estimation. The M.I.T. Press, Cambridge,
MA, 1974.
18] Donald E. Kirk. Optimal Control Theory. Dover Publications, Mineola, NY,
2004.
BIBLIOGRAPHY
191 Anthony Lawrence. Modern Inertial Technology: Navigation, Guidance and Control. Mechanical Engineering Series. Springer, New York, second edition, 1998.
[10] C.F. O'Donnell, editor. Inertial Navigation: Analysis and Design. McGraw-Hill,
New York, 1964.
111
George R. Pitman Jr., editor. Inertial Guidance. University of California Engineering and Physical Sciences Extension Series. John Wiley & Sons, New York,
1962.
1121 A.L. Rawlings. The Theory of the Gyroscopic Compass. Macmillan, New York,
1944.
113] Robert M. Rogers, editor. Applied Mathematics in Integrated Navigation Systems. AIAA Educational Series. AIAA, Inc., Reston, VA, second edition, 2003.
[141 Laurie Tetley and David Calcutt. Electronic Navigation Systems. ButterworthHeinemann, Boston, 2001.
I151 Walter Wrigley, Walter M. Hollister, and William G. Denhard. Gyroscopic Theory, Design, and Instrumentation. The M.I.T. Press, Cambridge, MA, 1969.
1161 Paul Zarchan and Howard Musoff, editors. Fundamentals of Kalman Filtering.
AIAA Progress in Astronautics and Aeronautics. AIAA, Inc., Reston, VA, 2000.
List of Figures
12
1-1
SEM photo of MEMS Tuning Fork Gyroscope . ............
1-2
SEM photo of MEMS accelerometer . .................
2-1
3-D Trace of a 2:1 Baseball Stitch Slew . ................
2-2
Plots of gimbal angles and angular rates of a 60 second Baseball Stitch
28
Contributions of different error sources to the overall rotation error for
.
the Baseball Stitch Slew ...........................
2-4
. .. .
. ..................
Plots of gimbal angles and angular velocities for Baseball Stitch Slew
.
34
. ................
Contributions of different gyroscope error sources to the change in ve36
locity error for the Baseball Stitch Slew with Reversal .........
2-9
33
Contributions of different error sources to the overall rotation error for
the Baseball Stitch Slew with Reversal
2-8
31
32
. .......
with Reversal ..................................
2-7
.
The total contribution to rotation error from gyroscope errors for a
non-rotating body . . ..................
2-6
30
The total contribution to rotation error from gyroscope errors for the
Baseball Stitch Slew
2-5
12
17
.....................................
Slew ......
2-3
.
Plots of gimbal angles and angular velocities for Baseball Stitch Slew
with Reversal Extended
...................
......
37
LIST OF FIGURES
2-10 Contributions of different error sources to overall rotation error for
Baseball Stitch with Reversal Extended . ................
38
2-11 Contributions of different gyroscope error sources to velocity error for
Baseball Stitch with Reversal Extended . ..........
...
.
40
2-12 Total Contribution to angle error from gyroscope errors for Baseball
Stitch Slew with Reversal Extended . ..................
41
3-1
Current MEMS IMU Azimuth Covariance for Baseball Stitch Slew . .
62
3-2
Current MEMS IMU Azimuth Covariance Comparison for Baseball
Stitch Slew and 2-Position Gyrocompassing
3-3
. .............
63
Current MEMS IMU Azimuth Covariance Comparison for Baseball
Stitch Slew and ±3600 turns ...............
3-4
......
Current MEMIS IMU Azimuth Covariance for Baseball Stitch Slew referenced in two different coordinate frames
3-5
. ..............
...................
..
67
Improved MEMS IMU Azimuth Covariance Comparison for Baseball
Stitch Slew and 2-Position Gyrocompassing
3-7
66
Current MEMS IMU Azimuth Covariance Comparison with zupt update performed every 5 seconds
3-6
64
. .............
68
Azimuth Covariance Error after 10 minutes for Varying Measurement
Noise ......
.............
................
74
3-8
Selected Azimuth Covariance Error Contributions over time
3-9
Lever Arm Error
...................
....
.
.........
75
78
3-10 Navigated solution compared to Azimuth Covariance when 3 cm lever
arms are uncompensated ....
.....
...........
.....
. .
80
3-11 Navigated solution compared to Azimuth Covariance when 10 cm lever
arms are uncompensated ...................
......
81
3-12 Increase in Azimuth Covariance when implementing Sub-optimal gains
for varying lever arm lengths ...............
.......
82
LIST OF FIGURES
95
3-13 Increase in Azimuth Covariance when implementing Sub-optimal gains
for varying outer gimbal angular rates . .................
84
96
LIST OF FIGURES
THIS PAGE INTENTIONALLY LEFT BLANK
List of Tables
...........................
.
45
.
60
2.1
Navigation Filter States
3.1
Initial Error States ...............................
3.2
Error Budget Comparison of Azimuth Error contributions in degrees
for Baseball Stitch Slew and 2-position Gyrocompassing ........
72
A.1 Design Specifications for the Current and Improved Model MEMS IMUs 89