1 Lecture Module 1 Topics: Introduction to Spacecraft Attitude Dynamics; Space environment; Environmental Torques Lectures 1 - 4 Introduction A satellite is an object artificially placed in space for various mission objectives, for example, remote sensing, telecommunication, weather monitoring, space exploration, space surveillance, etc.. A majority or all of mission objectives require that a satellite be oriented (post injection into an intended orbit from a launch vehicle) with respect to an object to capture maximum information about the object continually over a long period of time. For example, a communication satellite network requires a satellite to gather (receive) information from one source and transmit information to the end user. On-board solar panels facing Sun to collect maximum radiation energy to power a satellite may require orientation of the satellite in a particular fixed direction. In a fixed orbit, at a fixed location, a satellite may be required to be re-orientated towards another object of interest thus calling for an attitude maneuver (shifting from one attitude to another) with the help of on-board control systems. Several mission objectives thus require attitude (or orientation) of the satellite to be either kept fixed or manipulated in a controlled manner. This calls for continuous attitude measurement of a satellite. Attitude measurement sensors are especially designed for satellites with high precision and accuracy, else we might end up facing a complete loss of signal (or blackout) for our Television antenna or experience poor transmission leading to blurred images and/or broken sound and disrupted data transfer. Any deviation from intended attitude read by sensors has to be corrected either in a passive or in an active manner. The attitude of a satellite in space can be disturbed by external torques present in space environment where the satellite is placed. Torques offered by moving parts inside a satellite can also change its attitude. While small attitude changes are related to stability properties and small disturbance torques, large attitude changes, during a maneuver, for example, require large disturbance torques. A satellite is usually equipped with various mechanisms to counter undesired small and large disturbance torques. Space environment offering various torques can either be seen as disturbance torques and detrimental to functioning of satellite or they can be usefully manipulated for various purposes. Prof. Nandan K Sinha Aerospace Engineering IIT Madras 2 Geometric and inertia properties of a satellite are factors which define its stability characteristics. Through, rigid body motion analysis of a satellite using equations of attitude dynamics and kinematics, a configuration with desired stability properties can be arrived. Simulation via numerical integration of the equations of motion, eigenvalue analysis, and energy based methods for motion analysis are important tools adopted for satellite motion analysis. Major goal of this lecture series is to introduce readers to various aspects related to satellite attitude technology. The lectures are broadly categorised into five modules, covering: 1. Space environment 2. Spherical geometry, introduction to axes systems 3. Satellite attitude dynamics 4. Satellite attitude determination techniques 5. Satellite attitude stabilization and control Following books were referred to while preparing this lecture series: 1. Spacecraft Attitude Determination and Control, Edited by James R Wertz, Kluwer Academic Press, Boston, 1978. 2. Modern Spacecraft Dynamics and Control by Marshall H. Kaplan, John Wiley and Sons, NY, 1976. 3. Spacecraft Dynamics and Control an Introduction by Anton H.J. De Ruiter, Christopher J. Damaren, and James R. Forbes, John Wiley and Sons, NY, 2013. Space Environment Space environment offers many challenges to smooth operation of satellites. Environments offer sometimes known and at other times unknown disturbances to satellite in term of torques which changes satellite’s attitude from the desired orientation. Knowledge and modelling of these attitude disturbance torques are important for accurate prediction of a satellite attitude (or orientation) using mathematical models (or governing equations of motion). Such predictions further help in designing control systems for attitude stabilization and maneuvering. Prof. Nandan K Sinha Aerospace Engineering IIT Madras 3 Major sources of attitude disturbance torques are: a. Earth’s magnetic and gravitational fields, b. Solar radiation pressure, c. Aerodynamic drag, and d. Magnetic disturbance. 1.1 Gravity-gradient torque: Gravity gradient torque is particularly significant for non-symmetrical objects due to variation in Earth’s gravitational force over the object. Geometric center Center of mass r'i ri dmi RG Ri satellite of arbitrary shape Earth Figure 1.1: An arbitrary shaped satellite with distinct center of mass and geometric center experiencing gravitational force of Earth. The gravitational force dFi acting on the elemental mass dmi of the satellite is given by dFi dm i R i Ri3 Torque about geometric center due this force is: d N i r i d F i ( r i ) d F i ' Prof. Nandan K Sinha (1.1) Aerospace Engineering IIT Madras 4 The gravity gradient torque on the entire spacecraft is obtained by integrating Eq. (1.1). Thus, N GG ( r i ) ' dm i R i R 3 i M ˆ 3 RG 3 2 RG RG r i Rˆ G r i .Rˆ G dm i (1.2) Where GM E ; G is the gravitational constant of Earth and ME is the mass of Earth. M is the mass of the satellite. When the center of mass and the geometric center coincide, i.e., 0 , the expression for gravity gradient torque simplifies to N GG 3 RG3 r i 3 Rˆ G r i .Rˆ G dm i 3 Rˆ G I .Rˆ G RG (1.3) Where I is the inertia tensor. Assuming satellite body fixed axis system to be the principal axis system, so that, I IP I1 0 0 0 I2 0 0 0 I 3 and R0 be the absolute distance between the satellite center of mass and, following Eq. (1.3), components of Gravity Gradient Torque about satellite axes can be determined to be G1 3 ( I 3 I 2 ) sin 2 cos 2 2 R 03 G2 3 ( I 3 I 1 ) sin 2 cos 2 R 03 G3 3 ( I 1 I 2 ) sin 2 sin 2 R 03 (1.4) , in Eq. (1.4) are Euler angles. Example 1.1: For a spherical body with double symmetry, I 1 I 2 I 3 , gravity gradient torque G1 G 2 G 3 0 . Prof. Nandan K Sinha Aerospace Engineering IIT Madras 5 Example 1.2: Assuming small angles , , such that, sin( angle) angle, cos( angle) 1 and product of angles being of negligible value and insignificant, G1 3 3 3 ( I 3 I 2 ) sin 2 cos 2 ( I 3 I 2 ).2 .1 3 ( I 3 I 2 ) 3 3 2 R0 2 R0 R0 G2 3 3 3 ( I 3 I 1 ) sin 2 cos ( I 3 I 1 ).2 .1 3 ( I 3 I 1 ) 3 3 2 R0 2 R0 R0 G3 3 3 ( I 1 I 2 ) sin 2 sin ( I 1 I 2 ).2 . 0 3 2 R0 2 R03 (1.5) Example 1.3: For a body in circular orbit of radius R0, lateral velocity of the satellite R0 , and its angular orbital velocity also called orbital rate or frequency, v 0 v / R0 R03 , Eq. (x.5) can be re-written as G1 3 3 ( I 3 I 2 ) sin 2 cos 2 ( I 3 I 2 ).2 .1 3 02 ( I 3 I 2 ) 3 3 2 R0 2 R0 G2 3 3 ( I 3 I 1 ) sin 2 cos ( I 3 I 1 ).2 .1 3 02 ( I 3 I 1 ) 3 3 2 R0 2 R0 G3 3 3 ( I 1 I 2 ) sin 2 sin ( I 1 I 2 ).2 . 0 3 2 R0 2 R03 (1.6) Homework Exercise 1: For a cylindrical object with two plane of symmetry determine the Gravity-gradient torque. Some observations (from Eq. (1.3) with the approximation, that is, 0 ): The torque is normal to the local vertical, The torque is inversely proportional to the cube of the geometric distance, and The torque vanishes for a spherically symmetric spacecraft. For a spin stabilized satellite or a satellite with a composite of inertial and moving components, orbital parameters also need to be considered to arrive at gravity-gradient torque [see Ref. 1 for more details.]. Prof. Nandan K Sinha Aerospace Engineering IIT Madras 6 3.1 Solar radiation torque: Satellite’s surface is subjected to solar radiation pressure (radiation force per unit area equal to the vector difference between the incident and reflected momentum flux). Near Earth, magnitude of this pressure is around 4.5 10 6 N / m 2 . Solar radiation pressure on a satellite or spacecraft in Earth orbit is independent of the altitude of the satellite above Earth because of the large distance from Sun. Three dominant factors determining the solar radiation torque on a satellite are: The intensity and spectral distribution of the incident solar radiation, The geometry and optical properties of the satellite surface, and The intensity of the Sun vector relative to the satellite. In the following, the effect due to direct Solar radiation is considered. Mean momentum flux P that is also the solar radiation pressure acting on satellite surface normal to solar radiation is given by, P Fe c , where Fe is the solar constant which is wavelength dependent and c is the speed of light. Solar radiation incident on satellite surface Figure 1.2: Solar radiation incident upon surface of a satellite at an angle . For the part of incident radiation that is absorbed by the surface, the differential radiation force (for elemental area dA) which is momentum transferred per unit time is given by d F absorbed PC a cos SˆdA Prof. Nandan K Sinha (0 90 o ) Aerospace Engineering (1.7) IIT Madras 7 Where Sˆ is the unit vector from satellite to Sun and C a is the absorption coefficient. Part of the radiation which is specularly reflected (in the direction ( Sˆ 2 Nˆ cos ) ) or diffused (in all directions) from the satellite surface results in following differential forces, d F specular 2 PC s cos 2 Nˆ dA (0 90 o ) 2 d F diffuse PC d cos Nˆ cos Sˆ dA 3 (1.8) (0 90 o ) (1.9) Where C s and C d are coefficients of specular and diffusion reflections, respectively. Thus, total differential force is given by 1 F total P 1 C s Sˆ 2 C s cos C d 3 ˆ N cos dA (0 90 o ) (1.10) Where C a C s C d 1 . The solar radiation torque acting on the spacecraft is given by N solar R d F total (1.11) In Eq. (1.11), R is the distance between the center of mass of the satellite and the point at which resultant of force due to solar radiation (integrated over the exposed satellite surface area) act. The other two major sources of external torques that a satellite can experience as disturbance torques in space are Aerodynamic torque (in Low Earth orbit) and Magnetic torque due to interaction between a magnetic component placed anywhere on the satellite and Earth’s magnetic field. Some details about these torques follow. 4.1 Aerodynamic Torque For spacecraft in low Earth orbit (below 400km altitude), the aerodynamic torque is a dominant environmental disturbance torque acting on the spacecraft/satellite. The aerodynamic force acting on the satellite is not due to relative wind hitting the satellite surface, but due to momentum exchange due to molecules arriving at the surface. Therefore, continuum model of atmosphere do not apply here. Prof. Nandan K Sinha Aerospace Engineering IIT Madras 8 V dA C.M. Figure 1.3: Aerodynamic force acting on a small elemental area of the satellite. The force, d F aero , on a surface element dA with outward normal Nˆ is given by, 1 d F aero C D V 2 ( Nˆ .Vˆ )VˆdA . 2 (1.12) Vˆ is the unit vector in the direction of the relative velocity of the incident airstream (stream of air molecules), is the atmospheric density, and C D is the drag coefficient which is a function of the local angle of attack. An expression for the aerodynamic torque acting on the spacecraft thus can be arrived at as N aero r c d F aero (1.13) where r c is the distance between the center of mass of the spacecraft and the satellite surface element dA . For a spinning spacecraft the total velocity of the element dA with respect to the airstream is given by V V c rc (1.14) Where V c is the translational velocity of the center of mass of the spacecraft relative to the airstream, and is the angular velocity of the spacecraft. The expression for aerodynamic torque including the spin motion of spacecraft can be obtained to be N aero Nˆ . r c Vˆ c r c 1 1 C D Vc2 Nˆ .Vˆ c Vˆ c r c dA C D Vc dA 2 2 Nˆ .Vˆ c r c r c Prof. Nandan K Sinha Aerospace Engineering (1.15) IIT Madras 9 A satellite consisting of different parts of different shapes can be decomposed into some basic shapes. Aerodynamic force on various basic shapes can be thus found easily from empirical relations and aerodynamic torques due to each individual component integrated over the whole body of satellite to arrive at the final expression for aerodynamic torque. Some errors due to interference of different parts are expected in this way and must be accounted for. Shadowing of one part due to another is another source of error, which must be accounted for. Expressions for aerodynamic force for some simple geometric shapes are given below. 1 2 Sphere of radius R: F aero C D V 2R 2Vˆ 1 Plane with surface area A: F aero C D V 2 A Nˆ .Vˆ Vˆ , where Nˆ is the normal unit 2 vector. Right circular cylinder of length L and diameter D: 2 1 F aero C D V 2 DL 1 lˆ.Vˆ Vˆ 2 , (1.16) where lˆ is the unit vector along the length of the cylinder. 4.2 Magnetic Disturbance Torque Magnetic disturbance torques results from interaction between spacecraft’s residual magnetic field and geomagnetic field. Sources of spacecraft magnetic field are: Eddy currents Hysteresis Spacecraft’s magnetic moments The magnetic disturbance torque due to spacecraft magnetic moment is given by N mag M B (1.17) Where M is the total magnetic moment (in A.m2) due to permanent and induced magnetism and spacecraft generated current looks, B is the geocentric magnetic flux density (Wb/m2). Prof. Nandan K Sinha Aerospace Engineering IIT Madras 10 Torques created due to eddy currents and hysteresis are attributed to spacecraft’s spinning motion in the geomagnetic field. Expression for this torque is given by N Eddy k e B B (1.18) Where is the spacecraft’s angular velocity vector and ke is a constant coefficient which depends upon spacecraft geometry and conductivity. ke for some geometric figure of satellite (or its parts) with conductivity are: 2 4 r t 3 Thin spherical shell of radius r, thickness t: k e Circular loop of radius r and cross-sectional area A located in a plane containing the spin axis: k e Thin 4 walled r 3 A cylinder with length L, radius r, and thickness t: 2t L k e r 3 Lt 1 tanh L 2t Magnetic torque due to hysteresis only appreciable for very elongated soft magnetic material is given by N Hysterisis E H 2 t (1.19) Where t is the time over which the torque is being evaluated and E H is the energy loss over one rotation period given by E H V H .d B (1.20) V is the volume of the permeable material, H is the magnetic field of the surrounding medium, and dB is the induced magnetic induction flux in the material. Prof. Nandan K Sinha Aerospace Engineering IIT Madras
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