«Laboratory Instruction in Undergraduate Astronautics Christopher D. Hall Aerospace and Ocean Engineering Virginia Polytechnic Institute and State ...»
Laboratory Instruction in Undergraduate Astronautics
Christopher D. Hall
Aerospace and Ocean Engineering
Virginia Polytechnic Institute and State University
One signiﬁcant distinction between the “standard” educational programs in aeronautical and astronautical engineering is the extent to which experimental methods are incorporated into the curriculum. The use of wind tunnels and their many variations is ﬁrmly established in the aeronautical
engineering curricula throughout the United States. In astronautical engineering, however, there do not appear to be any standard experimental facilities in wide use. This is understandable, given the unique environment in which spacecraft operate; however, there are several facilities which could ﬁll this role, some of which are already in place at universities with a strong space emphasis. The purpose of this paper is to describe some of these facilities and their uses in teaching undergraduate astronautics.
We begin by describing the topics in astronautics that are distinct from other topics in aerospace engineering. We then describe a variety of ﬁeld exercises and laboratories that can be used to enrich the teaching of astronautics. These exercises focus on satellite “observation,” both visually and using amateur radio receivers. Additional laboratories described include a Spacecraft Attitude Dynamics and Control Simulator, and a “design, build, and ﬂy” project to be launched in late 2001.
Topics in Astronautics Some topics in aerospace engineering, such as structures, are common to both aeronautics and astronautics, so that related laboratories beneﬁt both parts of the curriculum. There are however some space-speciﬁc topics that typically have no laboratory component, primarily related to the motion of spacecraft. Satellite motion is a complicated combination of the orbital motion of the satellite around the earth and the attitude, or pointing, motion of the satellite platform. The overall motion is affected by gravity, controlled thrusters, material outgassing, motion of internal components of the satellite, solar radiation pressure, atmospheric drag, and other forces. The study of satellite dynamics and control is typically divided into astrodynamics and attitude dynamics, with additional applied material on spacecraft design.
Kepler (1571–1630) and Newton (1642–1727) laid the foundations for the subject of astrodynamics as it is taught today. Kepler’s three laws were formulated from curve-ﬁtting of the carefully
recorded astronomical observations of Tycho Brahe (1546–1601):
1. The orbit of each planet is an ellipse with the Sun at one focus.
2. The line joining the planet to the Sun sweeps out equal areas in equal times.
3. The square of the period of a planet is proportional to the cube of its mean distance to the sun.
Newton subsequently invented the differential and integral calculus and stated the law of gravitation and his three laws of motion, which he used to derive Kepler’s laws of planetary motion.
Both Kepler’s and Newton’s laws are included in essentially all textbooks on astrodynamics. The standard presentation includes development of the vector differential equation describing orbital motion, development of the solution to this equation, and detailed study of a variety of applications such as orbit determination, orbit transfers, and interplanetary trajectories. Typical textbooks for this subject include the affordable but out-dated Bate, Mueller, and White1 and the more recent book by Vallado.2 A fundamental concept in astrodynamics is that an orbit can be described by a set of six orbital elements, based on the fact that an orbit is a conic section. One set of six orbital elements includes semimajor axis, eccentricity, inclination, right ascension of the ascending node, argument of periapsis, and time of periapsis passage. Typical exercises in astrodynamics include a variety of numerical problems, such as Given the position and velocity of a satellite, determine its orbital elements Given a satellite’s orbital elements at a particular time (or epoch), determine the position and velocity at a later time Given a launch site and a desired orbit, determine the correct launch heading and velocity to achieve the orbit These and other problems can be solved by hand calculations when speciﬁcs are given, and they are certainly appropriate for implementation as computer subroutines that can be used in solving more complex problems. These are valuable exercises, but not all students develop a clear understanding of the signiﬁcance of the calculations or the results. Graphical tools for visualizing orbital motion are helpful, including FreeFlyer,3 Satellite ToolKit,4 and WinOrbit.5 The latter two have the distinction of being freely available for download off the internet. In addition, we have developed a suite of MatLab6 functions that students can use to perform calculations and to visualize orbital motion. A commercial MatLab “toolbox” is also available.7 Attitude dynamics is more complicated than astrodynamics in that it involves the rotational motion of reference frames instead of the comparatively simple translational motion of a point. The foundations of this topic were established by Euler (1707–1783), whose Euler angles, Euler axis, and Euler parameters are still standard topics. A standard though advanced reference for the material is the monograph edited by Wertz.8 An introductory text exclusively on attitude dynamics is available in Rimrott,9 whereas Hughes13 provides an advanced treatment of attitude dynamics.
More commonly, texts include both astrodynamics and attitude dynamics; e.g., Sidi,10 Wiesel,11 and Wie.12 Typical problems developed in attitude dynamics texts involve kinematics, dynamics, and control.
Kinematics problems involve working with rotation matrices and various parameterizations of rotation matrices, including Euler angles and Euler parameters (also known as quaternions). Attitude determination is treated in some texts (notably Refs. 8, 10, and 14), but is typically not covered at all. Rigid body dynamics is central to all attitude dynamics texts, with special attention to Euler’s equations of motion and the special solutions for the case of torque-free axisymmetric rigid bodies.
Other special cases include the use of thrusters, momentum wheels, or control moment gyros to control the rotational motion of a rigid body. As with astrodynamics, these problems can be studied using hand calculations or by using computer subroutines. The latter are certainly more useful in dealing with more advanced problems. Visualization of attitude motion is an important ability that students need in order to understand attitude dynamics. Satellite ToolKit4 has commercial modules that permit the visualization of attitude motion. We have also developed MatLab routines that students can use. A commercial MatLab toolbox is also available,15 and is compatible with Satellite ToolKit.
Space systems design is typically taught during the senior year. At Virginia Tech, space design is an alternative to aircraft design (a required one-year capstone design course), and is typically chosen by a signiﬁcant fraction of seniors. This course is usually taken after an astrodynamics course, but often students in space design have not had an attitude dynamics course. Typically, students develop a detailed spacecraft design in response to a Request for Proposal (RFP), written either by the instructor or a participating outside agency. Several textbooks on space design have appeared in the past few years, including Pisacane and Moore,14 Agrawal,16 Grifﬁn and French,17 Fortescue and Stark,18 and Larson and Wertz.19 A distinguishing characteristic of the latter text is its emphasis on the relationship between mission requirements and design, with a single non-trivial example mission being used throughout the text to illustrate the design process.
As we have described above, astronautics education typically focuses on applied numerical solutions to speciﬁc problems in orbital and attitude dynamics and control, with a capstone spacecraft design course to provide the students with some “practical” experience. In the remainder of the paper we describe some ﬁeld and laboratory work that can be used to enhance this standard approach.
Visible Satellite Tracking One way to engage students with the relatively
idea of satellites hurtling through outer space at 8 kilometers per second is to get them outside watching the satellites go by. As the astronomer and science ﬁction author, Sir Frederick Hoyle (1915– ) put it, “Space isn’t remote at all. It’s only an hour’s drive away if your car could go straight upwards.” The relative closeness of “space” means that we should be able to see some of the objects that are up there. Of course, we can see stars and planets and the moon, but with a little planning, we can also see many of the artiﬁcial moons in low Earth orbit (300–1000 km altitude). The hobby of satellitewatching is well enough established that there are websites available to tell you when and where Table 1: Example of GSOC Satellite Visualizations Homepage Output
to look to see such artiﬁcial moons as the Hubble Space Telescope, the Mir Space Station, and the International Space Station. For example, the German Space Operations Center (GSOC) maintains a Satellite Visualizations Homepage20 that allows a user to enter a particular Earth location and bookmark the resulting page for future use. This page provides the user with tables of visibility for speciﬁc satellites. For example, on Saturday, March 13, 1999, the International Space Station is visible from Blacksburg, Virginia as described in Table 1. The station rises in the northwest at about 7:43 PM local time and goes into the Earth’s shadow in the southeast about 6 minutes later.
This example illustrates some important aspects of satellite observing. The satellite must pass over the ground site; normally for low-Earth orbit satellites, the “pass” will last just a few minutes, and will occur relatively infrequently. Additionally, it must be dark at the ground site but the satellite must be in sunlight in order for the observer to see it. Thus satellite observing opportunities are typically just after sunset or just before sunrise.
An entertaining exercise for students in astrodynamics involves ﬁnding out when a particular satellite will be visible locally and then going out to see it. This can be combined with an essay-writing assignment on the particular spacecraft. Of course, local weather conditions can interfere with the observing part of this assignment. Furthermore, this exercise requires relatively minimal application of the concepts taught in astrodynamics. An enhanced version requires the student to duplicate the calculations that are performed automatically by the software at the Satellite Visualizations Homepage.20 Calculation of satellite visibility is based on the fundamental concepts taught in astrodynamics, speciﬁcally position vectors and orbital elements. There are, however, some additional details that must be covered, including the effects of Earth rotation and Earth oblateness, and the use of range, azimuth, and elevation as coordinates for describing position. These topics are covered in the standard astrodynamics texts. Another topic that is necessary is the interpretation and use of “two-line element sets” or TLEs. The classical orbital elements mentioned above are useful for describing basic orbits, but are inadequate for long-term prediction of future satellite position.
The reason for this inadequacy is that the classical orbital elements are based on several ideal assumptions about the orbital environment that are not strictly valid for Earth-orbiting satellites.
These include gravitational perturbations due to the Sun and Moon, the effects of Earth oblateness, and the effects of aerodynamic drag. The TLE is the standard format for communicating the orbital elements as well as information about the perturbations. The format description and an example TLE for the ISS are given in Table 2.
Further information about TLEs is available in Vallado,2 and at the CelesTrak website.21 The latter is also a useful source for current TLEs for many satellites, and includes links to useful software for working with satellites. Using “fresh” TLEs is important, since the effect of perturbations is to cause the orbital elements to change over time.
Using current TLEs for Earth-orbiting satellites, students can apply the basic algorithms of astrodynamics to determine when these satellites pass over a particular ground site, what the azimuth of the rise and set will be, and how the elevation will vary during the “pass.” Their calculations can be compared with trusted calculations such as those provided by the Satellite Visualizations Homepage.20 Furthermore, packages such as Satellite ToolKit4 and WinOrbit5 work with TLEs and provide similar information.
A simple MatLab application, Visible.m, has been developed at Virginia Tech to compute passes. This application allows a user to select a speciﬁc TLE (saved in a text ﬁle), a speciﬁc date, and speciﬁc ground site latitude and longitude, using a graphical user interface. The program then computes all the passes during a 24-hour period and allows the user to view the passes in several different graphical formats. One example is shown in Fig. 1. Note that this application does not take into account lighting conditions at the ground site or at the satellite, so that it is not quite as useful for visual satellite observing as the Satellite Visualizations Homepage20 is.
However, the purpose of Visible.m is not really to enable students to “see” satellites, but rather to enable them to predict when satellites will be above the horizon with respect to a particular ground site. This is a slightly more general concept of “visibility,” which we will call “accessibility.” Accessibility is in fact of more practical interest to satellite operators than visibility.
Amateur Satellite Tracking and Telemetry