flightModel.asv

function [h_end,v_end,Ae,q] = flightModel(phi,n_st,n_e,M_prop,M_str)

%% AN ANALITICAL APPROACH OF A ROCKET TRAJECTORY
% Jan Canet, Fabio Meloni

parameters;

% % Definition of parameters
% 
% phi_pl = 5;             % Payload fairing diameter [m]
% L_pl = 12;              % Payload fairing length [m]
% th_pl = 0.005;          % Payload fairing thickness [m]
% 
% M_pl = 7000;            % Payload mass [kg]
% 
% M_eng_vac = 500;        % Vacuum engine mass [kg]
% M_eng_sl = 470;         % Sea level engine mass [kg]
% L_eng_vac = 6;          % Longitude of vacuum engine [m]
% L_eng_sl = 3;           % Longitude of sea level engine [m]
% rho_prop = 1000;        % Density of the propellant [kg/m^3]
% 
% alpha = deg2rad(0.5);   % Angle of attack [deg->rad]
% g0 = 9.81;              % Gravitational acceleration at Earth's surface [m/s^2]
% thrust_1e = 900e3;      % Thrust of one SL engine [N]
% SF = 2;                 % Safety factor
% 
% rho = 2550;             % Structural material density [kg/m^3]
% E = 77e9;               % Young modulus [Pa]
% sig_y = 210e6;          % Yield strength [Pa]
% sig_s = 200e6;          % Shear strength [Pa]
% 
% h_tar = 300e3;          % Target orbital altitude [m]
% v_tar = 7.73e3;         % Target orbital velocity [m/s]


%% 1) Initial parameters and variables
% Me
Me = sym('Me','real');
gamma = 1.1488;
Ae_At = 21; % Exit to throat area ratio
rel_arees = Ae_At == (2/(gamma+1))^(0.5*(gamma+1)/(gamma-1))*1/Me*(1+(gamma-1)/2*Me^2)^(0.5*(gamma+1)/(gamma-1));
Me_V = vpasolve(rel_arees,Me);
Me_V = double(Me_V); % Exit Mach number

% CF & At
pa_pc = 0.0094; % Initial pressure ratio
CF_vac = (2/(gamma+1))^(0.5*(gamma+1)/(gamma-1))*(gamma*Me_V+1/Me_V)/sqrt(1+(gamma-1)/2*Me_V^2);
MFP_Me = sqrt(gamma)*Me_V/(1+(gamma-1)/2*Me_V^2)^(0.5*(gamma+1)/(gamma-1));
MFP_Mt = sqrt(gamma)*1/(1+(gamma-1)/2*1^2)^(0.5*(gamma+1)/(gamma-1));
CF = CF_vac-pa_pc*MFP_Mt/MFP_Me;

Wp = M_prop*g0; % Initial propellant weight [N]
Ws = M_str*g0; % Initial structural weight [N]
% Wu = 400*g0; % payload weight [N]
Wu = M_pl*g0; % payload weight [N]
W = sum(Wp)+sum(Ws)+Wu; % Initial weight [N]
F = thrust_1e; % Initial thrust [N]
Pc = 10.8e6; % Chamber pressure [Pa]
At = F/(Pc*CF); % Throat area [m^2]
Ae_sl = At*Ae_At;
S = pi*(max(phi)/2)^2; % Rocket's maximum cross-section [m^2]

% MASS FLOW RATE
R = 8.314472; % Universal gas constant [J/molK]
MM = 22.686; % Molecular mass [g/mol]
Rg = R/(MM/1000); % Gas constant [J/kgK]
Tc = 3655.73; % Chamber temperature [K]
m_dot = MFP_Mt*Pc*At/sqrt(Rg*Tc);

%% 2) Initial conditions
h0 = 0.001; % [m]
v0 = 0.001; % [m/s]
ang0 = deg2rad(90); % [deg->rad]

%% 3) Time span and time step
t_span1 = Wp(end)/(m_dot*g0*n_e); % [s]
t_step = 1e-3; % [s]

%% Optimize kick_time and kick_angle

% KICK ANGLE STUFF WHICH NEEDS TO BE AUTOMATED so that the flight path angle reaches 0 before the end of the burn (and stays at zero)!!!
% kick_time = t_span1*0.145; % time before starting gravity turn [s] 35
% kick_angle = deg2rad(9); % kick angle induced for gravity turn [deg->rad] 35

% % Define parameters for the objective function and constraints
% parameters.g0 = g0;
% parameters.W = W;
% parameters.m_dot = m_dot;
% parameters.gamma = gamma;
% parameters.Rg = Rg;
% parameters.CF_vac = CF_vac;
% parameters.Pc = Pc;
% parameters.MFP_Me = MFP_Me;
% parameters.MFP_Mt = MFP_Mt;
% parameters.At = At;
% parameters.S = S;
% parameters.n_e = n_e;
% parameters.t_span1 = t_span1;
% parameters.t_step = t_step;
% parameters.h0 = h0;
% parameters.v0 = v0;
% parameters.ang0 = ang0;

% Initial guess for [kick_time, kick_angle]
x0 = [t_span1 * 0.145, deg2rad(9)];

% % Set optimization options
% options = optimoptions('fmincon', 'Display', 'iter', 'Algorithm', 'interior-point');
% Set optimization options for fminunc with Newton's method
options = optimoptions('fminunc', 'Display', 'iter', 'Algorithm', 'trust-region');

% % Define lower and upper bounds for [kick_time, kick_angle] for fmincon
% lb = [1, deg2rad(1)]; % lower bounds
% ub = [t_span1, deg2rad(60)]; % upper bounds

% Run the optimization
obj_flightangle = @(x) objectiveFlightAngle(x, phi,n_st,n_e,M_prop,M_str);

% % Optimize with fmincon
% x_opt = fmincon(obj_flightangle, x0, [], [], [], [], lb, ub, [], options);
% Run the optimization using fminunc
x_opt = fminunc(obj_flightangle, x0, options);

% Extract optimal values
kick_time_opt = x_opt(1);
kick_angle_opt = x_opt(2);

% Display optimal values
disp(['Optimal kick time: ', num2str(kick_time_opt)]);
disp(['Optimal kick angle: ', num2str(rad2deg(kick_angle_opt))]);

% Use the optimized kick_time and kick_angle in the main script
kick_time = kick_time_opt;
kick_angle = kick_angle_opt;

%% 4) Solver
% FIRST STAGE
t0 = 0;
[t_1st_initial, y_1st_initial] = ode45(@(t,y) Fsyst(t,t0,y,g0,W,m_dot,gamma,Rg,CF_vac,Pc,MFP_Me,MFP_Mt,At,S,n_e), 0:t_step:kick_time, [h0;v0;ang0]);
% Extract final values
final_values = y_1st_initial(end,:);  % Final [altitude, velocity, angle]
% New initial conditions for the second integration
h0_new = final_values(1);
v0_new = final_values(2);
ang0_new = ang0-kick_angle;
% ang0_new = final_values(3);
% Solve second system
t0 = kick_time;
[t_1st_final, y_1st_final] = ode45(@(t,y) Fsyst(t,t0,y,g0,W,m_dot,gamma,Rg,CF_vac,Pc,MFP_Me,MFP_Mt,At,S,n_e), kick_time:t_step:t_span1, [h0_new;v0_new;ang0_new]);
% Concatenate time and results for plotting
t_1st_combined = [t_1st_initial; t_1st_final];
y_1st_combined = [y_1st_initial; y_1st_final];

% SECOND STAGE
% Me
Me = sym('Me','real');
gamma = 1.1488;
Ae_At = 116; % Exit to throat area ratio
rel_arees = Ae_At == (2/(gamma+1))^(0.5*(gamma+1)/(gamma-1))*1/Me*(1+(gamma-1)/2*Me^2)^(0.5*(gamma+1)/(gamma-1));
Me_V = vpasolve(rel_arees,Me);
Me_V = double(Me_V); % Exit Mach number

% CF & At
pa_pc = 0.0094; % Initial pressure ratio
CF_vac = (2/(gamma+1))^(0.5*(gamma+1)/(gamma-1))*(gamma*Me_V+1/Me_V)/sqrt(1+(gamma-1)/2*Me_V^2);
MFP_Me = sqrt(gamma)*Me_V/(1+(gamma-1)/2*Me_V^2)^(0.5*(gamma+1)/(gamma-1));
MFP_Mt = sqrt(gamma)*1/(1+(gamma-1)/2*1^2)^(0.5*(gamma+1)/(gamma-1));
CF = CF_vac-pa_pc*MFP_Mt/MFP_Me;

F = thrust_1e; % Initial thrust [N]
Pc = 10.8e6; % Chamber pressure [Pa]
At = F/(Pc*CF); % Throat area [m^2]
Ae_vac = At*Ae_At;
S = pi*(max(phi(1:end-1))/2)^2; % Rocket's maximum cross-section [m^2]

% MASS FLOW RATE
R = 8.314472; % Universal gas constant [J/molK]
MM = 22.686; % Molecular mass [g/mol]
Rg = R/(MM/1000); % Gas constant [J/kgK]
Tc = 3655.73; % Chamber temperature [K]
m_dot = MFP_Mt*Pc*At/sqrt(Rg*Tc);
t_span23 = Wp(1:end-1)./(m_dot*g0); % [s]

second_stage_initial = y_1st_final(end,:);  % Final [altitude, velocity, angle]
% New initial conditions for the integration of second stage
h0_2nd = second_stage_initial(1);
v0_2nd = second_stage_initial(2);
ang0_2nd = second_stage_initial(3);
% Solve system of 2nd stage
W = W - Ws(end) - Wp(end);
t0 = t_span1;
[t_2nd, y_2nd] = ode45(@(t,y) Fsyst(t,t0,y,g0,W,m_dot,gamma,Rg,CF_vac,Pc,MFP_Me,MFP_Mt,At,S,1), t_span1:t_step:(t_span1+t_span23(1)), [h0_2nd;v0_2nd;ang0_2nd]);

if (n_st == 3)
    % THIRD STAGE
    third_stage_initial = y_2nd(end,:);  % Final [altitude, velocity, angle]
    % New initial conditions for the integration of second stage
    h0_3rd = third_stage_initial(1);
    v0_3rd = third_stage_initial(2);
    ang0_3rd = third_stage_initial(3);
    % Solve system of 3rd stage
    W = W - Ws(end-1) - Wp(end-1);
    t0 = (t_span1+t_span23(end-1));
    [t_3rd, y_3rd] = ode45(@(t,y) Fsyst(t,t0,y,g0,W,m_dot,gamma,Rg,CF_vac,Pc,MFP_Me,MFP_Mt,At,S,1), (t_span1+t_span23(1)):t_step:(t_span1+sum(t_span23)), [h0_3rd;v0_3rd;ang0_3rd]);

    inertial_stage_initial = y_3rd(end,:);
    h0_inr = inertial_stage_initial(1);
    v0_inr = inertial_stage_initial(2);
    ang0_inr = inertial_stage_initial(3);
    [t_inr, y_inr] = ode45(@(t,y) Fsyst2(y), (t_span1+sum(t_span23)):t_step:(t_span1+sum(t_span23))+20, [h0_inr;v0_inr;ang0_inr]);
end


% 5) Postprocess
figure
subplot(3,1,1);
plot(t_1st_combined, y_1st_combined(:,1)/1000,'b') 
hold on
plot(t_2nd, y_2nd(:,1)/1000,'g')
if (n_st == 3)
    plot(t_3rd, y_3rd(:,1)/1000,'r')
    plot(t_inr, y_inr(:,1)/1000,'--k')
end
yline(h_tar/1000, 'y--', 'LineWidth', 1.5);
xlabel('Time [s]')
ylabel('Altitude [km]')
title('Altitude over time')
grid on

subplot(3,1,2);
plot(t_1st_combined, y_1st_combined(:,2)/1000,'b')
hold on
plot(t_2nd, y_2nd(:,2)/1000,'g')
if (n_st == 3)
    plot(t_3rd, y_3rd(:,2)/1000,'r')
    plot(t_inr, y_inr(:,2)/1000,'--k')
end
yline(v_tar/1000, 'k--', 'LineWidth', 1.5);
xlabel('Time [s]')
ylabel('Velocity [km/s]')
title('Velocity over time')
grid on

subplot(3,1,3);
plot(t_1st_combined, y_1st_combined(:,3) * 180/pi,'b')
hold on
plot(t_2nd, y_2nd(:,3) * 180/pi,'g')
if (n_st == 3)
    plot(t_3rd, y_3rd(:,3) * 180/pi,'r')
    plot(t_inr, y_inr(:,3) * 180/pi,'--k')
end
yline(0, 'k--', 'LineWidth', 1.5);
xlabel('Time [s]')
ylabel('Flight path angle [°]')
title('Flight path angle over time')
grid on

if (n_st == 3)
    h_end = y_3rd(end,1);
    v_end = y_3rd(end,2);
    Ae = [Ae_vac,Ae_sl];
else
    h_end = y_2nd(end,1);
    v_end = y_2nd(end,2);
    Ae = [Ae_vac,Ae_sl];
end


% Maximum Dynamic Pressure
if (n_st == 3)
    altitude = [y_1st_combined(:,1); y_2nd(:,1); y_3rd(:,1)];
    velocity = [y_1st_combined(:,2); y_2nd(:,2); y_3rd(:,2)];
    time = [t_1st_combined; t_2nd; t_3rd];
else
    altitude = [y_1st_combined(:,1); y_2nd(:,1)];
    velocity = [y_1st_combined(:,2); y_2nd(:,2)];
    time = [t_1st_combined; t_2nd];
end
for i = 1:length(altitude)
    [~, density(i), ~] = Compute_P_rho_Text(altitude(i));
end
dyn_pressure = 0.5*density.'.*velocity.^2;
q = max(dyn_pressure);

figure
plot(time,dyn_pressure/1000)
xlabel('Time [s]')
ylabel('Dynamic Pressure [kPa]')
title('Dynamic pressure over time')
grid on