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MIT OpenCourseWare http://ocw.mit.edu 16.346 Astrodynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 19 The State Transition Matrix #9.5-9.6 Linearization of the Equations of Motion • Deviations from reference • Equations of motion r(t) = rref (t) + δ (t) v(t) = vref (t) + ν (t) drref dδ = vref =ν dt dt =⇒ dν dvref = g(r) − g(rref ) = G(rref )δ = G(t)δ = g(rref ) dt dt � ∂ g �� δ + · · · = g(rref ) + G(rref )δ + O(δ 2 ) g(r) = g(rref ) + ∂ r �r=rref dr =v dt dv = g(r) dt since • State vector representation � � � �� � d δ O I δ = G(t) O ν dt ν � �� � � �� � x(t) F(t) dx = F(t)x dt or The State Transition Matrix • Deﬁne � � Φ(t, t0 ) = x1 (t) x2 (t) . . . x6 (t) � � Φ(t0 , t0 ) = x1 (t0 ) x2 (t0 ) . . . x6 (t0 ) = I • Matrix diﬀerential equation d Φ(t, t0 ) = F(t)Φ(t, t0 ) dt • Fundamental property with Φ(t0 , t0 ) = I x(t) = Φ(t, t0 )x(t0 ) Symplectic Matrices • Deﬁnition An even-dimensional matrix A is symplectic if T A JA = J where � O J= −I I O � Note: Since J2 = −I , the J matrix is analogous to the imaginary algebra. • Inverse of a Symplectic Matrix Postmultiply by A−1 and premultiply by J to obtain A−1 = −JA T J 16.346 Astrodynamics Lecture 19 √ −1 in complex The State Transition Matrix is Symplectic • Symplectic Property of Φ(t, t0 ) dΦT dΦ d T Φ (t, t0 )JΦ(t, t0 ) = JΦ + ΦT J dt dt dt = ΦT FT JΦ + ΦT JFΦ = ΦT [FT J + JF]Φ � � T O T G(t) − G (t) =Φ Φ O I−I =O since the gravity-gradient matrix G = G T is symmetric. Therefore Φ T (t, t0 )JΦ(t, t0 ) = constant = Φ T (t0 , t0 )JΦ(t0 , t0 ) = I T JI = J • Inverse of Φ(t, t0 ) � Φ1 (t, t0 ) Φ(t, t0 ) = Φ3 (t, t0 ) From the partitions the inverse is −1 Φ Φ2 (t, t0 ) Φ4 (t, t0 ) � ΦT 4 (t, t0 ) (t, t0 ) = Φ(t0 , t) = −ΦT 3 (t, t0 ) � −ΦT 2 (t, t0 ) T Φ1 (t, t0 ) � Fundamental Perturbation Matrices For the discussion of linear deviations from a reference orbit, deﬁne � � � � δr(t) δr0 x(t) = and x0 = δv(t) δv0 so that � � ⎡ � � � δr0 δr(t) = Φ(t, t0 ) δ v0 δv(t) � � � δ r0 δr(t) = Φ(t0 , t) δv(t) δv0 where ⎤ ⎢ ∂r ⎢ ∂r ⎢ 0 Φ(t, t0 ) = ⎢ ⎢ ∂v ⎣ ∂r0 ∂r ⎥ ⎥ ∂v0 ⎥ ⎥ ∂v ⎥ ⎦ ∂v0 ⎡ ⎤ ⎢ ∂ r0 ⎢ ∂r where Φ(t0 , t) = ⎢ ⎢ ∂v ⎣ 0 ∂r ∂r0 ∂v ∂v0 ∂v � � � R(t) R(t) = � V(t) V(t) ref ⎥ ⎥ ⎥ ⎥ ⎦ � T V (t) = � T (t) −V ref since Φ(t, t0 )Φ(t0 , t) = I 16.346 Astrodynamics and Φ(t0 , t) = Φ−1 (t, t0 ) Lecture 19 −RT (t) � T (t) R � Navigation Matrix Let t0 ≤ t ≤ t1 and deﬁne � � R(t) Φ(t, t0 ) = � V(t) Φ(t, t0 ) R(t) V(t) Guidance Matrix � and Φ(t, t1 ) � � � (t) R (t) R Φ(t, t1 ) = � V (t) V (t) Then dR(t) dR (t) = V(t) = V (t) R(t0 ) = O R (t1 ) = O dt dt dV(t) dV (t) V (t1 ) = I V(t0 ) = I = G(t)R(t) = G(t)R (t) dt dt The diﬀerential equations for the “tilde” matrices are the same but with initial conditions � (t ) = I R 1 � V (t ) = O � (t ) = I R 0 � V(t ) = O 1 0 Furthermore, � −1 � R � = V C Hence � � Φ(t, t1 )� t=t0 =Φ −1 � ∂v �� = ∂r � v1 =constant � � (t, t0 ) � t=t1 = ⇒ −1 C =V R � ∂v �� = ∂r �r1 =constant � � T � � (t ) R (t ) V (t1 ) R 0 0 = � T (t ) � (t ) V (t ) −V V 1 0 0 � −RT (t1 ) � T (t ) R 1 Diﬀerential Equation for the C Matrix C R = V and C−1 = R V−1 or C−1 V = R Diﬀerentiate the ﬁrst expression: d V dR dC = R + C dt dt dt =⇒ dC R + C V = GR dt Finally, postmultiply by R−1 to obtain dC 2 + C = G dt Since G is symmetric, then C is symmetric. Because C (t1 ) is inﬁnite, it is better to use the equation for the inverse matrix. dR dC−1 dV V + C−1 = dt dt dt =⇒ dC−1 V + C−1 GR = V dt Hence dC−1 + C−1 GC−1 = I dt with C−1 (t1 ) = O Note: For the constant gravity case we have G = O so that C−1 = (t − t1 )I . 16.346 Astrodynamics Lecture 19 Midcourse Orbit Corrections #11.6 • Fixed-Time-of-Arrival Correction � � � � (t) δr(t) R δx(t) = Φ(t, t1 ) δx(t1 ) =⇒ = � δv(t) V (t) Hence R (t) V (t) �� 0 δv(tA ) � δr(t) = R (t) δv(tA ) δv(t) = V (t) δv(tA ) Eliminate δv(tA ) δv(t) = V (t)R −1 (t) δr(t) = C (t) δr(t) Velocity correction ∆v(t) = δv(t+ ) − δv(t− ) = C (t) δr(t) − δv(t− ) • Variable-Time-of-Arrival Correction � � � �� � � (t) R (t) δr(t) R δr(tA ) = � δv(t) δv(tA ) V (t) V (t) � � Multiply through by −C (t) I �� � � � � � (t) R (t) � � δr(t) � � R δr(tA ) −C (t) I = −C (t) I � (t) V (t) δv(t) δv(tA ) V Hence, [using the starred form of Eqs. (9.57)], we have � (t) + V � (t)] δr(t ) −C (t) δr(t) + δv(t) = [−C (t)R A � �� � = −R −T (t) δv(t) = C (t) δr(t) − R −T (t) δr(tA ) or 1. Choosing δr(tA ) rp (tA + δt) = rp (tA ) + vp (tA ) δt r(tA + δt) = r(tA ) + v(tA ) δt =⇒ δr(tA ) = r(tA ) − rp (tA ) = −vr (tA ) δt Then r(tA + δt) = rp (tA + δt) where vr (tA ) = v(tA ) − vp (tA ) . Hence δv(t) = C (t) δr(t) + R −T (t)vr (tA ) δt �� � � = w(t) 2. Choosing δt to minimize |∆v (t)| δtA = − ∆v · w w·w 16.346 Astrodynamics and or ∆v (t) = ∆v(t) + w(t) δt � wwT � Min ∆v = I − T ∆v = M ∆v w w � �� � Projection operator M Lecture 19