Cynthia Vaskis
SLM521 Spring 2004
Dropin #3 Assignment
File: dropin3.htm
Star Tracking and Star Matching to
Determine Space Object Orientation or Attitude
Lesson grade level: Any middle or high school
student can appreciate the general discussion on the top lesson page and would
enjoy seeing the example web sites for Star Maps, Star Trackers, the Hubble
telescope and other ground based telescopes.
The math calculations discussed will use basic trigonometric operations
such as the sine and cosine functions and is for 11th or 12th
grade students who have had some basic trigonometry. The concepts of linear algebra using Dot
products, Cross Products and unitizing vectors by finding their magnitude (or
length) is discussed and shown in diagrams.
Lesson Topics
1. Due to the great distances between the Earth
and the stars, the viewing angles to locate the stars in the celestial sky is
the same for both the Earth and the Star Tracker in orbit.
2. The sequence of steps to calculate the
pointing vector of the onboard Star Tracker is shown in diagrams.
3. Examples show a simple combination of vector
definition, translation and rotation operations that allow the space object to point
toward a newly selected star.
4. Discuss other math applications where an
object needs to point toward something and must rotate in order to do so.
Suggestions for teaching the lesson
The students will learn more
if they discover it for themselves rather than have someone read it to
them. It would be good to have them read
the questions below first and then go on an “information hunt” through the text
and sub pages to find the answers. They
do not need to calculate anything in this lesson but just become familiar with
the terminology and see the types of math used to determine an object’s
orientation and orbital position vector.
If they work in small groups, some can take a few questions each and
then at the end of the session have a share time where they explain their
answers based upon what they learned.
The point of this lesson is for them to see how math is used in space
applications. This lesson builds upon
the first two in that they need to be familiar with what a vector is and what
is meant by adding two vectors (translation) from lesson 1 (drop-in 1) and how
a vector is rotated using a rotational matrix from lesson 2 (drop-in 2).
See Suggestion
1 about a game to play.
See Suggestion
2 about a demonstration model to build.
Topic 1 - Different ways that space objects find
their orientation in space
One method for finding an
object’s orientation in space is for that object to look at a pair of stars
that are easily identifiable in a star map (printed paper) or star catalog
(software files). This is done by
pointing a star tracker toward a known pair of stars to see if the star tracker
can see them and track them. Very good
star catalogs (computer files) are available for personal use as well as for
commercial use by space objects. Some
space objects have a star tracking device onboard. Another method to determine a space object’s
orientation is to use an onboard gravity gradient orientation system that
senses the Earth’s gravity gradient, or the direction of gravity’s pull, to
know which direction is toward the Earth and thus its orientation in 3D
space. When the space object knows its
orientation, it can point its antennas toward Earth to communicate with Mission
Control or a ground station. Solar
arrays point toward the Sun to maintain the space object’s electrical power
supply.
When the space object has
star trackers, it uses the tracker to point toward selected star pairs to
determine, or adjust, the space object’s orientation (or attitude) in 3D
space. There are about 50 or so
brightest stars that are frequently used by star trackers to locate and, thus,
determine their space object’s orientation.
Usually, the star tracker will move its Field Of View (FOV) center axis
toward one bright star, track it for awhile until it gets a good reading or
sighting in its sensor arrays inside the tracker and then slew around toward
the next selected star to track. Most
often the two stars used for a single tracking operation will not both be
within the first pointing position’s Field Of View (FOV). In other words, the two stars are far enough
apart that the star tracker will have to look at one first and then move to
look at the next. The tracker keeps a
record of where it has been pointing so that the computer algorithms can
determine the tracker’s orientation in space once it has a match on the two
stars with its onboard star catalog.
Typically, the star tracker will have some limited slewing, or movement,
ability apart from the space object’s ability to rotate about its axes.
Topic 2 - Assumptions about Star Tracker and Earth’s
position in 3D space relative to the great distance to the stars
There is a sequence of math
calculations that enable a space object and the Mission Control people (their
computer tracking programs) to determine the space object’s orientation. First, there is a set of assumptions that the
scientists make because of the great distances between the stars being tracked
and the Earth with the orbiting space object.
1. The position of the Star
Tracker, onboard the space object, is approximately the same distance from any
stars it might point toward as the Earth is from those same stars. Since this distance to the stars is so much
larger than the distance between Earth and the star tracker in orbit, the star
tracker and the Earth are essentially at the same position in 3D space. For example, if the star tracker and the Earth
emitted a light beam toward the same star, the two light beams would
essentially be parallel lines very close together considering the great
distance to the star.
The quote, “Aristarchus
considered that the radius of the sphere of the fixed stars was infinitely
large compared with the orbit of the earth (about the Sun).”, is from the entry
on Aristarchus
at the MacTutor
History of Mathematics Web site. He
hints at the fact that the distance of the Earth to the stars is so great that
the orbit of the Earth around the Sun is insignificant when compared in size to
the distance to the stars. Even smaller
is the orbit of a space object around the Earth when compared to the distance
to the stars. Therefore, we shall
consider that the direction and distance to the stars is the same for the Earth
as for any object orbiting the Earth.
This allows us to use the same star catalog onboard the Star Tracker as
on the surface of the Earth since the direction to the stars should be the
same.
2. Because of assumption 1 above, both the star
tracker and the Earth having the same position in 3D space, scientists consider
the two objects, the star tracker and the Earth, as having the same viewing
angles toward the stars which simplifies the calculations considerably. This allows us to use the same star catalog
data for the star tracker (no matter where it is located in its orbit around
the Earth) as someone on the Earth’s surface looking up at the stars. The star tracker probably can see more of the
celestial hemisphere than someone on the ground though. The star catalog’s recorded angles for any
star (from the Earth to the star) are the star’s angle of right ascension and
angle of declination. The star catalog
also records the brightness, or magnitude, of the star according to the
spectral response of the detector used in obtaining that data. There are approximately 50 or so brightest
stars in the catalog in order to cover the whole sky for the purposes of the
star catalog. The angle of declination
is the vertical measurement (like latitude) and ranges from zero in the
equatorial plane to positive 90 degrees (above the equatorial plane) and to
minus 90 degrees (below the equatorial plane).
The angle of right ascension is the measurement around the Earth (like
longitude) which measures zero pointing toward the first point of Aries and up
to 24 hours (in time using hours, minutes, seconds) around the Earth’s equator
heading toward the western hemisphere.
3. The addition of vectors (or “translation of
vectors” as defined in Dropin 2’s lesson) can be
applied here when we want to locate the origin of the coordinate system used by
the Star Tracker. We first define the
state (or position) vector from the Earth to the space object’s coordinate
system’s origin. Then we add the vector
in the space object’s coordinate system that points to (and touches) the origin
of the Star Tracker’s coordinate system.
Then we can perform some vector addition to obtain the new vector from
the Earth’s center (origin of ECI coordinate system) to the origin of the Star
Tracker’s coordinate system. See the
diagram state vector again and notice the new vector
E2T that is the result of adding the STATE vector with the Translation
vector. As was stated in a history
of vectors , the parallelogram law for the addition of vectors is so
obvious and old that no one knows the originator. The Translation vector obtains the Star
Tracker’s position in ECI coordinates on the orbital path.
Topic 3 - Sequence of math calculations
to determine the space object’s orientation in 3D space
Below
is the sequence of steps required to move a Star Tracker to point to a new star
or pair of new stars in order to find the actual orientation of the Star
Tracker in 3D space and consequently the orientation of the space object in 3D
space.
1.
Find the star in the
Star catalog that you would like the Star Tracker to point toward. Each star will have an angle of right
ascension and angle of declination defined from the Earth’s center in Polar
coordinates which is described in a set of Polar
equations. Convert the new star’s Polar coordinates into ECI coordinates and unitize to get the resulting vector NSECIU.
2.
Click here to see how
the STATE or position
vector is defined. See a diagram of
the state vector to visualize how it is
obtained. Use the Kepler equations
to calculate the space object’s position on its orbital path or STATE vector in
the Earth’s ECI coordinate system.
3.
Add the STATE vector to
the established Translation vector that was defined when the Star Tracker was
attached to the space object at the time it was built (before launch). The resulting vector is one from the center
of the Earth’s coordinate system to the center of the Star Tracker’s coordinate
system (see E2T vector in diagram for state vector).
4.
Determine the Star Tracker’s
field of view (FOV) center (STFOVC) vector’s current pointing direction from
the Star Tracker’s coordinate system and move it back into the Earth’s ECI
coordinate system by subtracting the E2T vector from the E2T vector plus Star
Tracker FOV center vector or simply perform the inverse
matrix that takes a vector in the Star Tracker’s coordinate system and
places it in the Earth’s ECI coordinate system keeping the same orientation in
3D space (See diagram Star Tracker to ECI coordinates
and the STFOVECI vector starting from the Earth’s origin or center).
5.
Then, unitize this Star
Tracker Field of View Center (STFOVECI) vector in ECI coordinates to get the
Star Tracker Field of View Center Unitized (STFOVECIU) vector (see Unitize Vector).
6.
Determine the angle
between the STFOVECIU vector and the New Star Unitized (NSECIU) vector by
taking the Dot product between the vectors to result
in the angle (ALPHA) between them. This
angle ALPHA will be the one used with the quaternion vector below to adjust the
orientation of the Star Tracker’s pointing vector (FOV center vector) or the
space object’s attitude control (or orientation) direction.
7.
Create the quaternion
vector by taking the Cross
Product between the Star Tracker Field of View Earth Center Inertial
Unitized (STFOVECIU) vector and the New Star Earth Center Inertial Unitized (NSECIU)
vector which produces an orthogonal vector to the plane that contains the
STFOVECIU vector and the NSECIU vector.
The values in a quaternion vector are used to
control the Star Tracker’s motor system to move the position of the Star
Tracker to point toward the new star. If
the new star is not within the Star Tracker’s field of regard (the fixed limits
of the Star Tracker’s viewing area) then a space object rotation matrix must be
generated (using the quaternion and angle ALPHA) to change the orientation of
the whole space object to allow the Star Tracker to view the newly selected
star.
The
ideal situation would be if the new star is already in the FOV of the current
orientation of the Star Tracker. If the
new star is not visible to the Star Tracker, then, the next effort should be to
just slew the tracker within its own movement and visibility limits to see the
newly selected star. If the star still
cannot be seen, then a matrix rotation of the whole space object will be
necessary for the star to come into view of the Star Tracker. Once the Star Tracker has been able to track
one star and match it with a known star in the star catalog, it will need to
slew to track another selected star since a pair of tracked stars is necessary
in order to determine the space object’s orientation in 3D space. The Star Tracker can remember where it was
pointing as a baseline direction when it made the first star match with the
star catalog. It uses that direction (or
heading) when it makes the second star match and can then figure out which
direction it is pointing in 3D space.
Topic 4 - Different types of orbital paths
(geocentric, geostationary and asynchronous)
Math equations describe an
object’s orbital path. All orbits are ellipses. See the ellipse math equation and
ellipse animated drawing for your general information (don’t need to read
it all now). An object’s velocity is
calculated which tells Mission Control the exact position of the object on its
orbital path at any point so that they can plan when to communicate with it.
The circular, or
geosynchronous, orbit is a special case of an ellipse when the two foci points
are at the same point (in this case, the Earth’s center). Geosynchronous orbits are used by many
telecommunication satellites as they hover in a fixed spot (geostationary)
several hundred miles above the Earth’s equator. There is a network of communication
satellites orbiting in a set pattern and of equal distance from each other that
keep our long distance telephone lines relaying information around the
world. In fact, the World Wide Web, or
Internet, would not work without them.
Another type of elliptical
orbital path that navigation and military satellites use is an asynchronous
orbit that is highly elliptical and the satellite does not stay in one spot but
moves around the Earth several times a day.
The Navigation (NAVSAT) satellites (24 of them) are on a 63 degree
inclination orbit above the equatorial plane.
There are three separate orbital paths with eight satellites equally
spaced on each of them. This ensures
that at least one satellite is in view at a time over most of the Earth’s
oceans for ships to find their location and receive weather reports. Geo-positioning (where you are on the Earth’s
surface) requires communication from at least two satellites at the same time.
Topic 5 - Math is used to locate the Space Object on
its orbital pathway.
A math model of the space
object must be created that defines its center, its shape and its orientation
in respect to the Earth’s center. When
this object is placed in an orbit around the Earth, the object’s location must
also be defined as a position (or state) vector (x, y, z) in the Earth Centered
Inertial (ECI) coordinate system.
Math equations describe the
object’s orbital path and its current position on that path. By knowing its exact position at any point in
time in the future, the Mission Control people can plan when to communicate
with it and send command signals to control it.
Antennas on the ground receive telemetry (operational data) generated by
the object which tells Mission Control how the object is operating (health
data) and what it has been doing to accomplish its tasks (mission data).
Every orbital path is an
ellipse in a plane that passes through the Earth’s center. The apogee is the highest point in the orbit
and is the farthest position of the object from the Earth. The perigee is the lowest point in the orbit
and closest to the Earth.
Some objects are located on
the Earth’s surface such as ground-based telescopes in observatories and the
ECI coordinate system is used to define where they are on the Earth’s
surface. The stars the ground-based telescopes
look for are defined in star catalogs and use two angles to define where the
stars are in the universe relative to the ECI coordinate system. The two angles that define a star’s location
are the angle of declination and the angle of right ascension. A star’s magnitude, or brightness, is also
recorded with its two angles in a star catalog.
Topic 6 - Math pointing algorithms used in space and
on the ground or ocean.
One example where math
pointing algorithms are useful is in maritime navigation. Antennas onboard ocean ships must point
toward the different communication satellites for geo-positioning and for
weather reports. Any space object that
has lost communications with the Earth’s ground station must reestablish its orientation
in space so that it knows what direction to point its communication antennas
toward Earth. The Hubble telescope must
adjust its orientation every time it orbits the Earth if it is going to take in
more picture data from the same location in deep space every time that area
comes back into view after the Hubble has gone through the back side of the
Earth in its orbit through perigee. The
International Space Station must adjust its orbit, and thus its orientation, to
keep from reentering the Earth’s atmosphere before the mission is complete.
Questions
.
1. What name is used to describe an orbital path
that is circular around the Earth and what type of satellite uses this kind of
orbit?
2. Does translating (or moving) a vector change
its magnitude (length)?
3. How do you calculate the length of a vector?
4. What linear algebra function would you use to
determine the angle between two vectors?
5. What linear algebra function finds an
orthogonal vector to the two initial vectors used as input to that function?
6. If the Star Tracker cannot slew its motors to
look at a new star because the star is too far out of its field of view, what
must happen to allow the Star Tracker to be able to view the newly selected
star?
7. Approximately how many brightest stars are
typically in a star catalog available for selection for star tracking purposes?
8. What are the names of the two angles recorded
in a star catalog for each star?
9. What does the word “magnitude” of the star
(not vector) mean?
10. List one other application in the real world,
besides those mentioned above, that needs math algorithms to point, or
orientate, an object in a certain direction in order for it to do its work or
function. See Hint3.
See the Answers.
The
answers can all be found within the text and sub pages listed above. The Web sites below are just for your
interest and viewing later.
Orbital and Star Tracking Web Sites
Satellite
Times Columns: Computers & Satellites – A list
of documents describing the mathematical calculations to determine the orbital
coordinates of objects orbiting the earth.
I recommend this site to anyone who
might want to work in astronomy or space exploration to learn the math behind
sending up a space shuttle or orbiting space vehicle. The topics describe in layman’s terms what is
happening with the calculations. I think
any adult or older student would be fascinated to learn how orbits are really
calculated and orbiting vehicle or satellites are observed from earth.
Date visited –
http://celestrak.com/columns/index.shtml
CelesTrak Orbital Coordinate
Systems, Part I - This page introduces the coordinate systems used to
define where satellites are located in their orbit around the earth. It covers the Earth-Centered Inertial (ECI)
coordinate system and conversions from an observer’s latitude and longitude
position on the earth’s surface into an ECI position vector. Even though this site is listed under the
previous bibliography reference, it contains crucial information to
understanding how orbits are modeled mathematically so I gave it its own
reference. This reference will be used
in the dropin2 discussion of coordinate systems.
Date visited –
http://celestrak.com/columns/v02n01/
CelesTrak Orbital Coordinate
Systems, Part II - This page focuses on determining the position of an observer
on the surface of the earth as they look at the stars in the Earth-Centered
Inertial (ECI) coordinate system. It is
a little too complicated for our students but some professional engineers or
astronomers may find it useful. The text
describes the problems in making these calculations and provides some insight
into how math is used extensively in the field of astronomy.
Date visited –
http://celestrak.com/columns/v02n02/
CelesTrak Orbital Coordinate
Systems, Part III - This page readjusts the earth model in calculating
where an observer on the surface would look for orbiting satellites. The earth is not perfectly spherical so this
page describes how the earth model is manipulated mathematically to be more
realistic with the earth’s pear shape.
This page gives the reader an idea about how complicated the math
algorithms can be to get an accurate description of the location of an observer
on the earth’s surface. I would
recommend that older students read through the text but not worry about
understanding the calculations until they have to deal with them in a job
situation. Every company that deals with
this stuff has their tried and true routines for making these calculations and
it is unlikely that someone would have to develop one from scratch.
Date visited –
http://celestrak.com/columns/v02n03/
Star Matching/Tracking algorithms
Hubble Telescope Site - The Hubble
telescope site shows the latest and best images gathered after hundreds of
orbits around the earth and one million seconds of exposure time. The pointing controls must be very accurate
in order to look at the same part of space to see the same stars. The telescope’s attitude control (or its
orientation) is critical if the project requires that a star, or set of stars,
be located. Once a star is located it
must be matched with a star map’s description (star identification or star
matching algorithm).
Date visited –
http://hubblesite.org/newscenter/
Skymaps.com – This Web site offers
information on how to purchase publication quality sky maps and star charts,
obtain beginning astronomy books or star atlases and where to purchase
telescopes and astronomy software. It
features a Night Sky Planisphere which shows the stars and constellations that
can be seen for any date and time (on left column menu). You can download either the northern or
southern hemisphere’s sky map of the month (see downloads menu choice) for free
and print only one copy for personal use.
They offer a free subscription to skymaps.com in order for you to get
information in your email. I would
recommend this site to anyone who has an interest in the stars but with
parental supervision for what is downloaded from the site and for any
subscription information entered.
Date visited –
http://www.skymaps.com/
Starchart - Star mapping software
– This site draws star maps of the sky overhead for any time and location. It permits the labeling of stars with their
name, number or letter and/or their magnitude.
The documentation menu choice has a nicely organized outline of
information about charting the stars.
You can read about how the program works and learn a lot about observing
the stars. Use the “next” and “previous”
menu choices at the bottom of the page to maneuver through the documentation
once you have initially selected a topic in the documentation outline. Check out the sample map from the menu
choices. I would recommend this web site
to any adult or older student who is interested in learning how to make star
charts.
Date visited –
http://starchart.sourceforge.net/
Astronomy
Astronomy Remote Control Telescopes,
Observatories, View the Universe – SLOOH.com – This web site is a company
who offers membership in a group that controls two ground based telescopes in
two different observatories in the Canary Islands. The group members vote as to which part of
the universe (their mission) they most want to see next through the telescopes
there. Some solo mission time comes with
the membership where an individual member can determine where the telescopes
are pointed next and more solo time can be purchased. A “Sneak Peek” button on the top web page
shows you the controls available to you to point the telescope. They offer a free 15-day trial period but
only after you have signed up with your member information including how you
plan to pay for the membership. Only
adults, 18 years and older, can become members but it would be fun to look at
the stars and galaxies real-time from personal computer with your kids. This web site received a great review by the
New York Times newspaper.
Date visited –
http://www.slooh.com/homejs.jsp
Zane Publishing – Search for Isaac Asimov’s
Universe Collection (7 titles) on astronomy.
These software packages include topics on astronomy, space exploration,
space speculation, the inner planets, the outer planets, the solar system, and
the universe. The site contains a lot of
other educational software mostly in history.
I would recommend this site for any parent or teacher who wants to
purchase software in order to motivate their child or student to study these
topics.
Date visited –
http://www.zane.com