Cynthia Vaskis

SLM521 Spring 2004

Dropin #2 Assignment

4/2/04

File: dropin2.htm

 

Lesson Title:  Math Calculations to Rotate an Object in 3D Space

 

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 the International Space Station orbital paths.  The exercise problems will use basic trigonometric functions such as sine and cosine and is for 11th or 12th grade students who have had some basic trigonometry.

 

Lesson Topics

 

1)         Explain why space object rotations are necessary.

2)         Show how math is used to calculate object rotations in three dimensional (3D) space and explain how rotations are physically performed onboard.

3)         Show how to mathematically calculate an object’s rotated position using rotational matrices and object point vectors (can be briefly discussed by teacher and used as homework exercises later).

4)         Provide actual real-time web site examples of orbital position and attitude data for the Space Shuttle (if there is an active mission), the International Space Station (ISS), and various types of communication, weather, and military satellites orbiting the Earth.

5)         Students are asked to think of some robotic applications that would require rotations of the robotic bodies or appendages in 3D space.

 

Why rotate an object in 3D space?

 

In the field of space exploration, scientists and engineers have created objects that orbit the Earth.  Some of these are the Hubble telescope, the Space Shuttles, the International Space Station (ISS), and various communication, navigation, weather, and military satellites.  In order to mathematically represent these objects in three dimensional space, a math model of each object must be created that defines the object’s rotational center (origin of its own coordinate system), the object’s shape and the object’s orientation (or attitude) relative to the Earth Centered Inertial (ECI) coordinate system which is a right-handed Cartesian coordinate system with the origin at the center of the Earth.  For our examples we will use the Earth Centered Inertial (ECI) coordinate system to define where objects are in 3D space, either in orbit above the Earth or on the surface of the Earth.

 

When a space vehicle or 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 Earth Centered Inertial (ECI) coordinates as defined by orbital position equations (take quick look to notice the math but don’t dwell there).  The mathematician who created these equations also worked with orbital data of the planets moving about the sun.

 

Once we know where an object is located on its orbital path, we must find its orientation in 3D space in relationship to the Earth.  It is not enough just to place an object in orbit, but we must also be able to control its movement or rotation about its axes.  The space object’s design engineers and Mission Control personnel know where everything is located on the space object that is needed to perform its mission.  Important parts of any space object are the antenna(s) for communication with Earth’s ground stations and Mission Control, the solar array(s) for electrical power and to recharge its batteries, and the star tracker to look for stars to determine the object’s orientation in 3D space.  The object must be rotated at different times so that the antennas point toward the Mission Control ground station to receive commands and transmit data (telephone/Internet communication signals).  Sometimes the object needs to rotate to keep the solar arrays in the fullest view of the sun although the solar arrays do rotate independently of the object’s orientation.  The star tracker is usually fixed to the object and needs the object to rotate if it is going to search for a newly selected pair of stars that are matched against a star map to determine the object’s orientation in 3D space.

 

How does a space object rotate about one or more of its coordinate axes?

 

Each object has its own coordinate axes that are usually represented in a Cartesian coordinate system with the center located at or near the object’s center of mass.  See more definitions about body (object) positions and orientations from MathWorks and the right-hand rule in the Glossary for reference later (too detailed to read it all now).  

 

In order for the object to perform its mission and maintain its health, it must be able to rotate around its own X, Y and Z coordinate axes.  Mission Control can command pairs of small reaction jets (or nozzles) onboard to release or squirt out gases that rotate the object around one or more of its axes.  The nozzles usually work in pairs pointing in opposing directions so that the rotation can also be stopped when the correct orientation is achieved.  The three directions of rotation are about the object’s X coordinate axis (usually called a Roll maneuver), about its Y axis (a Pitch maneuver), and about its Z axis (a Yaw maneuver).  For our purposes we will not be concerned about whether these reaction jet burns affect the orbital path.  Onboard gyroscopes can also be used to change the attitude (or orientation) of the object.

 

Show that a rotational matrix is used to perform Roll, Pitch and Yaw maneuvers

 

This lesson is to show how math is used to rotate an object in 3D space.  A rotational matrix (or set of trigonometric functions) is used to determine the new X, Y, and Z axes locations in 3D space.  The rotational matrix is applied to each of the initial X, Y, and Z coordinate axes to produce new X, Y, and Z coordinate axes.

 

To rotate an object around its X, Y and Z axes, use the three rotation matrix equations in (5), (6), and (7) (click and move down to see the rotating boxes about each axis and matrix equations (5), (6), and (7)).  The first box rotates about the X axis, the second box rotates about the Y axis, and the third box rotates about the Z axis.  The rotation matrices, listed in matrix equations (5), (6), and (7), are listed again within the exercises below.

 

To rotate an object about one axis the set of equations is written in the form called a rotational matrix written as matrix A or M(A).  A matrix is really a list of simultaneous equations with the coefficients taken out and placed in a box structure.

 

To rotate an object, we must rotate all the vectors (or points in 3D space) that define that object.  The object represented in our example has boundary Points defined by (x, y, z) vectors in 3D space.

 

To multiply a matrix times a vector you first write the matrix down and then write the vector in a vertical position to the right of the matrix.  Look here for basic matrix multiplication operations.

Since a rotational matrix can rotate a vector about a coordinate axis, if applied to all the vectors of an object, it then has rotated that whole object about the coordinate axis.

If there are several rotational matrices that need to be applied to a vector (or set of vectors that define an object), the matrices can first be multiplied together and the resulting matrix used to multiply the vectors.

 

Exercises in rotational matrix calculations (for homework later or for a quick overview by educator):

 

These exercises teach the student how to perform matrix multiplications with vectors.  It may be too time consuming to perform during a classroom session but could be assigned for homework.  The first calculations are already done (rotation about X axis) and serve as a template, or example, for the rest of the calculations.  The word “observe” below means the calculation is written out for a student to learn how to perform the matrix multiplication.  The word “compute” means there are some calculations left for the student to try.

 

For these exercises use a triangle with corner points of:

Point 1 = (x1, y1, z1) = (1, 1, 1)

Point 2 = (x2, y2, z2) = (2, 2, 2)

Point 3 = (x3, y3, z3) = (1, 1, 2). 

 

The following exercises show you how to multiply a rotational matrix with one of these Point vectors.  That ends up rotating the Point vector about the coordinate axis selected for the rotational matrix.  Each Point vector is multiplied by the rotational matrix and results in rotating the triangle about the selected axis the amount of that matrix’s rotational angle.  The rotation about the X axis is the angle alpha.  The rotation about the Y axis is the angle beta and about the Z axis is the angle gamma.

 

1.  Observe the rotation about the X axis (Roll maneuver).

2.  Observe and compute the rotation about the Y axis (Pitch maneuver).

3.  Compute the rotation about the Z axis (Yaw maneuver).

4.  Use graph paper to draw the final Point vectors for the fully rotated triangle.

 

Knowledge Review

 

The shape of all orbits is an ellipse.

The object’s position vector is called a state vector defined by Euler’s equations and converted into ECI coordinates.

Rotate the object by creating a set of rotational matrices, one about each axis of rotation.

To rotate an object about an axis of the object, a rotational matrix is created and each vector in the object’s description is multiplied by that rotational matrix.

If all three rotational matrices are multiplied together, then, the resulting matrix can be used to perform one full rotation of the object’s Point vectors.

 

Web sites that show space object orbital position, tracking data and orientation (or attitude) for the International Space Station and other space objects/satellites

 

Human Space Flight (HSF) – Orbital Tracking – Select the REALTIME DATA menu choice and view the orbital path of the International Space Station (ISS) with live data.  The real-time tracking data shows its latitude, longitude, and altitude position over the earth.  The data shows its speed in miles per hour (mph), kilometers per hour (kph), and meters per second (mps).  The Space Station’s attitude or orientation (roll, pitch, and yaw0 are listed.  Internal Space Station environment measurements as temperature, humidity, and air pressure are also displayed.  The viewer can select “Sighting Opportunities” to see when your location may be able to view the Space Station in the sky.  The Space Station orbits the Earth about every 90 minutes.  Different times around the world are shown at the top of the display (Houston for Mission Control, GMT in England, and Moscow, Russia).  Anyone can have fun watching the Space Station orbit the global map of the Earth and see where its path will be in the next two orbits.

Date Visited - 4/3/04

http://spaceflight.nasa.gov/realdata/tracking/index.html  

 

CelesTrak: NORAD Two-Line Element Set Format – NORAD’s Two-Line Element Set format which describes the general orbital parameters (Euler) used to generate the orbital path data for different space objects orbiting the Earth.  This site is too difficult to understand how the parameters are used but it is interesting to note what they are called.  This site is for just a quick look by anyone.

Date Visited - 4/3/04

http://celestrak.com/NORAD/documentation/tle-fmt.shtml

 

CelesTrak: NORAD Current Two-Line Element Set Format - Describes the actual orbital parameters (Euler) for a selected space object that is actively orbiting the Earth.  You can select one of the satellites on the list and see what its current parameters are as described in the previous Web site for NORAD’s Two-Line Element Set format.  Even though the parameters that describe the orbit are not easily understood, it is fun just to see real data for what is used to calculate the satellite’s orbit.  This site is for just a quick look by anyone.

Date Visited - 4/3/04

http://celestrak.com/NORAD/elements

 

BBC – Nature & Science – Space – International Space Station – A Web site (from England) that lists information related to the International Space Station and images from space.  Anyone can view the “Life on the Space Station” activity, watch the short preview of the IMAX File about the Space Station, take the “Interactive Tour” or learn about the experiments onboard the Station.

Date Visited - 4/3/04

http://www.bbc.co.uk/science/space/exploration/iss/index.shtml

 

BBC - Nature & Science – Space – Space Station Film – Anyone would enjoy the short preview of the recently released 3D IMAX movie about the International Space Station where astronauts filmed the building of it.

Date Visited - 4/3/04

http://www.bbc.co.uk/science/space/exploration/iss/imax.shtml

 

Jonathan’s Space Report – The Web site lists historical information about all space objects that have been launched since the 1960’s.  It is interesting to see how many there were and when they were launched.  Any middle school or older student interested in the space program would enjoy reading the list.

Date Visited - 4/3/04

http://www.planet4589.org/space/jsr/jsr.html

 

Spacelink – current satellite tracking elements – Satellite tracking data or values used to compute where the satellite will be in orbit so that people can observe it and scientists can communicate with it.  Recent Space Shuttle missions to build the Space Station are listed where you can look at more details and NASA’s Space Shuttle Home page is listed as well.  Anyone middle school and up would enjoy reading about the Space Shuttle missions to build the Space Station.

Date Visited - 4/3/04

http://spacelink.nasa.gov/Instructional.Materials/Multimedia/Satellite.Tracking/Current.Satellite.Tracking.Elements/.index.html

 

Questions and Answers

 

a)  What geometric shape are all orbital paths?

b)  List three reasons that mission control people would want to rotate the space object.

c)  What is the name of the vector that defines an object’s location on its orbital path and what mathematician defined the orbital parameters needed to find an object’s location at any point in time?

d)  What can you do with the three X, Y, and Z axes rotational matrices to get one full rotational matrix that does all of the rotations at one time to the object’s point vectors?

e) List some robotic applications in industry or space applications today where they may need to use rotational matrices to calculate movement of the object or its appendages.

 

Look here for the Answers