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TesseralQGFr.m
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TesseralQGFr.m
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function [W, dWdX, didGWgdidHWh, dWdletadWdg, d2WdX2,Wghx,Wlgx] = TesseralQGFr(coe,f0,n,m,C20,Cnm,Snm,mu,we,Re,tol,HessianOn,QuadTesseralsOn)
% Closed-form Short-period and m-daily tesseral generating function in the
% Earth-fixed frame. Uses mean anoamaly as the independent variable for
% computing Jacobians and Hessians.
% Author: Bharat Mahajan (https://github.com/princemahajan)
% classical elements
a = coe(1);
e = coe(2);
i = coe(3);
hr = coe(4);
g = coe(5);
f = coe(6);
eta = sqrt(1 - e^2);
L = sqrt(mu*a);
G = L*eta;
mm = sqrt(mu/a^3);
detade = -e/eta;
delta = we/mm;
ddeltada = we/sqrt(mu)*3/2*sqrt(a);
d2deltada2 = we/sqrt(mu)*3/4/sqrt(a);
% dfdl = (1 + e*cos(f))^2/eta^3;
% dldf = 1/dfdl;
% dfde = (2+e*cos(f)).*sin(f)/eta^2;
% d2ldf2 = 2*e*eta^3*sin(f)/(1+e*cos(f))^3;
% d2ldedf = ((1+e*cos(f))^2*3*eta^2*detade - eta^3*2*(1+e*cos(f))*(cos(f)-e*sin(f)*dfde))/(1+e*cos(f))^4;
%
% d2fdlde = -3/eta^4*(-e/eta)*(1+e*cos(f))^2 + 2*(1+e*cos(f))/eta^3*(cos(f)-e*sin(f)*dfde);
% d2fdldf = 2*(1+e*cos(f))/eta^3*(-e*sin(f));
% select proper C and S order
if mod(n-m,2) == 0
CS1 = Cnm;
CS2 = Snm;
else
CS1 = -Snm;
CS2 = Cnm;
end
% intermediate terms
C = -2/C20^2*Re^n*sqrt(mu)/a^(n-1/2)/eta^(2*n-1);
dCda = -2/C20^2*Re^n*sqrt(mu)*(-n+1/2)*a^(-n+1/2-1)/eta^(2*n-1);
dCde = -2/C20^2*Re^n*sqrt(mu)/a^(n-1/2)*(-2*n+1)*eta^(-2*n)*detade;
Caa = -2/C20^2*Re^n*sqrt(mu)*(-n+1/2)*(-n+1/2-1)*a^(-n+1/2-2)/eta^(2*n-1);
Cae = -2/C20^2*Re^n*sqrt(mu)*(-n+1/2)*a^(-n+1/2-1)*(-2*n+1)/eta^(2*n)*detade;
Cea = Cae;
Cee = -2/C20^2*Re^n*sqrt(mu)/a^(n-1/2)*(-2*n+1)*(-eta^(-2*n-3)*(2*n*e^2 + 1));
dCdX = [dCda, dCde, zeros(1,4)];
d2CdX2 = [Caa, Cae, 0,0,0,0;
Cea, Cee, 0,0,0,0;
zeros(4,6)];
W = 0;
dWdX = zeros(1,6);
d2WdX2 = zeros(6);
didGWgdidHWh = 0;
dWdletadWdg = 0;
Wghx = zeros(1,6);
Wlgx = zeros(1,6);
% P Summation
for p = 0:n
if f < 0
period = -2*pi;
else
period = 2*pi;
end
E = mod(2*atan(sqrt((1-e)/(1+e))*tan(f/2)),period) + fix(f/period)*period;
l = E - e*sin(E);
% beta and partials
beta = (n-2*p)*g + m*(hr + delta*l);
cbeta = cos(beta);
sbeta = sin(beta);
dbetada = m*ddeltada*l;
dbetadg = n-2*p;
dbetadh = m;
dbetadl = m*delta;
betax = [dbetada, 0, 0, dbetadh,dbetadg, dbetadl];
betaxx = [m*d2deltada2*l, 0, 0, 0, 0, m*ddeltada;
zeros(1,6);
zeros(1,6);
zeros(1,6);
zeros(1,6);
m*ddeltada, 0,0,0,0,0];
% The culprit integral and partials
coeI = [a,e,f];
if QuadTesseralsOn == true
[I, dIdX, dIdlg1, dIdlg2,I1xx,I2xx,dI1lgdX,dI2lgdX] = qCulpritIntegrals(coeI,f0,n,m,p,delta,ddeltada,d2deltada2,tol, HessianOn);
else
[I, dIdX, dIdlg1, dIdlg2,I1xx,I2xx,dI1lgdX,dI2lgdX] = eCulpritIntegrals( delta,ddeltada,d2deltada2,e,f,l,n,m,p,HessianOn);
end
I1x = dIdX(1,:);
I2x = dIdX(2,:);
% Inclination function
[ F, Fd, Fbysi2, Fdd, Fbysi2d] = IncFunc( n,m,p,i );
dFdX = [0,0,Fd,0,0,0];
d2FdX2 = [zeros(2,6);
0,0,Fdd,0,0,0;
zeros(3,6)];
% W
Vp = (CS1*(cbeta*I(1) - sbeta*(I(2))) + CS2*(cbeta*I(2) + sbeta*I(1)));
W = W + C*F*Vp;
% Partials of W w.r.t classical elements
Vpx = @(Y) CS1*(cbeta*(I1x(Y)-betax(Y)*I(2)) - sbeta*(I2x(Y) + betax(Y)*I(1))) ...
+ CS2*(cbeta*(betax(Y)*I(1) + I2x(Y)) + sbeta*(I1x(Y) - betax(Y)*I(2)));
for ctr = 1:6
dWdX(ctr) = dWdX(ctr) + dCdX(ctr)*F*Vp + C*dFdX(ctr)*Vp + C*F*Vpx(ctr);
end
% didG*dWdg+didH*dWdh term
if (n-2*p) == m
Wghcoeff = -m*tan(i/2)/G;
Fform = F;
else
Wghcoeff = 1/(G*2*cos(i/2))*(cos(i)*(n-2*p)-m);
Fform = Fbysi2;
end
Vgh = CS1*(-cbeta*I(2) - sbeta*(I(1))) + CS2*(cbeta*I(1) - sbeta*I(2));
didGWgdidHWh = didGWgdidHWh + C*Wghcoeff*Fform*Vgh;
% 1/e*(dWdl - 1/eta*dWdg) term
Vlg = (CS1*(cbeta*(-(n-2*p)*dIdlg1(2) - (n+1)/eta^3*dIdlg2(1)) ...
+ sbeta*(-(n-2*p)*dIdlg1(1) + (n+1)/eta^3*dIdlg2(2))) ...
+ CS2*(cbeta*((n-2*p)*dIdlg1(1) - (n+1)/eta^3*dIdlg2(2)) ...
+ sbeta*(-(n-2*p)*dIdlg1(2) - (n+1)/eta^3*dIdlg2(1))));
dWdletadWdg = dWdletadWdg + C*F*Vlg;
if HessianOn == true
dI1lgadX = dI1lgdX(1,:);
dI1lgbdX = dI1lgdX(2,:);
dI2lgadX = dI2lgdX(1,:);
dI2lgbdX = dI2lgdX(2,:);
Vpxy = @(X,Y) CS1*(cbeta*(I1xx(X,Y) - betax(X)*(betax(Y)*I(1) + I2x(Y)) - betax(Y)*I2x(X) - betaxx(X,Y)*I(2)) ...
- sbeta*(I2xx(X,Y) + betax(X)*(-betax(Y)*I(2) + I1x(Y)) + betax(Y)*I1x(X) + betaxx(X,Y)*I(1))) ...
+ CS2*(cbeta*(I2xx(X,Y) + betax(Y)*(-betax(X)*I(2) + I1x(X)) + betax(X)*I1x(Y) + betaxx(X,Y)*I(1)) ...
+ sbeta*(I1xx(X,Y) - betax(Y)*(betax(X)*I(1) + I2x(X)) - betax(X)*I2x(Y) - betaxx(X,Y)*I(2)));
FVpx = @(X) dFdX(X)*Vp + F*Vpx(X);
FVpxy = @(X,Y) d2FdX2(X,Y)*Vp + dFdX(X)*Vpx(Y) + dFdX(Y)*Vpx(X) + F*Vpxy(X,Y);
% compute each element of Hessian of W
for row = 1:6
for col = 1:6
d2WdX2(row,col) = d2WdX2(row,col) + d2CdX2(row,col)*F*Vp + dCdX(row)*FVpx(col) ...
+ dCdX(col)*FVpx(row) + C*FVpxy(row,col);
end
end
% Partials of terms for equinoctial variations
% didG*dWdg+didH*dWdh term
if (n-2*p) == m
Wghcoeffx = [m*tan(i/2)/(2*a*G), -m*tan(i/2)*e/(eta^2*G), -m/G/(1+cos(i)), 0, 0, 0];
Fformx = dFdX;
else
Wghcoeffx = [-1/(a*G*4*cos(i/2))*(cos(i)*(n-2*p)-m), e/(eta^2*G*2*cos(i/2))*(cos(i)*(n-2*p)-m), ...
-1/(2*G)*((n-2*p) + (m+n-2*p)/(1+cos(i)))*sin(i/2), 0, 0, 0];
Fformx = [0,0,Fbysi2d,0,0,0];
end
Vghx = @(X) CS1*(sbeta*betax(X)*I(2) - cbeta*I2x(X) - cbeta*betax(X)*I(1) - sbeta*I1x(X)) ...
+ CS2*(-sbeta*betax(X)*I(1) + cbeta*I1x(X) - cbeta*betax(X)*I(2) - sbeta*I2x(X));
for ctr = 1:6
Wghx(ctr) = Wghx(ctr) + dCdX(ctr)*Wghcoeff*Fform*Vgh + C*Wghcoeffx(ctr)*Fform*Vgh ...
+ C*Wghcoeff*Fformx(ctr)*Vgh + C*Wghcoeff*Fform*Vghx(ctr);
end
% 1/e*(dWdl - 1/eta*dWdg) term
Vlgx = @(X) CS1*(-sbeta*betax(X)*(-(n-2*p)*dIdlg1(2) - (n+1)/eta^3*dIdlg2(1)) ...
+ cbeta*(-(n-2*p)*dI2lgadX(X) - (n+1)*(3*e/eta^5*dIdlg2(1) + 1/eta^3*dI1lgbdX(X))) ...
+ cbeta*betax(X)*(-(n-2*p)*dIdlg1(1) + (n+1)/eta^3*dIdlg2(2)) ...
+ sbeta*(-(n-2*p)*dI1lgadX(X) + (n+1)*(3*e/eta^5*dIdlg2(2) + 1/eta^3*dI2lgbdX(X)))) ...
+ CS2*(-sbeta*betax(X)*((n-2*p)*dIdlg1(1) - (n+1)/eta^3*dIdlg2(2)) ...
+ cbeta*((n-2*p)*dI1lgadX(X) - (n+1)*(3*e/eta^5*dIdlg2(2) + 1/eta^3*dI2lgbdX(X))) ...
+ cbeta*betax(X)*(-(n-2*p)*dIdlg1(2) - (n+1)/eta^3*dIdlg2(1)) ...
+ sbeta*(-(n-2*p)*dI2lgadX(X) - (n+1)*(3*e/eta^5*dIdlg2(1) + 1/eta^3*dI1lgbdX(X))));
for ctr = 1:6
Wlgx(ctr) = Wlgx(ctr) + dCdX(ctr)*F*Vlg + C*dFdX(ctr)*Vlg + C*F*Vlgx(ctr);
end
end
end
end
% function [ J ] = FiniteDiffIxx(X, cdtol, f0,n,m,p,delta,ddeltada,d2deltada2,tol, HessianOn)
%
% J = zeros(8,6);
%
% [~, f1] = KeplerEqSolver(X(6), X(2), 1e-11);
% [~, f2] = KeplerEqSolver(X(6) + cdtol, X(2), 1e-11);
%
% for ctr = 1:6
%
% dx1 = X;
% dx2 = X;
% dx1(ctr) = X(ctr);
% dx2(ctr) = X(ctr) + cdtol;
%
% dx1(6) = f1;
% dx2(6) = f1;
%
% if ctr == 6
% dx2(6) = f2;
% end
%
% [I, dIdX, dIdlg1, dIdlg2,I1xx,I2xx,dI1lgdX,dI2lgdX] = qCulpritIntegrals([dx1(1:2),dx1(6)],f0,n,m,p,delta,ddeltada,d2deltada2,tol, HessianOn);
% dy1 = [dIdX(1,:), dIdlg1(1), dIdlg2(1)]';
% [I, dIdX, dIdlg1, dIdlg2,I1xx,I2xx,dI1lgdX,dI2lgdX] = qCulpritIntegrals([dx2(1:2),dx2(6)],f0,n,m,p,delta,ddeltada,d2deltada2,tol, HessianOn);
% dy2 = [dIdX(1,:), dIdlg1(1), dIdlg2(1)]';
%
% J(:,ctr) = (dy2 - dy1)/cdtol;
%
% end
%
% end