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!