Other useful resources:
1. Introduction
I have found that I can easy resolve very complicated science and engineering
problems using my software. But it is very difficult to teach others. I am
trying different methods of teaching. One of them is "live parallel" with well
known software. Here I describe live parallel with
The Incredible Machine. I do not need dithyrambs. I am
looking for readers which have strong interest to my ideas. My ideas are not
fastfood. My articles are also my excercises explanation of my ideas. And
feedback of readers is are very helpful. So I am writing this article in real
time mode
2. Background
I find that The Incredible Machine very useful game. When I was a
child I did not have computer. But I had constructed Grandiose Projects in my
mind. These projects were very close to
The Incredible Machine tasks. These tasks were not
realistic. Then I had learned math, physics and could solve realistic tasks. My
software is realistic. But I also is being use unrealistic samples for teaching.
Although The Incredible Machine is not realistic it is very useful for
education. Here I will compare my software with The Incredible Machine.
3. Common features of Framework and The Incredible Machine
3.1 Qualitative variety of phenomena
The Incredible Machine contains a large qualitative variety of phenomena. It
includes geometry, mechanics, electricity, pneumatic, explosions and even
psychology of animals. A lot of advanced engineering software do not contains
such variety. For example LabVIEW
contains a lot of components, but LabVIEW do not have mechanical components.
Similarly the Framework has mechanics, electricity et cetera.
3.2 Interaction of phenomena
The Incredible Machine has interaction of phenomena. For example, if a ball is
close to fan than air flow acts on the ball. Typical Framework example is an
action of Earth's magnetic field on spacecraft motion.
3.3 Declarative approach
For simulation of air flow action on ball we need set ball position only. We do
not need following imperative: "Flow, act on the ball". Action is provided
implicitly. The same words we can say about action of Earth's magnetic field on
spacecraft motion.
3.4 Relation to Grandiose Projects
Grandiose Project has a lot of qualitatively different phenomena. For example if
electrical device is near heat source then electrical processes depend of heat.
So there is interaction between phenomena. Declarative approach makes software
development robust and easy.
4. Difference between Framework and The Incredible Machine
The incredible machine devoted to unrealistic phenomena only. The Framework is
used for complicated science and engineering problems.
5. Getting Started
This article will become really interesting when I will fulfill it by samples. I
am writing it mostly at weekends. And first sample will appear in next weekend.
One objective of this article is research of popularity. This article does not
contain any code yet. But current number of visits is my own record. I think
that such popularity caused by article name. But I strictly comply following
CodeProject rule: "Article should contain code". I think this rule do not
contradicts my research. Moreover other members of CodeProject can enlarge their
popularity by usage of good article name. I find that it is not interesting
write article about soft that was developed later, since the Framework can
resolve unpredictable tasks. So I will be thankful to reader who asks me to
resolve unpredictable task. It would be a good test for the Framework. Maybe I
will include some of suggested tasks into this article. If one of readers would
construct a sample of incredible machine then it would be excellent. I shall
include this sample into this article. I had constructed incredible machines
many years and do not have very interest for this occupation. I would like that
another developers be able construct incredible machines.
5.1 Kinematics
5.1.1 Very simple sample
This sample is "Hello world" of the Framework. It is presented in following
picture:
We have a point Motion which is moving relatively point
Base. We can simulate this picture by the Framework:
Kinematics is defined by
Math object. It has following
properties:
Derivation order is equal to 1 for calculation of
relative velocity.
The Math object is used by Motion object.
Properties of Motion object are presented below:
This means that x  coordinate of Motion object is defined by
Formula 1. Other coordinates and Q_{1}, Q_{2},
Q_{3} components of
orientation quaternion are defined by Formula
2. The Q_{0} component is defined by Formula 3. Since Formula 1
= t, Formula 2 = 0 and Formula 3 = 1 we have
following 6D motion law:
x(t) = t;
y(t) = 0;
z(t) = 0;
Q_{0}(t) = 1;
Q_{1}(t) = 0;
Q_{2}(t) = 0;
Q_{3}(t) = 0;
The Measurements object measures parameters of Motion object
motion relatively the Base object. The Graph
object indicates parameters of relative motion:
Red curve indicates relative distance and blue one indicates relative velocity
5.1.2 Sample with medium frame
Kinematic picture of this sample is presented below:
We have three frames. The frames are named as Base,
Medium and Motion. The framework shows this picture by
the following way:
The L 1 and L 2 are geometric links. The
L 1 links means that motion of Medium frame is
considered relatively to Base frame. Similarly link L 2
means that motion of Motion frame is considered relatively
Medium frame.
The Medium frame has following properties:
These properties mean that relative coordinates of Medium frame
are equal to:
x = 0;
y = 0.5;
z = 0;
Properties of Motion are the same as in chapter 5.1.1. Arrows between
Measurements  Base and Motion 
Measurements mean that Measurements provides parameters or
Motion frame motion with respect to Base. The
Chart object indicates relative motion parameters:
Red curve is relative distance and blue one is relative velocity. Note that
differentiation of distance is implemented implicitly. We only put frames and
automatically obtained distance derivation. The Framework performs symbolic
differentiation. But differentiation is performed implicitly. Formulas do not
contain functions which are presented in above charts. These charts are defined
by relative positions of frames. But engineer have a lot of other problems which
are outside calculation of explicit dependencies.
5.1.2 Sample with rotated frame
Readers. Excuse me please during construction of next sample I has found a bug.
The bag is fixed. If you have already downloaded Aviation project you should
download it once again. The Framework is being continuously developed. Previous
samples have been tested.
Kinematics of this sample is presented below:
This sample contains four frames. Properties of their coordinat systems are
presented below:
Frame

Frame center

Coordinates

Base 
B 
x_{B},y_{B} 
Medium 
Med 
x_{Med},y_{Med} 
Rotated 
Rot 
x_{Rot},y_{Rot} 
Motion 
Mot 
x_{Mot},y_{Mot} 
The framework picture is presented below:
Base  Medium  Rot 
Motion
The link
Medium  Rot
is not rigid. Its motion low is presented below:
x(t) = 0;
y(t) = 0;
z(t) = 0;
Q_{0}(t) = cos(at);
Q_{1}(t) = 0;
Q_{2}(t) = 0;
Q_{3}(t) = sin(at);
where t is time and a is constant.
It means that Rot is rotated with respect to Medium.
Angular velocity is constant.
Following chart presents relative motion of Motion frame with
respect to Base frame:
Writing temp of this article is very slow. But my experience tell me that
understanding of my ideas is more slow. Readers which have strong interest to my
ideas can download above samples and try to develop other. I will accept any
questions and suggestions.
Wait next weekend.
5.1.3 A complicated set of frames
Set of frames can be more complicated. Complicated set of frame is presented
below:
The framework diagram which correspond to this set of frames is presented below:
Now we can calculate relative distance and velocity between frame 1.3
and 2.3. We have following complicated curves:
This sample can be regarded as too artificial. But I know a lot of engineering
problems with similar configuration. Reader can find these samples in my
articles
Virtual Reality at Once and "Control systems. Processing of signals"
5.1.4 Trajectory import
Today (02/07/2008) I have received a message about lack of code. Thanks. Indeed
this article is not interesting without code samples. So I have prepared a
little code for this article. You can download it:
This code is used in framework for kinematics. Third party software is used by
IObjectFactory
interface. Sample code contains class
ComplicatedTrajectory
that implements this interface:
string[] IObjectFactory.Names
{
get { return new string[] {"Complicated Trajectory"}; }
}
ICategoryObject IObjectFactory.this[string name]
{
get { return new ComplicatedTrajectory(); }
}
The Names
property contains names of exported objects. Property
this
returns object by name. Class ComplicatedTrajectory
impelements IObjectTransformer
interface:
void IObjectTransformer.Calculate(object[] input, object[] output)
{
double a = (double)input[0];
double b = (double)input[1];
double t = (double)input[2];
double x = a * t;
double y = x * t;
double z = y * t;
double phi = Math.Atan(b * t);
double Q0 = Math.Cos(phi / 2);
double Q1 = 0;
double Q2 = 0;
double Q3 = Math.Sin(phi / 2);
output[0] = x;
output[1] = y;
output[2] = z;
output[3] = Q0;
output[4] = Q1;
output[5] = Q2;
output[6] = Q3;
}
This code calculates parameters of trajectory. The t is time, a
and b are constants. Meaning of output parameters is evident. So we
have third party project and we would like use it for kinematics. Following
picture demonstrates this usage:
The Import is imported object. It has following properties:
One of properties is *.dll filename. Second one is object name. Note that *.dll
is saved in file *.cfa and then project does not depend on initial *.dll file
existence. Full diagram enable us to define Motion object
trajectory. The Motion  Base object performs relative motion
measurements:
Red, blue and green curves a relative x, y and z
coordinates respectively. Note that we cannot define velocities since imported
library do not contain derivations. User shoud implement a set of interfaces for
definition of velocities.
5.2 Dynamics
Realistic mechanics requires differential equations of dynamics. The framework
has differential equatioins' component. This component can be used for
mechanics. Here we consider a set of samples.
5.2.1 Mechanical oscillator
This sample is classical. It is presented below:
This sample is described by following differential equations:
dx/dt = v;
dv/dt = av  bx;
where x is coordinate and v is velocity. The "bv" is caused by
damping
factor and "bx" is caused by force of string. The framework
picture is presented below:
Here the Equations object contains dynamical differential
equations. Chart of relative distance and velocity is presented below:
We have damping oscillations as it have been expected.
5.2.2 Jump
The incredible machine contains jumps. Adequate realistic description of jumps
provides Dirac delta function. The framework has it. Framework
sample of oscillations with jump is presented below:
The Jump object contains Dirac delta function:
So we have jump at following charts:
At this chart red curve is a distance and blue curve is a velocity. The distance
is a
continuous function. But its derivation (distance) has a
point of discontinuity (Jump) as it has been expected.
5.2.3 Realistic Sample. Earth artificial satellite
Hello everybody. I have updated source code of Aviation project ad uploaded it.
Let us consider realistic sample. Earth artificial satellite had already been
considered in
article 6 and
atricle 12. Here I would like to compare imperative
approach with declarative one. This sample requires import of third party
software (external libraries). The
Asrtoframe
project user guide (1.94 MB) contains import instruction. Also reader can
download source code of third patry software:
The task is presented in following picture:
We have a Satellite and would like simulate its motion
parameters with respect to Ground Frame.
5.2.3.1 Imperative approach
Imperative picture is presented below:
The Gravity and Atmosphere are imported
objects. These objects are used in Motion equations object
which contains
ordinary differential equations of satellite. The
Distance and Velocity objects contain explicit formulas of relative distance and
velocity:
The a, b and c are coordinates of Ground
Frame. These coordinates are constants which belong to Distance
object. Values of constants you can find in Distance
properties:
Result of simulation is presented in following charts:
These charts show relative distance and relative velocity respectively.
5.2.3.2 Declarative approach
The picture of declarative approach is presented below:
This picture do not contain explicit formula of distance. Instead we have
reference frames. The Motion frame is reference frame of
satellite. This frame uses parameters Motion equations object.
The Frame 1 is ground frame. Its coordinates are presented
below:
This coordinates are equal to coordinates of Ground Frame in
imperative approach. We also have Frame 2. This frame is
presented for demonstration of advantages of declarative approach. Objects
Measurements 1 and Measurements 2 perform
relative measurements "Motion  Frame 1" and "Motion 
Frame 2" respectively. Simulation result is presented in following
charts:
Every chart contains two curves which correspond to Frame 1 and
Frame 2 respectively. Coordinates of Frame 1
are equal Ground Frame coordinates to imperative approach
coordinates. So one curve on each chart is the same as in imperative approach.
5.2.4 Mechanics and differentiability. Declarative approach once again
Differentiability has important role in mechanics. However engineer should not
be too overloaded by differentiability issue. A lot of work should be performed
declaratively. Let us consider following mechanical task. We have a moved object
and its coordinates are time functions x(t), y(t), z(t).
If we would like to install (virtual)
inertial navigation system on this object then
these functions should be twice differentiable. However second derivations can
be
discontinous. If these functions are not twice differentiable then attempt
of inertial navigation system installation results to exception. Twice
differentiable function can be obtained by natural way. For example they can be
obtained from following dynamics equations:
In these equations F_{x}, F_{y}, F_{z}
can be discontinous. But V_{x} , V_{y},
V_{z} are continuos with discontinous derivations. Functions
x(t), y(t), z(t) are twice differentiable with
discontinous second derivations. Let us consider how framework operates with
differentiability. Suppose that we would like to solve following system of
differential equations:
Here a(t) is discontinous function:
Solution of this problem by framework is presented below:
Here Input component contains above discontinous function. The
Diff Equations components is solver of above differential
equation system:
Now we would like consider x as twice differetiable function. We can do
it by defining following properties of variables:
Here x is twice differentiable and y is once differentiable.
If we try increase order of derivavative of every variable then exception will
be thrown since x is twice differentiable and y is once
differentiable only. The Diff Test component performs test of this sample by
(twice) differentiation of x:
If we try to define third derivation of x, then exception will be thrown.
Test result is presented in following picture:
Red curve is x(t), green and blue curves are its first and second devivations
respectively.
5.3 Physical fields
Physical fields are implemented abstractly. It enables us use them in
radiophysics, magnetism, pneumatics. Moreover physical fields provides
advantages of declarative approach. Development becomes more laconic. Let us
consider some examples.
5.3.1 Pneumatic
Let us consider a "Fan and Ball" sample:
We have Base reference frame. The Y and
Z are its axes of reference. Axis X is
perpendicular to Y and Z. The picture shows
wind rose of fan. We would
like simulate pneumatics as physical field. So we have constructed following
diagram:
This picture has three parts: Fan, Ball and
Bridge. Fan is a source of physical field.
Ball is a consumer of field. The Bridge performs
interoperability. We have field interoperability and kinematic interoperability.
Let us consider all subjects of this picture.
5.3.1.1 The Fan
Fan has a reference frame (
Field Frame) and physical field (
3D
Field). The field is strongly linked with frame:
The
3D Field has following properties:
Field parameters can have different types. Here field parameters are
Field
Result.Formula_1. It means that parameters are provided by
Field
Result object.
Formula_1 provides 3D vector. So we have vector
field. The field has "Covariant" flag. Following picture explains meaning of
this word.
If 3D vector is not covariant then its components depend on sensor position
only. Covariant vector components depend on both orientation and position.
Values of components are projections of geometric vector to sensor's axes of
reference. The picture above presents two orientations of sensor: blue and
green. Projections of field vector
A are different for these
different orientations. Besides covariant vectors Framework supports covariant
tensors. Such
tensors can be used in
Gravimetry. The
Field Result object
properties are presented below:
Right pane shows that types of
k and
x are
Double
and
Double[3]
respectively. So
kx is scalar  vector
product. It this formula is used by
3D Field. The
x is
relative vector and
k is wind rose coefficient defined by formula
where phi is angle between reference line and z  axis of
Fan.
r is a distance between field source (Fan) and sensor
.
5.3.1.2 Bridge
Bridge contains common Base reference frame and declarative
link Field between 3D Field (of Ball) and
Sensor. This link means that Sensor is a sensor of 3D
Field. Any field can have a set of sensors which have different
positions and orientations.
5.3.1.3 The Ball
Configuration of the Ball is presented below:
The ball has reference frame (Ball) and a sensor (Sensor).
Ball and Sensor are rigidly linked. Sensor is a sensor of Fan
pneumatic field. Output of Sensor is used (through Disass) in
differential equations Equation. Otherwise Equation result is
used for the reference frame Ball motion simulation.
5.3.1.4 Declarative approach
Declarative approach enables us simulate a lot of properties by changing of few
parameters. For example we can change orientation of Field Frame
(frame of Fan). In result the Ball trajectory will be changed. Two different
Field Frame orientations are presented below:
These orientations are defined by
transformation matrix. Change of orientation results to following change of
the Ball trajectory. You can compare two charts below:
These charts shows time evolution of z  coordinate of the Ball.
However the Fan position and orientation do not force only the Ball trajectory.
Relative position of Ball with respect Fan depends on position of Fan.
5.3.2 Dipole radiation + Kinematics
Here we use kinematics picture which is considered in 4.1.2. Besides kinematics
we will consider
dipole radiation. Full picture is presented below:
We have three parts: Dipole, Bridge and Receiver. Let us describe them.
5.3.2.1 Dipole
Dipole uses kinematics which is used in 4.1.2. The Field Calculation
object contains calculation of dipole field parameters: amplitude and phase.
Amplitude formula is presented below:
Here m is dipole moment, r is distance, x is relative
radius vector. The cross product means
vector
cross product. In fact this formula coincides with one
dipole radiation
formulas.
The Field object contains two components:
These components are Field Calculation.Formula_1 and Field
Calculation.Formula_2. First one is vector and second one is scalar. First
component physically means amplitude. Second one means phase.
5.3.2.2 Bridge
Bridge contains the Base object and the Link
link. The Base is common reference frame. The Link
is declarative link between source of field and receiver.
5.3.2.3 Receiver
Receiver contains Sensor of field and Receiver
object. The Receiver object performs processing of Sensor information by
following expression:
Here x, y, z are components of Field amplitude, o
is
angular frequency, t is time and p is phase. Note that
this formula is rather reasonable interpretation than realistic receiver
formula.
In result we have following receiver signal:
Let us provide its qualitative explanation. First of all we have amplitude peak.
Also we have monotone decrease of frequency. These phenomena can be explained by
motion. Motion chart is presented below:
Red curve is distance between field source and receiver. Blue curve is relative
velocity. Amplitude peak is caused by distance extremum. Monotone decrease of
frequency is caused by monotone increase of relative velocity.
5.3.3 Dipole radiation + Kinematics + Envelope detector
Let us add to previous situation
envelope detector.
The envelope detector can be implemented by following circuit:
This circuit is simulated by the following way:
The Rectifier object contains formula of rectifier. The
Filter object simulate dynamic object with following
transfer function:
where
.
The Filter object is in fact a
lowpass
filter with following
frequency response:
Following chart shows result of envelope detector:
Red curve is output of Rectifier. Blue curve is output of
Filter. So blue curve is in fact envelope (with coeffcient).
Reader need Aviation + Control Systems project for this sample:
5.3.4 Dipole radiation + Kinematics + Frequency detector
Frequency dectector can be implemeted by usage of
highpass filter. Scheme of filter is presented below:
Filter has following transfer function:. Frequency response and
phase
response of the filter are presented below:
Following scheme is used for amplitude variation compensation:
This scheme contains envelope detector (see 5.3.3). Result of scheme bottom
branch is divided by output of envelope detector.
The framework picture of frequency detector is presented below:
Left part contains envelope detector (see 5.3.4). Right part contains highpass
filter (Diff) rectifier (Rectifier 1) ald
lowpass filter (Filter 1). The Detector
object divides output of Filter 1 by output of Filter.
Output of Detector is proportional to frequency. It is
presented below:
Such behavoir of frequency can be easy explained by following chart of relative
velocity:
Reader need Aviation + Control Systems project for this sample:
5.3.4 Dipole radiation + Kinematics + Phase detector
Here we consider application of
phase
detector. Situation is presened in following picture:
Dipole radiator (Dipole) is moved. We have two receivers 1 and
2. Distances D_{1 }and D_{2} are different.
The d parameter is called base. So we have phase difference. Here we
will define this difference by correlation method. This method can be expessed
by following formulas:
where I_{xx }, I_{yy }, I_{xy }
are correlation integrals which can be calculated by the following way:
where x(t) and y(t) are singnals of
recievers.
Calculation is divided on two steps. First step is simulation and recording of
signals. Second step contains calculation by correlation method.
5.3.4.1 Simulation and recording
Configuratiton of this step is very similar to configurations whis was
considered in 5.3.2 and 5.3.3. But here we have two receivers istead one.
Fragment of the situation is presented below:
Every receiver has own reference frame (Frame 1 and
Frame 2). We can change parameter d by changing coordinates of
these frames. Simulation gives following signals of receivers:
Red curve is signal of Receiver 1, blue curve is signal of
Receiver 2. Now we can record these signals and go to next
step.
5.3.4.2 Calculation by correlation method
Calculation scheme is presented below:
Left part squares contain recorded signals. Red/Blue curve is signal of
first/second receiver. Recursive element calulates necessary integals. Right
object calulates cosine of phase difference. In result we have following chart
of the cosine:
It is clear that this chart corresponds to considered kinematics. If we multiply
d (base) by 2 then we will obtain following chart of phase cosine:
This result is evident.
5.4 Regression
Nonlinear regression in
statistics is the problem of fitting a model:
to multidimensional x,y data, where f is a
nonlinear function of x, with regression
parameter θ. I find that pure regression software is not quite effective. It
should be integrated into more common framework. I had been considered
application of regression to
determination of orbits of artificial satellites
problem. The standalone regression soft could not resolve such problems. I had
considered regression in
article 2. Here I will consider new essential
feature. It is Fdistribution. Why Fdistribution? We need it for
looking for appropriate math model. Statisics knows that more complicated model
is not more adequate. For example if secection contains 20 points then usage of
19  degeree polynom leads to zero residue. So we can think that 19  degree
polynom is ideal model. But in fact this polynom reflects rather errors than
real physical picture. Here we will consider following sample:
Top part of this picture contains selection. Left part contains 4degree
polynomial model, right part contains 5degree one. The residuals of 5degree
model is less than 4degree one. It is presented in following picture:
Blue curve is 4degree polynomial approximation and red curve is 5degree
polynomial approximation. The 0.383037446029811 of Fisher is
significance level of Fdistribution
statistical
criteria. Roughly speaking this number is chance that 5degree polynom is
better than 4degree one. Usually statistics prefer the 4degeree polynom in
this case.
5.5 Algebraic topology
At the beginnig of XX century
General relativity and
Quantum
mechanics had being factastic ideas reather reality. But these brunches of
science are everyday engineering tools now. Present day
theoretical physics operates with XX and XXI centuries math. This math
includes algebraic topology. Scientist and futurist
Dr. Michio Kaku told about
making what was once considered impossible technology into
reality. I had begun implement this idea.
Category Theory project is devoted to this purpose. I
know a lot of cases when
General Relativity and
Classical Mechanics had been used in the single project. So I think that
kinematics, dynamics, field theory and algebraic topology should have universal
software framework. Framework should be common for all branches of science.
Algebraic topology could not be undrestood at once. This chapter is rather
inspiration. Reader need Category Theory project and its user guide:
Here I would lilke define some invariants of space X which is direct
product of real projective space and
Klein bottle
.
Later I will denote projective space by R.
Anatoliy Fomenko expressed these spaces by following way:
5.5.1 Calculation of homology
Homology can be calculated by chain complexes. These
complexes are presented below:
Chain complexes of projective space and Klein bottle are well known. We would
like calculate it for direct product. Chain complex of Klein bottle is presented
below:
Framework represents this chain complex by following way:
Chain complex of projective space is presented below:
Properties of objects and arrows of this diagram can be edited by right mouse
click on squares. So let us calculate homology groups of X. First of
all we will construct chain complexes of projective space and Klein bottle:
Top complex is complex of Klein bottle. Bottom complex is complex of projective
space. Gray squares K and R are complexes as
single objects. Chain complex of X is tensor product of these
complexes. We can construct it (see user guide).
Now we have chain complex of X:
Now we can calculate homology of X (see user guide):
Top complex is chain complex of X. Bottom row of squares are homology
groups of X.
5.5.2 Homology with coefficients in Z/2Z
Homology with coefficients can be calculated by
application of tensor product
functor
. Calculation scheme is prezented below:
Top complex is chain complex of X. Bootom part contains complex of
X as single object and tensor product functor. We perform
tensor product functor
.
5.5.3 Calculation of cohomology
Cohomology
can be calculated by application of Hom(Z,)
contravariant
functor. Framework presents this functor by following way:
Left square represensts Z group. Right square represents
functor. Arrow between squares means association. Application of this functor
provides
cochain complex. Cohomolgy are homology of this cochain complex. User can
download full calculation of cohomolgy of X below:
6. Advanced Sample. Controlled spacecraft
A spacecraft is a very good example of the incredible machine. It has a lot of
systems and aggregates. The picture belows presents
Mir orbital station:
This station is a subject of History. But building blocks of the framework
enable us construct new spacecrafts and orbital stations. The station is a very
complicated mechanical object. It is not rigid body. It has
solar cell
panels. These panels are elastic. The station is stabilizied by
gyros.
Moreover station configuration is not constant. I had already considered
mechanical aspects in
article 9. This chapter contains interoperability of
mechanics and other phenomena.
6.1 Mechanics
We will use aggregate designer for mechanics simulation. Why aggregate designer?
Indeed mechanical equations are well known long time ago. But software
development for simulation of complicated mechanical objects is not quite easy
task. Aggregate designer make this task much easier. Aggregate designer is
integrated into framework. This fact enables us provide interoperability of
mechanics with physical fields. So it is easy to simulate action of magnetic
fields on mechanical objects. I will consider this task below. Now we would like
to create mechanical model of spacecraft from models of its modules. Typical
spacecraft module is schematically presented below:
This module has own coordinat system OXYZ. Also it has places of
connections. We can connect other modules to this module. Behavior of module is
defined by following kinematic parameters:
 Radius vector r;
 Velocity V;
 Orientation quaternion Q;
 Angular velocity
.
But module is not rigid in general. And these parameters are not parameters of
module. These parameters are rather parameters of one point of module. In this
article we suppose that these parametres are parameters of origin of OXYX
coordinat system. Since module is not rigid it has additional
degress of freedom. These degrees of freedom can be
interpreted as generalized coordinates q_{i} (i=
1,...,n). Instant state of module is defined by following parameters:
I will call them state variables. Parameters:
will be called accelerations. Mechanical equations define accelerations by state
parameters. Accelerations near connecton can be defined by following way:
where i is number of connection. Other variables are matrixes which
depend on state variables. Let us connect two modules:
Both modules have equal acceleration near connection. First module acts to
second one by force F_{12 }and mechanical momenum M_{12.
}Similarly second module actcs to first one by force F_{21
}and mechanical momenum M_{21}. Following equations:
F_{12}=F_{21};
M_{12}=M_{21};
are well known. Mechanical equations of module can be represented by the
following way:
In these expessions accelerations are independent variables. F_{i}
(M_{i}) is force (mechanical momentum) of i h
connected module. Other vector and matrix parameters denend on state variables.
Adding following evident expressions:
results to linear by accelerations system of equations. This system enables us
to find all acceleratios. So we have mechanical equations. It is worth to note
that this system is redundant. If n_{1}(n_{2})
is number of freedom degrees of first (second) module then system has n_{1}+n_{2}
degrees of freedom. However aggegate has n_{1}+n_{2}6
degrees of freedom. The framework can avoid this redundancy. But I will not
describe it in this arctile.
There are a lot of module types. Prorammatically all of them implement following
interface:
public interface IAggregableMechanicalObject
{
int Dimension
{
get;
}
int NumberOfConnections
{
get;
}
double[] State
{
get;
}
double[] InternalAcceleration
{
get;
}
double[] this[int numOfConnection]
{
get;
set;
}
double[,] GetAccelerationMatrix(int numOfConnection);
double[,] GetForcesMatrix(int numOfConnection);
double[] GetInternalAcceleration(int numOfConnection);
double[] GetConnectionForce(int numOfConnection);
Dictionary<IAggregableMechanicalObject, int[]> Children
{
get;
}
bool IsConstant
{
get;
}
IAggregableMechanicalObject Parent
{
get;
set;
}
}
I will consider samples of modules below. Here I describe this interface.
Meaning of this interface members is presented in following table:
Number

Member 
Comment (meaning) 
1 
Dimension 
Dimension of differential equation system 
2 
NumberOfConnections 
Number of connections with other aggregates 
3 
State 
State vetor (contains independent variables of differential equation) 
4 
InternalAcceleration 
Accelerations of generalized coordinates

5 
this[int numOfConnection] 
State of connection 
6 
GetAccelerationMatrix 
Acceleration matrix of connection 
7 
GetForcesMatrix 
Forces matrix of onnection 
8 
GetConnectionForce 
Connection force and momentum 
9 
Children 
Children aggregates 
10 
IsConstant 
The is constan flag 
11 
Parent 
Parent object 
So this interface reflects parameters of above formulas. These parameters
contains necessary information for aggregate construction. It is worth to note
that if matrix of linear equations is constant then we can invert it one time
and this fact enable us simplify solution Therefore this interface has
IsConstant
flag.
6.1.1 Aggregate library
Complicated spacecraft a lot of aggregates. Software models of these aggregates
should be developed by different specialists. These software models should also
reflect phenomena which do not belong mechanic domain. I has developed simple
aggregate library especially for this article.
Here I describe it.
6.1.1.1 Rigid body
Equations of rigid body are well known and I will not present them here. Rigid
body of this library has variable number of connections. Its editor of
properties is presented below:
We can change number of connections their positions and orientations.
6.1.1.2 Elastic console
Elastic console body is a mechanical system of infinite degrees of freedom.
Usually math model of this object contains finite degrees of freedom with finite
set of valuable
harmonic oscillations. Every harmonic oscillation can be described by
following second order system of ordinary differential equation:
We can edit number of harmonics and their properties.
5.1.1.3 Aggregation
We have to types of objects. Let us aggregate them. This operation looks like:
Properties of Conn arrow are presented below:
It means that first connection place of Elastic console is
connected to second connection place of Rigid. In result we
have following spececraft:
6.1.1.4 Vibration testing. Regression once again
Technological processes have defects. So we need tests of our details for
identifiation of their properties. Here
vibration
testing of elastic console is presented. This is passive vibration testing.
We have a sensor which measures following value:. Here q_{i} are generalized coordinates and
a_{i} is are unknown constants. These constants are called
nuisance variables. These variables should be also identified. Vibration
test proessing is presented below:
The Seletion object contains testing result. The
Elastic console is math model of tested object. The
Coefficients object contains unknown coefficients. The Regression
object performs identification. This object has following properties:
Left part means that we would like identify parameters c and Eps
of Elastic console. Right part means that we compare math
calculation with Y components of Selection object. Middle panel
presents math processing result (Formula_1 of Function).
Identification process changes properties of Elastic console.
Following picture presents these properties before and after identification
respectively:
Now Elastic console with idetified properties can be exported
for further development. This sample shows that engineering software shoud be
unified.
6.1.1.5 Rigid body with vector interface
"Rigid body with vector interface" component provides vector representation of
external forces and momentums. This representation is more laconic. Here I show
a sample of this representation. The sample is devoted to "Sputnik  1"
spacecraft. The "Sputnik  1" mission is "it just works". Therefore it
was unconrolled. It has own magnitic momenum caused by equipment currents and
other factors. So it was forced by Earth's magnetic field:
Let us consider simulation of this satellite. Linear motion of artifical
satellite nad already been considered in 4.2.3.1. The Greenwich reference frame
had been used. Here I consider usage of inertial reference frame. It is more
convenient for some tasks. Relation between these reference frames is shown at
following picture:
The OXYZ is inertial frame and OX'Y'Z' is Greenwich one.
Greenwich frame is rotated with respect to inertial one. We should
transform acceleration vector g from Greenvich frame
to inertial frame. Calculaton of g in inertial
reference frame has two features. Satellite coordinates with respect to OXYZ
are not equal to coordinates with respect to OX'Y'Z'. Moreover
projections of g on OXYZ axes of coordinates
are different to projections on OX'Y'Z' axes of coordinates.
Declarative approach enables us to resolve both problems at once. Usage of
covariant fields provides solution of both problems (See 5.3.1.1). Simulation of
linear satellite motion is presented below:
Here Earth's Frame component represents Geenwich reference
frame. In 4.2.3.1 Gravity component is used for motion
equations as data source. Here this component is used as physcial field (in
Eatrh's fields). We can say the same about Atmosphere
component. Physial field is linked to Greenwich reference frame. Gravity field
is marked as covariant. These circumstances provide solution of both above
problems. The Sensor is linked to Spacecraft frame.
Its orientation coinsides with orientation of inrertial reference frame. Sensor
results are used in Motion equations of spacecraft. Otherwise
Motion equations results are used by Spacecraft frame.
So we have math model of spacecraft linear motion.
Above picture is not convenient for further development since it contains a lot
of squares. Some of squares should be
encapsulated. Following picture shows container designer
which provides encapsulation:
User checks public components on left panel and sets their positions on right
panel. In result we have little black squares instead big squares. Only public
components are visible and we can identify them by tooltip texts.
Some of aerospase tasks require Earh's magnitic field. Let us develop new
component that includes Earh's magnitic field. Design of this comopnent is
presented below:
In this picture we have early developed model of spacecraft motion and model of
magnetic field (Magnetic Field). The Magnetic Field
is covariant and linked to Greenwich reference frame. Now we can install new
components. In result components will be appeared in pallette.
Now we can simulalte motion action of magnetic field on uncontrolled spacecraft:
The F link between Sensor and Earth's
magnetic field means that the sensor is the sensor of the field. The G
link means that Sensor is geomerically linked with spacecraft
reference frame. The Momentum component calucales
dipole
mechanical momentum by the following way:
Here we have cross product of m and b. The b
is magnetic field vector provided by Sensor. The m is
magnetic momentum of spacecraft. This product is used in Angular Motion
by the following way:
This picture means that Angular Motion (it is Rigid body with
vector interface) uses Formula_1 of Momentum as
mechanical monentum vector. Vector notation is more laconic in this case.
6.1.1.6 Flywheel
Flywheels are used in spin stabilization systems of spacecrafts. Following
documents contain informaion devoted to spin stabilization systems:
Flywheel is forced by reversible engine (See piture below).
Otherwise flywheel acts to engine by momentum M_{x} . Since
engine is attached to spacecraft this momentum is transferred to spaceraft. This
momentum is used for spaeraft stabilization. Sine flywheel is rotated we have
additional gyro momentum M_{gyro} . Gyro momentum can be
calculated by following expression:
where J_{F } is inertial momentum of flywheel,
is angular veloictiy of flywheel and
is
angular velocity of engine (and also spacecraft). Total momentum M is
equal to geometric sum M = M_{x} + M_{gyro} ;
Gyro momentum is undesirable factor. Stabilization system should require
following condition M_{gyro} << M_{x} .
However since engine acts to flywheel value of
is being inreased by the time. Increasing of
compensated by other devices which acts to spcecraft. In this article
eletromagnetic devices will be considered. Previous components did not
essentially use function GetConnectionForce
, since forces and
momentums was trivial. Return of this function was zero vector (array). But
return of this function in Flywheel component contains notrivial total momentum.
Whole stabilization system need know angular velicity of flywheel. So flywheel
component implements IMeasurements interface which provides measure of angular
veloity. Typical sample of flywheel usage is presented below:
In this example Input calulates control momentum M_{x}
of Flywheel. The angular velocity of Flywheel
is used by Output.
6.1.2 Mechanical model of controlled spacecraft
In this article spececraft with two consoles and three flywhells is considered.
Its construction is presented below:
Numbers 1  5 are numbers of connections places of spacecraft. Flywheels
attached to 3, 4, 5 connection places realize angular stabilization of spaecraft
with respect to axes X, Y, Z. Besides flywheels spacecraft containts three
electromagnets for stabilization. Mehanical model of spacecraft is presented in
following picture:
Nunmbers 15 of links are numbers of spaceraft connections. Components of this
model had been included into single container as well as components of linear
motion model (See 5.1.1.5): Main component and flywheels are
public. Consoles are private. In result we have following container:
6.1.3 Vibration test of spacecraft
Vibration tests provide identification of mechanical model. Let us consider
following vibtration test:
Hydraulic cylinders impact on spacecraft. In result spacecraft is forced by
momentum. The
PID control
law of momentum has been used:
Where
M_{x} is mechanical momentum,
is X
 pojection of spacecraft angular veloctity, of spacecraft,
is
rotation angle of spacectaft with respect to X  axis. Parameters
K_{1},
K_{2},
K_{3} are constants.The test purpose is
definition of spaceraft transformation function. Its definition can be obtained
by response on harmonic input. The
chirp input singnal has been used:
Followig scheme has been used for simulation of vibration test:
Top squares of this scheme provide necessary formulas. The
Recursive
component calculates integral for PID control. Response of consoles and
spacecraft is presented below:
Response of console has two strong resonance peaks. This situatioon is typical
for space technology which use light weight constructions. However these
resonances made control task very complicated. Resonances of consoles impact on
spacecraft motion. Above pictures shows this impact. This sample requires usage
of containers.
6.2 Space aerodynamic
The key feature of space aerodynamics is that spacecraft interacts with
molecules which do not collide with each other. Therefore aerodynamic force
depends on visible area and does not depend on other parameters of spacecraft
shape. We can use
digital image processing for space aerodynamic calculation. Digital image
processing for science and engineering has been considered in my
article 8. Let us consider it for space
aerodynamic. I had bad feeling that this article contains photo of Mir station
but the photo is not used. Now the photo is used for calculation of space
aerodynamic by following way:
Reader can compare Prototype photo and Result
of digital image processing:
Now we can obtain visible area by counting of white pictures on Result
image. This image is dirty. But user can find good images in my
article devoted to virtual reality. We have
interesting picture. Vitrual reality and digital image processing are being used
in motion equations of spacecraft. To say true I do not have strong interest to
space technology as itself. Space technology is rather training ground for my
ideas. Reader need Astronomy Express version for this sample.
6.3 Celestial Navigation
Classical feedback control scheme is presented below:
It requires sensor. We use astrosensor for spacecraft control. Asrtosensor
enables to define orientation of spacecrafts. There are a lot of types of
astrosensors. I provide one of possible shemes. Suppose that we have equipment
that provides celestial images and star catalogues. Comparation of image and
catalogue enable us to define orientation of equipment. So we can define
orientation of spacecraft. Algorrithm of this sensor is presented below:
Left part of this scheme contains image processing. Right part represents usage
of star catalogue. Brigde compares both of them and in result we have parameters
of spacecraft orientation. Let us consider details.
6.3.1 Image processing
Suppose that equipment provides following celestial image:
This image contains interfering information. We need filtration for its
exclusion. Nonlocal digital image processing is being used for this purpose.
Sheme of this prosessing is presented below:
This scheme contains Initial image (Source image) obtained by
equipment. Little squares provides necessary math. It result we have
Filtered image (Filtration result). Both images are presented below:
Following picture explains filtration algorithm:
If we have 9 closed white pixels then we replace it by one blak pixel. Every
other pixels are white. Result of filtration enables us to obtain X and Y
coorginates of black pixels. Then this numbers will be compared with star
catalogue. The Stat component extrats this numbers from
Filtered image.
6.3.2 Star catalogue usage
Star catalogue is stored in database. Necessary information can be extracted by
Sql qurery:
Query statement is presented below:
++++++++++++++
SELECT RAdeg, DEdeg FROM hip_main WHERE RAdeg > @RAMIN AND RAdeg < @RAMAX AND
DEdeg > @DEMIN AND DEDeg < @DEMAX AND BTmag > @BTMIN ORDER BY RAdeg
++++++++++++++
This statement has following meaning. First of all we consider limited area of
sky.
Declination and right ascension belong to small intervals. Secondly we
consider only susch stars which
magnitudes exceed defined constant (in this sample the
constant is equal to 9). Qurey result provides following chart:
We would like compare this chart with filtered image. This operation requires a
set of math transformations. Full scheme of these transformations is presented
below:
Essential feature of these transformations is euclidean transformation:
Parameters a, b, and
are
unknown. Comparation of star catalogue and filtered image enable us to define
these parameters. Using these parameters we can define orientationn of
spacecraft.
This sample requires Sql Server Express and Astronomy project. It also requires
star Tyho and Hipparcos star catalogue that reader can download:
6.4 Development of control system
6.4.1 Identification
Development of control system requires math model of controlled object. This
model will be obtained by processing of vibration test. Here nonparametric and
parametric
identification will be considered.
6.4.1.1 Nonparametric identification. Envelope detector once again. Phase
detector once again
The natural way for obtaining transfer function is usage of envelope detector
and phase one. This theme has been considered in 5.3.4 and 5.3.3. You an
download modifications of this detectors adopted to recorded singnals:
Frequency responce and phase characteristics are presented below:
6.4.2.2 Parametric identification. Regression once again
In this section transfer function of object will be obtained. Control systems
specialists use logarithmic scale for frequency resopnce. This function provides
clear picture of control object. So we transform functions of previous sections
to logarithmic scale. In result we have following charts:
The Y axis of frequency resoponse is also logarithmic. Control system
specialist that such chars correspond to following transfer function:
Parameters
k,
a,
b,
d,
f,
g,
h,
l,
m can be defined by nonlinear regression.
Regression scheme is presented below:
Charts in the left part of this picture represents approximated functions
(Frequeny responce, sine and cosine of phase). Other squares contain necessary
math. In result we have following approximation of our charts:
6.4.2 Doubleloop stabilization system
Stabilization system could not use flywheels only (See 6.1.1.6). Otherwise there
is an obstacle for constuction of stabilization system which uses electromagnets
only. Electromagnetic momentum is always perpendicular to magnetic induction
B. But stabilization requires all directions of control momentums. So space
technology uses doubleloop systems which use both flywheels and electromagnets.
Let us consider both loops of this system. First loop (I name it high frequency
loop) is presented in followng scheme:
We use celestial navigation
for definition of orientation and
optical gyroscope for definition of angular velocity. In result we have
following parameters
. First three parameteres are angle deviations with respect to
desired axes X, Y, Z. Following three parameters are
angulular velocities with respect to same axes. Control momentums are obtained
by flywheels
.
We have identified spacecraft angular motion model in 6.4.1. In accrordance to
identified model we develop control law. I drop details here. Maybe my book will
contain details. Here I note that control law of high frequency loop is designed
in compliance with
control theory.
Second loop scheme is presented below:
This loop purpose is limitation of flywheels' angular velocities. Sensor of this
loop are tachometers
of flywheels. Necessary momentum is provided by eletromangets
. It is possible to use different control laws for this loop.
But main idea of these laws is single In this article I have used following
control law. Let be H_{gyro} is total
angular momentum.
Then momentum of electromagnets is opposite to H_{gyro}. But
electomagets are not always switched on. If H_{gyro} is too
small then electromagnets are switched of. Otherwise if angle between Earth's
magnetic induction Hgyro and H_{gyro} is too small then
electromagnets cannot provide substantionally large momentum that is opposite
H_{gyro}. Therefore if angle between H_{gyro}
and B is too small then electromagnets are also switched off. Since
action of electromagnets is not contionous I name this loop low frequency loop.
Here I explain how magnetic momentum reduces angular velocities of flywheels.
Magnetic momemtum causes deviation of spaceraft orientation. High frequency loop
tries to eliminate this deviatrion by changing of angular velocity of flywheels.
High frequency loop tries to eliminate this deviatrion by changing of angular
velocity of flywheels. Since electromagnetic momentum is opposite to H_{gyro}
changing of angular velocities is reducing of thier values. Full control picture
is presented below:
This picture corresponds to following situation:
In this picture we have complicated spacecraft with electromagnets which
interact with Earth's magnetic field. Geometrical picture is presented below:
The X,Y,Z is Greenwich reference frame and X', Y', Z' is inertial
reference frame. Center of X', Y', Z' coinsides with center of Earth. We also
have X", Y", Z" reference frame. Center of this frame coincides with mass center
of spacecraft and axes X", Y", Z" are collinear to X', Y', Z'. We also have
desired reference frame of spacecraft X_{d}, Y_{d},
Z_{d} and reference frame of spacecraft X_{s},
Y_{s}, Z_{s}. Stabilization system should
provide orientation of spacecraft near desired reference frame. Some of these
frames are contained in designed above components. Following picture explains
it:
Spacecraft body is instance of reference frame. It is X_{s}, Y_{s},
Z_{s} reference frame. Full layout of frames is presented
below:
The L 1 link means that we consider desired frame relatively
frame X", Y", Z" frame. Its relative orientation are defined by following
transformation matrix
The X_{s}, Y_{s}, Z_{s}
reference frame is considered as relative with respect to X", Y", Z" frame.
However X_{s}, Y_{s}, Z_{s}
is frame of mechanical object. So this frame could not be considered relatively
any frame. Coordinates of parent frame should be twice differentiable. The
desired reference frame has constant relative position and orientation. So
sufficient condition of twice differentiability of its coordinates is twice
differentiability of its parent frame X", Y", Z". Otherwise X", Y", Z" uses
Motion equation component. We require that coordinates x,
y, z of Motion equation should be twice
differentiable by following way:
But we do not simply require it. This requirement should not contradict with
structure of Motion equation. This structure prohibits twice
differentiability of u, v, w. The framework
automatically checks all these requirements. The layout also contains
Sensor of Earth's magnetic field. The L 2 link means
that Sensor is installed on spacecraft body. The F
link means that sensor is installed on spacecraft. The sensor is being used for
geomgnetic control. The Relative component defines relative
position and orientation of spacecraft with respect to desired reference frame.
This information is being used in high frequency control loop. Following picture
explains this usage:
Components Q_{1}, Q_{2}, Q_{3} of relative orientation
quaternion are parameters x, y, z of following
control law:
Parameters u, v, w of above expressions are
components of angular velocity of spacecraft body. So spacecraft body is
provider of data. Parameters u, v, w are linked to
components of angular velocity by following way:
Flywheels are consumres of data. They are being use high frequency loop output
by following way:
However flywheels are not data consumers only. They are data provides. Their
output data are their (angular) velocities. This output is being used in low
frequency loop by the following way:
This output and output of Sensor is used for geomagnetic control. Spacecraft
body is not data provider. It is also data consumer. It is being use output of
low frequency control loop by the following way:
Following picture shows mechanical momentums of high frequency and low frequency
loops:
We have evident correlation between both momentums. Transition processes are
presented below:
We also have correlation.
6.5 Spacecraft mission. Astronomical observations
Two telescopes are bieng used for spacecraft mission. The telescopes are
installed on spacecraft body:
Telescope 1 is rigidly attached to spacecraft.
Telescope 2 is rotated relatively spacecraft. This situation looks
artificial. However I would like to show advantage of usage of relative referene
frames. Following presents simulation of this mission.
Left part of this picture repesents simulation of spacecraft motion. Right part
repesents mission components. Let us consider these components in details:
Sinse Telescope 1 is installed diretly on spaecraft. So
Telescope 1 do not use additional frames. Telescope 2
is installed on Rotation frame. Otherwise Rotation
frame is installed on spacecraft body. Both telescopes observes single
collection of stars (Stars component). This collection is
genenerated by following way. First of all Query component
performs SQL query of star catalogue. Then other components perform necessary
math transformations. Result of these transformations is used by Stars
component. Properties of this component are persened below:
These properties have following meaning. Star coordinates X, Y,
Z are equal to Formula_1, Formula_2 Formula_3
(of X Y Z component) respectively. Similarly color and size of
star indication are defined. This sample requires Astronomy Express project and
star catalogue (Hipparcos and Tycho).
This sample is animated. Use "Control" pane for its animation.
7. GIS Satellite. Web based application
Here we will consider sample of satellite which is used for
Geographic Information System
(GIS). Since we are interested in geographical coordinates we will use Greenwich
reference frame as well as in 5.2.3. Full model of satellite is presented below:
We have added the GEO coordinates component to old model. This
component calculates geographical coordinates of the satellite (longitude and
latitude).
Red and blue curve represent longitude and latitude respectively. Simulation
model has been saved in file SatelliteCoordinates.cfa and then this file can be
used as resource in other applications. Here this file is used with GIS service.
In this article the Environmental Systems
Research Institute (ESRI) service has been used. ESRI provides GIS web
services. Using these services I have developed following animated WPF demo
application:
User can enter initial coordinates X, Y, Z and components of velocity V_{x},
V_{y}, V_{z} of velocity then we have animated view of Earth
from the satellite. Animation has low frequency since we use web service. I have
chosen update interval to 10 s. This application used real time (system time).
If you would like to compile this demo application then you should download
lbraries from ESRI site and add
ESRI.ArcGIS.dll reference to ESRI.ControlLibrary project
8. Future. Calibration of optical gyroscope
Previous text is rather history. Now I would like exhibit one of future
problems. This problem is described below
8.1 Rationale
Optical gyroscope can have a lot of applications.
For example it can be used in
inertial navigation system (INS). Otherwise
inertial navigation system can be used for Mars
atmospheric
reentry. During interplanetary flight to Mars parameters of can be
essentially changed by unpredictable factors. Maybe we should calibrate
parameters of optical gyroscope. We do not have laboratory at interplanetary
spacecraft. So we are looking for other ways for calibration. These ways will be
described below.
8.2 Research proposal
First of all it worth to note that usage of optical gyroscope is not necessary
right way. If we cannot calibrate optical gyroscope with accepted accuracy then
it should be rejected. In this case we should use for example
mechanical
gyroscope. But mechanical gyroscope can have mass that exceeds mass of
optical gyroscope. Power consumption and size of mechanical gyroscope also can
exceed corresponding parameters of optical gyroscope. Maybe this research would
have negative result. But a lot of venture research also have negative results.
So we have proposed venture research and begun to perform it.
8.3 Method outlook
Any calibration requires independent measurements. Spacecraft can have
additional astrosensors. These astrosensors would not be used for gyroscope
calibration only. However optical gyroscope defines angular velocity of
spacecraft. Astrosensors define its orientation. In case of spacecraft rotation
we have dependence between angular velocity and orientation. Qualitative picture
of this dependence is presented below:
We have rotated spacecraft. Astrosensor defines its position. Measurements of
astrosensor and optical gyroscope enable us to calibrate the gyroscope.
8.4 Document structure
Present day software is nonlinear. So it is not easy to undestand using linear
documents only. Different types of
UML
diagrams help us to catch meanig of software structure. Similarly following
diagram explains structure of this research.
Every green arrow means usage of theory. Blue arrow means data import.
8.5 Theory in brief
8.5.1 Nonlinear regression
In statistics, nonlinear regression is a form of regression analysis in which
observational data are modeled by a function which is a nonlinear combination of
the model parameters and depends on one or more independent variables. The data
are fitted by a method of successive approximations. More theoretical details
readers can find
here. A lot of applications of nonlinear regression are described in
following CodeProject articles:
It is clear that nonlinear regeression can be used for optical gyroscope
calibration.
8.5.2 Sagnac effect
Sagnac
effect is key physical principle of optical gyroscope. Following well
known relation is used in this research:
Where
is angular velocity and
U is output signal of optical
gyroscope. Parameters
a and
k depends on technological
properties of optical gyroscope. These parameters can be unstable and we should
calibrate them. Indeed inertial navigation system contains more than one
gyroscope. Typical inertial navigation system contains three gyroscopes which
are oriented along orthogonal axes
X,
Y,
Z. So we
have three following equations:
Every subscript means axis of hyroscope. Above expressions have inverse ones:
These expressions enable us calculate components of angular velocity.
8.5.3 Qaternion applications for rigid body kinematics
Following referenes are useful for studying this subject:
A lot of information can be found in my CodeProject articles. If orientation is
described by following quaternion q=q_{0}+iq_{1}+jq_{2}+kq_{3}
then q_{0}, q_{1}, q_{2},
q_{3} satisfy following system of ordinary differential equations:
8.5.4 Rigid body dynamics
Dynamics of absolutely rigid body is described by
Euler's equations. However spacecraft dynamics could not be always
described as absolutely rigid body. This article contains a lot of relevant
examples Different variants can be considered in single framework. Usage of
bridge
pattern for spacecraft dynamics is presented in following diagram:
This diagram explains usefulness of bridge pattern in space techonology
simulation. We need 7 types of objects instead 12 ones.
8.5.5 Random processes
Simulating
a continuoustime random signal is only used in this research.
Simulating
a continuoustime random signal can be performed by usage of control
theory transfer
functions. Other applications of transfer functions are described in my
article: "Universal Framework for Science and Engineering  Part 3:
Control systems. Processing of signals".
8.6 Implementation
8.6.1 Calibration algorithm
8.6.1.1 Outlook
Calibration algorithm is based on nonlinear regreesion. Nonlinear regression
have input and output parameters. Input parameters contains output data of
optical gyroscopes and its calibraion parametes. Usage of these parameters
enable us to integrate kinematics equations. In result we have quaternion as
time functions q(t). In particular we have values q(t_{1}),
..., q(t_{n}) of quaternion for fixed t_{1},
... , t_{n}. In our case dependences output data of optical
gyroscopes is considered as fixed data. Input paramaters for regression contains
calibtation parameters only. Output data contains q(t_{1}),
..., q(t_{n}).
8.6.1.2 Implementation
Spacecrafts' design begins from design of
payload. So this description begins from
component which performs calibration. Later we describe subordinate components.
Calibration is performed by regression component
. These component has high level of absraction. So it provides
high flexibility. Regression component support definition of very different
parametres. Input parameters can be also very different. Calibration algorithm
scheme implementaion is presented below:
In this scheme regression component is named as Processor. Its
properties are prerented below:
Panels of this properties corresspond to terms of regression
. Left, middle and right panel corresponds to x, f(x)
and y respetively. Here components of x are parameters a,
b, c, k, l, m of Calibr
component. Reressiion function f contains Formula_1,
Formula_2, Formula_3 of Regression component. Now
we have three suborditate branches x, f and y of
regression. Let us consinder them.
8.6.1.2.1 First branch. Defined parameters
Branch x is provided by Calibr component. This
component contains inverse calibration formulas:
The framework do not contain Greek idetifiers. So Greek symbols are replaced by
Latin aliases. Mapping between equation symbols and identifiers is reflected in
following comments of Calibr component:
Parameters x, y, z are imported from omega
component. The omega is adapter. Input of omega
contains three functions in
settheoretical definition style. The omega provides
calculation of this functions. Roughly speaking omega calulates
value f(t) of f function. Such indirect way caused
that we can use f in following expression f(t+a)
etc. Propreties and comments of omega object are
presented below:
Functions f,g,h above are imported from
omega(x), omega(y), omega(z)
respeively. These functions corresspond to U_{x}, U_{y},
U_{z} of Sagnac inverse formulas. Properties of
omega(x) are presented in following picture:
Properties of omega(y), omega(z) objects are
similar to omega(x) one. Import of U_{x},
U_{y}, U_{z} shall be considered below. First
branch is fully described now.
8.6.1.2.2 Second branch. Regression function calculation
Roughly speaking first branch is x and second branch is f(x).
So second branch depends on second on first one. The Dynamics
object solves kinematics differential equations (see 8.5.3). The
Dynamics object has following properties and comments:
The Vector object assemblies output of Dynamics
to vector. The Vector object has following properties:
Rougly spaking Vector input contains four scalars q,
x, y, z and Vector output is 4D
vector Q=(q, x, y, z). Components
of this vector are components of orientation quaternion. According our alhorithm
we need values of this quaternion in fixed time moments of quaternion for fixed
time moments t_{1}, ... , t_{n}. We will
perform following operaions for this purpuse. Indeed Vector
calculates values of 4D vector Q(t) for fixed time t.
We will transform this calculation to
settheoretical function Q. The Accumulator
performs this transformation. Properties of Accumulator are
presented below:
These properies mean that time function is being calculated from time = 0 with
step = 0.01. Step count is equal to 1800. The Regression object
imports this function and calculates its values. Properties and comments of The
Regression object are presented below:
.
In fact a,b,c are fixed time moments t_{1},
t_{2}, t_{3}. The Regression
component output is used by Proccesor .
8.6.1.2.3 Third branch. Seletions (Experimental data)
Selections data should be independent from first and second branches. In these
case selections are contained in Selection 1, Selection
2 and Selection 3 objects. These selections correspond
to orientation quatenion in moments t_{1}, t_{2},
t_{3}.
8.6.2 Definition of realistic angular velocity
Calibration accuracy depends on partial derivatives of measured parameters with
respect to calibration ones. It is clear that these derivatives depend on
angular velocities as time functions. So we need do define realistic values of
the angular velocities. We would like to separate definition of angular velocity
from calibration algorithm. This separation is in fact a version of the
bridge
pattern.
This scheme enables us construct 9 combinations from 6 objects. Since there are
a lot of spacecrafts' types and a lot of spacecrafts' systems and subsystems the
bridge pattern usage provides tremendous effect. Here we will consider
spacecraft which can be considered as rigid body and values of mechanical
momentum can be discrete.
Component 
+ 
0 
 
M_{x} 
a 
0 
a 
M_{y} 
b 
0 
b 
M_{z} 
c 
0 
c 
This table has following meaning. Mechanical momentum with respect to X
axis can be equal to a or a etc. This scheme can be
implemented by usage of reactive engines. Scheme of engines is presented below:
Simultaneous force of engines 1 and 4 results to mechanical momentum with
respect to X axis. We will denote this momentum by a.
Similarly simultaneous force of engines 2 and 3 to mechanical momentum with
respect to X. But now momentum is equal to a. This scheme
looks trivial. However since we can switch engines at different time moments
this scheme provides infinite set of variants. We can implement optimal and
suboptimal algorithms. But usually very simple algorithms are being considered
at first. Let us consider following physical arguments. We should calibrate
three optical gyroscopes which measures angular velocities with respect to X,
Y, Z axes. It is known that the good way of parameters'
estimation is such way where we have no correlation between parameters. Note
that calibration procedure includes astrosensors. It is known that accuracy of
astrosensor depends on angular velocity. High accuracy can be reached in case of
small angular velocity. So it is reasonably to use following contol scheme.
Red, green and blue lines on this chart represent M_{x}, M_{y}
and M_{z} respectively. Let us explain these scheme. The Mx,
My and Mz components provide M_{x},
M_{y} and M_{z} functions of time. These functions are
defined by following expression
This expression has C++ style. It reflects piecewise constant function.
Objects Mx, My and Mz
correspond to maximal moments a, b and c
respectively. These objects generate three momentum functions those presented in
following picture:
These functions are being used in Dynamics component. The
Dynamics component integrates equations of rigid body dynamics.
Properties of this component are presented below:
The Decalibr component transforms angular velocities to
measurements of optical gyroscopes. Properties of Decalibr are
presented below:
Following chart presents time dependencies of angular velocities and
measurements of optical gyroscopes
More bright curves on chart present angular velocities and more dark lines
present optical gyroscopes' measurements
8.6.4 Simulation of random errors of optical gyroscope
We need error model for estimation of accuracy of our method. Error model of
optical gyroscopes will be considered here. In accordance to Bridge pattern
error model we have independent error model. The scheme of error model is
presented below:
This scheme contains generator of random numbers (
Random)
normailizing element (
Norm) and filter (
Filter).
Normalizing element transforms interval of random numbers from (0,1) to (1,1).
In result we have centalized sequence of random numbers. The
Filter
implements following transformation function:
8.6.5 Analysis of errors
He we develop automation equipped working place of analysis of errors. We have
calibration algorithm described in 8.6.1 and algorithm of simulation of random
errors (8.6.4). We would like to develop automation equipped working place which
used both algorithms. Files Calibration_algorithm.cfa and
Angular_Velocity_Error_Model.cfa correspond to calibration algorithm and error
model respectively. We use this files as calculation resources. Foolowing
picture presents usage of these files as resources.
Then this resources are use looded and are being used in project. Following code:
byte[][] bytes = new byte[][] { Properties.Resources.Calibration_algorithm,
Properties.Resources.Angular_Velocity_Error_Model };
foreach (byte[] b in bytes)
{
PureDesktopPeer d = new PureDesktopPeer();
d.Load(b);
ldesktops.Add(d);
}
presents loading of resources
Screenshots of automation equipped working place are presented below:
Input of these application contains statistical parameters of input errors an
other parameters. Output contains covariance matrix and constant bias. This
application is specially developed for this article only. I would like to show
how the framework can be used as calculation resource.
So I had finished this article since it is too large. But I will continue some
themes in my following articles
Points of interests
I think that my ideas are more important than my code. Otherwise ideas are
useless without good explanation. This article is a milestone in explanation
way.
History
I had written a series of articles about my ideas. Then I have understood that
high rating is not objective criterion of my work. For example rating of my
article about Category Theory is equal to 4.89. However
I do not know even one reader which understands it. But I need understanding. So
I had begun to test it. I had installed
5 propeller blades on Apace helicopter. All my
articles contain a lot of similar tests. My work will be good only when my tests
will be noticed. Then I have decided to inspire negative reaction to my work.
This is standard way for better understanding. I wrote
special
article for this purpose. So "my vote of 1" looks a symphony for me. The
best history is written in real time history. I am writing history of this
article in real time. One of my readers told me that my example are very
complicated for users which are not familiar with Framework. So I decided to
include simple samples. Maybe text of this article will be very large. Then I
will separate it into parts. Today (01/24/2009) one of my readers has proposed
me include pneumatic effect similar to The incredible machine. I shall do it in
this article. I also obtained very useful critique. I shall analyze it next
week. Today (02/07/2009) I have received message about lack of code. Thanks. I
have added relevant sample (part 4.1.4). Abu Mami encouraged me and today
(02/19/2009) I had written chapter devoted algebraic topology. Yesterday
(04/17/2009) is remarkable data of the framework. I repeated some errors which
can cause damage of spacecraft. These errors were not special. They were my own
errors. I have experience of damage prediction. But I did not predict such
valuable damage as spacecraft destroying. I hope that I will predict very
valuable damage. I am being asked: "How can you work so hard?" My standard
answer is: "I am taking easy blame." My article have become too large and I have
begun write new article. I would like to make refferences to new article. Today
(09/07/2009) I have decided to finish this article. Latest editions will contain
only bug fixing in code and grammar.
Prospects
I intend to write a book about my ideas. My recent article
"Topdown Paradigm in
Engineering Software Integration" is in fact announcement of this book. The
draft of book will be shared on my site for discussion. I also promised to
write article about meteorology. But I do not know when I will implement my
plans. Now I am engaged in
obstacles of The Universe collapse..