Lecture Module 2: Spherical Geometry, Various Axes Systems

Transcription

1 1 Lecture Module 2: Spherical Geometry, Various Axes Systems Satellites in space need inertial frame of reference for attitude determination. In a true sense, all bodies in universe are in motion and inertial frame of reference can only be defined relatively. This lecture module gives an overview of various axes systems used in satellite dynamics and spherical geometry, which are integral part of this subject. Lectures The spacecraft-centered celestial sphere A sphere known as celestial sphere with the satellite at the center is shown in the figure below. In space, due to extremely large distances involved, measurements are made in terms of arc length, rotation angle, and solid angle. In figure 2.1, the celestial sphere is a sphere of unit radius. A straight line joining the satellite and Earth thus marks a point E on the surface of the celestial sphere. Similarly a straight line joining the satellite and Sun marks a point S on the surface of sphere and satellite attitude axis marks a point A. The great circle which divides the celestial sphere in two equal parts (hemispheres) defines the equatorial plane. Any other circle on the celestial sphere is a smaller circle. Similar to the points E, S, and A on the upper hemisphere, the anti points E -1 (zenith), S -1 (anti solar), and A -1 are also marked on the surface of the celestial sphere in the lower hemisphere through the extended lines. Following terminologies are useful to get acquainted with spherical geometry: a. Spherical triangle: a triangle on the surface of a sphere such as EAS b. Arc segment: Segment of curves on the sphere such as ES, EA, or SE. c. Arc length (or angular separations): Separation between two arc segments S A E E -1 A -1 S -1 Figure 2.1: Satellite at the center of a celestial sphere.

2 2 In figure 2.1, arc lengths on the celestial sphere are denoted by the angle triplet (,, ), and included angles between the arc segments are denoted by the angle triplet (,, ), all of them having units radian or degree. The size of the spherical triangle EAS can be measured in terms of an area which is in terms of the solid angle formed at the satellite joining the vertices of the triangle EAS. Unit of solid angle is square degree or steradian. Note: Although arc lengths and included angle between arc lengths have same units of angles, they are different quantities and angles must not be interchanged. Three fundamental relations hold between the angular separations of points on a sphere (arc lengths) and included angles between them. They are: The law of sines: sin sin sin, (2.1) sin sin sin The law of cosines for sides: cos cos cos sin sin cos, (2.2) And the law of cosines for angles: cos cos cos sin sin cos. (2.3) A right spherical triangle is one in which one of the included angles between the two sides (arc lengths) or the rotation angles is 90deg. A spherical triangle in which one of the sides is 90deg is known as quadrantal spherical triangle. Rules of the spherical trigonometry are considerably simplified for these two case (Homework: Verify!). A 16 th century Scottish mathematician John Napier developed these rules for the special case of right and quadrantal spherical triangles. Laws of sines and cosines are very useful in attitude analysis. We will show an example of spherical trigonometry a little later, let us first look at various axis systems used in spacecraft attitude determination and understand the various terminology associated with a celestial sphere.

3 Spacecraft centered coordinate systems: Reference or prime meridian North celestial pole Meridian Parallels P Ecliptic Vernal equinox Celestial equator : right ascension declination Figure 2.2: Coordinate systems on a spacecraft centered celestial sphere. In Figure 2.2: Ecliptic: plane of rotation of Earth s orbit about Sun Vernal equinox: point on the celestial equator where ecliptic crosses Coordinate of any point P on the celestial sphere is measured with respect to vernal equinox in terms of two angles, (measured from 0 to 360 deg), which is known as right ascension or azimuth/longitude, and (measured from -90 deg to +90 deg) which is known as declination or latitude. Vernal equinox a specific reference point (with coordinates 0 0, 0 0 ) in this case is also the point of intersection of reference or prime meridian and celestial equator. The great circles through the poles and perpendicular to the equator are called meridians and the small circles above or below the equator are called parallels. A parallel at declination angle is a small circle of angular radius Example 1: Location of P on celestial sphere: (, ) (40,60 ) Example 2: Distance between two points on a celestial sphere can be measured in terms of the difference between the angles.

4 4 Exercise Problem: Locations of two stars S 1 and S 2 are measured by a satellite to be S : (, ) (25,60 ), : (, ) S terms of angular separation on satellite centered celestial sphere. Example 3: At the poles where 0 0 (125,30 ). Determine the distance between them in 0 90, parallels have zero radius and azimuth becomes undefined. Example 4: An example problem based on spherical trigonometry. On a satellite centered celestial sphere, six optical sensors (For example Sun sensors, we learn about them in Lecture Module 4) are placed in such a way that optical axes of four of them lie equally spaced on the celestial equator (points A, B, C, D), and optical axes of two of the sensors point towards the poles (points E, F). A sketch is provided below for clarity. Problem is to find a point on the celestial sphere that is at a maximum distance (angle) from the axis of the closest sun sensor. E C D A B Figure 2.3: Sensors placed on the satellite at the center of the celestial sphere. F Let us consider the spherical triangle EAB. By symmetry, it is clear that the one such farthest point lies on a meridian halfway between the meridians passing through EA and EB. Further, this point P (Fig. 2.4) lies on the triangle EAB such that, (360 / 8) deg 45deg; 90deg ; 1 2 Now applying the rule of cosine for sides for spherical triangle EPB, we see that cos 1 cos 3 cos sin 3 sin cos 1

5 5 E P A B Figure 2.4: Spherical triangles EAB, EAP, EPB. Substituting the values we get cos cos cos(90deg) sin sin(90deg) cos( 45deg) 0 (1/ 2)sin 54.73deg. z Reference or prime meridian North celestial pole y Reference point x Celestial equator Figure 2.5: Celestial sphere with poles.

6 6 2.3 Spacecraft-Fixed coordinates: In spacecraft fixed coordinate system, the spin axis of the spacecraft is the z-axis joining the celestial (north and south) poles as shown in Fig An axis drawn from satellite to the intersection point (reference point on celestial sphere) of prime meridian (an arbitrarily chosen one in this case) and the celestial equator (spin plane of satellite) is the x-axis, and the y-axis is such that it completes a right handed orthogonal coordinate system. Attitude measurement sensor hardware mounted on the satellite (irrespective of where they are located on the satellite) measure the satellite s attitude with respect to certain star in space and Earth. For nonspinning satellites, no standard orientation (axis system) is defined. 2.4 Inertial co-ordinates: The universe being dynamic and one with motion all the time, in general, it is not possible to define an inertial co-ordinate system. But for practical purposes, different time scales of motion that different planetary bodies have are useful for defining an inertial co-ordinate system. In the inertial co-ordinate system, reference point is fixed as the vernal equinox as described earlier with respect to Fig [Note: Vernal equinox usually slides along the ecliptic due to Earth s spin axis undergoing precession with respect to the ecliptic poles, but this motion is too slow (period 26,000years!) to make any significant difference on attitude determination.] 2.5 Orbit defined co-ordinates: In this system of co-ordinates (called l-b-n), plane of the spacecraft orbit (say around Earth) is the equatorial plane. Axis l is parallel to the line from center of Earth to the ascending node (point of intersection of equatorial plane of Earth and equatorial plane of spacecraft s orbit in the ascending phase south to northward motion), n axis is parallel to the orbit normal (perpendicular to the plane of spacecraft orbit), and b axis is such that for unit vectors along the axes, bˆ nˆ lˆ. If the spacecraft is stationary in the orbit, this co-ordinate system is inertial. [HomeWork Exercise: Figure out this axis system with respect to the Figure below.]

7 7 APOGEE SATELLITE ORBIT R Y P EARTH EQUATORIAL PLANE ASCENDING NODE PERIGEE Figure 2.6: Satellite orbit around Earth. Another system of co-ordinates that maintains its orientation relative to Earth while the spacecraft is in motion in an orbit around Earth is known as roll, pitch, yaw or R-P-Y axis system. In this system, yaw axis Y is directed toward Earth center (nadir), the pitch axis P is directed towards negative orbit normal, and the roll axis R is perpendicular to P and Y such that unit vectors along these axes satisfy the relation, Rˆ Pˆ Yˆ. Note that, order of multiplication of vectors must be obeyed here. 2.6 Nonspacecraft-centered coordinate systems: Other than spacecraft centered co-ordinate systmes, which is useful for satellite attitude related work, sometimes use of nonspacecraft centered coordinate systems become important. For example, geocentric inertial coordinates with the center of the coordinate system at the center of Earth. This system of coordinate is useful for orbit related work and for determining reference vectors such as the magnetic field vector or position vectors to the objects seen by the spacecraft. Coordinates centered at Sun to determine position of planets within the solar system is known as heliocentric coordinate system. Heliocentric longitude and latitude are defined with respect to the ecliptic plane and vernal equinox as reference node. Selenocentric coordinates (centered at Moon) are used for satellites in lunar orbit.