Pointing Control Definitions target target desired pointing direction estimate true actual pointing direction(mean) estimate estimate of true(instantaneous pointing accuracy (long-term) stability(peak-peak motion) true knowledge error control error a= pointing accuracy =attitude error s=stability=attitude jitter ource G mosi NASA GSFC
Pointing Control Definitions Pointing Control Definitions target desired pointing direction true actual pointing direction (mean) estimate estimate of true (instantaneous) a pointing accuracy (long-term) s stability (peak-peak motion) k knowledge error c control error target estimate true c k a s Source: G. Mosier NASA GSFC a = pointing accuracy = attitude error a = pointing accuracy = attitude error s = stability = attitude jitter s = stability = attitude jitter
Attitude Coordinate systems North celestial Pole) GCI: Geocentric Inertial Coordinates Cross product AA Geometry: Celestial Sphere YEZXX dihedral A Y VERNAL X EQUINOX a: right ascension Inertial Coordinate δ: Declination ystem X andy are in the plane of the ecliptic
Attitude Coordinate Systems Attitude Coordinate Systems X Z Y ^ ^ ^ Y = Z x X Cross product Cross product ^ ^ ^ Geometry: Celestial Sphere Geometry: Celestial Sphere : Right Ascension : Right Ascension : Declination : Declination (North Celestial Pole) Arc length dihedral Inertial Coordinate Inertial Coordinate System GCI: Geocentric Inertial Coordinates GCI: Geocentric Inertial Coordinates VERNAL EQUINOX EQUINOX X and Y are in the plane of the ecliptic
Attitude description notations &=Coordinate system P=Ⅴ ector P P=Position vector w.r. t. A 100 Unit vectors of (A]=XA YA Z0 1 0 001 Describe the orientation of a body (1)Attach a coordinate system to the body 2)Describe a coordinate system relative to an inertial reference frame
Attitude Description Notations Attitude Description Notations Describe the orientation of a body: (1) Attach a coordinate system to the body (2) Describe a coordinate system relative to an inertial reference frame ZA ˆ X A ˆ YA ˆ Position vector w.r.t. { } Vector { } Coordinate system P A P A = = ⋅ = P A Py Px Pz = z y x A P P P P [ ] = = 0 0 1 0 1 0 1 0 0 Unit vectors of { A } X A YA Z A ˆ ˆ ˆ
Rotation matrix A (A=Reference coordinate system B B= Body coordinate system Rotation matrix from b to Aj B 后R=|xB^BA2p A Special properties of rotation matrices B (1)Orthogonal RR=R=R (2 ) Orthonormal R=1 R=0 cos0-sin0 (3)Not commutative 0 sin] cosb RPR≠RRR
Rotation Matrix Rotation Matrix Rotation matrix from {B} to {A} Jefferson Memorial ZA ˆ XA ˆ YA ˆ {A} = Referencecoordinatesystem XB ˆ YB ˆ ZBˆ {B} = Body coordinate system [ B B B ] A A B R = Xˆ AYˆ AZˆ Special properties of rotation matrices: 1 , − R R = I R = R T T R =1 (1) Orthogonal: R R R R B A C BC AB ≠ B Jefferson Memorial ZA ˆ XA ˆ YA ˆ XB ˆ YB ˆ ZBˆ θ θ R = AB 0 sin cos 0 cos -sin 1 0 0 (2) Orthonormal: (3) Not commutative
Euler angles(1) Euler angles describe a sequence of three rotations about different axes in order to align one coord system with a second coord system Rotate about za by a Rotate about yb by B Rotate about Xc by y ZG C B B B C coSa -sina cosb 0 sinB 100 R= sina cosa 0 R=010 D R=0 coSy -siny inB 0 cosB 0 siny cosy AR-BR ER SR
Euler Angles (1) Angles (1) Euler angles describe a sequence of three rotations about different axes in order to align one coord. system with a second coord. system. = 0 0 1 sin cos 0 cos -sin 0 α α α α RAB Rotate about Z ˆ A byα Rotate about Y ˆ B by β Rotate about XC by γ ˆ ZA ˆ XA ˆ YA ˆ XB ˆ YB ˆ ZB ˆ α α ZB ˆ XB ˆ YB ˆ XC ˆ YC ˆ ZC ˆ β β ZC ˆ XC ˆ YD ˆ X D ˆ YC ˆ ZD ˆ γ γ = β β β β -sin 0 cos 0 1 0 cos 0 sin RBC = γ γ γ γ 0 sin cos 0 cos -sin 1 0 0 RCD R R R RCD BC AB AD =