ECE210a_lecture10_small

# ECE210a_lecture10_small - Rotations Rotations(in 2...

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Unformatted text preview: Rotations Rotations (in 2 dimensions) Consider rotating the vector u to get the vector v Hull = llvll = a This can be expressed as: v= [:2] a(cos 0 cos ¢ — sin 0 sin qb) a(sin0 cos 925 + cos 0 sin qb) a(cos 0 cos <15 — sin 0 sin (1)) a(si110 cos ([5 + cos 0 sin d9) cos0 —sin0 acosqﬁ sin9 cos0 a sin 45 _ cos 0 — sin 0 U1 _ sin 0 cos 0 uz W P Roy Smith, ECE 210a, 10: 1 Rotations Rotations (in 2 dimensions) P = [cose —s1n6] sin 6 cos 0 Note that PT is a rotation in the opposite direction so PT 2 P’1 Therefore PT P = I and so P is unitary Roy Smith, ECE 210a, 10:2 Rotations Rotations (in 3 dimensions) Rotation about a coordinate axis (2) Projection onto the x,y plane: V 1 0 0 PM = 0 1 0 y 0 0 0 i) nyv gives the 2-D version of P from before, cos6 —sin6 P _ [sin6 cos6] Now the 2 component in unchanged and we get, cos 6 — sin 6 0 P2 (6) = sin 6 cos 6 0 0 0 1 Roy Smith, ECE 210a, 10:3 Rotations Rotations (in 3 dimensions) The complete set of coordinate rotations in 3-D are: 1 0 0 Pg, (6) = 0 cos 6 — Sin 6 Rotation about the x axis 0 sin 6 cos 6 cos 6 0 sin 6 P7, (6) = 0 1 0 Rotation about the y axis — sin 6 0 cos 6 cos 6 — sin 6 0 PZ (6) = sin 6 COS 0 0 Rotation about the z axis 0 0 1 Roy Smith, ECE 210a, 10:4 Rotations Rotations (in 3 dimensions) We can describe any rotation in 3-D by a rotation about each of the axes: P = Py(0)PZ(¢) Rotations are not commutative. In general, they can be described by Euler angles: (¢, 9,1/1) To specify a rotation we must specify the axes, the order, and whether the axis also moves. Roy Smith, ECE 210a, 10:5 Rotations Rotations Z /' Euler angle rotations: \ P = PzW) PM) Pz(¢>) y The second and third rotations X are about the new x and z axes. There are 12 possible conventions for these angles: 6 of the z-x-z type 6 of the x-y-z type Properties of rotation matrices (in 3-dimensions) eig(P) = {Laijﬂ}, trace(P) 2 14—20050, det(P) = 1. Reference: wikipedia: Euler angles Roy Smith, ECE 210a, 10: 6 Rotations Rotations We can also consider the case where the rotation axes remain ﬁxed. There is an equivalent ﬁxed axis set of rotations. If (and only if) the z and z’ axes start aligned, these rotations are: P : P2(¢’)Px(0,)Pz(¢l) with, W — (b, 0’ = 0, ¢' = 1&- Roy Smith, ECE 210a, 10: 7 Rotations Rotations Euler’s theorem: A113-D rotations can be described by a single rotation about a given vector, u. Three parameters are required to uniquely specify a rotation (u can be normalized). Roy Smith, ECE 210a, 10:8 Rotations Rotations: Aerospace conventions roll This is useful in aerospace applications for describing the effects of body-axis rotations. "Euler angles:” successive rotations to move the vehicle carried local vertical to the body (or wind) frame. P = P’s/(0) PZ(¢) qb Azimuth angle 0 Elevation angle 1/} Bank angle The body and wind frames are related by the rotations: Py (a) (angle-of—attack) and P2 (—ﬂ) (sideslip). Reference: Etkin, “Dynamics antmosphBris Flight,” Wiley, 1972 Roy Smith, ECE 210a, 10:9 Rotations Plane Rotations I 0 c s <— z'th row Deﬁne P as: Pij = I —s c <— jth row 0 I with c2 + 82 = l (for example: cos 0, sin 0) Then for a given x, 331 371 0:12, + sac]- :L' 2 PM :1) = 2:71 —sa:,,- + cxj \$n Roy Smith, ECE 210a, 10:10 Rotations Plane Rotations 11:1 0127-, + sac]- Pij \$ = —s:ri + 0an \$7L . 302‘ 957' If xi aé 0 and acj aé 0, then the ch01ce, c = — 3 = —7 gives, as? + as? + \$1 + P. t I = _ Note that we can use this to selectively put zeros m 1 into a vector. 0 11.77, Roy Smith, ECE 210a, 10:11 Rotations Plane Rotations Applying this once gives, P12 :3 : Applying it to the next component gives, P13P12 x llivll 0 And after n-l rotations, PM . . . P13P12 a: = Roy Smith, ECE 210a, 10:12 Plane Rotations Notethat, T IIijII=Hm1 x§+x§ 0 95,] andso, ||Pij|| =1 and,infact,P,-jisunitary 13512:]- = I, P77 = Pfl 1.] U We can use plane rotations to orthogonalize a matrix and get a QR decomposition. llwll 0 P1n---P13P12-T Rotations Roy Smith, ECE 210a, 10:13 Rotations Givens Rotations Recall the ﬁrst step of the Householder reﬂection method: For the Givens case use H1 2 PM . . . P13P12. The next step in the Householder procedure used, 1 0 HP [0 ml For the Givens case use7 H2 2 [PgnP2n_1 . . . P23] _v_/ n — 2 plane rotations Roy Smith, ECE 210a, 10:14 Rotations QR decompositions Consider the computation effort (in terms of the number of multiplications) Given, AERW" A = QR Gaussian elimination (scaled, partial pivoting): n3/ 3 Gram-Schmidt (classical & modiﬁed): n3 Householder: 2713 / 3 Givens: 4713 / 3 Givens is very useful if A is sparse. Roy Smith, ECE 210a, 10:15 Rotations QR decompositions Q1 R1 = Q2 R2 Is the QR decomposition unique? A Deﬁne U = ngl ( =Q'2TQ2R2RI1) = Rng1 Note that R2 R1— 1 is also upper-triangular and has positive elements on the diagonal. Consider the ﬁrst column of U U11 0 U>k1 : . But = 1 2} U11 2 2:1. As R2 Rfl also has positive diagonal elements, um = 1 Roy Smith, ECE 210a, 10:16 QR decomposition QR decompositions Now consider the second column, U12 U22 U*2 = 0 But UEUQ = 0 (orthogonal) => U12 = 0- 0 Again U22 = :|:1 and the positivity of the diagonal of R2R1—1 implies that 1422 = 1. Repeat this argument for the other columns, U=I andso U=Q2TQ1=L This implies that Q; = Qfl = Q? SO Q1 : Q2 and R1 = = : R2_ Roy Smith, ECE 210a, 10:17 Rotations Computational stability Forward stability: Do small errors in the problem imply a Close answer? Very small perturbation matrix: F << F Consider a series of unitary operations, P = Pk . . . P1 “(Pk - - - P1)(A + E)||p = IIPA + PEllp llerrorllp = llPEllp = llEllp << llAllp = llAllp Backward stability: Does the answer correspond to a “Close” problem? For the QR decomposition A = Q R Q+E = Q, R+F = 1%. This is a QR decomposition for some Q R = fl Is A a: A ? Roy Smith, ECE 210a, 10:18 Rotations Computational stability Backward stability: Is A w A ? A = QR = (Q+E)(R+F) QR + QF + ER + EF EF is negligible Now bound each of the error terms: “QFHF = “FIIF “AHF = IIQRIIF = “RHF “ERHF S llEllpllRllp = llEllpllAllp This gives, llA-fillp S “FHF + HEIIFHAIIF + “EHFHFIIF SoAzA Roy Smith, ECE 210a, 10:19 ...
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ECE210a_lecture10_small - Rotations Rotations(in 2...

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