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I've been doing a comparison of the plots for roll, yaw, and pitch of a satellite in GG orbit and I've been unable to get my nonLinear function to relatively match my Linearized equation. I've tried my variant of the equations and the provided variants while only getting the linearized equation to come out correctly. As the linearized equation is just a simplified version of the nonlinear equation, and I've tried known good equations without success, I suspect I am doing something wrong in how I set up the code.

Here is the base equation's that are being worked off off of. The first longer one is set equal to the smaller second one to make the end result that is used in the code for the nonlinear function. This is the same equation set that when linearized, works just fine in ODE45. Since the first one is so long, this is why I broke it down into a smaller equation by using alphabetical variable for the solver.

To start off, the three equations are dependent on each other a little bit so I simplified the equations to alphabetical variables, and then had MATLAB solve them for me. I then just add in what the variables are equal to once in the the ODE45 function. I must be missing something here. Below is the code for my derivation of the equations, plotting results are similar to the know correct ones.

% Simplified nonLinear equations

eq1 = ((A-B+C-D*x6d)/(-F))-x2d;

eq2 = ((G+H*I+L*x6d)/(-K))-x4d;

eq3 = ((M-N*O-P*x4d)/(-R))-x6d;

% Solve 2nd order ODE's to get in terms of x2d, x4d, and x6d

[sol1, sol2, sol3] = solve([eq1, eq2, eq3],[x2d,x4d,x6d]);

% Initial conditions of 1 degree in yaw, roll, pitch

X0 = [1 0 1 0 1 0];

% 1.6 times the rotation period of satellite in seconds

t = 0:60:5676.98*1.6;

[time_lin, solution_lin] = ode45(@(t,x) Lin_func(t,x), t, X0);

[time_nonlin, solution_nonlin] = ode45(@(t,x) nonLin_func(t,x), t, X0);

% X has [X(1), X(2), X(3), X(4), X(5), X(6)]

function d_dt = Lin_func(t,X)

% Constants

Ix = 6;

Iy = 8;

Iz = 4;

mu = (3.986*10^14);

Rad = 6878140;

w = sqrt(mu/Rad^3);

% Generate ODE's

d_dt = X*NaN;

d_dt(1) = X(2);

d_dt(2) = (w*(Ix+Iz-Iy)*X(6)-4*(w^2)*(Iy-Iz)*X(1))/Ix;

d_dt(3) = X(4);

d_dt(4) = 3*(w^2)*(Iz-Ix)*X(3)/Iy;

d_dt(5) = X(6);

d_dt(6) = ((w^2)*(Ix-Iy)*X(5)-w*(Iz+Ix-Iy)*X(2))/Iz;

end

function d_dt = nonLin_func(t,X)

% Constants

Ix = 6;

Iy = 8;

Iz = 4;

mu = (3.986*10^14);

Rad = 6878140;

w = sqrt(mu/Rad^3);

% Equation chunks

A = (Iy-Iz)*(w*cosd(X(5))*sind(X(1))-w*cosd(X(1))*sind(X(3))*sind(X(5))-sind(X(1))*X(4)+cosd(X(3))*cosd(X(1))*X(6))*(w*cosd(X(1))*cosd(X(5))+w*sind(X(3))*sind(X(1))*sind(X(5))-cosd(X(1))*X(4)-cosd(X(3))*sind(X(1))*X(6));

B = Ix*w*cosd(X(3))*cosd(X(5))*X(6);

C = Ix*X(4)*(w*sind(X(3))*sind(X(5))-cosd(X(3))*X(6));

D = Ix*sind(X(3));

E = (3*mu/(2*Rad^3))*(sind(2*X(1))*(cosd(X(3))^2)*(Iy-Iz));

F = Ix;

G = (Iz-Ix)*(w*cosd(X(5))*sind(X(1))-w*cosd(X(1))*sind(X(3))*sind(X(5))-sind(X(1))*X(4)+cosd(X(3))*cosd(X(1))*X(6))*(w*cosd(X(3))*sind(X(5))-X(2)+sind(X(3))*X(6));

H = Iy;

I = (-w*cosd(X(5))*sind(X(3))*sind(X(1))*X(6)+w*cosd(X(1))*sind(X(5))*X(6)+X(2)*(w*cosd(X(5))*sind(X(1))-w*cosd(X(1))*sind(X(3))*sind(X(5))+cosd(X(3))*cosd(X(1))*X(6))-sind(X(1))*X(4)*(w*cosd(X(3))*sind(X(5))+X(2)+sind(X(3))*X(6)));

J = (3*mu/(2*Rad^3))*(sind(2*X(3))*cosd(X(1))*(Ix-Iz));

K = cosd(X(1))*Iy;

L = Iy*cosd(X(3))*sind(X(1));

M = (Iy-Ix)*(-w*cosd(X(3))*sind(X(5))+X(2)-sind(X(3))*X(6))*(-w*(cosd(X(1))*cosd(X(5))+sind(X(3))*sind(X(1))*sind(X(5)))+cosd(X(1))*X(4)+cosd(X(3))*sind(X(1))*X(6));

N = Iz;

O = (w*cosd(X(1))*cosd(X(5))*sind(X(3))*X(6)+w*sind(X(1))*sind(X(5))*X(6)+cosd(X(1))*X(4)*(w*cosd(X(3))*sind(X(5))+X(2)+sind(X(3))*X(6))-X(2)*(w*cosd(X(1))*cosd(X(5))+w*sind(X(3))*sind(X(1))*sind(X(5))-cosd(X(3))*sind(X(1))*X(6)));

P = sind(X(1))*Iz;

Q = (3*mu/(2*Rad^3))*(sind(2*X(3))*sind(X(1))*(Iy-Ix));

R = cosd(X(3))*cosd(X(1))*Iz;

% Generate ODE's

d_dt = X*NaN;

d_dt(1) = X(2);

d_dt(2) = -(D*G*P + D*K*M + A*L*P + A*K*R - B*L*P - D*J*P - B*K*R + C*L*P + C*K*R - D*K*Q - E*L*P - E*K*R + D*H*I*P - D*K*N*O)/(F*(L*P + K*R));

d_dt(3) = X(4);

d_dt(4) = -(G*R - L*M - J*R + L*Q + H*I*R + L*N*O)/(L*P + K*R);

d_dt(5) = X(6);

d_dt(6) = -(G*P + K*M - J*P - K*Q + H*I*P - K*N*O)/(L*P + K*R);

end

darova
on 9 Dec 2019

YOu have to express , and . Solve the system

function du = nonlin(t,y)

fi = y(1);

psi = y(2);

theta = y(3);

% shortcuts

cf = cosd(fi);

cp = cosd(psi);

ct = cosd(theta);

sf = sind(fi);

sp = sind(psi);

st = sind(theta);

% coefficients of fi'', psi'' and theta''

A = [1 -st 0

0 ct*sf cf

0 ct*cf sf];

B(1,1) = long expression1;

B(2,1) = long expression2;

B(3,1) = long expression3;

X = A\B;

du(1:3,1) = y(4:6);

du(4:6,1) = X;

end

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