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/***********************************************/
/**
* @file integralEquation.cpp
*
* @brief Functions for the (short arc) integral equation approach.
*
* @author Torsten Mayer-Guerr
* @date 2005-02-24
*
*/
/***********************************************/
#include "base/import.h"
#include "files/fileInstrument.h"
#include "classes/earthRotation/earthRotation.h"
#include "classes/tides/tides.h"
#include "classes/gravityfield/gravityfield.h"
#include "integralEquation.h"
/***********************************************/
IntegralEquation::IntegralEquation(UInt integrationDegree, UInt interpolationDegree)
{
init(integrationDegree, interpolationDegree);
}
/***********************************************/
void IntegralEquation::init(UInt integrationDegree, UInt interpolationDegree)
{
try
{
this->interpolationDegree = interpolationDegree;
this->integrationDegree = integrationDegree;
if(integrationDegree%2 == 0)
throw(Exception("polnomial degree for integration must be odd."));
// Es gibt grad Gleichungssysteme
// grad/2 am Anfang des Bahnbogens (es fehlen Punkte zum integrieren)
// grad/2 am Ende des Bahnbogens (es fehlen Punkte zum integrieren)
// 1 fuer den Rest des Bogens
W.resize(integrationDegree);
UInt half = (integrationDegree-1)/2;
for(UInt k=0; k<half; k++)
{
W.at(k) = Matrix(integrationDegree+1, integrationDegree+1);
for(UInt i=0; i<W.at(k).columns(); i++)
for(UInt n=0; n<W.at(k).rows(); n++)
W.at(k)(n,i) = ((n==0) ? 1.0 : pow(static_cast<Double>(i)-k, n));
inverse(W.at(k));
}
for(UInt k=0; k<=half; k++)
{
W.at(k+half) = Matrix(integrationDegree+1, integrationDegree+1);
for(UInt i=0; i<W.at(k+half).columns(); i++)
for(UInt n=0; n<W.at(k+half).rows(); n++)
W.at(k+half)(n,i) = ((n==0) ? 1.0 : pow(static_cast<Double>(i)-half-k, n));
inverse(W.at(k+half));
}
}
catch(std::exception &e)
{
GROOPS_RETHROW(e)
}
}
/***********************************************/
/***********************************************/
/***********************************************/
IntegralEquation::Arc IntegralEquation::integrateArc(const OrbitArc &orbit, const std::vector<Rotary3d> &rotEarth, GravityfieldPtr gradientfield,
const const_MatrixSlice &g, const const_MatrixSlice &accCalibration, Bool computeVelocity, Bool computeAcceleration) const
{
try
{
UInt posCount = orbit.size();
Double T = (orbit.at(posCount-1).time - orbit.at(0).time).seconds();
// Gravitationsgradient = Tensor
// (im raumfesten System)
// -----------------------------
std::vector<Matrix> tensor(posCount);
for(UInt i=0; i<posCount; i++)
{
Time time = orbit.at(i).time;
Vector3d posEarth = rotEarth.at(i).rotate(orbit.at(i).position);
tensor.at(i) = rotEarth.at(i).inverseRotate(gradientfield->gravityGradient(time, posEarth)).matrix();
}
Matrix A(3*posCount, 3*posCount + 6 + g.columns());
MatrixSlice VPos (A.column(0, 3*posCount));
MatrixSlice VPosBoundary(A.column(3*posCount, 6));
MatrixSlice vPos (A.column(3*posCount+6, g.columns()));
// Integral mit dem Kern K(tau, tau')
// und Drehung in das erdfeste System
// ----------------------------------
if(IntegrationPos.rows()!=3*posCount)
IntegrationPos = integrationMatrixPosition(T, posCount);
for(UInt i=0; i<posCount; i++)
matMult(1., IntegrationPos.slice(3, 3*i, 3*posCount-3, 3), rotEarth.at(i).matrix().trans(), VPos.slice(3, 3*i, 3*posCount-3, 3));
// Referenzpositionen und Randwerte
// --------------------------------
Vector vBound(3*posCount);
Vector vGpsPos(3*posCount);
Vector3d posStart = orbit.at(0).position;
Vector3d posEnd = orbit.at(posCount-1).position;
for(UInt i=0; i<posCount; i++)
{
Double tau = static_cast<Double>(i)/(posCount-1.0);
vGpsPos(3*i+0) = orbit.at(i).position.x();
vGpsPos(3*i+1) = orbit.at(i).position.y();
vGpsPos(3*i+2) = orbit.at(i).position.z();
vBound(3*i+0) = (1-tau) * posStart.x() + tau * posEnd.x();
vBound(3*i+1) = (1-tau) * posStart.y() + tau * posEnd.y();
vBound(3*i+2) = (1-tau) * posStart.z() + tau * posEnd.z();
}
// Referenzbeschleunigungen integrieren
// Reference = (Gravity*g-(vPos-vBound))
// -------------------------------------
matMult(1., VPos, g, vPos);
vBound -= vGpsPos;
for(UInt j=0; j<g.columns(); j++)
vPos.column(j) += vBound;
// partielle Ableitungen nach den Randwerten berechnen (Boundary)
// --------------------------------------------------------------
for(UInt i=0; i<posCount; i++)
{
Double tau = i/(posCount-1.0);
VPosBoundary(3*i+0, 0) = VPosBoundary(3*i+1, 1) = VPosBoundary(3*i+2, 2) = (1-tau); // r_anf
VPosBoundary(3*i+0, 3) = VPosBoundary(3*i+1, 4) = VPosBoundary(3*i+2, 5) = tau; // r_end
}
// Indirekten Effekt beruecksichtigen
// (Aenderung der Position aendert die Feldstaerke entlang der Bahn)
// (I-K*tensor)^-1
// ----------------------------
// Einheitsmatrix
Matrix Inv(3*posCount, 3*posCount);
for(UInt i=0; i<Inv.rows(); i++)
Inv(i,i) = 1.0;
// I +- K*Tensor
for(UInt i=0; i<posCount; i++)
matMult(-1., IntegrationPos.slice(3,3*i,3*posCount-3,3), tensor.at(i), Inv.slice(3,3*i,3*posCount-3,3));
// (I-K*tensor)^-1(Gravity*g+vBound-vPos)+pos
// wobei A = (Gravity, Boundary, Reference)
// und Reference = (IntegrationPos*g+Randwerte-pos)
// ------------------------------------------------
solveInPlace(Inv, A);
// Referenzpositionen
// Reference = (I-K*tensor)^-1(Gravity*g+vBound-vPos)+pos
// --------------------
Matrix vDeltaPos = vPos;
for(UInt j=0; j<vPos.columns(); j++)
vPos.column(j) += vGpsPos;
Arc arc;
arc.vPos = vPos;
arc.VPos = VPos;
arc.VPosBoundary = VPosBoundary;
// =============================================================================
// Beobachtungsgleichungen für Vektorielle Beschleunigungen aus Beschleunigungen
// -----------------------------------------------------------------------------
if(computeVelocity || computeAcceleration)
{
arc.VAcc = Matrix(3*posCount, 3*posCount);
// gedrehte Einheitsmatrix
for(UInt i=0; i<posCount; i++)
copy(rotEarth.at(i).matrix().trans(), arc.VAcc.slice(3*i,3*i,3,3));
// Referenzbeschleuigungen
// -----------------------
arc.vAcc = arc.VAcc * g;
// indirekter Effekt
for(UInt i=0; i<posCount; i++)
matMult(1., tensor.at(i), vDeltaPos.row(3*i,3), arc.vAcc.row(3*i,3));
// indirekter Effekt
// -----------------
for(UInt i=0; i<posCount; i++)
matMult(1., tensor.at(i), arc.VPos.row(3*i,3), arc.VAcc.row(3*i,3));
// Randwerte (nur indirekter Effekt)
// ---------------------------------
arc.VAccBoundary = Matrix(3*posCount, 6);
for(UInt i=0; i<posCount; i++)
matMult(1., tensor.at(i), arc.VPosBoundary.row(3*i,3), arc.VAccBoundary.row(3*i,3));
}
// =============================================================================
// Beobachtungsgleichungen für Vektorielle Geschwindigkeiten aus Beschleunigungen
// ------------------------------------------------------------------------------
if(computeVelocity)
{
// Integral mit dem Kern (d/dt K(tau, tau'))
if(IntegrationVel.rows()!=3*posCount)
IntegrationVel = integrationMatrixVelocity(T, posCount);
arc.VVel = IntegrationVel * arc.VAcc;
// Randwerte
// ---------
arc.VVelBoundary = Matrix(3*posCount, 6);
for(UInt i=0; i<posCount; i++)
{
arc.VVelBoundary(3*i+0, 0) = arc.VVelBoundary(3*i+1, 1) = arc.VVelBoundary(3*i+2, 2) = -1/T; // r_anf
arc.VVelBoundary(3*i+0, 3) = arc.VVelBoundary(3*i+1, 4) = arc.VVelBoundary(3*i+2, 5) = 1/T; // r_end
}
// indirekter Effekt
matMult(1., IntegrationVel, arc.VAccBoundary, arc.VVelBoundary);
// Referenzgeschwindigkeiten
// -------------------------
arc.vVel = IntegrationVel * arc.vAcc;
// Randwerte - pos
for(UInt i=0; i<posCount; i++)
for(UInt j=0; j<arc.vVel.columns(); j++)
{
arc.vVel(3*i+0,j) += (arc.vPos(3*posCount-3,j) - arc.vPos(0,j))/T; //(1/T) * pos1End.x() - (1/T) * pos1Start.x();
arc.vVel(3*i+1,j) += (arc.vPos(3*posCount-2,j) - arc.vPos(1,j))/T; //(1/T) * pos1End.y() - (1/T) * pos1Start.y();
arc.vVel(3*i+2,j) += (arc.vPos(3*posCount-1,j) - arc.vPos(2,j))/T; //(1/T) * pos1End.z() - (1/T) * pos1Start.z();
}
}
// =============================================================================
// Fit integrated orbit to given orbit
// -----------------------------------
Matrix B(3*posCount, 6+accCalibration.columns());
copy(VPosBoundary, B.column(0,6));
if(accCalibration.size())
matMult(1., VPos, accCalibration, B.column(6,accCalibration.columns()));
Vector x = leastSquares(B, vDeltaPos);
if(arc.vPos.size()) matMult(-1, arc.VPosBoundary, x.row(0,6), arc.vPos);
if(arc.vVel.size()) matMult(-1, arc.VVelBoundary, x.row(0,6), arc.vVel);
if(arc.vAcc.size()) matMult(-1, arc.VAccBoundary, x.row(0,6), arc.vAcc);
if(accCalibration.size())
{
Vector y = accCalibration*x.row(6,accCalibration.columns());
if(arc.vPos.size()) matMult(-1, arc.VPos, y, arc.vPos);
if(arc.vVel.size()) matMult(-1, arc.VVel, y, arc.vVel);
if(arc.vAcc.size()) matMult(-1, arc.VAcc, y, arc.vAcc);
}
return arc;
}
catch(std::exception &e)
{
GROOPS_RETHROW(e)
}
}
/***********************************************/
// interpolate to observations times
// ---------------------------------
void IntegralEquation::interpolateArc(const std::vector<OrbitArc> &pod, const OrbitArc &orbit, const Arc &arc, Matrix &l, Matrix &VPos, Matrix &VPosBoundary) const
{
try
{
// observations
l = Matrix(3*pod.at(0).size(), arc.vPos.columns());
for(UInt j=0; j<l.columns(); j++)
for(UInt k=0; k<pod.at(j).size(); k++)
{
l(3*k+0,j) = pod.at(j).at(k).position.x();
l(3*k+1,j) = pod.at(j).at(k).position.y();
l(3*k+2,j) = pod.at(j).at(k).position.z();
}
Polynomial polynomial(orbit.times(), interpolationDegree);
auto timesPod = pod.at(0).times();
l -= polynomial.interpolate(timesPod, arc.vPos, 3);
VPos = polynomial.interpolate(timesPod, arc.VPos, 3);
VPosBoundary = polynomial.interpolate(timesPod, arc.VPosBoundary, 3);
}
catch(std::exception &e)
{
GROOPS_RETHROW(e)
}
}
/***********************************************/
/***********************************************/
Matrix IntegralEquation::integrationMatrixPosition(Double T, UInt posCount) const
{
try
{
Matrix D(3*posCount, 3*posCount);
UInt unt = posCount-1; // Anzahl Integrationsintervalle
UInt integrationDegree = W.size();
for(UInt i=1; i<posCount-1; i++)
{
Double tau = i/static_cast<Double>(unt);
Vector factor(posCount);
UInt nr = 0;
UInt nullpunkt = 0;
for(UInt k=0; k<unt; k++) // Integrationsintervall
{
for(UInt j=0; j<W.at(nr).columns(); j++) // Position im Interpolationspolynom
for(UInt n=0; n<W.at(nr).rows(); n++) // Grad im Interpolationspolynoms
if(i>k)
factor(k+j-nullpunkt) += -T*T/(unt*unt) * (1-tau) * (1./(n+2.) + k /(n+1.)) * W.at(nr)(j,n);
else
factor(k+j-nullpunkt) += T*T/(unt*unt) * tau * (1./(n+2.) - (unt-k)/(n+1.)) * W.at(nr)(j,n);
// Polynommatrix auswaehlen
if(k<integrationDegree/2)
nr++, nullpunkt++;
if(k>=unt-integrationDegree/2-1)
nr++, nullpunkt++;
}
for(UInt k=0; k<posCount; k++)
D(3*i+0,3*k+0) = D(3*i+1,3*k+1) = D(3*i+2,3*k+2) = factor(k);
}
return D;
}
catch(std::exception &e)
{
GROOPS_RETHROW(e)
}
}
/***********************************************/
Matrix IntegralEquation::integrationMatrixVelocity(Double T, UInt posCount) const
{
try
{
Matrix D(3*posCount, 3*posCount);
UInt unt = posCount-1; // Anzahl Integrationsintervalle
UInt integrationDegree = W.size();
for(UInt i=0; i<posCount; i++)
{
Vector factor(posCount);
UInt nr = 0;
UInt nullpunkt = 0;
for(UInt k=0; k<unt; k++) // Integrationsintervall
{
for(UInt j=0; j<W.at(nr).columns(); j++) // Position im Interpolationspolynom
for(UInt n=0; n<W.at(nr).rows(); n++) // Grad im Interpolationspolynoms
if(i>k)
factor(k+j-nullpunkt) += T/(unt*unt) * (1./(n+2.) + k /(n+1.)) * W.at(nr)(j,n);
else
factor(k+j-nullpunkt) += T/(unt*unt) * (1./(n+2.) - (unt-k)/(n+1.)) * W.at(nr)(j,n);
// Polynommatrix auswaehlen
if(k<integrationDegree/2)
nr++, nullpunkt++;
if(k>=unt-integrationDegree/2-1)
nr++, nullpunkt++;
}
for(UInt k=0; k<posCount; k++)
D(3*i+0,3*k+0) = D(3*i+1,3*k+1) = D(3*i+2,3*k+2) = factor(k);
}
return D;
}
catch(std::exception &e)
{
GROOPS_RETHROW(e)
}
}
/***********************************************/
|
