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11. Spacecraft Dynamics MAE 342 2016

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2/12/20 Spacecraft Dynamics Space System Design, MAE 342, Princeton University Robert Stengel • Angular rate dynamics • Spinning and non-spinning spacecraft • Gravity gradient satellites • Euler Angles and spacecraft attitude • Rotation matrix • Precession of spinning axisymmetric spacecraft Copyright 2016 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE342.html 1 1 Angular Momentum of a Particle • Moment of linear momentum of differential particles that make up the body – Differential mass of a particle times component of its velocity that is perpendicular to the moment arm from the center of rotation to the particle dh = (r × dmv) = (r × v m )dm = (r × (vo + ω × r))dm 2 2 1 2/12/20 Angular Momentum of an Object Integrate moment of linear momentum of differential particles over the body h = r × v + ω × r dm ∫ ( ( o )) Body = r × v dm + r × ω × r dm ∫ ( o ) ∫ ( ( )) Body Body = 0 − ∫ (r × (r × ω))dm = − ∫ (r × r)dm × ω Body Body ⎡ ⎤ ≡ − (r!r!)dmω ⎡ ⎤ ω x ∫ hx ⎢ ⎥ Body ⎢ ⎥ ω = ⎢ ω y ⎥ h = ⎢ hy ⎥ ⎢ ω ⎥ ⎢ h ⎥ ⎣⎢ z ⎦⎥ ⎣⎢ z ⎦⎥ 3 3 Angular Momentum with Respect to the Center of Mass Choose center of mass as origin about which angular momentum is calculated (= center of rotation) Use cross-product-equivalent matrix to define the inertia matrix, I h = − ∫ r! r! dm ω Body x y z ⎡ 0 −z y ⎤⎡ 0 −z y ⎤ max max max ⎢ ⎥⎢ ⎥ = − ∫ ∫ ∫ ⎢ z 0 −x ⎥⎢ z 0 −x ⎥ρ(x, y,z)dx dydz xmin ymin zmin ⎢ −y x 0 ⎥⎢ −y x 0 ⎥ ⎣ ⎦⎣ ⎦ ⎡ (y2 + z2 ) −xy −xz ⎤ ⎢ ⎥ 2 2 = ∫ ⎢ −xy (x + z ) −yz ⎥dm ω = I ω Body ⎢ 2 2 ⎥ ⎢ −xz −yz (x + y ) ⎥ 4 ⎣ ⎦ 4 2 2/12/20 Inertia Matrix 2 2 ⎡ ⎤ ⎡ (y + z ) −xy −xz ⎤ I xx −I xy −I xz ⎢ ⎥ ⎢ ⎥ 2 2 −I I −I I = ∫ ⎢ −xy (x + z ) −yz ⎥dm = ⎢ xy yy yz ⎥ Body ⎢ 2 2 ⎥ ⎢ ⎥ −xz −yz (x + y ) −I xz −I yz Izz ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ Moments of inertia on the diagonal Products of inertia off the diagonal If products of inertia are zero, (x, y, z) are principal axes ⎡ I 0 0 ⎤ ⎢ xx ⎥ IP = ⎢ 0 I yy 0 ⎥ ⎢ 0 0 I ⎥ ⎣⎢ zz ⎦⎥ For fixed mass distribution, inertia matrix is constant in body frame of reference 5 5 Inertial-Frame Inertia Matrix is Not Constant if Body is Rotating Newton’s 2nd Law applies to rotational motion in an inertial frame Rate of change of angular momentum = applied moment (or torque), m ⎡ M ⎤ Chain Rule ⎢ x ⎥ dh d(Iω) d = = M = ⎢ M y ⎥ (Iω) dh d I dt dt = = ω + Iω! ⎢ M ⎥ dt dt dt ⎣⎢ z ⎦⎥ Inertial-frame solution for In an inertial frame angular rate d d I I ≠ 0 Iω! = M − ω dt dt −1 ⎛ d I ⎞ ω! = I ⎜ M − ω⎟ ⎝ dt ⎠ 6 6 3 2/12/20 How Do We Get Rid of dI/dt in the Angular Momentum Equation? Write the dynamic equation in a body-referenced frame § Inertia matrix is ~unchanging in a body frame § Body-axis frame is rotating § Dynamic equation must be modified to account for rotation 7 7 Expressing Vectors in Different Reference Frames • Angular momentum and rate are vectors – They can be expressed in either the inertial or body frame – The 2 frames are related by the rotation matrix (also called the direction cosine matrix) B HI : Rotation transformation from inertial frame ⇒ body frame I HB : Rotation transformation from body frame ⇒ inertial frame B I hB = HI hI hI = HBhB B I ω B = HI ω I ω I = HBω B 8 8 4 2/12/20 Euler Angles • Body attitude measured with respect to inertial frame • Three-angle orientation expressed by sequence of three orthogonal single-angle rotations Inertial ⇒ Intermediate1 ⇒ Intermediate2 ⇒ Body ψ : Yaw angle ψ : Yaw angle1 θ : Pitch angle θ : Pitch angle φ : Roll angle φ : Yaw angle2 • 24 (±12) possible sequences of single-axis rotations • Aircraft convention: 3-2-1, z positive down • Spacecraft convention: 3-1-3, z positive up 9 9 Reference Frame Rotation from Inertial to Body: Aircraft Convention (1-2-3) Yaw rotation (ψ) about zI axis ⎡ ⎤ ⎡ x ⎤ ⎡ cosψ sinψ 0 ⎤⎡ x ⎤ xI cosψ + yI sinψ ⎢ ⎥ ⎢ ⎥ 1 ⎢ y ⎥ ⎢ y ⎥ x sin y cos r = H r ⎢ ⎥ = ⎢ −sinψ cosψ 0 ⎥⎢ ⎥ = ⎢ − I ψ + I ψ ⎥ 1 I I ⎢ z ⎥ ⎢ 0 0 1 ⎥⎢ z ⎥ ⎢ z ⎥ ⎣ ⎦1 ⎣ ⎦⎣ ⎦I ⎣ I ⎦ Pitch rotation (θ) about y1 axis ⎡ x ⎤ ⎡ cosθ 0 −sinθ ⎤⎡ x ⎤ ⎢ ⎥ ⎢ ⎥ 2 2 1 2 y = ⎢ ⎥ y r = H r = ⎡H H ⎤r = H r ⎢ ⎥ ⎢ 0 1 0 ⎥⎢ ⎥ 2 1 1 ⎣ 1 I ⎦ I I I ⎢ z ⎥ ⎢ sinθ 0 cosθ ⎥⎢ z ⎥ ⎣ ⎦2 ⎣ ⎦⎣ ⎦1 Roll rotation (ϕ) about x2 axis ⎡ x ⎤ ⎡ 1 0 0 ⎤⎡ x ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ B B 2 1 B y 0 cosφ sinφ y rB = H2 r2 = ⎣⎡H2 H1 HI ⎦⎤rI = HI rI ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢ z ⎥ ⎢ 0 −sinφ cosφ ⎥⎢ z ⎥ ⎣ ⎦B ⎣ ⎦2 ⎣ ⎦ 10 10 5 2/12/20 Reference Frame Rotation from Inertial to Body: Spacecraft Convention (3-1-3) Yaw rotation (ψ) about zI axis ⎡ ⎤ ⎡ x ⎤ ⎡ cosψ sinψ 0 ⎤⎡ x ⎤ xI cosψ + yI sinψ ⎢ ⎥ ⎢ ⎥ 1 ⎢ y ⎥ ⎢ y ⎥ x sin y cos r = H r ⎢ ⎥ = ⎢ −sinψ cosψ 0 ⎥⎢ ⎥ = ⎢ − I ψ + I ψ ⎥ 1 I I ⎢ z ⎥ ⎢ 0 0 1 ⎥⎢ z ⎥ ⎢ z ⎥ ⎣ ⎦1 ⎣ ⎦⎣ ⎦I ⎣ I ⎦ Pitch rotation (θ) about x1 axis ⎡ x ⎤ ⎡ 1 0 0 ⎤⎡ x ⎤ ⎢ ⎥ ⎢ ⎥ 2 2 1 2 y = ⎢ 0 cos sin ⎥ y r = H r = ⎡H H ⎤r = H r ⎢ ⎥ ⎢ θ θ ⎥⎢ ⎥ 2 1 1 ⎣ 1 I ⎦ I I I ⎢ z ⎥ ⎢ 0 −sinθ cosθ ⎥⎢ z ⎥ ⎣ ⎦2 ⎣ ⎦⎣ ⎦1 Yaw rotation (ϕ) about z2 axis ⎡ x ⎤ ⎡ cosφ sinφ 0 ⎤⎡ x ⎤ ⎢ ⎥ B B 2 1 B ⎢ y ⎥ ⎢ y ⎥ r = H r = ⎡H H H ⎤r = H r ⎢ ⎥ = ⎢ −sinφ cosφ 0 ⎥⎢ ⎥ B 2 2 ⎣ 2 1 I ⎦ I I I ⎢ z ⎥ ⎢ 0 0 1 ⎥⎢ z ⎥ ⎣ ⎦B ⎣ ⎦⎣ ⎦2 11 11 Rotation Matrix from I to B Aircraft Convention (1-2-3) B B 2 1 HI (φ,θ,ψ) = H2 (φ)H1 (θ)HI (ψ) # 1 0 0 &# &# cosψ sinψ 0 & % ( cosθ 0 −sinθ % ( 0 cos sin % ( = % φ φ (% 0 1 0 (% −sinψ cosψ 0 ( % ( % ( $ 0 −sinφ cosφ '$% sinθ 0 cosθ '($ 0 0 1 ' # cosθ cosψ cosθ sinψ −sinθ & % ( = % −cosφ sinψ +sinφ sinθ cosψ cosφ cosψ +sinφ sinθ sinψ sinφ cosθ ( % ( $ sinφ sinψ + cosφ sinθ cosψ −sinφ cosψ + cosφ sinθ sinψ cosφ cosθ ' 12 12 6 2/12/20 Rotation Matrix from I to B Spacecraft Convention (3-1-3) B B 2 1 HI (φ,θ,ψ) = H2 (φ)H1 (θ)HI (ψ) ⎡ cosφ sinφ 0 ⎤⎡ 1 0 0 ⎤⎡ cosψ sinψ 0 ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ 0 cos sin ⎥ ⎢ −sinφ cosφ 0 ⎥⎢ θ θ ⎥⎢ −sinψ cosψ 0 ⎥ ⎢ ⎥ 0 −sinθ cosθ ⎢ ⎥ ⎣ 0 0 1 ⎦⎣⎢ ⎦⎥⎣ 0 0 1 ⎦ ⎡ cosφ cosψ − sinφ cosθ sinψ cosφ sinψ + sinφ cosθ cosψ sinφ sinθ ⎤ ⎢ ⎥ = ⎢ −sinφ cosψ − cosφ cosθ sinψ −sinφ sinψ + cosφ cosθ cosψ cosφ sinθ ⎥ ⎢ sinθ sinψ −sinθ cosψ cosθ ⎥ ⎣ ⎦ 13 13 Properties of the Rotation Matrix B ⎡ ⎤ h11 h12 h13 B ⎢ ⎥ HI (φ,θ,ψ ) = ⎢ h21 h22 h23 ⎥ ⎢ ⎥ h31 h32 h33 ⎣ ⎦I B B 2 1 HI (φ,θ,ψ ) = H2 (φ)H1 (θ)HI (ψ ) B −1 B T I ⎣⎡HI (φ,θ,ψ )⎦⎤ = ⎣⎡HI (φ,θ,ψ )⎦⎤ = HB (ψ ,θ,φ) Orthonormal transformation Angles between vectors are preserved r s Lengths are preserved rI = rB ; sI = sB ∠(rI ,sI ) = ∠(rB ,sB ) = x deg 14 14 7 2/12/20 Rotation Matrix Inverse Inverse relationship: interchange sub- and superscripts B rB = HI rI B −1 I rI = (HI ) rB = HBrB Because transformation is orthonormal Inverse = transpose Rotation matrix is always non-singular I B −1 B T I 1 2 HB = (HI ) = (HI ) = H1H2HB I B B I HB HI = HI HB = I 15 15 Vector Derivative Expressed in a Rotating Frame I hI (t) = HB (t)hB (t) Effect of Chain Rule body-frame rotation ! I ! ! I hI (t) = HB (t)hB (t) + HB (t)hB (t) Rate of change expressed in body frame Alternatively h! = HI h! + ω × h = HI h! + ω" h I B B I I B B I I ⎡ 0 −ω ω ⎤ ⎢ z y ⎥ ω! = ⎢ ω z 0 −ω x ⎥ ⎢ ⎥ −ω y ω x 0 16 ⎣⎢ ⎦⎥ 16 8 2/12/20 Angular Rate Derivative in Body Frame of Reference Angular momentum change ! B ! ! B B ! hB = HI hI + HI hI = HI hI − ω B × hB = HBh! − ω" h = HBM − ω" I ω I I B B I I B B B h! M " I B = B − ω B Bω B Constant body-axis inertia matrix ! hB (t) = IBω! B (t) = MB (t) − ω" B (t)IBω B (t) Angular rate change −1 ω! B (t) = IB ⎣⎡MB (t) − ω" B (t)IBω B (t)⎦⎤ 17 17 Euler-Angle Rates and Body-Axis Rates ⎡ ⎤ ω x Body-axis angular rate ⎢ ⎥ ω = ω vector (orthogonal) B ⎢ y ⎥ ⎢ ⎥ ω z ⎣⎢ ⎦⎥B ⎡ φ ⎤ ⎡ ψ ⎤ Form a non-orthogonal ⎢ ⎥ ⎢ ⎥ or vector of Euler angles Θ = ⎢ θ ⎥ ⎢ θ ⎥ ⎢ ψ ⎥ ⎢ φ ⎥ ⎣ ⎦3−2−1 ⎣ ⎦3−1−3 ⎡ φ! ⎤ ⎡ ψ! ⎤ ⎡ ω ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ x ⎥ Euler-angle ! ! Θ = ⎢ θ ⎥ or ⎢ θ! ⎥ ≠ ⎢ ω y ⎥ rate vector ⎢ ⎥ ψ! ⎢ φ! ⎥ ⎢ ω ⎥ ⎣ ⎦3−2−1 ⎣ ⎦3−1−3 ⎢ z ⎥ ⎣ ⎦I18 18 9 2/12/20 Relationship Between (1-2-3) Euler-Angle and Body-Axis Rates • ψ˙ measured in Inertial Frame ! $ ! $ p φ ! 0 $ ! 0 $ ˙ # & # & # & • θ measured in Intermediate Frame #1 B B 2 # & # q & = I3 # 0 &+ H2 # θ &+ H2 H1 0 • φ˙ measured in Intermediate Frame #2 # & # r & "# 0 %& "# 0 %& # ψ & " % " % €€ ! p $ ! 1 0 −sinθ $! φ $ € # & # &# & B # q & = # 0 cosφ sinφ cosθ &# θ & = LI Θ # & # 0 sin cos cos &# & " r % " − φ φ θ %" ψ % Inverse transformation [(.)-1 ≠ (.)T] $ ' φ $ 1 sinφ tanθ cosφ tanθ '$ p ' & ) & )& ) I & θ ) = & 0 cosφ −sinφ )& q ) = LBωB & ψ ) & )& ) % ( % 0 sinφ secθ cosφ secθ (% r ( 19 19 Relationship Between (3-1-3) Euler-Angle and Body-Axis Rates ˙ ⎡ ⎤ • ψ measured in Inertial Frame ω x ⎡ 0 ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ 0 0 • θ˙ measured in Intermediate Frame #1 ⎢ ⎥ B ⎢ ⎥ B 2 ⎢ ⎥ ⎢ ω y ⎥ = I3 0 + H2 θ! + H2 H1 0 ˙ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ • φ measured in Intermediate Frame #2 ⎢ ⎥ ⎢ φ! ⎥ ⎢ 0 ⎥ ⎢ ψ! ⎥ ⎢ ω z ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦B €€ ⎡ ⎤ ω x ⎡ sinθ sinφ cosφ 0 ⎤⎡ ψ! ⎤ € ⎢ ⎥ ⎢ ⎥⎢ ⎥ ω ! B ! ⎢ y ⎥ = ⎢ sinθ cosφ −sinφ 0 ⎥⎢ θ ⎥ = LI Θ ⎢ ⎥ ⎢ cos 0 1 ⎥⎢ φ! ⎥ ⎢ ω z ⎥ ⎣ θ ⎦⎣ ⎦ ⎣ ⎦B Inverse transformation [(.)-1 ≠ (.)T] ⎡ ⎤ ⎡ ψ! ⎤ ⎡ sinφ cosφ 0 ⎤ ω x ⎢ ⎥ 1 ⎢ ⎥⎢ ⎥ I ⎢ θ! ⎥ = cosφ sinθ −sinφ sinθ 0 ⎢ ω y ⎥ = LBω B sinθ ⎢ ⎥ ⎢ ! ⎥ ⎢ ⎥⎢ ⎥ φ −sinφ cosθ cosφ cosθ sinθ ω z ⎣ ⎦ ⎣ ⎦⎣⎢ ⎦⎥B 20 20 10 2/12/20 (1-2-3) Euler-Angle Rates and Body-Axis Rates 21 21 Options for Avoiding the Singularity at θ = ±90° § Don’t use Euler angles as primary definition of angular attitude § Alternatives to Euler angles - Direction cosine (rotation) matrix - Quaternions (next lecture) Propagation of rotation matrix (1-2-3) (9 parameters) H I h = ω HI h B B I B B Consequently ⎡ 0 −r(t) q(t) ⎤ ⎢ ⎥ ! B B B HI (t) = −ω" B (t)HI (t) = − ⎢ r(t) 0 − p(t) ⎥ HI (t) ⎢ ⎥ ⎢ −q(t) p(t) 0(t) ⎥ ⎣ ⎦B B B HI (0) = HI (φ0 ,θ0 ,ψ 0 ) 22 22 11 2/12/20 Body-Axis Angular Rate Dynamics [(3-1-3) convention] −1 ω! B = IB [MB − ω" BIBω B ] ⎡ ω ⎤ ⎢ x ⎥ ω B = ⎢ ω y ⎥ ⎢ ⎥ ω z ⎣⎢ ⎦⎥B For principal axes ⎡ ⎤ ⎡ ⎤ ⎡ − ω t ω t / ⎤ ω!
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    Euler Quaternions

    Four different ways to represent rotation my head is spinning... The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why det( R ) = ± 1 ? − = − Rotations preserve distance: Rp1 Rp2 p1 p2 Rotations preserve orientation: ( ) × ( ) = ( × ) Rp1 Rp2 R p1 p2 The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why it’s a group: • Closed under multiplication: if ∈ ( ) then ∈ ( ) R1, R2 SO 3 R1R2 SO 3 • Has an identity: ∃ ∈ ( ) = I SO 3 s.t. IR1 R1 • Has a unique inverse… • Is associative… Why orthogonal: • vectors in matrix are orthogonal Why it’s special: det( R ) = + 1 , NOT det(R) = ±1 Right hand coordinate system Possible rotation representations You need at least three numbers to represent an arbitrary rotation in SO(3) (Euler theorem). Some three-number representations: • ZYZ Euler angles • ZYX Euler angles (roll, pitch, yaw) • Axis angle One four-number representation: • quaternions ZYZ Euler Angles φ = θ rzyz ψ φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1. Rotate φ about z axis 0 0 1 θ θ 2. Then rotate θ about y axis cos 0 sin θ = ψ Ry ( ) 0 1 0 3. Then rotate about z axis − sinθ 0 cosθ ψ − ψ cos sin 0 ψ = ψ ψ Rz ( ) sin cos 0 0 0 1 ZYZ Euler Angles φ θ ψ Remember that R z ( ) R y ( ) R z ( ) encode the desired rotation in the pre- rotation reference frame: φ = pre−rotation Rz ( ) Rpost−rotation Therefore, the sequence of rotations is concatentated as follows: (φ θ ψ ) = φ θ ψ Rzyz , , Rz ( )Ry ( )Rz ( ) φ − φ θ θ ψ − ψ cos sin 0 cos 0 sin cos sin 0 (φ θ ψ ) = φ φ ψ ψ Rzyz , , sin cos 0 0 1 0 sin cos 0 0 0 1− sinθ 0 cosθ 0 0 1 − − − cφ cθ cψ sφ sψ cφ cθ sψ sφ cψ cφ sθ (φ θ ψ ) = + − + Rzyz , , sφ cθ cψ cφ sψ sφ cθ sψ cφ cψ sφ sθ − sθ cψ sθ sψ cθ ZYX Euler Angles (roll, pitch, yaw) φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1.
  • A Guide to Space Law Terms: Spi, Gwu, & Swf

    A Guide to Space Law Terms: Spi, Gwu, & Swf

    A GUIDE TO SPACE LAW TERMS: SPI, GWU, & SWF A Guide to Space Law Terms Space Policy Institute (SPI), George Washington University and Secure World Foundation (SWF) Editor: Professor Henry R. Hertzfeld, Esq. Research: Liana X. Yung, Esq. Daniel V. Osborne, Esq. December 2012 Page i I. INTRODUCTION This document is a step to developing an accurate and usable guide to space law words, terms, and phrases. The project developed from misunderstandings and difficulties that graduate students in our classes encountered listening to lectures and reading technical articles on topics related to space law. The difficulties are compounded when students are not native English speakers. Because there is no standard definition for many of the terms and because some terms are used in many different ways, we have created seven categories of definitions. They are: I. A simple definition written in easy to understand English II. Definitions found in treaties, statutes, and formal regulations III. Definitions from legal dictionaries IV. Definitions from standard English dictionaries V. Definitions found in government publications (mostly technical glossaries and dictionaries) VI. Definitions found in journal articles, books, and other unofficial sources VII. Definitions that may have different interpretations in languages other than English The source of each definition that is used is provided so that the reader can understand the context in which it is used. The Simple Definitions are meant to capture the essence of how the term is used in space law. Where possible we have used a definition from one of our sources for this purpose. When we found no concise definition, we have drafted the definition based on the more complex definitions from other sources.
  • Theory of Angular Momentum and Spin

    Theory of Angular Momentum and Spin

    Chapter 5 Theory of Angular Momentum and Spin Rotational symmetry transformations, the group SO(3) of the associated rotation matrices and the 1 corresponding transformation matrices of spin{ 2 states forming the group SU(2) occupy a very important position in physics. The reason is that these transformations and groups are closely tied to the properties of elementary particles, the building blocks of matter, but also to the properties of composite systems. Examples of the latter with particularly simple transformation properties are closed shell atoms, e.g., helium, neon, argon, the magic number nuclei like carbon, or the proton and the neutron made up of three quarks, all composite systems which appear spherical as far as their charge distribution is concerned. In this section we want to investigate how elementary and composite systems are described. To develop a systematic description of rotational properties of composite quantum systems the consideration of rotational transformations is the best starting point. As an illustration we will consider first rotational transformations acting on vectors ~r in 3-dimensional space, i.e., ~r R3, 2 we will then consider transformations of wavefunctions (~r) of single particles in R3, and finally N transformations of products of wavefunctions like j(~rj) which represent a system of N (spin- Qj=1 zero) particles in R3. We will also review below the well-known fact that spin states under rotations behave essentially identical to angular momentum states, i.e., we will find that the algebraic properties of operators governing spatial and spin rotation are identical and that the results derived for products of angular momentum states can be applied to products of spin states or a combination of angular momentum and spin states.
  • Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22

    Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22

    Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation