AN AUTOMATED TOOL FOR FORMULATING
LINEAR COVARIANCE ANALYSIS PROBLEMS
by
Andrew W. Lewin
S.B., Aeronautics and Astronautics
Massachusetts Institute of Technology, 1991
Submitted to the Department of
Aeronautics and Astronautics in Partial
Fulfillment of the Requirements for the
Degree of
Master of Science in
Aeronautics and Astronautics
at the
Massachusetts Institute of Technology
September 1992
© The Charles Stark Draner Laboratory, Inc. 1992
Signature of Author
Department of Aeronautics And Astronautics
August 19, 1992
Approved by
John J. Turkovich
Technical Supervisor, Charles Stark Draper Laboratory
Certified by
Professor Joseph F. Shea
Thesis SuDervisor
Accepted by
Prot~sor Harold Y. Wachman
Chairman, Department Graduate Committee
MASSACHUSETTS INSTITUF
OF TECHNnI f"'
rSE P 22 1992
A
. .. ..
AN AUTOMATED TOOL FOR FORMULATING
LINEAR COVARIANCE ANALYSIS PROBLEMS
by
Andrew W. Lewin
Submitted to the Department of Aeronautics and Astronautics
on August 21, 1992 in partial fulfillment of the requirements
for the Degree of Master of Science in Aeronautics and Astronautics
ABSTRACT
This thesis describes the design and development of an automated tool for
developing linear covariance analysis simulations. A generic specification
language was developed for navigation linear covariance analysis problems.
This language served as the basis for the user interface of the automated
tool. The ability to convert linear covariance analysis specifications to Ada
code was demonstrated with the aid of ASTER, a software development
tool that generates source code from functional specifications.
Although work is ongoing, the linear covariance analysis tool has
demonstrated that using a design language as the basis of the computer code
can save valuable development time, while increasing reliability and
maintainability of the resulting code. Automation also allows the engineer
to take advantage of inherent properties of the analysis equations to improve
computational efficiency.
Thesis Supervisor: Mr. John J. Turkovich
Title: Staff Engineer, The Charles Stark Draper Laboratory, Inc.
Thesis Supervisor: Professor Joseph F. Shea
Title: Adjunct Professor of Aeronautics and Astronautics
Acknowledgements
Many people at Draper Lab have made my stay a very enjoyable one. I want to offer my
thanks to Milt Adams for his stimulating conversation and interest in my work despite his
many duties. I also thank Eva Moody for her assistance in everyday survival at the Lab.
She makes everyone's job much easier. Thanks also to Professor Shea, who took me on
and became an insightful advisor with many excellent suggestions, and to Ken Spratlin and
Stan Shepperd who taught me linear covariance analysis and who provided me much
guidance in the design of the automated tool.
I appreciate the long hours put in on my behalf by Dave Chamberland and Rick Brenner. I
offer a special thanks to Dave for not killing me, despite the fact that I was consistently
interrupting his work with bug reports or requests for new features. I must offer my
greatest thanks to John Turkovich, who brought me to the Lab three years ago and has
given me exciting projects and a nurturing environment ever since.
I thank my parents for helping me to get to this point, and for dealing with all the
difficulties it caused. Most of all, I want to thank my fiancee Ellen, whose love and
support enabled me to successfully complete this thesis.
This thesis was prepared at the Charles Stark Draper Laboratory by company sponsored
research (CSR-103), internal research and development (IR&D 462), and the MIT Space
Grant Program.
Publication of this thesis does not constitute approval by Draper or MIT of the findings or
conclusions contained herein. It is published for the exchange and stimulation of ideas.
I hereby assign my copyright to The Charles Stark Draper Laboratory, Inc., Cambridge,
Massachusetts.
/1
Andrew W. Lewin
Permission is hereby granted by The Charles Stark Draper Laboratory, Inc., to the
Massachusetts Institute of Technology to reproduce any or all of this thesis
Table of Contents
A bstract ............................................................................
Acknowledgem ents.................................................
Chapter 1: Introduction............................................
..................................................
......................................................
3
5
.................................................. 9
Chapter 2: Linear Covariance Analysis...........................................
13
2.1 Linear Covariance Analysis Elements ......................................................... 13
2.1.1 Celestial Bodies ..................................................................
14
2.1.2 Platform s ........................................... ......................................... 14
2.1.3 Sensors ............................................................................................. 14
2.2 Mission Specification Procedure.........................................16
2.1.1 Specifying the M ission.. ............................................................ 16
2.2.2 Specifying the Measurements .....................................
..... 17
2.2.3 Simulation Parameters, Initial Conditions, and Output Modes........ 17
2.2.4 Output Evaluation ................................................. 18
18
.........
2.3 Performing Linear Covariance Analysis...............................
2.3.1 Propagating the Covariance Matrix and the Platform and
the Platform and Celestial Body State Vectors.............................20
....... 21
2.3.2 Measurement Updating..................................
2.4 Applying Linear Covariance Analysis to Navigation Problems..................22
2.5 Improving the computational Performance of the Algorithm......................24
2.5.1 Computational Properties of the Linear Covariance
........ 24
Analysis Algorithm....................................
2.5.2 Improving Computational Efficiency...............................26
Chapter 3: Linear Covariance Analysis Natural Design Language..............
3.1 Linear Covariance Analysis Elements .......................................... 34
....................... 38
3.2 Mission Specification...............................................
33
Chapter 4: Automated Linear Covariance Analysis Tool..................................
43
4.1 Mission Specification Language Implementation...................44
4.1.1 Defining Linear Covariance Analysis Elements...........................45
4.1.1.1 Celestial Bodies.................................................................45
4.1.1.2 Platforms ............................................
...... 45
4.1.1.3 Sensors ................................................... 46
4.1.2 Specifying the Mission..............................
............
55
4.1.3 Analysis Aides..................................................................................63
4.2 A STER .......................................................... ............................................ 63
4.3 Internal Data Representation.......................................... 66
4.3.1 Common Classes..............................................
................. 68
4.3.2 Linear Covariance Analysis Elements ............................................. 72
4.3.3 Mission Specification Classes................................
..... 78
4.3.4 Output Forms Classes............................................79
4.4 Code Generation.................................................. ...................................... 79
4.4.1 Initialization ..............................................
........ 84
4.4.2 Main Simulation Loop .........................................
...... 88
Chapter 5: Results and Conclusions...........................................................................93
Chapter 6: The Next Step ..........................................................................................
97
References ............................................................................................................
99
Chapter 1: Introduction
Improvements in computer hardware over the last decade have produced orders of
magnitude improvements in both speed and memory capabilities. These advances have led
to revolutionary changes in software as well. Software began with machine languages
which provided only the most basic commands. Nearly trivial processes required an
engineer to write many lines of code. As a result, assembly languages were developed.
These were somewhat easier to use than machine language, and the use of words or
abbreviations of words was certainly helpful. However, assembly language commands
were still prohibitively primitive. Therefore, higher level languages were developed that
were much easier to use. The commands were actual english words, and the command
structure resembled sentences.
At the same time, software systems have been growing exponentially in size and
complexity. Advances in microprocessor speed and memory capabilities have lead system
designers to ever increasing demand for software functionality. The Apollo 11 astronauts
successfully landed on the moon guided by a computer with only 38,000 words of
memory. Now, 23 years later, many aerospace software systems require several millions
of lines of code.
The purpose for creating high-level programming tools is to reduce the time necessary to
design, build, and test, and maintain a software system. As software systems become
increasingly complex, design and coding capabilities must evolve accordingly. These gains
are accomplished at the expense of memory utilization, execution speed, and number of
lines of machine-language code. However, the lower efficiencies are far outweighed by the
cost and schedule benefits of the reduced development time.
Research and commercial efforts are now developing the next revolution in software
development technology. Systems are being developed that raise software specification to
design specifications rather than computer code. This new generation of compiler offers
substantial improvements in development time for computer systems in three ways.
First, the development time is dramatically reduced. The engineer does not need to spend
time either writing code himself or describing the design to a computer programmer who
probably does not understand the design language the engineer naturally uses to describe
the design or the engineering aspects of the design.
Second, the software development process is made more reliable. An automated system,
once matured, generates code nearly error-free. Although the use of automated software
generators does not eliminate the need for testing, it can greatly decrease the number of
errors present in the system, and thereby reduce the amount of time spent debugging.
Furthermore, enabling an engineer to specify the design in familiar terms rather than the
sometimes complex workings of a programming language reduces the amount of work
required by the engineer, which reduces the opportunity for errors to be introduced into the
system.
Third, automating the software development process makes the software system much
easier to maintain. Early systems were small enough that maintenance was of little
concern. However, as software systems have become more complex, maintenance has
become very expensive. By specifying a design in terms of a design language rather than
computer code, changes can be made much more easily. The person responsible for
making the change does not have to decipher the code. Rather, he or she just has to look at
the design and make the necessary modifications.
One application which can particularly benefit from automation is linear covariance
analysis. The purpose of linear covariance analysis is to evaluate the performance of
dynamic systems. One common application of linear covariance analysis is error modelling
for guidance and navigation systems. Guidance and navigation accuracy is a major design
concern for both military and civilian applications. Linear covariance analysis enables an
engineer to evaluate how a particular set of instruments and navigation aids affect the
accumulation of error throughout a mission. Therefore, linear covariance analysis plays a
critical role in the design tradeoff between additional or higher performance instruments and
the additional cost associated with these items.
This thesis details the development of a linear covariance analysis software development
tool. Chapter 2 describes the linear covariance analysis algorithm and its application to
navigation systems. Chapter 3 describes the development of a design language for linear
covariance analysis problems. This process was necessary because the navigation system
analysis community has not yet developed a design language standard. Chapter 4 describes
the details of the automated linear covariance analysis tool that was developed. Chapter 5
evaluates the system that was developed, and Chapter 6 identifies desired improvements to
the tool to extend its functionality and better serve its users.
Chapter 2: Linear Covariance Analysis
Linear covariance analysis is a technique for analyzing error performance of dynamic
systems. The error is described relative to given nominal performance. The linear
covariance analysis begins by describing the elements that comprise a mission. These
elements are then combined to create a particular mission. The linear covariance analysis
then performs a simulation of the specified mission and determines how errors propagate
through the system during the course of the mission.
2.1 Linear Covariance Analysis Elements
Analysis of a mission must consider three basic elements: celestial bodies, platforms, and
sensors. A celestial body supports ground-based beacons and determines the equations of
motion of orbiters and maneuvering vehicles. Celestial bodies include the sun, planets, and
moons. Platforms are bodies which carry sensors. Platforms include navigational aids
such as beacons and satellites as well as maneuvering bodies such as rovers and vehicles.
Finally, sensors are the instruments placed on board the platforms which provide
navigation information through external measurements.
The core of any design process is modelling the physical process taking place. For linear
covariance analysis, this requires the development of appropriate models for the celestial
bodies, platforms, and sensors that comprise the mission to be evaluated. In order to
design an automated tool for building linear covariance analysis simulations, it is necessary
to determine which physical properties must be modelled. This involves identifying the
relevant properties of the building blocks of a mission.
2.1.1 Celestial Bodies
Celestial bodies form the basis of any mission, since the purpose of the mission is to guide
and navigate vehicles about these objects. The linear covariance analysis algorithm requires
information about the bodies that determines how they interact with the vehicles and
navigational aides involved in the mission. Obviously, the mass is a critical parameter,
since it governs the gravitational motion of bodies in the vicinity of the body. However,
the distribution of the mass also plays an important role in determining the motion of
orbiting satellites, both natural and man-made. Other important quantities include the
radius and atmosphere height, which determine occultations of an orbiting platform by the
body. For problems involving multiple celestial bodies, the orbital elements of the body
are required in order to evaluate the relative position and velocity of the bodies. Similarly,
the rotation rate and rotation axis must be known to calculate the orientation of the celestial
body relative to inertial space. This rotation is especially important for determining the
location of surface-based platforms such as beacons and rovers.
2.1.2 Platforms
By definition, a platform is any entity which can carry sensors. For typical aerospace
linear covariance analysis problems, there are four main platform types: vehicles, rovers,
orbiters, and beacons. A maneuvering vehicle is an object capable of powered flight, and
its motion is given by a predefined trajectory. It is described only by its trajectory. A rover
is also capable of independent movement, but it is constrained to traverse a planetary
surface. Like the maneuvering vehicle, a rover is also modelled by a trajectory. An orbiter
moves under gravitational influences only with no powered maneuvers beyond simple
stationkeeping. The motion of an orbiter is described by its orbital elements. Finally,
beacons are navigational aids which are placed on the surface of a celestial body and have
no means of independent locomotion. In addition to its location, a beacon model includes
an elevation angle above the horizon requirement for communication with orbiting
platforms. This parameter models surface irregularities or atmospheric effects such as
refraction.
2.1.3 Sensors
Since the primary purpose of linear covariance analysis is to predict error performance of a
guidance or navigation system, it is not surprising that the sensor models are the most
complicated. Instrument or sensor modelling involves the specification of four primary
elements: error terms, measurement sensitivity, measurement variance, and constraints.
The sensor error terms provide a mathematical model of the physical characteristics of an
instrument. Error term dynamics are expressed as differential equations. The terms of the
differential equations are functions of the modelled errors present in the system as well as
position and velocity errors of the host platform. However, linear covariance analysis
requires that the sensor error dynamics be expressed as a series of linear first-order
differential equations. Therefore, non-linear terms must be linearized or ignored, and
higher-order differential equations must be reduced to a system of linear first-order
differential equations.
Although the error terms can provide a very accurate model of the sensor performance, no
model can ever be perfect. As a result, each differential equation usually includes a zero
mean white noise error term. The purpose of this term is to capture the effects of known
modelling deficiencies as well as unknown error sources, thereby preventing the filter from
becoming overly optimistic.
The second sensor modelling parameter is the measurement sensitivity, b. The
measurement sensitivity describes how the measurement would vary as a function of small
changes in the elements of the state vector. Mathematically,
b= ()-
where q is the sensor measurement (usually scalar) and e is the full error state vector.
Measurement variance, the third sensor parameter, is used to identify the accuracy of a
sensor measurement. The measurement variance is expressed as a single equation. In
general, this equation is comprised of two parts. The first part is a sensor noise term. The
noise term is used to model the accuracy of the instrument itself. However, this quantity
may then be modified by a term relating to the conditions of the measurement to get the
measurement sensitivity. For example, a sensor measurement may degrade with the
measurement range or perhaps even the temperature of the sensor. By combining these
two factors, an equation describing the accuracy of the measurement, the measurement
variance, is determined.
Finally, constraints are used to identify whether a measurement can actually be performed.
One common constraint for sensors is that they be able to "see" the target sensor. For
example, radio communication is impossible through a planet, so a two-way range
measurement using radio waves requires a line of sight constraint. Another constraint is a
maximum operating range. If two sensors are too far apart, they may not have enough
power to successfully take a measurement.
2.2 Mission Specification Procedure
Once the celestial body, sensor, and platform characteristics have been determined, the
specification of a linear covariance analysis problem is broken down into four major steps.
First, the celestial bodies, platforms, and sensors in the problem are specified. The next
step is to select the measurements to be taken and their operating characteristics. Third, the
simulation parameters, initial conditions, and desired outputs are determined. Finally, the
output is evaluated and the process is repeated.
2.2.1. Specifying the Mission
The first step of linear covariance analysis is to specify the mission to be undertaken. This
involves selecting the celestial bodies in the simulation. This may include sun, planets, or
moons that affect motion or bodies such as stars which are used for navigational purposes.
Then, the the vehicles, orbiters, rovers, and navigational aids are specified. For vehicles
and rovers this involves providing the trajectory. Orbiters must be given an orbit about one
of the bodies, and navigational aids such as beacons are placed at a fixed location on a
planet or moon.
Once the platforms have been identified, the instruments aboard each platform are
specified. First, the engineer selects the sensors to be placed on board each platform.
Then, he or she specifies the modelling fidelity for each sensor. For example, an sensor
specification may include as many as 50 or 100 error terms. However, they may not all be
necessary, so the engineer specifies which error terms to use. The engineer can also
declare an error term to be a consider state. Consider states are included in the covariance
matrix, but they do not benefit from measurement update information. This is the
equivalent of examining an error term which is not included in the actual filter. The state is
not being estimated; it is only propagated.
2.22 Specifying the Measurements
Once the mission has been specified, the measurements taken by each instrument are
determined. Sensors such as an inertial measurement unit (IMU) do not require this step
because the measurement being taken does not depend upon other platforms in the mission.
Placing the IMU on a platform is sufficient to determine what measurement to take.
However, most instruments involve objects other than the host platform when taking a
measurement. For example, an altimeter measures the distance from the surface of a
celestial body. For the altimeter, specifying the measurement to be taken means identifying
the celestial body to which the distance is measured.
In addition to determining the measurements to be taken, an engineer must decide how
frequently the measurements are taken. Measurement frequency plays an important role in
navigational errors. When more measurements are taken, the errors will naturally be
reduced; however, each additional measurement adds to the amount of throughput required
of the avionics system. Furthermore, each additional measurement has diminishing
returns. Also, sensors may be turned on or off at specific times during a mission. On the
Apollo missions, unused sensors had to be turned off because they overloaded the tiny
flight computer. Another use of this feature is to model fault tolerance of the navigation
system. By turning off sensors at specific times, an engineer can simulate the failure of
that sensor.
2.23 Simulation Parameters, Initial Conditions, and Output Modes
Once the mission has been determined, several parameters involved in the linear covariance
analysis must be specified. First, the initial covariance matrix must be given to provide an
initial condition for the integration and propagation of the error covariance. Since the
covariance matrix is often very large, this can be a time-consuming process; however, the
covariance matrix is symmetric.
Another useful simulation parameter is measurement underweighting. Measurement
underweighting provides a mechanism for not taking full advantage of the information
provided by a measurement. This is particularly useful when a reduced order system is
being used or in a region where the system is particularly non-linear. If the measurement
underweighting is not used, the algorithm can become overly optimistic.
The final step of the mission specification process involves defining parameters that affect
the simulation. These include the starting and ending times as well as the time step. The
time step plays an important role in the quality of the simulation. A small time step yields
an accurate simulation, but it requires many calculations which can cause the analysis
process to take a long time to perform. For some missions, it may be desirable to change
the time step during the mission. The time step is chosen to be very small during phases
where the dynamics are undergoing rapid change and longer during relatively quiet periods.
2.2.4. Output Evaluation
Once the mission has been defined and the linear covariance simulation has been
performed, the true analysis begins. Linear covariance analysis is simply a tool that
enables an engineer to simulate the error propagation of a navigation system. With the data
gleaned from the results of the linear covariance analysis algorithm, an engineer must
determine whether or not the current mission scenario satisfies the given requirements. If
not, he or she must examine the data to determine what caused the mission to not meet the
requirements. Then the engineer uses this information to modify the mission and repeat the
entire process. In order to make this process effective, the engineer must have a variety of
ways to examine the output of the linear covariance analysis. Selecting the desired outputs
and the method of viewing them constitutes the final step in the linear covariance analysis
process.
2.3. Performing Linear Covariance Analysis
The purpose of linear covariance analysis is to provide a model for determining the error
performance of a navigation system. The basis of the linear covariance analysis algorithm
is the assumption that instrument errors and vehicle motion can be modelled as a set of
linear differential equations driven by zero mean white noise:
S= Fe+Gu
(1)
where e is the vector of state errors and u is a vector of white noise. The matrix F, or state
dynamics matrix, describes the linearized dynamics of the platforms and sensors involved
in the analysis. The matrix G, or noise distribution matrix, models the impact of the noise
on each of the error terms. If the noise sources are independent for each platform and
sensor error term, G will be diagonal or block diagonal, depending upon how the noise
affects vector-valued errors. In most cases, G will be an identity matrix.
Linear covariance analysis is not a deterministic simulation; rather, it is a statistical
simulation. The primary output of the algorithm is the covariance matrix. The covariance
matrix represents the state error variances and covariances, not the errors for an actual
mission. Mathematically, the covariance matrix is given by the equation:
E = expected-value(e eT) = oxy
Oay
The diagonal terms of the covariance matrix are called the variances. The square root of the
variance is the standard deviation. The standard deviation is a useful measure because it has
a simple physical interpretation. For an actual mission, the error can be expected to be less
than the standard deviation 68.27% of the time, less than double the standard deviation
95% of the time, and less than three standard deviations 99.73% of the time.
The off diagonal terms determine the correlation between two states. The correlation
coefficient is given by:
xy
xy
Oxay
(-1< xy
1)
ýxy = 0 means that states x and y are not statistically correlated, whereas
that they are 100% correlated.
,xy = ±1
means
The linear covariance analysis algorithm consists of two primary tasks: 1)propagating the
covariance matrix and the position and velocities of the platforms and celestial bodies in the
mission, and 2) updating the covariance matrix when measurements are taken. Before this
process can begin, the covariance matrix and state vector must be initialized. These
quantities represent the cumulative effects of everything that has happened prior to initiation
of the simulation. Therefore, the linear covariance algorithm describes the accumulated
error performance for all time up to the end of the simulation, not just the short window
actually simulated.
2.3.1 Propagating the Covariance Matrix and the Platform and Celestial Body State
Vectors
Once the simulation begins, the first major step is to update the error covariance matrix and
the platform states. For a linear time-varying differential equation driven by white noise
such as equation (1), the covariance matrix dynamics are given by the Lyapunov
differential equation:
E=FE +EFT +Q
(2)
where E is the covariance matrix, F captures the dynamics of the error state vector e, and Q
is the noise intensity or process noise matrix. The process noise matrix is the product of
the noise distribution matrix G and the noise vector u.
The method of determining platform states depends upon the type of platform. Beacons are
fixed to a planet, so their position and velocity can be determined from the translation and
rotation of the planet. Orbiters are incapable of self-propulsion, so their motion is
calculated by integrating the gravitational acceleration vector to get the velocity, and the
velocity vector to get the position. Rovers and maneuvering vehicles, on the other hand,
are capable of providing forces other than gravitational attraction. Therefore, the position
and velocity vectors for these platforms as a function of time are given as inputs to the
algorithm.
When a mission specification contains more than one celestial body, all celestial body
position and velocity vectors must also be maintained. This is accomplished by simply
integrating their state vectors in the same way as for orbiters. This process is unnecessary
when only one celestial body is included in a mission because that body is used as the
inertial center of the computational system.
Accuracy is extremely important for the systems being evaluated by the linear covariance
analysis algorithm. Therefore, accuracy of the simulation is equally important. Experience
has shown that a 4th-order Runge-Kutta integration scheme is sufficiently accurate for all
state and covariance integrations. During periods such as vehicle coasting, where changes
are not happening as rapidly, lower-order integration schemes may also be acceptable.
20
2.3.2 Measurement Updating
The second step is to process all the measurements. For simplicity, the updating has been
restricted to scalar updates, in other words, the measurements are processed sequentially,
even when several measurements have been scheduled for the same time. The
measurements still occur at the correct time step, but they use a covariance matrix which
has already been updated by previously processed measurements from the same time step.
To begin, the algorithm determines whether a measurement is scheduled to take place. If
so, the algorithm must then ensure that the sensor is on and all operating constraints are
met. If all of these conditions are met, a measurement is performed. For each
measurement, the covariance matrix is updated to reflect the new information. This is
accomplished by the following series of equations:
First, the temporary vector U and the measurement residual variance,
calculated:
esidual are
U=Eb
residual = bT U
The residual is a scalar measure of the states with respect to the scalar measurement. If the
residual is small, the states are well-known. Then, the measurement weighting vector is
formed:
(1 + Iuw) 0residual + (eas
where [3uw is the measurement underweighting and a•eas is the measurement variance.
The weighting vector identifies the relative importance of the various error terms in
updating the covariance matrix. If an error term is only a consider state, it is not used to
update the state vector, so the appropriate element of the weighting vector is set to zero.
The array C identifies whether each error term is a consider state.
If C(i) is true, Q(i) is set to 0
The update proceeds with the equation:
U
W=
residual +
meas
If consider states or underweighting are used, w and w will not be in the same direction.
For optimal filters, (no underweighting or consider states) the covariance matrix is updated
using the equation:
Enew = (I - wbT) E
For sub-optimal filters (those with underweighting and/or consider states), the Joseph form
is used:
T
Enew = (I - wbT) E (I - wbT)T + 4easW'
Here, it can be seen that by introducing consider states or underweighting, Wis smaller
than it otherwise would be, so I - wbT is larger, which makes E larger, indicating that the
vehicle has more uncertainty than it would without the sub-optimalities. Finally, the
symmetry of the covariance matrix is retained through the equation
E=E+ET
2
Without this equation, the covariance matrix can lose symmetry due to computer round-off
errors.
2.4 Applying Linear Covariance Analysis to Navigation Problems
Although the terminology used to describe the linear covariance algorithm in Section 2.3 is
specific to navigational linear covariance analysis problems, the technique is not. The
linear covariance analysis algorithm must be applied to the objects described in Section 2-1.
The first part of linear covariance analysis is propagating the covariance matrix and state
vectors. The covariance matrix propagation involves two inputs, the dynamics matrix F
and the noise intensity matrix Q. The dynamics matrix is created by creating a system of
linear equations that describe the platform state error dynamics and the sensor error terms.
Celestial body state error dynamics are not modelled. These linear equations contains terms
based on error terms in the linear covariance analysis as well as the white noise associated
with each of the sensors. The terms of the differential equations based upon current errors
are expressed in matrix form. This matrix is the dynamics matrix F. The noise terms are
also converted to a matrix form, creating the noise intensity matrix Q.
Some of the differential equations include terms which are a function of the position or
velocity of platforms or celestial bodies in the mission. Therefore, these state vectors must
be propagated in time. The propagation is defined by simple integration of acceleration to
get velocity and velocity to get position. The propagation requires an initial condition to be
provided for both the position and velocity of each object.
The second step of the linear covariance analysis is to perform measurements and update
the covariance matrix to account for the new information provided by those measurements.
Before this can occur, the algorithm must determine which measurements should be
attempted; then, the update occurs for any successful measurements. This process involves
three checks: the sensor must be on, a measurement must be scheduled, and the
environment must meet all sensor constraints. Once a mission is specified, the
measurements to be performed must also be identified. This includes the data involving
when a sensor in on or off and how often the sensor measurements will be taken. The
constraints, on the other hand, are a property of the sensor.
When a measurement is performed, three inputs are required: the measurement sensitivity
vector b, underweighting factor Puw, and the consider state array. Each sensor has a
measurement sensitivity specification. This specification includes values for terms in the
error state vector that relate to the sensor itself, state errors of the platform it resides upon,
and state errors of any other platforms involved in the measurement. These specifications
are used to define the appropriate elements of the measurement sensitivity vector. The
remaining elements are set to zero. The measurement underweighting factor is used to
prevent the linear covariance analysis from becoming too optimistic in its predictions of
error performance. It is a global constant used in all calculations.
Finally, the algorithm allows an engineer to select consider states, which are error terms
that are propagated, but the information captured by those error terms is not used by the
vehicle to improve its estimation of the vehicle states. These terms are identified as part of
describing the desired modelling fidelity of a sensor on board a platform.
2.5 Improving the Computational Performance of the Algorithm
Since a sensor can be required to perform many measurements during a time step, the
update process may be repeated several times for a sensor in the same time step. An
example of this situation is a satellite constellation containing 20 orbiters, each carrying a
range measurement device. This instrument is commanded to estimate the range from the
host orbiter to all the other satellites in the constellation. Therefore, on the time step when a
measurement is scheduled to occur, the algorithm checks to see which of the other 19
satellites meet the constraints for the range measurement. In this case, the constraint would
probably be that the central planet is not between the two orbiters. For each measurement
which is actually taken, the covariance matrix must be updated. Therefore, the covariance
matrix update equations could be evaluated as many as 19 times in one time step for just
one sensor. If all orbiters were scheduled to take measurements at the same time, as many
as 380 updates could be required.
As this example illustrates, linear covariance analysis problems can be computationally
intensive. Therefore, it is worthwhile to examine whether an automated tool can be used to
reduce the computation. This study determined that several inherent properties of the linear
covariance problem can be exploited to dramatically reduce the number of calculations
required on each iteration.
2.5.1 Computational Properties of the Linear Covariance Analysis Algorithm
Before any progress can be made, the sources of the calculation time must be identified.
The majority of the calculations occur in the Runge-Kutta integration and covariance matrix
update for each measurement. The calculations involved in these two processes are
evaluated for an error state vector containing n elements.
Runge-Kutta Integration of Lyvapunov differential equation E = FE + EFT + Q
The Runge-Kutta integration algorithm is as follows:
Given: Initial covariance matrix
Noise matrix
Vector
Eo
Q
k = (0.0 0.5 0.5 1.0)
loop from i=l to 4
if i=1,
let E = EO
otherwise,
let E = EO + ki * AEi-I
let AEi = At (FE + EFT + Q)
end loop
E = Eo + (AE1 + 2*(AE2 + AE3) + AE4)/6
E=E+ET
2
As given, this algorithm has the following computational properties:
Eauation
E = Eo + ki * Ei-1
(executed 3 times)
AEi = At (FE + EFT + Q)
Operations
n2
n2
FE:
n3
EFT:
n2(n-1)
n3
(executed 4 times)
n2(n-1)
rest:
E = EO + (AE1 + 2*(AE2 + AE3) + AE4)/6
Total:
3(n2 ) + 4(2n 3 + n2) + n2
n2
3(n2) + 4(2n3) + 4n 2
16n 3 + 16n 2
2n2
n2
n2
n2
4n2
Type
multiplication
addition
multiplication
addition
multiplication
addition
addition
multiplication
multiplication
division
addition
multiplication
division
addition
floating-point operations
Measurement updates
Every time a measurement takes place, the covariance matrix must be updated to account
for errors introduced by that measurement. The process for updating the covariance matrix
has the following computational properties:
Eoua itnn
in
(Lr.Q
4,L
~~~~
~
U=Eb
residual = bT
=1 + iw)
U
Oresidua
e
i + ceas
Enw = (I - ;bT)E (I - ObT)T +
neas"dTr
2
• & LLlkJtI
M
b]
n
n(n-l)
n
n-1
multiplication
addition
multiplication
addition
n
multiplication
2n 3 + 2n 2
2n 3 - n2
multiplication
addition
subtraction
3n2
Total:
I
avtWp
ions
r
O
2n 3 + 3n 2 + 2n
2n 3 - 1
multiplication
addition
3 n2
Rubtraction
4n 3 + 6 n 2 + 2n - 1
floating-point operations
At first glance, this computation may seem trivial compared the Runge-Kutta integration of
the covariance matrix since the integration is O(n 3) whereas with some manipulation the
update equations are O(n 2). However, this computation is repeated as many times as there
are measurements. For some problems, the number of measurements can become a
significant percentage of the number of elements in the error state vector, so the two
calculations may be at the same order. Furthermore, improvements in the integration
algorithm can reduce it to O(n2 ) as well.
2.5.2 Improving Computational Efficiency
By taking advantage of the physical properties of the problem being modeled by the linear
covariance analysis, the amount of computation can be dramatically reduced. In particular,
the dynamics matrix and measurement sensitivity vectors are almost always sparse.
Furthermore, the location of these zeros is well defined. This knowledge is sufficient to
allow changes in the algorithm to greatly decrease the required computation time.
Since the dynamics matrix captures the differential equations describing the propagation of
errors, the formulation of the error state vector directly determines the format of the
dynamics matrix. The error state vector is arranged by platform. The first terms in the
vector for each platform are the position and velocity errors for that platform. These are
followed by the error terms for each sensor on board the platform. Sensors such as inertial
measurement units (IMU) which affect the velocity error of the host platform and sensors
whose error terms depend upon the position and velocity error of the host platform are
placed first. This configuration is shown in Figure 2-1.
error state vector
platform
r
)ensor
platform
0-'ý
;ensor
platform
sensor
error term
error term
error term
*404ýý
Figure 2-1: Formulation of the error state vector
This approach leads to a major advantage. The propagation of errors for most sensors are
solely dependent upon errors in that sensor. Therefore, the dynamics for each sensor will
appear in a block along the diagonal of the dynamics matrix. A few sensors either depend
upon or affect the position and velocity errors of the host platform. These sensors have
additional dynamics in the off-diagonal locations corresponding to the host platform
position and velocity errors. By placing these sensors first, the dynamics for these sensors
and the host platform position and velocity errors are contained in a larger block, while the
remaining sensors have individual blocks along the main diagonal of the dynamics matrix.
The form of the dynamics matrix is shown in Figure 2-2.
dynamics matrix
ar
ros
av
14
IMU
sensor
f
platform
sensor
e=
sensor
zeros
10
Figure 2-2: Dynamics matrix in block diagonal form.
With the dynamics matrix in this form, the multiplication of the dynamics matrix and the
covariance matrix is much simpler. For example, if the n x n dynamics matrix is broken
into m equal size matrix blocks along the main diagonal, the number of calculations is n3/m
multiplications and n2(n-m)/m additions. The full matrix case described earlier corresponds
to m = 1, yielding n3 multiplications and n2 (n-1) additions. At the other extreme, a
diagonal dynamics matrix, m = n, requires only n2 multiplications and no additions.
In practice, the savings can approach the order of magnitude reduction seen with the
diagonal matrix case. Many error terms are modeled as first-order Markov processes, so
the error term dynamics are solely dependent upon the error term itself. Therefore, the only
contribution to the dynamics matrix lies directly on the main diagonal. This enables the
blocks associated with each matrix to be broken down even further to gain the maximum
benefit.
The other major improvement occurs in the update equations. This improvement takes two
forms, manipulating the update equation to reduce the order of the computations as well as
the number of mathematical operations required. This begins with a modified version of
the Joseph form developed by Stan Shepperd 1 :
Enew
=
T
T + k( V- w)(Q - W)
E - kwW
where k = residual + meas
In this form, 2n 2 multiplications and 2n 2 additions are required as compared to the original
O(n3) equation. This yields the following computational properties:
Eouation
Operations
U=Eb
n2
residual = bT U
W
1
(1 + Puw) o2esidual
multiplication
n(n-1)
n
n-1
addition
multiplication
n
multiplication
2n 2 + 2n
multiplication
addition
U
ma
Enew = E - kwwT + k(j - w)(Q -
w)T
n22
n
3n 2 + 4n
2n 2 - 1
n2
Total:
I~
addition
subtraction
multiplication
addition
subtraction
floating-point operations
6n 2 + 4n - 1
Sheppard's form completely eliminated 4n 3 computations at the expense of just 2n. It
should also be noted that the order of operations can be very important. In the kwwT term,
multiplying k and w and then multiplying wT requires n2 + n calculations whereas
multiplying wwT uses 2n 2 calculations. This is because kw is a vector multiplication and
k(wwT ) is a matrix multiplication. A similar situation arises with k(W - w)(V - w)T. The
performance properties given above assume that the more efficient computation is used and
that %- w is performed only once.
However, Sheppard's equation can be further reduced with some additional manipulations.
To begin, replace w and Wwith their definitions, yielding
Enew
=
1 2 UUT
E- 1UUT+k(
k
(1 +
3uw)
Oesidual +
meas
This step yields an important restriction. This equation is only valid for a system with no
consider states, because when a state is considered, it is zeroed in Wbut not in w, so w and
w are in different directions, which prevents the factoring out of UUT. Most linear
covariance problems do include consider states; however, pure underweighting is common
in filtering problems.
Next, multiply k into the expression in parenthesis and combine the UUT coefficients:
k
=
E +
Enew
=E
[((1+
N
-
1
2
sidual +
1
UU
jw)
k
~eas
In this form, both the multiplications and additions are cut in half. Thus, in final form, the
update equations have the following computational properties (ignoring zero order
calculations):
Eauation
Operations Type
U=Eb
n2
multiplication
n(n-1)
addition
n
multiplication
n-1
addition
1
addition
esidual
=
bT U
k = residual +
Enew =E+
eas
k
-12 1 UUT
(1+juw) 0sidual +4eas I k n2
n2
Total:
2n 2 + n
2n 2
4n 2 + n
multiplication
addition
multiplication
addition
floating-point operations
This is down from 4n 3 + 6n 2 + 2n floating point operations required for the original form
of the measurement update equations, and 6n 2 + 4n - 1 from using Shepperd's form of the
Joseph update equation. However, this set of equations is only valid for simulations
without consider states. Nonetheless, this form has added elegance because it can be
reduced to the same form as the optimal filter. First, define the constant K such that
EEnew =wK
1 UUT
k
Then, replace U with EbT and - U with w so
Enew = E -KwbTE
Factoring out E, this gives the same functional form as the optimal case with the addition of
the constant K:
Enw = (I- cwbT) E
Additional reductions in the number of computations can be obtained because of zeros in
the measurement sensitivity vector. When a measurement is taken, the only non-zero
elements in the measurement sensitivity vector are the ones corresponding to the sensor
taking the measurement, any sensors it is communicating with, and the position and
velocity errors associated with the platforms housing each sensor. When multiplying the
measurement sensitivity vector and the covariance matrix, the columns that would be
multiplied by the zeros can be ignored. Therefore, if m out of n elements of the
measurement sensitivity vector are non-zero, only m2 multiplications and m(m-1) additions
are necessary as compared to n2 multiplications and n(n-1) additions for the full matrix
case.
When the full measurement sensitivity vector is used to update the covariance matrix, the
number of calculations rises dramatically as new measurements are added. This is because
each new sensor increases the number of calculations in two ways. First, the sensor
increases the number of measurements, which raises the number of times the update
equations must be performed by an equal amount. Second, the additional sensor increases
the size of the covariance matrix, which means that more computations must take place for
each update cycle for all previously defined measurements as well as the new ones. By
taking advantage of the zeros, the number of calculations associated with a single
measurement is invariant with the total size of the covariance matrix. Therefore, the only
additional computation required is the extra update of the covariance matrix.
One other improvement is possible, although it does not offer the same magnitude of
improvement as the other options. The covariance matrix is symmetric, so only a triangular
matrix including the main diagonal must be calculated. This can reduce both the integration
and update calculations by nearly a factor of two and also potentially save an equal amount
of memory, but implementing a series of calculations to operate only on the upper or lower
triangular portion of the matrix is relatively complicated as compared to the gain.
However, if computational performance is a problem, this feature may be exploited.
Chapter 3: Linear Covariance Analysis Natural Design
Language
A key element of the design process is peer evaluation, especially in large projects. Thus,
an engineer must be able to effectively communicate his or her design to other engineers. A
good design language provides a common format for design data that helps to ensure clarity
and reliability in this communication. The design specification consists of the essential
elements of the design; no additional information is introduced through the specification
language.
A good specification language gets at the heart of how an engineer thinks about the problem
at hand. The specification language is then designed to mimic this thought process. Once
again, extraneous information must be ignored, leaving only the essence of the design.
The problem with developing a design language for linear covariance analysis is that the
only analysis tool currently available is computer code. In fact, many linear covariance
analysis algorithms are generated by putting together old pieces of code and making
modifications as necessary. This approach obscures the true application design with issues
related to the implementation of the design on a computer such as the language, compiler,
computation environment, and similar concerns that have nothing to do with linear
covariance analysis. Worse, over time, these computer related issues become a part of the
design process of the engineer developing a linear covariance analysis problem.
In order for a design specification technique to be successful, this cycle must be broken.
This can be accomplished by having an application engineer step back and think about the
elements of the linear covariance analysis problem and how these elements fit together to
form a mission specification. Many years of experience with thinking about the problem
one way are not easily replaced with a different one. The engineer must be frequently
reminded to avoid any concerns about the implementation of the algorithm on a computer.
In this way, the pertinent design information can be extracted from computer-related design
issues and an application specification language can be developed.
The purpose of the design language is to capture the information necessary to perform the
linear covariance analysis. Therefore, any language which accomplishes this task
efficiently is acceptable. However, a good design language captures how an engineer
thinks about the problem as well as the design data. This narrows the specification options
significantly; on the other hand, it also implies that the optimum design language will be
different for each individual. However, the differences between individual tastes are
usually minor and do not pose a major threat to the development of a good design
language.
Many languages also allow room for minor modifications to meet these differences. For
example, in control block diagrams, the symbols
all represent addition. Although a design language may select any one of these three
symbols, the basic concept of block diagram connectivity and functionality is unaffected.
This chapter provides a design language for linear covariance analysis, with the
understanding that small changes may eventually be desirable until the engineering
community as a whole decides upon a common specification language.
3.1 Linear Covariance Analysis Elements
Linear covariance analysis missions are comprised of three primary elements: celestial
bodies, platforms, and sensors. Celestial bodies are the stars, planets, moons, and other
gravitational bodies. Platforms are vehicles, rovers, orbiters, and beacons which carry
sensors. Sensors are the instruments which perform measurements and provide
information about the environment of the platforms they reside upon.
The only design information for a celestial body is a set of modelling parameters used to
describe the body. For this purpose, a simple list of the parameters and their values is
sufficient. A sample celestial body is shown.
Body: Earth
Physical properties:
Gravity constant (pL):
3.986*1014 m3/s2
Radius:
6378 km
Atmosphere height:
Rotation rate:
25 km
7.2921*10-5 rad/s
Orbital properlies:
Central body:
Sun
Semi-major axis:
Eccentricity:
Inclination:
1.0 a.u.
0.01
8"
Longitude of the ascending node:
Argument of pericenter:
True anomaly:
0'
00
147'
Platforms
The properties of the different platforms are defined within the automated system, so no
specification language is necessary. The development of a design language must consider
the general properties of a platform. Some design parameters include whether the platform
must rest on a surface, whether the platform is capable of self-propulsion, what external
forces act on the body, and so forth.
Sensors
A sensor specification consists of four major parts, error terms, constraints, and
measurement sensitivity, and measurement variance. The first step is to define the error
terms that model the sensor performance. This is done by simply listing the system of
linear first-order differential equations. A list of the variables and their physical
significance is provided as well, although the analysis could proceed without it. The
following specification is a simple model for an inertial measurement unit.
eR = ev
ev = GReR + UV + Mx(S)ew + d'bi's + Mdiag(S)esfc"
+ Ugyro)
'ro = MI.BMB.NB(e4e
=_
kaacel
= bias0_el
sf
=
)bias
,.accel
aelbsf
= -
+
+b ccel
+
+
basca
csf
where:
eR
ev
4yro
egyro
eV
ebias
sfcel
Platform position error
Platform velocity error
Gyro misalignment
Gyro drift
Accelerometer bias
Accelerometer scale factor error
GR = (3iRiT - I)'
Mx(V)=
0-Vz
Vy
Vz
0
-Vy Vx
Vx 0 0
-Vx
0
Mdiag(V) = 0 Vy 0
L00 Vz
S
Specific acceleration on the vehicle (excluding gravity)
Like the error terms, the constraints are most easily expressed in a mathematical form, so a
list of inequalities is sufficient for many sensors. Some well understood text descriptions
may also be given. For example, the mathematical meaning of a clear line-of-sight is
known to any linear covariance applications engineer.
Since the IMU does not take measurements with respect to other platforms, it has fewer
constraints and no measurement sensitivity, so a different sensor is used to show a sample
constraint list. The measurement involves two platforms, so words such as range and
relative acceleration are assumed to be with respect to the other platform. Three sample
constraint specifications are:
Clear line-of-sight
Range < 2500 km
Relative acceleration < 5.0 g
The elements of the measurement sensitivity vector have a one-to-one correspondence with
the error state vector. Therefore, the measurement sensitivity depends upon the defined
error terms. The measurement sensitivity vector is simply specified in traditional vector
notation. The error terms are provided first to put the measurement sensitivity vector into
perspective. The specification shown is for a 2-way radio range measurement with clock
bias and clock drift.
= ev
gR
ev = GReR
e=seD
eD = - eD + UD
Q
-uhs
b
=
-UO
0
1
0
where:
uLm = unit(r2 - r1)
The measurement sensitivity calculation has non-zero elements only for the sensor taking
the measurement, the host platform position and velocity errors of the sensor taking the
measurement, and the position and velocity errors of any platform the sensor is
communicating with.
The measurement variance is simply an equation that describes the accuracy of the
measurement being taken. This may be a constant or a function of the conditions under
which the measurement is performed. For example, a range measurement may decrease in
accuracy as the bodies get farther apart, so a sample equation is:
aOeas = 10-6 1r2- r 1 0 2
where 0 2 is a measurement of the accuracy of the instrument. The measurement accuracy
variance, ame, is the overall measurement accuracy which degrades linearly with
increasing distance between the two bodies involved in the measurement. This type of
measurement variance is characteristic of a constant power system.
3.2 Mission Specification
The goal of a design specification language for a linear covariance analysis mission is to
simply and clearly specify the mission. The language does not need to be restricted to
pencil and paper. Rather, it can and should take advantage of advancing knowledge
capture and information management technology. For linear covariance analysis problems,
using hierarchical objects yields a much more elegant design language.
A hierarchical object language consists of a root object, in which several other objects are
defined. Each of these objects then becomes a root object for another set of objects. Each
sub-object may capture design information in a different way. For example, one object
may simply capture a list of equations while another takes video data. Thus, the
hierarchical approach is the most natural method of capturing the inherently hierarchical
nature of a linear covariance analysis mission while retaining full flexibility in specific
knowledge capture techniques. A diagram of the mission specification object hierarchy is
shown in Figure 3-1.
mission
platforms
trajectory
data
celestial bodies
sensors
on board
definition
constraints
error
terms
measurement
sensitivity
definition
modelling
modelling
implementations
measurement
variance
error
terms
used/
considered
Figure 3-1: Linear covariance analysis mission definition design language object hierarchy.
For linear covariance analysis, the root object is a mission. Each mission is comprised of
the platforms and celestial bodies used in the design. The basic mission components are
best represented graphically as shown in Figure 3-2. This method clearly identifies the
platforms and celestial bodies. For missions with few orbiters and maneuvering vehicles,
the trajectories may even be included in the diagrams. The graphical information also
provides some hints as to the relationship among the bodies, although the specific data is
included as a part of the body objects.
GPS
f fGPS
GPS
0
Moon
In 1
GPS
Figure 3-2 Mission specification design language.
Each celestial body is comprised of a celestial body definition and modelling selections.
The purpose of the celestial body definition is to provide the necessary information to
model a celestial body such as Earth. In some cases, data for several different models may
be provided. For example, a celestial body definition may include many harmonics of a
celestial body which are unnecessary in a particular problem. Therefore, the celestial body
object used in a mission must consist of a definition as well as modelling selections such as
the gravity model. The celestial body definition is described in Section 2.1.1. As Figure
3-2 shows, the celestial body definition is identified by the graphical symbol used to
represent the body. The additional modelling information is simply listed. For the scope
of the linear covariance analysis specification problem under consideration, the only
additional modelling parameter is the gravity model.
The second mission object, platforms, include the type of platform, sensors on board, and
trajectory data. Each platform is identified by a name which is displayed along with the
platform graphical symbol which shows the type of the platform. The method of
specifying the trajectory data depends upon the type of platform. For maneuvering vehicles
and rovers, the trajectory is given by a mathematical equation or a series of vectors that
contain the state information for each time step. The orbiter trajectory is defined by a series
of initial orbital elements and a central body which are specified in the same manner as
those of the celestial body definition. Finally, latitude, longitude, and altitude and host
planet information for beacons are used to determine their trajectory. Beacons also have a
unique modelling parameter, elevation angle. The elevation angle is used to determine
when sensors on board the beacon can communicate with other sensors both on the planet
or above it. A sample beacon specification is provided:
Name: Beacon-1
Type: Beacon
Sensors on board:
Central body:
Latitude:
Longitude:
Altitude:
Elevation angle
Radio
2-way range
Earth
41.75"
-125.12'
1.342 km.
8"
Litton A-190
Honeywell HT-525
The list of sensors on board is comprised of sensor objects which form another level of the
hierarchy. Like celestial bodies, the sensor object is made up of a sensor definition and
additional modelling information. This includes the sensor implementation, error term
modelling, consider states, and measurement definitions. Since sensor definitions are used
only to capture the functional form of the differential equations that define the error
propagation, implementations are used to gather the specific performance parameters for a
particular piece of hardware. Since the sensor on board the platform is a real piece of
hardware, both a definition and an implementation are required. The design also may not
require as high a modelling fidelity for a sensor as is provided in the definition, so the
sensor object includes information telling which error terms should actually be included in
the model. These consider states are error terms which are propagated but not used to
improve measurement accuracy. A sample sensor object specification is shown below:
Name (Definition): 2-way range
Implementation: Litton CC-121
Fully modelled error terms: Clock bias
Delay bias
Consider states:
Transceiver clock drift
The final sensor object element is a measurement definition. This definition is itself an
object that identifies which measurements are taking place and when they should occur.
Each sensor can only take one measurement. Therefore, the only information about a
measurement that needs to be entered is the sensor taking the measurement, and the actual
bodies that are referred to by the associated body definitions. Measurement timing
information includes the measurement frequency and a list of times the sensor is turned off.
Estimate range to:
Measurement frequency:
Shut off:
Space Shuttle Atlantis
Beacon-1
TDRSS-1
TDRSS-2
TDRSS-4
GPS-1
GPS-4
GPS-11
5 hz.
4.0 min. to 5.2 min.
Chapter 4: Automated Linear Covariance Analysis Tool
The purpose of developing an automated system for linear covariance analysis is twofold.
First, the specification of a high-level design language as compared to low-level computer
code is much easier for an engineer. As with any process, this specification simplicity
leads to a more reliable process. Second, automation dramatically reduces the time
necessary to produce simulation code and even evaluate the results. Thus, the ease of
specification, increased reliability, and faster development of simulation code and analysis
all contribute to an ability to produce linear covariance analysis simulation code in a more
cost-effective way.
Automation of the linear covariance analysis tool begins by specifying the system design
through a user interface. This interface is designed to be as close as possible to the natural
design language described in Chapter 3. The implemented interface is discussed in Section
4-1. The linear analysis tool then reads the data entered by the engineer and places it into a
set of data structures. These structures are defined in detail in Section 4-3. Finally, an
understanding of linear covariance analysis and its application to navigation system
evaluation is combined with the user specified data to produce the actual simulation code.
The code generation process is accomplished with the help of an engineering software
development tool called ASTER (Automation Software Technology for Engineering
Reliability). ASTER is capable of taking equations and block diagrams and converting
them into computer code. The choice to use ASTER rather than a code generator built
specifically for linear covariance analysis is twofold. First, it eliminates the need to write a
code generator, thereby allowing more time to be spent developing a powerful linear
covariance analysis specification tool. Using the ASTER code generators serves the long-
term interests of the linear covariance analysis tool as well. As improvements are made to
the ASTER code generation process, the linear covariance analysis tool automatically
acquires these benefits. Furthermore, it reduces the amount of maintenance required by the
two tools. The software development process for linear covariance analysis problems is
shown in Figure 4-1. The figure also delineates responsibilities of the engineer, linear
covariance analysis tool, and ASTER.
design
criteria Ndesign
$
E
conceptualization
Sdesign
/
,*
language language
tool
Engineer
design
specification
Linear
L
Covariance
Analysis
language
•
•
Tool
internal
representation
4
code
generator ,
input
*
code
4
ASTER
Figure 4-1: Software development process for linear covariance analysis code.
4.1 Mission Specification Language Implementation
The user interface for the linear covariance analysis tool is designed to allow an application
engineer to specify linear covariance analysis missions using the language described in
Chapter 3. However, some modifications to the design have been necessary in order to
simplify development and ease of use. Despite these changes, the implementation of the
natural design language in the graphical user interface successfully approximates the
language requirements.
Since the specification automation and code generation processes dramatically reduce the
amount of time required to develop a linear covariance analysis simulation, the application
engineer will now spend the majority of his or her time analyzing results. Therefore, any
further reductions in the time required to perform linear covariance analysis must relate to
this analysis. For this reason, the user interface also includes capabilities to aid in the
analysis process. In particular, the engineer can specify outputs that are to be examined in
a variety of formats. Graphical formats such as plots and charts are valuable for examining
time traces, while a textual format is used to manipulate and view the covariance matrix.
The following sections describe the user interface for the automated linear covariance
analysis tool. The windows were designed to correspond to the design language as closely
as possible. However, some differences are unavoidable due to the processes by which
data can be entered into a computer.
4.1.1 Defining Linear Covariance Analysis Elements
Before any mission can be specified, the properties of the elements of the mission must be
defined. When applying linear covariance analysis to navigation systems, three main
elements are used, celestial bodies, platforms, and sensors.
4.1.1.1 Celestial Bodies
The celestial body dialog box is used to specify the properties of a celestial body. The first
data input is the name of the celestial body. Then, the physical properties of the body are
specified. These include the gravitational attraction constant g, radius, atmospheric height,
rotation rate, and the model of the body to be used for gravitational calculations. The
second section is used to describe the orbital properties of the body. This information is
used when more than one celestial body is included in a mission specification. The six
orbital elements required by the tool are the semi-major axis, eccentricity, inclination,
longitude of the ascending node, argument of pericenter, and true anomaly. In addition,
the central body for this orbit must also be given. Finally, the engineer can enter a detailed
description of the models used for deriving the numbers specified to the linear covariance
analysis tool. The celestial body definition dialog box is shown in Figure 4-2.
4.1.1.2 Platforms
In order to limit the scope of this effort, platforms are the only basic linear covariance
analysis element that cannot be specified by the user. Since the user cannot define the
properties of a platform type, no user interface is necessary. However, the linear
covariance analysis tool provides four types of platforms: maneuvering vehicles, rovers,
orbiters, and beacons.
Figure 4-2: Celestial body definition window.
Maneuvering vehicles have self-propulsion capabilities. Therefore, the applications
engineer must specify the trajectory of the vehicle. Rovers can also propel themselves, but
they must reside upon the surface of a celestial body. A trajectory must also be provided
for rovers. Orbiters move in a fixed orbit about a celestial body; they may have an attitude
control system, but the Av introduced by that system is ignored. Once the simulation has
begun, the orbit is determined by the gravity of the central planet, but an initial orbit must
be specified. Beacons are a navigational aide that reside at a fixed location on a celestial
body. A beacon is defined by its location on the body which includes its latitude,
longitude, and altitude.
46
4.1.1.3 Sensors
As expected, sensor models are the most difficult to specify since they are the most
complicated. The sensor model is actually a template for a class of sensors. In most cases,
the equations that describe the error terms of a sensor have the same functional form
regardless of the manufacturer of that sensor. The performance differences between the
sensors are captured by entering different performance constants for each vendor's
product. In the linear covariance analysis tool, the term sensor refers to the general
functional description of an instrument model, and implementation refers to the model for a
specific piece of hardware. This model consists of the general description combined with
the vendor-specific performance parameters.
Sensors introduce an interesting knowledge capture situation. Much of the modelling of a
sensor is described with equations or inequalities. To someone familiar with linear
covariance analysis, the equation
6 gwyro
-
e gyro + ugyro
quite clearly represents the differential equation for the scale factor error of a gyro.
Similarly,
alt =magnitude(ic - rit) - r
calculates the altitude of a spacecraft above the earth from the inertial position vectors and
the equatorial radius of the Earth. However, to the linear covariance analysis tool, these
equations are just a set of mathematical operations applied to some variables. The variables
have no physical meaning. Therefore, the meaning of each input of an equation must be
given. In other words, the user must tell the linear covariance analysis tool that egTyro is the
gyro scale factor error, risc is the inertial position of the spacecraft, earth is the inertial
position of the Earth, r. is the equatorial radius of the Earth, and so on.
When an equation is entered into the linear covariance analysis tool, the equation is parsed
and the inputs are displayed in a list box. Each input gets its name from the variable name,
but the type, units, frame of reference, and description are supplied by the user. The other
slot, source, is used to identify the meaning of each input. The allowable quantities are
constants, mission loads, state vectors of the host body or an associated body, celestial
body properties, and simulation parameters.
A constant is a global constant such as x, 6.0, or e. These only need to be defined once
per mission. A mission load is constant during a simulation, but the constant can have
different values for different implementations of the sensor. State vector information
includes the position, velocity, acceleration, attitude, angular velocity, and angular
acceleration vectors. The available celestial body properties are the gravitational attraction
constant gt, radius, atmosphere height, rotation rate, and the initial orbital parameters semimajor axis, eccentricity, inclination, longitude of the ascending node, argument of
pericenter, and true anomaly. Direction cosine matrices describe a vector expressed in one
frame of reference in another frame of reference. Finally, the simulation parameters include
the current simulation time, start time, end time, time step, and the measurement
underweighting. The sensor definition process requires several dialog boxes. The main
dialog box is used to enter identification information, create error terms and constraints,
and to call other dialog boxes for defining error terms, measurement sensitivity,
constraints, measurement variance, and contributions to the host platform velocity error.
The identification information available for a sensor are a name, label, and description. The
name is used in all the menus throughout the system to identify the sensor. The label is an
abbreviated version of the name used in the sensor palette on the right hand side of the main
window. The description is entered by clicking on the button "Sensor description..." at the
bottom of the dialog box. This pops up a dialog box that enables basic text editing. This
space can be used to describe the modelling process used to derive the inputs, the source of
the input data, or any other useful background information. The sensor dialog box is
shown in Figure 4-3 for a 2-way radio range sensor.
The first list box is used to create, copy, and delete error terms. When an error term is
selected from the list, its name and description are shown to the right of the list. In
addition, the engineer specifies the error term equation and the measurement sensitivity
equation. These items are selected from a list of options. For the error term definition, two
options are available, general equations or a first order Markov model. The first order
Markov model is supplied simply as a way of speeding data input. A Markov model can be
entered as a general equation. If a different general equation is desired, a new dialog box
appears on the screen.
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The second list box shows the operating constraints placed on the sensor. Constraints can
be created, copied, and deleted by using the buttons to the right of the list box. When a
constraint is selected from the list box, its name and description can be entered. Also, the
constraint itself is chosen. The linear covariance analysis tool provides three constraint
options: line-of-sight, range, and a general constraint expression. A line-of-sight
constraint is used to check whether the sensor has a clear line to the sensor it is trying to
communicate with. A range constraint allows the engineer to specify the minimum and
maximum distance between the sensor and either the body or sensor it is taking a
measurement with. Finally, the general equation can be used to enter any other constraints.
Both the line-of-sight and range constraints can be entered with the general equation
capability, but they are provided as special options to reduce specification time.
The final list box shows the associated bodies for this sensor. An associated body is used
as a placeholder for other objects during the specification process. When the sensor is
being modelled, the exact body the measurement will be taken from is not known.
Therefore, an associated body is used to identify the fact that another body is involved
without identifying which body it is. For example, an error term for an altitude
measurement device on board the space shuttle may require the distance from the shuttle to
Earth. This is computed simply by taking the magnitude of the difference between the
inertial location of the body and the inertial location of the center of Earth and subtracting
the Earth's radius. However, when the sensor is being defined, the linear covariance
analysis tool has no knowledge of a planet Earth, so an associated body is defined to
identify that a planet's properties will be involved in the sensor model. When a
measurement is defined, the associated bodies are replaced by the actual bodies used in the
measurement.
An associated body definition consists of three components, a name, description, and type.
The tool provides four types of associated bodies: celestial body only, platform with same
sensor only, platform only, and any. This is used to determine which bodies are eligible to
replace the associated body when the measurement is defined. For example, a radio cannot
communicate with a planet, it must communicate with another radio, so the associated body
type would be a platform with same sensor.
Finally, the main sensor modelling window has five buttons that enable the user to specify
the measurement variance, contribution to the velocity error of the host platform,
measurement sensitivity of the host position and velocity error, and a detailed description of
the sensor and its model.
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An error term component consists of a coefficient and the error it is multiplying. The error
term can be any error defined for the sensor as well as the position and velocity error of the
host platform. The coefficient can include any of the variables outlined at the beginning of
this section.
Each error term also has a noise component. Since the linear covariance analysis is a
statistical simulation, any noise must be white; however, the intensity of the noise can be
specified. The intensity is usually constant, but it can be a function of the physical
variables included in the linear covariance analysis. For example, the noise of a range
measurement may depend upon the distance between the two bodies. The error term and
noise definition dialog box is shown in Figure 4-4.
Measurement Sensitivity
The measurement sensitivity vector is used to relate the errors in a measurement to the error
terms in the error state vector. Each error term has a corresponding measurement
sensitivity. Sometimes, the measurement sensitivity is not easily expressed as a single
equation, so an area is provided to allow the engineer to specify any preliminary
calculations which are necessary to define the measurement sensitivity. The measurement
sensitivity equation can include the full range of physical variables, which are specified in
the same way as the variables in the error term and noise equations. A space is also
provided for comments about the measurement sensitivity calculation. The measurement
sensitivity dialog box is shown in Figure 4-5.
Constraints
The purpose of the constraints is to identify the conditions under which the sensor is able to
take a measurement. This dialog box is almost identical to the measurement sensitivity
dialog box. A constraint consists of an equality, inequality, or logical operation. The
terms in the comparison may be any of the physical variables or results of additional
calculations which can be specified in the supporting equations text field. These equations
may also use physical variables. The inputs are identified in the same way as for the other
equations. The constraint dialog box is shown in Figure 4-6.
Figure 4-5: Measurement sensitivity specification window.
To speed the specification process, two predefined constraint types are provided. The
range constraint sets a maximum distance between the host platform and another body.
This body is specified in a small dialog box which appears when the range constraint
option is selected. The second constraint type is a line-of-sight constraint. The line-ofsight constraint ensures that the host platform has an unobstructed view of another object.
This object is entered in an identical dialog box as for the range constraint. It must be
remembered that when a sensor is being defined, access to information about other bodies
is accomplished with associated bodies. Therefore, the range and line-of-sight dialog
boxes prompt for an associated body, not an actual platform or celestial body.
Figure 4-5: Measurement sensitivity specification window.
To speed the specification process, two predefined constraint types are provided. The
range constraint sets a maximum distance between the host platform and another body.
This body is specified in a small dialog box which appears when the range constraint
option is selected. The second constraint type is a line-of-sight constraint. The line-ofsight constraint ensures that the host platform has an unobstructed view of another object.
This object is entered in an identical dialog box as for the range constraint. It must be
remembered that when a sensor is being defined, access to information about other bodies
is accomplished with associated bodies. Therefore, the range and line-of-sight dialog
boxes prompt for an associated body, not an actual platform or celestial body.
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Figure 4-7: Measurement variance specification dialog box.
4.1.2 Specifying the Mission
The first step of the mission specification process is to identify the platforms and celestial
bodies involved in the mission. This is accomplished both graphically and with menus.
Celestial bodies are added by selecting the appropriate body from the Add Celestial Body
option in the Universe menu. This creates an icon of the body which is placed on the
workspace. Platforms are selected from a palette in the upper right-hand corner of the
linear covariance analysis window. Once these objects are placed on the screen they can be
edited, moved, or deleted with the mouse. The position of the objects on the screen is
unimportant to the linear covariance analysis tool, although proper orientation of the objects
can help convey physical information about the mission. The main specification window is
shown in Figure 4-8.
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For the celestial body objects, the only parameter not identified is the gravity model. This
is done by editing the appropriate celestial body and selecting the desired model from a
menu. The platform object specification mechanism varies slightly with the type of
platform. All platforms contain a list of sensors on board. A sensor is placed on a
platform by selecting the sensor from the sensor palette on the right-hand side of the main
window. The sensor is created by left clicking on the sensor in the palette and dragging the
sensor onto the desired platform. A sensor is removed from the platform by selecting it
from the sensors on board list in the platform object and hitting the Remove button.
The method of specifying the trajectory varies among the platform types, so each dialog
box is slightly different. Maneuvering vehicles trajectory information is given in the form
of a data file. The file name is entered, and the data included is identified by a series of
check boxes. The rover dialog box includes a menu of celestial bodies to be used as the
central body in addition to the trajectory information. If the trajectory file has the state
vectors in inertial coordinates, this slot is unnecessary. Orbiters also have a central body,
but the trajectory is specified in the orbiter dialog box. Finally, the beacon dialog box
contains a central body menu as well as slots to enter the latitude, longitude, and altitude of
the beacon on the central body. A slot is also provided for the elevation angle parameter.
Examples of design specifications for each of the platform types are shown in Figures 4-9
to 4-12.
The next level of the hierarchical design language is sensor specifications. These are
shown by selecting the Modelling button next to the sensors on board list in the platform
dialog box. The sensor model consists of three primary pieces of data, the sensor
definition, implementation of that definition, and error term modelling. The sensor
definition is determined by the sensor that is selected from the sensor palette. When a
sensor is selected, it uses the default implementation. This default can be changed by right
clicking on the sensor in the palette. This pops up a menu of the available implementations.
The implementation can also be selected in the sensor modelling dialog box. Error term
modelling consists of which error terms are included in the sensor model and which error
terms are used as consider states. The dialog box includes a list of all the error terms in the
sensor defminition. Each error term is followed by two check boxes, one to include the error
term in the sensor model and one to use it only as a consider state. Consider states must
have both boxes checked. The error term modelling dialog box is shown in Figure 4-13.
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Figure 5-13: Sensor modelling fidelity specification dialog box.
The sensor object also includes a definition of the measurements to be taken and the times
at which they should occur. However, the actual implementation breaks from the design
specification slightly. Rather than feeding directly from the sensor on the vehicle, the
measurements are defined in a dialog box called by the Measurement definition option in
the Simulation menu. The measurements to be taken are defined by identifying which
actual bodies replace the associated bodies in the sensor definition. The actual bodies
which can be selected are governed by the type of the associated body. The list is defined
by selecting from among the eligible objects which are displayed in a menu button. This
must be done for each associated body definition. The active associated body definition is
selected from a menu item in the dialog box. The measurement frequency is also entered in
this dialog box. Finally, spaces are provided for up to four periods where the sensor is
turned off.
4.13 Analysis Aides
Although an analysis definition is not formally developed here for a linear covariance
analysis problem, a user interface has been designed to define the types of outputs that will
be examined and the format in which they are viewed. The two most common output
modes are two-dimensional plots and textual representations of subsections of the
covariance matrix in a variety of coordinate frames.
The 2-D plot is a very general capability for defining graphs of data. The plot is defined by
providing equations for the data to be presented along each of the axis of the plot. These
equations may be simple parameters such as time, or complicated equations of the linear
covariance analysis variables. Any of the physical variables available for sensor and frame
specifications can be used in the plotting equations. In addition, the plot can include any of
the analysis variables such as the dynamics matrix, covariance matrix, noise matrix,
measurement residual variance, and others. The equation may also plot vector-valued
quantities. If one axis is a vector and the other is a scalar, separate graphs of each element
of the vector versus the scalar are shown. If both axis are vector valued, the elements of
each vector are plotted against the corresponding element of the other vector.
The second common output mode is used to examine the covariance matrix. This feature
has two forms. First, the engineer can browse the entire covariance matrix at a particular
point in time. However, the entire covariance matrix cannot be stored for each pass of a
simulation because it would take too much time and memory. Second, a particular submatrix of the covariance matrix can be viewed. The sub-matrix is identifyed by selecting
two error terms in the error state vector. It does not matter which one is the row and which
is the column because the covariance matrix is symmetric. However, the engineer can
specify the coordinate frame for each error term. This is important because the meaning of
the covariance matrix is usually much easier to determine in a particular frame.
4.2 ASTER
The Automated Software Technology for Engineering Reliability (ASTER) tool is an
ongoing effort at the Charles Stark Draper Laboratory to develop an automated environment
for engineering application software development, particularly for real-time systems. This
environment is designed to use automation to improve the design process and improve
reliability while dramatically reducing development time. 2
One of the key elements of the ASTER design is the array of application user interfaces
provided by the ASTER system. These interfaces allow engineers to specify designs with
a language specific to the problem they are solving. For example, the current system
includes three different input mechanisms: block diagrams, algebraic expressions, and
computer code. Each of these input formats is converted into a common representation
from which code is generated for either Ada or C. The overall ASTER automated software
development concept is depicted in Figure 4-14.
Figure 4-14: ASTER automated software development concept.
Since the linear covariance analysis tool uses ASTER to perform the actual code generation,
the input structure to the ASTER code generators plays an important role in the design of
the linear covariance analysis tool.
ASTER code generation is based upon control block diagrams. Although control block
diagrams are not the best way of describing the linear covariance analysis algorithm, using
the ASTER code generation capability offers two advantages over a system customized to
the linear covariance analysis simulation. First, the ASTER code generators already exist
and have acquired a level of maturity. Second, and more importantly, it is in the long term
interests of the automated linear covariance analysis tool to be part of a common code
generation system. If each application specification system had its own code generators, it
would be impossible to maintain and upgrade each one. By creating a common interface,
namely, the ASTER internal block diagram representation, the same code generators can be
used for the full range of application specification tools.
The ASTER block diagram interface is based upon block diagrams from classical control
theory. 3 The language has been extended to include elements for a larger body of
problems. 4 The block diagrams are organized into functional elements called transforms.
A transform takes a series of inputs and "transforms" them into a series of outputs. A
sample transform is shown in Figure 4-15.
Figure 4-15: Sample transform for the elevator command control logic for a Boeing 737
autopilot landing system.
This type of specification method is inconvenient for implementing equations. For
example, the calculation of the distance between two points (xl,yl) and (x2,y2) is given by
z = /(x2 - x) 2 + (Y2 -Yt1
However, in a control block diagram format, this function is much more complicated as
shown in Figure 4-16.
Figure 4-16: Distance equation in a block diagram format.
Despite these difficulties, ASTER provides several advantages. When a specification is
mapped into ASTER's block diagram form, the specification can be checked by ASTER for
consistency and completeness. Consistency checking includes type checking and ensuring
proper connectivity among the items in the block diagram specification. These features aid
the debugging process of the ASTER object generation system.
Overall, although the use of the ASTER code generators and the block diagram input
specification adds some difficulty to the development of the code generation process for
linear covariance analysis simulations, the availability, reliability, and long-term benefits
offered by the ASTER system make it worth the extra investment.
43 Internal Data Representation
Once the design specification language has been developed, a graphical user interface is
designed to enable an application engineer to describe a mission using this specification
language. The purpose of the user interface is to provide a mapping from the specification
language to an internal data representation. The internal representation of a design is highly
dependent upon the programming language used to implement the linear covariance
analysis formulation tool. In particular, the representation is constrained by the structures
provided by the implementation language. The closer the structures can match the user
interface structure, the easier the transformation process between the user interface and the
internal representation will be.
For the linear covariance analysis development tool, object-oriented programming
languages such as Common Lisp offer distinct advantages from both a programming and a
functional viewpoint. Object-oriented programming enables the functionality of the
automated linear covariance analysis tool to remain clear in the internal data representation.
Each distinct element of the linear covariance analysis becomes a separate object.
Inheritance allows the objects to be classified such that common characteristics such as
identification information can be shared among all objects while each objects can still
maintain special features unique to that object. 5 This classification system can include
many layers. For example, an orbiter might consist of information unique to orbiters in
addition to properties common to all platforms and information used by all database
objects.
In addition to the functional benefits, object-oriented programming greatly speeds
development time. By utilizing inheritance, one piece of code can be written to apply to all
objects that draw upon common information. For example, if position and velocity are two
properties of a platform, and vehicles, rovers, orbiters, and beacons inherit from platforms,
only one function must be written to handle the position and velocity for all types of
platforms.
In Common Lisp, data is stored in structures called objects. The properties of an object are
defined in a class. A class is comprised of named slots which hold data. A slot is
equivalent to a Lisp variable. It can hold anything including lists, strings, symbols, and
other objects. When a class is defined, it can inherit some of its slots from other classes.
So, if class A is defined to have slots Al and A2, and class B inherits from A and has its
own slot B I, class B will actually consist of slots Al, A2, and Bl. In this case A is a
superclass or superior to B, and B is a subclass or inferior to A. In a similar manner,
functions, called methods, that are written for class A are also then inherited by class B.
The structure of the Lisp objects is designed to mirror the hierarchical specification
structure of the natural design language discussed in Chapter 3. Conceptual elements such
as missions, sensors, celestial bodies, and platforms each have their own class. Therefore,
the mapping from the user interface displays to the Lisp objects is relatively
straightforward.
One minor complication does arise with the linear covariance analysis elements that may not
be evident in their description. Several of these elements have two classes associated with
them, definitions and manifestations. The difference between definitions and
manifestations is best described through an example. Each sensor has a single definition.
This definition describes the properties of the sensor including its error terms, measurement
sensitivity, and constraints. However, each time the sensor is placed on a platform, it is
modelled in a different way. Therefore, a sensor manifestation is used. The manifestation
object contains all the specific information relating to the use of that sensor on a specific
platform. This includes data such as which implementation is used, which error terms are
included, and which error terms are used as consider states. Each sensor definition may
have many manifestations.
4.3.1 Common Classes
The internal structure for the linear covariance analysis tool begins with objects available
throughout the project. This includes the simplest object containing name and description
information, graphical information, variables (signal), coordinate frames, and source data.
Source data is used to identify the physical meaning of a signal.
The basis of most objects in the linear covariance analysis is identification information such
as a name and description. This class forms the root for most of the linear covariance
inheritance hierarchy.
Object: id
Inherits from: saveable-object
string
Slots: name
description
string
The basic inheritance structures must be augmented with additional classes which are
inherited by different classes throughout the inheritance hierarchy. These classes include
signal, source and the classes which inherit from it, and orbit. All classes must inherit
saveable-object, which is used to enable the object to be written to a file.
Object: signal
Inherits from: id
Slots: aster-def
type
frame
units
source
ASTER object
ASTER type
<frame>
string
<source>
Equation used to generate the algebraic transform
Data type of the signal
Coordinate frame (only applicable to vectors)
Units of the signal
Physical meaning of the signal
The signal object is the equivalent of a variable in computer code. The aster-def slot is used
during ASTER object generation to point to the ASTER definition of the signal. The
source slot tells the linear covariance analysis tool what information the signal actually
contains.
Object: source
Inherits from: saveable-object
Slots: None
The source object is used simply as a superclass to help organize the source objects. The
objects which inherit from source are constant, mission-load, state-information, host-state,
associated-body-state, celestial-body-property, simulation-parameter, and analysisvariable.
Object: constant
Inherits from: source
Slots: constant-value
string, list of string, or list of list of string
Value of the constant
A constant is usually a perpetual constant such as 7 or e which does not change its value
between equations and is constant in time. The type of the constant-value slot depends
upon the type of the input that is constant.
Object: mission-load
Inherits from: source
Slots: None
A mission load is constant for a mission, but its value can be specified at the beginning of
the mission. Since almost all signal objects are created during the sensor definition phase,
the actual value is not known and is not needed. It is supplied by the implementation. This
feature enables the engineer to enter the functional form of an equation and specify a series
of actual values that correspond to the different versions of that sensor.
Object: state-information
Inherits from: source
Slots: platform
platform object
state-vector symbol
The platform whose state vector is desired
Identifies which state vector is desired
This object is used for getting state information about any platform. The platform slot must
be an actual platform object, not an associated body definition. Therefore, this is not
normally used until after a mission has been specified.
Object: host-state
Inherits from: source
Slots: state-vector symbol
Identifies which state vector is desired
The host state object is used to identify the a state vector of the host platform during sensor
definition. A platform slot is unnecessary because the error term the information is used to
define knows which sensor it is modelling, and the sensor knows which platform it is
placed on.
Object: associated-body-state
Inherits from: source
Slots: associated-body-definition
symbol
symbol
state-vector
The associated body whose state vector is
desired
Identifies which state vector is desired
Like host-state, the associated-body-state object is used to refer to state information of an
associated body. The actual body that is used is determined at code generation time.
Object: celestial-body-property
Inherits from: source
Slots: celestial-body
property
<associated-body-definition>
The celestial body whose physical property is desired
symbol
Identifies which property is desired
This object is also used during sensor definition, so the actual celestial body property
desired is not known. Therefore, the celestial-body slot is filled with an associated-bodydefinition object
Object: simulation-parameter
Inherits from: source
Slots: parameter
symbol
Desired simulation parameter
Simulation parameters include time, time step, start time, end time, and measurement
underweighting data. The desired property is indicated by the appropriate symbol in the
parameter slot.
Object: analysis-variable
Inherits from: source
Slots: variable
simulation
symbol
Desired variable from the linear covariance analysis
Analysis variables consist of all the data calculated during the linear covariance analysis
simulation that are not explicitly defined in the user interface. These include the dynamics
matrix, covariance matrix, noise matrix, and similar quantities.
Object: direction-cosine
Inherits from: source
Slots: from-frame <frame>
<frame>
to-frame
list of list of string
IC
Initial frame
Transformed frame
Initial value of the matrix
Direction cosine matrices are used to convert a vector from one coordinate frame to another.
The direction cosine matrix can be time-dependent.
Object: algebraic-transform
Inherits from: saveable-object
string
Slots: equation
algebraic-transform
inputs
Equation used to generate the algebraic
transform
ASTER algebraic-transform object
Algebraic transform block diagram
list of <signal>
Inputs to the algebraic transform
Algebraic transforms are used extensively throughout the system. An algebraic transform
is used to convert an equation into the appropriate engineering block diagram.
Object: frame
Inherits from: id
Slots: primary-dir
symbol
Coordinate axis being defined by the
primary-eq
<algebraic-transform>
secondary-dir
symbol
secondary-eq
<algebraic-transform>
primary equation
Equation for the primary direction
vector
Coordinate axis being defined by the
secondary equation
Equation for the secondary direction
vector
A frame is defined by two vectors. The primary vector is used to determine the first axis.
This axis is given in the primary-dir slot. The second coordinate axis, defined by the
secondary-dir slot, is defined by the part of the secondary equation normal to the primary
coordinate axis. The third axis is the cross-product of the first two vectors. The coordinate
frame is always right-handed.
Object: graphical-object
Inherits from: saveable-object
Slots: graphical-rep
highlighted-graphical-rep
label
x
y
size-x
size-y
<icon>
<icon>
string
integer
integer
integer
integer
Unhighlighted icon
Highlighted icon
Label placed below the object
x position of the object in the workspace
y position of the object in the workspace
width of the object in pixels
height of the object in pixels
The graphical-object object is used to define graphical representation of celestial bodies,
platforms, and sensors.
43.2 Linear Covariance Analysis Elements
Celestial bodies
A celestial body object consists of information to identify the celestial body, define the
orbit, capture the physical properties of the body, and describe the graphical properties of
the body.
Object: celestial-body
Inherits from: id, orbit, graphical-object
Slots: gravity-mu
string
radius
string
atmos-height
string
string
rotation-rate
gravity-model
symbol
frame-definition <frame>
Gravitational attraction of the body
Radius of the body
Atmospheric height of the body
Rotation rate of the body
Model of the body used for gravity computations
Definition of the body frame
Platforms
Platforms start with a general description of data common to all platform types. Then, each
type has its own object which inherits from the platform object which contains the
information specific to that platform type.
Object: platform
Inherits from: id, graphical-object
Slots: sensors-on-board list of <sensor-manifestation> Sensors on board the platform
Position error differential equation
<error-term-definition>
delta-r-dot
<error-term-definition>
Velocity error differential equation
delta-v-dot
Object: vehicle
Inherits from: platform
Slots: datafile
string
list of boolean
data-in-file
File containing the trajectory information
Boolean for each state vector being in the data file
Object: rover
Inherits from: platform
Slots: datafile
File containing the trajectory information
string
Boolean for each state vector being in the data file
list of boolean
data-in-file
host-body
<celestial-body> Body the rover is placed on
Object: orbiter
Inherits from: platform orbit
Slots: central-body <celestial-body> Body the orbiter is circling
The necessary orbital information for the orbiter is contained in the orbit class and the
related-bodies slot of the platform class which are inherited.
Object: beacon
Inherits from: platform
Slots: latitude
longitude
altitude
elevation-angle
host-body
string
string
string
Latitude of the beacon on the host body
Longitude of the beacon on the host body
Altitude of the beacon relative to the host
body radius
string
Viewing horizon of the beacon
<celestial-body> Body the beacon is placed on
Sensors
The sensor object is implemented in two parts, a definition and a manifestation. The
manifestation is the equivalent of the sensor object described in Section 4.###. The sensor
manifestation is the object actually placed on a platform. Many of the sensor-related objects
do not inherit from one another, however, a clear hierarchy of the object is evident. The
primary reason for this is that inheritance only provides one copy of the slots from the
superclass, whereas a sensor may require several copies. For example, error term data is
clearly a subset of a complete sensor definition; however, the error term class does not
inherit from the sensor definition class. This is because a sensor is comprised of many
error terms, so the class structure is somewhat implicit. The natural hierarchy is captured
in the natural design language for linear covariance analysis specifications, and an example
is shown in Figure 4-17.
t_
components
%.rr
AJA- eram
measurement-sensitivity
noise
constraints
sensor-definition
measurement-variance
assoc-bodies
imllplentaIl~l1tionsII
Figure 4-17: Natural information hierarchy for a sensor definition.
Figure 4-17 shows the information flow for a sensor definition. However, a sensor
definition is not placed on board any platform. Rather, a manifestation must be created.
Manifestations are necessary because each sensor can be modelled slightly differently, even
within a single sensor definition. For example, the actual bodies selected to replace the
associated body definitions are different for each sensor. Thus, using the concept of
manifestations greatly enhances the legibility and power of the linear covariance analysis
tool without burdening the user with reams of additional input requirements.
Object: sensor-manifestation
Inherits from: saveable-object
Slots: sensor-definition
host
<sensor-definition> Definition of the sensor
Platform the sensor is placed on
platform object
list of <error-term-manifestation>
Fully modelled error term
list of <associated-body-manifestation>
assoc-bodies
Actual associated bodies
<implementation> Sensor implementation to be used
implementation
string
meas-freq
Measurement frequency
list of list of string List of periods the sensor is
on-off
turned off
scheduling-vector-location integer
Used during code generation to
identify where this sensor is in the
vector that is used to determine
when a measurement should take
place
list of two integers Identifies the sensor location in
state-vector-location
the error state vector
error-term-manifestations
Object: sensor-definition
Inherits from: id graphical-object
list of <error-term-definition> Error term definitions
Slots: error-terms
measurement-sensitivity list of <measurement-sensitivity>
Measurement sensitivity vector
definition
dy-contribution
<error-term-definition> Contribution of the sensor to
the velocity error of the host
platform
constraints
list of <constraint>
Sensor operating constraints
measurement-variance <measurement-variance> Measurement variance of the
sensor
assoc-bodies
list of <associated-body-definition>
Associated body definitions
implementations
list of <implementation> Sensor implementations
default-implementation <implementation>
Default implementation used
when a sensor manifestation is
created
manifestations
list of <sensor-manifestation>
List of the sensor manifestations that have been created
Object: implementation
Inherits from: id
Slots: error-term-IC
mission-load-IC
hash table
hash table
Initial condition for the sensor error terms
Value of the mission loads
The error propagation requires that an initial condition be provided for the error state vector
to account for all previous dynamics. Although the initial error term values are not usually
a property of a sensor, it is included in the implementation to enable values to be stored.
The mission load values, on the other hand, determine the performance characteristics of
the sensor.
Object: error-term-manifestation
Inherits from: saveable-object
Slots: error-term-definition <error-term-definition>
on-sensor
<sensor-manifestation>
in-lin-cov
boolean
consider
state-vector-location
boolean
string or list of two
strings
Sensor this error term is
modelling
Manifestation of the sensor
definition this error term is
modelling
Is this term included in the
sensor model
Is this term a consider state
Used in the code generation
process to identify the location
of the error term in the error
state vector
Object: error-term-definition
Inherits from: signal
<sensor-definition>
Slots: sensor-definition
equation-type
symbol
list of <component>
components
noise
manifestations
Sensor this error term is modelling
Model to be used for the error-term
Components summed to form the
error term differential equation
<noise>
Noise term
list of <error-term-manifestation>
List of the manifestations that
reference this definition
Object: component
Inherits from: saveable-object
Slots: equation
string
algebraic-transform
ASTER object
error-term
<error-term-definition>
inputs
Object: noise
Inherits from: saveable-object
Slots: noise-type
equation
algebraic-transform
error-term
inputs
list of <signal>
symbol
string
ASTER object
<error-term-definition>
list of <signal>
Coefficient expression
Evaluated expression
Error term the coefficient is
multiplying
Inputs to the expression
Type of noise
Noise expression
Evaluated expression
Error term the noise affects
Inputs to the noise expression
Object: measurement-sensitivity
Inherits from: signal
<error-term-definition> Corresponding error term in the error
Slots: error-term
state vector
Used to define variables used in the
support-equations string
measurement sensitivity equation
when necessary
algebraic-transform <algeb raic-transform> Measurement sensitivity equations
Object: constraint
Inherits from: id
Slots: constraint-type
equations
constraint-eq
algebraic-transform
associated-body
inputs
sym bol
string
Type of constraint
Equations for computing variables used
in the constraint expression
strin g
Constraint expression
Evaluated expression
AST'ER object
<associated-body-definition>
Used for some pre-defined constraints to
indicate which body the constraint should
use in the equation
list of <signal> Inputs to the constraint equations
Object: measurement-variance
Inherits from: algebraic-transform
Slots: None
Object: associated-body-definition
Inherits from: id
Type of associated body
symbol
Slots: type
manifestations
list of <associated-body-manifestation>
List of the manifestations of this associatedbody
Associated bodies are used to identify other bodies during sensor definition. At the time a
sensor is defined, the other bodies such as platforms and planets that the sensor may
reference are unknown. Therefore, an associated body is used instead. When the mission
specification is complete, the actual bodies to be used are identified. Associated bodies
definitions can be one of four types, celestial body only, platform only, platform with same
sensor only, or any.
Object: associated-body-manifestation
Inherits from: saveable-object
Slots: assoc-body-definition <associated-body-definition>
Associated body definition
actual-bodies
list of platform object and/or <celestial-body>
List of the manifestations of this associatedbody
The type of object in the actual body list is determined by the associated body type.
Celestial body only types obviously only allow <celestial-body> objects to be used.
Platform only restricts the options to <vehicle>, <rover>, <orbiter>, and <beacon>
objects. Platform with same sensor only is used for 2-way communication devices which
require that both ends of the measurement have the same sensor. The same four platform
types are allowed, but only the ones that contain the sensor being defined. Finally, the any
option enables a user to use both platform and celestial body objects.
4.3.3 Mission Specification Classes
Since a mission is simply comprised of linear covariance analysis elements, very few data
structures are necessary for this information. In fact, a mission is contained in a single
object.
Object: mission
Inherits from: id
Slots: celestial-bodies-used
list of <celestial-body-definition>
platforms
list of platform objects
frames
output-forms
constants
list of <frame>
list of output form objects
list of constants
Object: simulation
Inherits from: id
Slots: start-time
string
end-time
string
delta-t
string
measurement-underweighting string
list of list of string
initial-covariance
integration-tech
string
Celestial bodies used
in the mission
Platforms defined in the
mission
Frame definitions
Desired outputs
Constants defined in the
mission
Simulation start time
Simulation end time
Simulation time step
Measurement underweighting number
Initial value of the covariance matrix
Integration technique to be used for
propagating the covariance matrix and the
state vectors
43.4 Output Forms Classes
One of the useful features of an automated linear covariance analysis tool is the ability to
define output forms. This enables the engineer to quickly and easily view the pertinent data
in a variety of formats. This capability is not completely flushed out, but some data
structures have been provided.
Object: output-form
Inherits from: id
Slots: None
This class is currently used primarily for organizational purposes. Although no slots are
currently defined, an information that is common to all output forms such as the
identification would be defined in this class.
Object: 2-d-plot
Inherits from: output-form
<algebraic-transform>
Slots: x-axis
<algebraic-transform>
y-axis
Equation defining the x-axis data
Equation defining the y-axis data
The 2-D plot data is created by evaluating the x-axis equation and plotting it against the yaxis equation. The equations can be simple such as just time, or complicated functions of
the variables and physical quantities included in the linear covariance simulation.
4.4 Code Generation
The code generation process is a joint effort between the linear covariance analysis tool and
the ASTER code generation system. As discussed in Section 4-2, the ASTER system has
two main input formats, mathematical equations and block diagrams. The mathematical
expression handler is used to analyze all equations entered into the linear covariance
analysis tool. Whenever an equation is entered by an engineer, it is sent to the
mathematical expression handler to determine its inputs. Then, at code generation time, the
equation is used to generate an equivalent block diagram for the ASTER system.
The ASTER system is also used directly by the linear covariance analysis tool to provide a
specification from which ASTER generates code. The linear covariance analysis algorithm
described in Chapter 2 is combined with the data entered by the engineer to generate the
ASTER block diagram input format. The interaction among the tools used for code
generation is shown in Figure 4-18.
Linear
covariance
analysis tool
Algebraic
expression
handler
User
Interface
Equation
analyzer
L
Block
diagram
generator
Block
diagram
generator
ASTER builder of internal
representations for block diagrams
ASTER
code
generator
code
Figure 4-18: Code generation process
The goal of the linear covariance analysis tool is to create source code from the original user
input and an understanding of linear covariance analysis as described in Chapter 2. In
order to demonstrate the link between user input, data structures, and the generated code,
the transformation process from user input to ASTER input is demonstrated for a relatively
simple example. The example is based upon a description of the differential equation for an
error term. Firgure 4-19 demonstrates the entire process from user input to Ada code.
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<gyro-drift-object>
sensor-definition: <IMU-object>
equation-type:
"Equation"
F
components:
(<component-object#1>)
noise:
<noise-object>
manifestations:
nil /
<noise-object>
,.Jýý
noise-type:
equation:
algebraic-transform:
error-term:
inputs:
-
rnal
Inte
Representation
<component-object#l>
equation:
"-1.O/taugyro drift"
algebraic-transform: <algebraic-expression-handler-object>
error-term:
<gyro-drift-object>
inputs:
(<signal#1>)
"White noise"
"ugyro-drift"
<algebraic-expression-handler-o bject>
<gyro-drift-object>
(<signal#2>)
<signal2>
<signal#2>
name:
description:
aster-def:
type:
frame:
units:
"u_
.gyro_drift"
....
nil
"Float"
<n -frame-object>
Pol
source:
<constant-object>
<signal#1>
name:
description:
aster-def:
type:
frame:
units:
source:
"tau_gyro_drift"
off
nil
"Float"
<no-frame-object>
oil
<mission-load-object>
Figure 4-19(a): Start of the software development process.
ASTER
Specification
Generated
Code
F := matrix_insert(F, 7..9, 7..9, -1.0/taugyrodrift*identity(3));
Q := matrixinsert(Q, 7..9, 7..9, ugyro drift*identity(3));
Figure 4-19(b): Completion of the software development process.
The right-hand side of the differential equation is entered in terms of components which can
be summed together to produce the time rate of change of the error term being specified.
Each component consists of a coefficient and an error term that it multiplies. This input
mechanism guarantees that the differential equation is linear. The data is placed into the
internal data representation as shown in the figure. The data structures involved in this
window are described in Section 4-3. Since the structure of the data objects closely
matches the user interface format, this storage process is not difficult.
The interal objects are then used to generate an ASTER specification as shown in the third
part of the process. Finally, the error term differential equation is included in the code.
The error terms appear in the form of the dynamics matrix, F. The dynamics matrix is
simply the expression of the error term differential equations in matrix form. The noise
objects are placed in the noise intensity matrix Q. This process is identical to the process
for setting the values of the dynamics matrix.
At the highest level, the simulation code consists of two parts, the initialization and the
simulation loop itself. Five different quantities must be initialized by this section:
covariance matrix, consider state array, celestial body positions and velocities, orbiter
positions and velocities, and the dynamics and noise matrices. The main simulation loop
performs two tasks, propagating the covariance matrix and the state vectors of the
platforms and celestial bodies and determining the measurements that must be performed
and updating the covariance matrix to account for the new information. The top-level
diagram is shown in Figure 4-20.
Initialization
.
* Main simulation loop
0
0
P
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0.
0
0
0
0
0
0
0
0
p
0
0
0
0
0
p
A
Figure 4-20: Top level functional diagram.
The first step of the ASTER object generation is to create the error state vector. As shown
in Figure 2-1, the error state vector is organized by platform. Within each platform, the
first six elements are the position and velocity errors. These are followed by the error
terms of sensors on board the platform with terms that depend upon the position and
velocity errors. Finally, error terms of sensors on board the platform that do not have error
terms that depend upon the position and velocity errors. Ideally, the first error terms after
the state errors would be just the error terms that depend upon those errors, not all of the
error terms in the sensor. This implementation was chosen for simplicity in the design of
the code generators and for clarity in the generated code. This process is repeated for all
platforms included in the mission.
4.4.1 Initialization
Initialization consists of five major components, initializing the covariance matrix, consider
state array, celestial body state variables, orbiter state variables, and the dynamics and noise
matrices.
The first procedure is the initialization of the covariance matrix. The initial covariance
matrix data is stored as a hash table in the initial-covariance slot of the simulation object.
The initial covariance matrix is created by using the following algorithm:
loop through the error state vector objects
(identifies the row of the covariance matrix)
loop through the error state vector objects
(identifies the column of the covariance matrix)
create an ASTER constant for the initial value
--this is found by looking in the hash table based upon the
two error terms
set the constant into the covariance matrix
--the row(s) of the covariance matrix are identified by the
state-vector-location slot of the outer loop error term
--the column(s) of the covariance matrix are identified by the
state-vector-location slot of the inner loop error term
connect the ASTER set object to the output
end loop
end loop
A sample block diagram generated by this algorithm is shown in Figure 4-21.
n 1*I
not
0
Figure 4-21: ASTER block diagram input for the initialization of the covariance matrix.
The symbol I stands for the identity matrix.
The first step is also responsible for creating the initial error state vector values. This is
done ina similar fashion. The algorithm isas follows:
loop through the error state vector objects
generate the ASTER constant
--this data is stored as a hash table based upon the error term
in the implementation object
create the ASTER set construct
--the element(s) of the error state vector to be set are stored in the
state-vector-location slot of the error term
connect the set to the output
end loop
The resulting block diagram is very similar to Figure 4-21 except that the ASTER set
objects have only one branch because they are setting elements of a vector instead of a
matrix.
The second step is to create the function that creates the consider state array. The consider
state array is an array of booleans that determine whether a term is to be "considered" when
formulating performance estimations. The consider state array does not necessarily have to
be the same size as the error state vector because each term of the error state vector has only
one corresponding term in the consider state array. However, for simplicity, the consider
state array is made to be the same size by setting vector-valued error terms to have three
elements with the same value.
The third function is to create the initial position and velocity vectors for any celestial
bodies included in the mission. Since the current version of the automated linear
covariance analysis tool is only designed to handle a single celestial body definition in a
mission, this celestial body is placed at the center of the inertial coordinate system.
Therefore, the position and velocity vector for the body are both zero. However, the
celestial body may still rotate. In terms of ASTER inputs, this is accomplished by creating
two vectors of zeros and connecting them directly to the two outputs of this transform.
The fourth initialization procedure is to create the initial state vectors for any orbiters
defined in the mission. The initial state vectors are input as a set of six orbital elements.
However, the linear covariance analysis algorithm requires the position and velocity vectors
of the orbiters instead. Therefore, they are converted using a set of equations developed by
Battin 6:
cos Q cos 0 - sin 0 sin 0 cos i
r = r sin
cos -cos
sin0cosi
sin 0 sin i
V= -
(3)
cos 9 (sin 0 + e sin o) + sin Q (cos 0 + e cos co) cos i
sin Q (sin 0 + e sin o) - cos Q (cos 0 + e cos o) cos i
- (cos 0 + e cos co) sin i
(4)
where:
p= a(1 - e2)
h = 1PA
r=
Parameter
Angular momentum
p
1 + e cos f
0=c0+f
Magnitude of the position vector
a
e
Semi-major axis
Eccentricity
i
Inclination
f1
Longitude of the ascending node
Argument of pericenter
True anomaly
f
These equations are then implemented in terms of block diagrams. This function is an
illustrative example of the disadvantage of using the block diagram input format. The block
diagrams for these equations are complicated, and they obscure rather than clarify the
design. Figure 4-22 shows the block diagram corresponding to equation (3). A block
diagram for equation (4) would be even more complicated.
Figure 4-22: ASTER block diagram for calculating the position vector from
the orbital elements.
The position and velocity of the orbiter must be initialized because the orbiter states are
propagated automatically by the linear covariance analysis algorithm. The state vectors for
maneuvering vehicles and rovers are read from a file so no initialization is required.
Beacons also do not require initialization because their state vectors are determined by the
celestial body they reside upon.
Finally, the dynamics and noise matrices are initialized. Each of these matrices can be timedependent; however, most of the terms are zero, and many of the remaining terms are
constant. Therefore, the linear covariance analysis object generator places constant terms in
each of the matrices in the initialization routine to prevent them from being re-computed on
each iteration. Otherwise, the mechanics of this process are similar to those of the other
initialization procedures.
4.4.2 Main Simulation Loop
The main simulation loop performs two primary tasks, propagating the covariance matrix
and state vector information and performing measurements.
The first major step of the main simulation loop is to update the covariance matrix and the
celestial body and platform state vectors. The covariance matrix dynamics are governed by
the Lyapunov equation (2) in Section 2.3.1. This equation is a function of the dynamics
and noise matrices. Before the integration takes place, the time-varying elements of both
matrices must be updated.
Currently, the only available integration scheme is a 4th-order Runge-Kutta algorithm.
This is used to integrate the covariance matrix and orbiter state vectors. Once multiple
celestial bodies can be included in a mission, the algorithm will be used to integrate their
state vectors as well. The Runge-Kutta integration technique is described in Chapter 2.
The corresponding block diagrams are shown in Figures 4-23 and 4-24.
t
A
i=1
l
.l
.
J
IF
If'4
Figure 4-23: Runge-Kutta integration block diagram for the covariance matrix.
i = 1..4
values of r and v
after the iteration
iscomplete
Figure 4-24: Runge-Kutta integration block diagram for orbiter state vectors. The block
diagram would be the same for integration of celestial body state vectors.
State vectors are also determined for maneuvering vehicles, rovers, and beacons.
Maneuvering vehicle state vectors are loaded directly from a file. The position and velocity
vectors are assumed to be in inertial coordinates. For rovers, the position vector describes
the position of the rover relative to the celestial body-fixed coordinate frame. This position
is converted to the inertial coordinate frame and added to the position of the host body
relative to the origin of the inertial coordinate frame. For single-body problems, the body
and the center of the inertial frame are collocated. Beacon dynamics are maintained in a
celestial body-fixed coordinate frame, so the position does not change. This greatly
simplifies maintaining the state vector, but it adds complications in other parts of the
system.
The second step is to determine which measurements should take place and then update the
covariance matrix to account for the new information provided by the additional
measurement. This process for this is detailed in Chapter 2. Once again, the algorithm
must be converted into a block diagram format.
The first part of the measurement procedure is to determine which measurements take
place. This begins by checking if a measurement is scheduled, since this test will fail most
frequently. The measurement schedule is an array with one entry for each sensor used in
the mission. If the element of the vector corresponding to the sensor taking the
measurement is less than the current simulation time, a measurement is performed. The
element is then incremented by the reciprocal of the measurement frequency. If a
measurement frequency is faster than the simulation frequency, the inverse of the
measurement frequency will be smaller than the time step, so a measurement will be
scheduled for each iteration. Regardless of the commanded measurement frequency, a
sensor will not repeat a measurement in a single time step.
Scheduling a measurement does not imply that it is actually performed; a measurement can
only be taken if the sensor is on and all operating constraints are met as well. Determining
whether the sensor is on is accomplished by checking the current simulation time against
any on-off pairs which have been defined for the sensor. This is shown in block diagram
form in Figure 4-25.
90
Figure 4-25: Block diagram for determining whether a sensor is on.
The measurement constraints are generated by ASTER by taking the equations entered by
the user and using the algebraic expression engine to convert the equations into block
diagrams. The resulting boolean variables are fed into an and gate to determine whether the
constraint should occur.
When a measurement is taken, the first step of the update process is to calculate the
measurement sensitivity vector. Most of the elements of the vector are zero. The only nonzero elements are those corresponding to the sensor taking the measurement, its host
platform position and velocity error vectors, and the position and velocity error vectors of
any associated bodies. The equations to determine the value of these terms are entered
through the user interface and translated into block diagrams by the ASTER algebraic
transform capability.
Finally, the update equations must be converted from the algorithmic format described in
Chapter 2 into block diagrams. The resulting block diagram is shown in Figure 4-26.
to
ine)
Figure 4-26: Block diagram for the measurement update equations.
After the functions are created, they are connected as shown in Figure 4-19. The final
product is then sent automatically to ASTER for code generation, and the process is
complete.
Chapter 5: Results and Conclusions
The overall goal of this project was to design and develop an automated tool for developing
linear covariance analysis code. The work that has been completed is sufficient to
determine whether the stated goal is possible, although evaluation of the tool and
subsequent refinement may be the topic of future research.
The first major accomplishment is the development of design specifications for a linear
covariance analysis tool. The navigation community does not have a standard for
specifying mission designs for linear covariance analysis. The hierarchical design language
developed in Chapter 3 is a first step towards creating that standard. It takes advantage of
advanced design specification techniques and lends itself well to automation as
demonstrated by the close resemblance of the design language to the user interface to the
linear covariance analysis tool. Recognizing that the navigation community may ultimately
adopt a standard that is different than the one proposed here, the automated linear
covariance analysis tool is designed such that the user interface can be easily changed to
conform to the new standards without major changes to the underlying data structures or
the ASTER object generation system.
The next major step is the completion of the specification user interface. A user interface
has been designed and built that successfully captures linear covariance analysis designs.
The specification language successfully matches the design language. Most differences
between the natural design language and the mission specification language make the
specification process easier for the user. For example, the list of error terms to be included
in the model is written by hand (or typed into a computer) in the design language, but in the
user interface, a list of all the error terms is provided and the user simply checks the terms
to be included. Obviously, this is inefficient to do by hand, but on the computer checking
the desired terms is faster than typing them in.
The final part of the tool development process is the code generation system. This system
takes the internal data structures and creates source code. The linear covariance analysis
tool uses a code generation system called ASTER. This system generates code from either
block diagram specifications or mathematical expressions. Therefore, the linear covariance
analysis tool is only required to produce the appropriate block diagrams or mathematical
equations rather than developing code.
The ability to take a mission specification, transform it into the linear covariance analysis
tool internal data representation, generate the ASTER input format, and generate Ada code
has been successfully demonstrated for a problem exercising a reduced set of the linear
covariance analysis tool's functionality.
Overall, the use of the ASTER code generation system has greatly speeded the development
of the linear covariance analysis tool. However, using the tool presented some difficulties.
One of the biggest problems is the block diagram specification technique. Block diagrams
are not a natural way to specify an algorithm such as linear covariance analysis. In the
absence of the specification tool, a navigation engineer would be unlikely to use ASTER to
specify a linear covariance analysis simulation despite the fact that ASTER would perform
the code generation.
In particular, ASTER should have an algorithmic specification mechanism similar to the
algebraic expression handler but with additional functionality such as loop and if-then-else
structures. This may seem to be a step backwards, since the specification looks more like
code, but this is not the case. Programming languages were developed to perform
algorithms, so the languages naturally attempt to parallel the algorithm specification
technique in the same way that the linear covariance analysis tool mission specification
language attempts to match the natural design language.
The purpose for automating linear covariance analysis simulation development is to reduce
development time while increasing reliability and maintainability, so these form the ultimate
evaluation criteria for the tool. The current system is sufficiently developed to provide
some suggestions as to the gains that can be expected.
The first step of the simulation development process is to specify the mission. In the
traditional development process, this involves pulling together code for old sensors,
platforms, and celestial bodies, and then modifying an existing linear covariance analysis
simulation to use these objects. If the objects do not already exist, they must be coded. In
the automated linear covariance analysis tool, the mission is created graphically. When
new models are included, they are entered through the user interface described in Chapter
4. Specifying sensor, celestial body, and platform models requires about the same amount
of time with either system, especially since the majority of the time is spent finding or
developing the model. However, putting together a mission is much easier in the
automated tool, so this part of the process will proceed much more quickly.
The next step of the process is developing the simulation code. In the traditional system,
the code development and mission specification are the same thing. However, once the
code is written, it must be debugged. This can be a very time-consuming process. In the
automated tool, code is generated by pressing a button. The code generation is completed
in a few minutes, and the resulting code requires far less debugging time than handgenerated code. This feature provides the largest single time savings offered by the
automated linear covariance analysis tool.
Finally, the generated code is compiled and run. Both the automated tool and handgenerated code can take advantage of Shepperd's form of the Joseph equation to reduce
computation for each measurement update, but implementing the additional improvements
outlined in Section 2-5 is error prone. Using the computational savings in hand-generated
code would make the resulting code operate as quickly as the automatically generated code,
but the development and maintenance time would be greatly increased, cancelling any
benefits gained in execution speed.
Although time restrictions precluded a detailed study of the output evaluation and iterative
design process, it appears that the automated tool can also be used to aid in the presentation
and analysis of the results from the linear covariance analysis simulation. Some basic ideas
for viewing data are discussed in Section 4.1.3, but much more research into analysis aides
is necessary.
Overall, the results demonstrated by the automated linear covariance analysis tool show that
automating linear covariance analysis problems specification and software development is
possible and promises large time savings throughout the software development cycle.
96
Chapter 6: The Next Step
The work presented in this document represents a preliminary design and implementation
of an automated system for navigation linear covariance analysis. The current tool provides
a framework for additional development which can greatly enhance the scope and utility of
the tool. These advances fall into two major categories: adding to the navigation linear
covariance analysis system, and increasing the breadth of the tool to apply to a wider range
of problems.
Before either of these advancements can be made, the functionality described in this
document must be fully implemented and tested. This will provide a solid framework from
which to add further capabilities.
The first category is to improve the navigation linear covariance analysis system. One
particularly beneficial capability is multiple celestial bodies in a mission. This introduces an
additional level of complexity to the linear covariance algorithm. However, many advanced
missions being studied today involve trips to the Moon and to Mars, so a multiple body
feature is very important.
Another major improvement is analysis aides. Since the automated linear covariance
analysis tool will reduce the development, testing, and execution time of a linear covariance
analysis simulation, the engineer will be spending a larger percentage of his or her time
analyzing the results. Therefore, the linear covariance tool should provide capabilities that
speed the analysis process as well. Some preliminary suggestions are discussed in Chapter
4, but a thorough investigation into the methods used for viewing and analyzing the data is
required.
The second group of improvements increase the breadth of problems covered by the linear
covariance analysis tool. These capabilities build on the current system. They do not
require major changes to the user interface or design methodology.
One potential extension is to enable the linear covariance analysis tool to develop the
navigation flight software. Many navigation software systems are based upon a Kalman
filter. The Kalman filter is essentially a reduced-order version of the linear covariance
analysis. Therefore, the flight software development tool can use the existing database
information with minimal additional user input. This could then be mated in ASTER with
control and guidance block diagram specifications to create a complete GN&C system.
Second, the measurement update equations are currently coded into the ASTER object
generation system. This should be changed such that the users can enter the equations
themselves. This provides two new capabilities. First, the current linear covariance
analysis is a statistical simulation. By allowing the user to specify the equations, a
deterministic simulation could be performed instead. Second, linear covariance analysis is
often applied to guidance systems. This process requires a different set of update equations
than those for navigation analysis, but they operate on essentially the same information, so
no major user interface changes would be required.
Overall, the current system provides the foundation for further improvements in
development capabilities for guidance and navigation problems. Even in its infancy, the
automated linear covariance analysis tool can save many hours of development time. As it
evolves, the tool can grow to meet ever expanding needs for analysis of guidance and
navigation systems.
References
IShepperd, Stanley W., Personal communication.
2Turkovich,
John J. "Automated Code Generation for Application Engineers."
Proceedingsof the AIAA DigitalAvionics Systems Conference, October 1990.
3Ogata,
4ALS
Katsuhiko, Modern Control Engineering,Prentice-Hall, Inc., Englewood Cliffs,
N.J., 1970.
CASE User's Manual, Charles Stark Draper Laboratory, Cambridge, MA, 1991.
5Booch, Grady.
Object-OrientedDesign with Applications. Benjamin-Cummings
Publishing Company, 1991.
6Battin,
Richard H., An Introductionto the Mathematics andMethods of Astrodynamics,
American Institute of Aeronautics and Astronautics, New York, N.Y., 1987.