Kinematical LEO Orbit Determination. From Pure Kinematical to Reduced. Titelmaster. Precise Orbit Determination (POD) Credit: European Space Agency

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1 Titelmaster From Pure Kinematical to Reduced Kinematical LEO Orbit Determination Akbar Shabanloui, Karl Heinz Ilk Session 4 Geodätische Woche Sep. 009, Karlsruhe, Deutschland Precise Orbit Determination (POD) Credit: European Space Agency

2 Precise Orbit Determination (POD) methods POD Precise Orbit Determination (POD) methods Geometrical POD : point-wise, positions!= Kinematical POD Kinematical POD : continous, positions, velocities and accelerations Dynamical POD : continous, positions, velocities and accelerations based on force function information POD 3 4

3 Kinematical Precise Orbit Determination (KPOD) Processing procedures 5 6

4 Processing procedures Kinematical POD methods Equation of motion w.r.t self-adjoint differential operator: ( ) L r ( t ) = a ( t ; r, r& ) corresponding Fredholm s integral equation: ( ) ( ) (, ) ( ;, ) r τ = r τ T K τ τ a τ r r& d τ τ = 0 Reference Position motion T = L ( t B r ( Integral τ t )) A t 0 kernel t τ = A T Force function 7 8

5 Short arc representation ellipse mode A satellite short arc: ( τ ) ( τ ) ( τ ) ( τ ) sin( πτ ) r = r + d = r + d = elliptical reference motion: sin µ ( τ ) sin µτ r ( τ ) = r + r µ µ A B sin sin difference function: = d ( τ ) d sin( πτ ) = Short arc representation Fourier series A satellite short arc can be represented: τ τ τ πτ r ( τ ) ( = r ) ( τ ) ( + d ) ( τ ) ( = r ) ( τ ) + sin( d sin( πτ ) ) r r = d = d = τ = 0 = d = d ( τ )sin( πτ )d τ e d ( τ ) with τ e e =, e =, K, N e e ]0,[ N τ + e N Reference e motion π e d d ( τ e )sin N e + e = N e + r r A B 9 0

6 Fourier analysis ellipse mode Amplitudes Remainders Short arc representation Euler-Bernoulli A satellite short arc can be represented: with Fourier amplitudes: Fourier series amplitudes: r ( τ ) = r ( τ ) + d sin( πτ ) τ = 0 = d = d ( τ )sin( πτ )d τ J j + ( ) [ j ] [ j ] d = [( ) d () d (0)] + j + ( π ) j = [ J + ] ( )sin( ) d J + τ = 0 + β τ πτ τ ( π ) d

7 Short arc representation Euler-Bernoulli d = d ( τ ) = d sin( πτ ) = F = e j E j ( τ ) b j + B j + ( τ ) d P j = j = = + = A satellite short arc can be represented: d sin ( πτ ) = e j E j + b j + B j + = j = j = ( τ ) ( τ ) Determination of Euler-Bernoulli coefficients A satellite short arc can be represented with the Euler-Bernoulli term up to degree J as: J J e + b r ( τ ) r ( τ ) = d ( τ ) E ( τ ) + B + ( τ ) j j j j j = j = Reference motion Euler-Bernoulli r r A B 3 4

8 Determination of Euler-Bernoulli coefficients 5 E.-B terms Remainders Short arc representation LEO orbit can be represented as: or r ( τ ) = r ( τ ) + d ( τ ) = r ( τ ) + d sin( πτ ) n = Gibbs effect! Precision! 6 J J r = r + d = r + e + b ( τ ) ( τ ) ( τ ) ( τ ) ( τ ) ( τ ) j E j j B j j j + + = = Solution? J J n r ( τ ) = r ( τ ) + e E ( τ ) + b B ( τ ) + d sin( πτ ) j j j + j + j = j = = fast convergence!

9 Kinematical POD ellipse mode, J=4 Amplitudes Remainders Processing procedures 7 8

10 Kinematical POD simulation (cm noise SST ) Position differences Velocity differences Acceleration differences Statistical values Processing procedures 9 0

11 KPOD real case Four short arcs (30 min.) ground track of CHAMP Kinematical POD short arc case (b) IGG - GFZ positions IGG - GFZ velocities Statistical values GPS-SST residuals

12 Processing procedures Reduced-Kinematical POD method Dynamical info. r ( τ ) = r ( τ ) + d sin( πτ ) n = T d % = sin( πτ ) ( τ ;, ) d τ a r r & π µ τ = 0 d % L d %, C % % ( ) Introduction of an approximate force function i j ( d il d j ) ( Fixing only some orbit parameters d % il d % j ), C ( i j ) d % L d % 0 C ( d % L d % n ) Down- or up weighting in relation to C ( ) d L d n 3 4

13 Processing procedures RKPOD simulation (Amp. -5, cm, N=300) Position differences Velocity differences Acceleration differences Statistical values 5 6

14 Conclusions and recommendations GNSS-LEO satellites configuration and geometrical strength play an important role in POD, Kinematical POD can be used to recover the Earth s gravity field model based on the POD methods, No gravity field and no force models have been used in the Geometrical and Kinematical modes (advantage), The proposed kinematical orbit determination method is very flexible. A smooth transition from kinematical to reduced kinematical and finally dynamical or vice-versa is possible.? 7

15 Reduced Kinematical POD method Kinematical observation equation Constraints x l = ( A A ), C x x l = ( 0 I ), C x l A A x C 0 =, C = l 0 I x l 0 C x ˆ A C l Q Q T - - ˆ ˆ ˆ ˆ - x x x x, ˆ = N T - - N = x A C l + C l Q x ˆ ˆ ˆ ˆ x Q x x Thank you for your attention 9