Usuari:Glm/proves

IRI

slides

$\mathbf {F} (\mathbf {u} )=0$

curs

$\left[{\begin{array}{c}y\\-x\\1\end{array}}\right]=\left[{\begin{array}{ccc}1&0&b\\0&1&-a\\0&0&1\end{array}}\right]\left[{\begin{array}{c}y'\\-x'\\1\end{array}}\right]$

$\left[{\begin{array}{c}y\\-x\\1\end{array}}\right]=\left[{\begin{array}{ccc}c&-s&0\\s&c&0\\0&0&1\end{array}}\right]\left[{\begin{array}{c}y'\\-x'\\1\end{array}}\right]$

$[E]=\left[{\begin{array}{ccc}c&-s&b\\s&c&-a\\0&0&1\end{array}}\right]$

$J=\left[{\begin{array}{ccc}y_{1}&y_{2}&y_{3}\\-x_{1}&-x_{2}&-x_{3}\\1&1&1\end{array}}\right]$

$J^{-1}={\frac {1}{detJ}}\left[{\begin{array}{c}a_{23}{\hat {s}}_{23}^{T}\\a_{31}{\hat {s}}_{31}^{T}\\a_{12}{\hat {s}}_{12}^{T}\end{array}}\right]$

$detJ=a_{23}{\hat {s}}_{23}^{T}{\hat {S}}_{1}=a_{31}{\hat {s}}_{31}^{T}{\hat {S}}_{2}=a_{12}{\hat {s}}_{12}^{T}{\hat {S}}_{3}$

$A=\left[{\begin{array}{ccc}y_{1}&0&y_{3}\\-x_{1}&1&-x_{3}\\1&0&1\end{array}}\right]$

$B=\left[{\begin{array}{ccc}y_{1}&y_{2}&y_{3}\\-x_{1}&-x_{2}&-x_{3}\\1&1&1\end{array}}\right]$

$C=\left[{\begin{array}{ccc}0&-1&0\\1&0&1\\0&0&0\end{array}}\right]$

$D=\left[{\begin{array}{ccc}1&0&\cos {\gamma }\\0&-x_{2}&\sin {\gamma }\\0&1&0\end{array}}\right]$

${\hat {T}}=\omega \left[{\begin{array}{c}y_{Q}\\-x_{Q}\\1\end{array}}\right]=\left[{\begin{array}{c}v_{Ox}\\v_{Oy}\\\omega \end{array}}\right]$

$\delta {\hat {D}}=\delta \phi \left[{\begin{array}{c}y_{Q}\\-x_{Q}\\1\end{array}}\right]=\left[{\begin{array}{c}\delta r_{Ox}\\\delta r_{Oy}\\\delta \phi \end{array}}\right]$

${\hat {\mathbf {T} }}=\mathbf {J} {\hat {\mathbf {\gamma } }}$

$\delta {\hat {\mathbf {D} }}=\mathbf {J} \delta {\hat {\mathbf {\theta } }}$

CNES

delta-v

$\Delta t={\frac {initial\_mass}{\dot {m_{e}}}}\left(1-e^{\frac {-\Delta v}{isp\cdot g_{0}}}\right)$

Stumpf functions

$C(z)={\begin{cases}{\frac {1-cos{\sqrt {z}}}{z}}&(z>0)\\{\frac {cosh{\sqrt {-z}}-1}{-z}}&(z<0)\\{\frac {1}{2}}&(z=0)\end{cases}}$ $S(z)={\begin{cases}{\frac {{\sqrt {z}}-sin{\sqrt {z}}}{({\sqrt {z}})^{3}}}&(z>0)\\{\frac {sinh{\sqrt {-z}}-{\sqrt {-z}}}{({\sqrt {-z}})^{3}}}&(z<0)\\{\frac {1}{6}}&(z=0)\end{cases}}$

quaternions

$\mathbf {q} ={\begin{bmatrix}\cos(\phi /2)\cos(\theta /2)\cos(\psi /2)+\sin(\phi /2)\sin(\theta /2)\sin(\psi /2)\\\sin(\phi /2)\cos(\theta /2)\cos(\psi /2)-\cos(\phi /2)\sin(\theta /2)\sin(\psi /2)\\\cos(\phi /2)\sin(\theta /2)\cos(\psi /2)+\sin(\phi /2)\cos(\theta /2)\sin(\psi /2)\\\cos(\phi /2)\cos(\theta /2)\sin(\psi /2)-\sin(\phi /2)\sin(\theta /2)\cos(\psi /2)\\\end{bmatrix}}$

${\begin{bmatrix}\phi \\\theta \\\psi \end{bmatrix}}={\begin{bmatrix}{\mbox{arctan}}{\frac {2(q_{0}q_{1}+q_{2}q_{3})}{1-2(q_{1}^{2}+q_{2}^{2})}}\\{\mbox{arcsin}}(2(q_{0}q_{2}-q_{3}q_{1}))\\{\mbox{arctan}}{\frac {2(q_{0}q_{3}+q_{1}q_{2})}{1-2(q_{2}^{2}+q_{3}^{2})}}\end{bmatrix}}$

$\mathbf {q} _{0}=\cos(\alpha /2)$

$\mathbf {q} _{1}=\sin(\alpha /2)\cos(\beta _{x})$

$\mathbf {q} _{2}=\sin(\alpha /2)\cos(\beta _{y})$

$\mathbf {q} _{3}=\sin(\alpha /2)\cos(\beta _{z})$

kp_kepler_gen:

$PSO_{M}=PSO_{E}-E_{x}sin(PSO_{E})+E_{y}cos(PSO_{E})$

clohessy-wiltshire

${\begin{bmatrix}X\\Y\\Z\\{\dot {X}}\\{\dot {Y}}\\{\dot {Z}}\end{bmatrix}}=M(\theta ){\begin{bmatrix}X_{0}\\Y_{0}\\Z_{0}\\{\dot {X_{0}}}\\{\dot {Y_{0}}}\\{\dot {Z_{0}}}\end{bmatrix}}+N(\theta ){\begin{bmatrix}\Delta \gamma _{X}\\\Delta \gamma _{Y}\\\Delta \gamma _{Z}\end{bmatrix}}$

exponential atmospheric model

$\rho =\rho _{0}e^{-h_{\rho }(Z-Z_{0})}$

circular drift

$At\ t=t0{\begin{cases}X=0\\Y=0\\Z=z_{0}\\{\dot {X}}={\frac {3}{2}}\omega z_{0}\\{\dot {Y}}=0\\{\dot {Z}}=0\\\end{cases}}$

keplerian covariance propagation

${\begin{matrix}M(\Delta t)={\begin{bmatrix}0&0&0&0&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\\-{\frac {3}{2}}{\sqrt {\mu }}a^{\frac {-5}{2}}\Delta t&0&0&0&0&0\\\end{bmatrix}}\\\\COV(t+\Delta t)=M\cdot COV(t)\cdot M^{t}\end{matrix}}$

Draw correlated values

${\begin{array}{lcl}COR=RR^{*}\ {\mbox{(Cholesky decomposition)}}\\{\underset {p\times n}{X}}\sim N(0,1)\\drawnValues=R\cdot X\cdot sd+meanValues\\\end{array}}$

Great circle distance

${\color {white}{\frac {\bigg |}{|}}|}GCdist=\arctan \left({\frac {\sqrt {\left(\cos \phi _{f}\sin \Delta \lambda \right)^{2}+\left(\cos \phi _{s}\sin \phi _{f}-\sin \phi _{s}\cos \phi _{f}\cos \Delta \lambda \right)^{2}}}{\sin \phi _{s}\sin \phi _{f}+\cos \phi _{s}\cos \phi _{f}\cos \Delta \lambda }}\right)\qquad {\mbox{(1)}}$

${\color {white}{\Big |}}GCdist=\arccos {\big (}\sin \phi _{s}\sin \phi _{f}+\cos \phi _{s}\cos \phi _{f}\cos \Delta \lambda {\big )}\qquad {\mbox{(2)}}$

${\color {white}{\frac {\bigg |}{|}}}GCdist=2\arcsin \left({\sqrt {\sin ^{2}\left({\frac {\phi _{f}-\phi _{s}}{2}}\right)+\cos {\phi _{s}}\cos {\phi _{f}}\sin ^{2}\left({\frac {\Delta \lambda }{2}}\right)}}\right)\qquad {\mbox{(3)}}$

Consumption

$\Delta m=m(1-e^{\frac {-\Delta v}{g_{0}isp}})$

$F=g_{0}isp{\frac {\Delta m}{\Delta t}}$

Local hour

$smra=tsid-2\pi (cjd-int(cjd))+\pi \qquad {\mbox{(1)}}$

$lon=ra-tsid\qquad {\mbox{(2)}}$

$mha=ra-smra\qquad {\mbox{(3)}}$

$tha=ra-smra\qquad {\mbox{(4)}}$

$hl={\frac {12}{\pi }}(h+\pi )\quad modulo\quad 24\qquad {\mbox{(5)}}$

heliosynchronism

${\frac {d(\Omega )}{dt}}={\frac {-3ncos(i)R^{2}J2}{2a^{2}(1-e^{2})^{2}}}\qquad {\mbox{(1)}}$

$e_{x}=ecos(\omega )\qquad e_{y}=esin(\omega )$

$e_{x}=ecos(\omega +\Omega )\qquad e_{y}=esin(\omega +\Omega )$

$i_{x}=2sin({\frac {i}{2}})cos(\Omega )\qquad i_{y}=2sin({\frac {i}{2}})sin(\Omega )$

parameters

${\begin{array}{lcl}{\mbox{keplerian parameters:}}\ (a,e,i,\omega ,\Omega ,M)\\\ a:\ {\mbox{semi major axis}}\\\ e:\ {\mbox{excentricity}}\\\ i:\ {\mbox{inclination}}\\\ \omega :\ {\mbox{periapsis argument}}\\\ \Omega :\ {\mbox{right ascension of ascending node}}\\\ M:\ {\mbox{mean anomaly}}\\\end{array}}$

${\begin{array}{lcl}{\mbox{circular adapted parameters:}}\ (a,e_{x},e_{y},i,\Omega ,\omega +M)\\\ a:\ {\mbox{semi major axis}}\\\ e:\ {\mbox{excentricity}}\\\ e_{x}=ecos(\omega )\qquad e_{y}=esin(\omega )\\\ i:\ {\mbox{inclination}}\\\ \omega :\ {\mbox{periapsis argument}}\\\ \Omega :\ {\mbox{right ascension of ascending node}}\\\ M:\ {\mbox{mean anomaly}}\\\end{array}}$

${\begin{array}{lcl}{\mbox{circular equatorial adapted parameters:}}\ (a,e_{x},e_{y},i_{x},i_{y},\omega +\Omega +M)\\\ a:\ {\mbox{semi major axis}}\\\ e:\ {\mbox{excentricity}}\\\ e_{x}=ecos(\omega +\Omega )\qquad e_{y}=esin(\omega +\Omega )\\\ i:\ {\mbox{inclination}}\\\ i_{x}=2sin({\frac {i}{2}})cos(\Omega )\qquad i_{y}=2sin({\frac {i}{2}})sin(\Omega )\\\ \omega :\ {\mbox{periapsis argument}}\\\ \Omega :\ {\mbox{right ascension of ascending node}}\\\ M:\ {\mbox{mean anomaly}}\\\end{array}}$

${\begin{array}{lcl}{\mbox{keplerian parameters:}}\ (a,e,i,\omega ,\Omega ,M)\\{\mbox{circular equatorial adapted parameters:}}\ (a,e_{x},e_{y},i_{x},i_{y},\omega +\Omega +M)\\\ a:\ {\mbox{semi major axis}}\\\ e:\ {\mbox{excentricity}}\\\ e_{x}=ecos(\omega +\Omega )\qquad e_{y}=esin(\omega +\Omega )\\\ i:\ {\mbox{inclination}}\\\ i_{x}=2sin({\frac {i}{2}})cos(\Omega )\qquad i_{y}=2sin({\frac {i}{2}})sin(\Omega )\\\ \omega :\ {\mbox{periapsis argument}}\\\ \Omega :\ {\mbox{right ascension of ascending node}}\\\ M:\ {\mbox{mean anomaly}}\\\end{array}}$

rotation matrix

$M{\vec {v}}_{R}={\vec {v}}_{S}$

drift j2

${\begin{array}{lcl}\ drpom={\frac {-3n\cdot j2\cdot er^{2}(1-5cos(i)cos(i))}{4(1-e^{2})^{2}a^{2}}}\\\ \\\ drgom={\frac {-3n\cdot j2\cdot er^{2}cos(i)}{2(1-e^{2})^{2}a^{2}}}\\\ \\\ drM=n+{\frac {3n\cdot j2\cdot er^{2}(3cos(i)cos(i)-1)}{4(1-e^{2})^{1.5}a^{2}}}\\\ \\\ dpom(1,:)={\frac {d(drpom)}{d(a)}}={\frac {-3.5\cdot drpom}{a}}\\\ \\\ dgom(1,:)={\frac {d(drgom)}{d(a)}}={\frac {-3.5\cdot drgom}{a}}\\\ \\\ dM(1,:)={\frac {d(drM)}{d(a)}}={\frac {2n-3.5\cdot drM}{a}}\\\ \\\ dpom(2,:)={\frac {d(drpom)}{d(e)}}={\frac {4e\cdot drpom}{1-e^{2}}}\\\ \\\ dgom(2,:)={\frac {d(drgom)}{d(e)}}={\frac {4e\cdot drgom}{1-e^{2}}}\\\ \\\ dM(2,:)={\frac {d(drM)}{d(e)}}={\frac {3e(drM-n)}{1-e^{2}}}\\\ \\\ dpom(3,:)={\frac {d(drpom)}{d(i)}}=5Ksin(2i)\\\ \\\ dgom(3,:)={\frac {d(drgom)}{d(i)}}=-2Ksin(i)\\\ \\\ dM(3,:)={\frac {d(drM)}{d(i)}}=3Ksin(2i)(1-e^{2})^{0.5}\\\ \\\ {\mbox{where }}K={\frac {-3n\cdot j2\cdot er^{2}}{4(1-e^{2})^{2}a^{2}}}\\\end{array}}$

ground track

${\begin{array}{lcl}\ gt=T_{k,n}\cdot {\frac {2\pi }{T_{si}-dgom}}\\\ \\\ {\mbox{where }}T_{k,n}{\mbox{ is the keplerian period or the nodal period}}\\\end{array}}$