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Trajectory_Run_TL_0407.m
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%This Script is used to run the Trajectory Function and find and graph the
%optimal trajectory
clear all
%Vandenberg: Lat = 34.60 deg.
%Kennedy Space Center: Lat = 28.50 deg.
%Output as X by 10 double that is columns for the following:
%scale_factor_1, scale_factor_2, scale_factor_3, pitch_kick, mleft_1, mleft_2, mleft_3, count, delta_v_total, delta_v_circularization
%-------- Variables for Inputs for Function defined here (in kg,m,s format)
One_dv_params = readtable('LVMasses\OnedVParameters.csv'); % load Mission 1 dv requirements csv into table
Two_dv_params = readtable('LVMasses\TwodVParameters.csv'); % load Mission 2 dv requirements csv into table
Latona1_params = readtable('LVTrajectory\Latona-1TrajReqs.csv');
Latona2_params = readtable('LVTrajectory\Latona-2TrajReqs.csv');
Minerva1_params = readtable('LVTrajectory\Minerva-1TrajReqs.csv');
Minerva2_params = readtable('LVTrajectory\Minerva-2TrajReqs.csv');
Zephyr1_params = readtable('LVTrajectory\Zephyr-1TrajReqs.csv');
Zephyr2_params = readtable('LVTrajectory\Zephyr-2TrajReqs.csv');
%--Zephyr Inputs--
grav_sea_level = 9.80665;
Mass_Payload_1 = 30;
Mass_Payload_2 = 95;
% Zephyr_1_Cd2 = 0.2;
% Zephyr_1_Radius2 = 0.45;
% Zephyr_1_mi2 = 1336.692;
% Zephyr_1_mf2 = 143.7361 + Mass_Payload_1;
% Zephyr_1_mprop2 = Zephyr_1_mi2 - Zephyr_1_mf2;
% Zephyr_1_T2 = 1.05 * (Zephyr_1_mi2) * grav_sea_level;
% Zephyr_1_Isp2 = 359.1;
%
% Zephyr_1_Cd1 = 0.2;
% Zephyr_1_Radius1 = 0.45;
% Zephyr_1_mi1 = 4338.224;
% Zephyr_1_mf1 = 447.2046;
% Zephyr_1_mprop1 = Zephyr_1_mi1 - Zephyr_1_mf1;
% Zephyr_1_T1 = 1.4 * (Zephyr_1_mi1 + Zephyr_1_mi2) * grav_sea_level;
% Zephyr_1_Isp1 = 296.1;
%
% %---
%
% Zephyr_2_Cd3 = 0.2;
% Zephyr_2_Radius3 = 0.42;
% Zephyr_2_mi3 = 426.4761;
% Zephyr_2_mf3 = 36.46237 + Mass_Payload_2;
% Zephyr_2_mprop3 = Zephyr_2_mi3 - Zephyr_2_mf3;
% Zephyr_2_T3 = 0.9 * (Zephyr_2_mi3) * grav_sea_level;
% Zephyr_2_Isp3 = 359.1;
%
% Zephyr_2_Cd2 = 0.2;
% Zephyr_2_Radius2 = 0.45;
% Zephyr_2_mi2 = 1306.692;
% Zephyr_2_mf2 = 143.7361;
% Zephyr_2_mprop2 = Zephyr_2_mi2 - Zephyr_2_mf2;
% Zephyr_2_T2 = 1.05 * (Zephyr_2_mi2 + Zephyr_2_mi3) * grav_sea_level;
% Zephyr_2_Isp2 = 359.1;
%
% Zephyr_2_Cd1 = 0.2;
% Zephyr_2_Radius1 = 0.45;
% Zephyr_2_mi1 = 4338.224;
% Zephyr_2_mf1 = 477.2046;
% Zephyr_2_mprop1 = Zephyr_2_mi1 - Zephyr_2_mf1;
% Zephyr_2_T1 = 1.4 * (Zephyr_2_mi1 + Zephyr_2_mi2 + Zephyr_2_mi3) * grav_sea_level;
% Zephyr_2_Isp1 = 296.1;
Zephyr_1_Cd2 = 0.2;
Zephyr_1_Radius2 = Zephyr1_params{2,2};
Zephyr_1_mi2 = Zephyr1_params{2,3};
Zephyr_1_mf2 = Zephyr1_params{2,4} + Mass_Payload_1;
Zephyr_1_mprop2 = Zephyr_1_mi2 - Zephyr_1_mf2;
Zephyr_1_T2 = 1.05 * (Zephyr_1_mi2) * grav_sea_level;
Zephyr_1_Isp2 = Zephyr1_params{2,8};
Zephyr_1_Cd1 = 0.2;
Zephyr_1_Radius1 = Zephyr1_params{1,2};
Zephyr_1_mi1 = Zephyr1_params{1,3};
Zephyr_1_mf1 = Zephyr1_params{1,4};
Zephyr_1_mprop1 = Zephyr_1_mi1 - Zephyr_1_mf1;
Zephyr_1_T1 = 1.4 * (Zephyr_1_mi1 + Zephyr_1_mi2) * grav_sea_level;
Zephyr_1_Isp1 = Zephyr1_params{1,8};
%---
Zephyr_2_Cd3 = 0.2;
Zephyr_2_Radius3 = Zephyr2_params{3,2};
Zephyr_2_mi3 = Zephyr2_params{3,3};
Zephyr_2_mf3 = Zephyr2_params{3,4} + Mass_Payload_2;
Zephyr_2_mprop3 = Zephyr_2_mi3 - Zephyr_2_mf3;
Zephyr_2_T3 = 0.9 * (Zephyr_2_mi3) * grav_sea_level;
Zephyr_2_Isp3 = Zephyr2_params{3,8};
Zephyr_2_Cd2 = 0.2;
Zephyr_2_Radius2 = Zephyr2_params{2,2};
Zephyr_2_mi2 = Zephyr2_params{2,3};
%disp(Zephyr2_params{2,3});
Zephyr_2_mf2 = Zephyr2_params{2,4};
Zephyr_2_mprop2 = Zephyr_2_mi2 - Zephyr_2_mf2;
Zephyr_2_T2 = 1.05 * (Zephyr_2_mi2 + Zephyr_2_mi3) * grav_sea_level;
Zephyr_2_Isp2 = Zephyr2_params{2,8};
Zephyr_2_Cd1 = 0.2;
Zephyr_2_Radius1 = Zephyr2_params{1,2};
Zephyr_2_mi1 = Zephyr2_params{1,3};
%disp(Zephyr2_params{1,3});
Zephyr_2_mf1 = Zephyr2_params{1,4};
Zephyr_2_mprop1 = Zephyr_2_mi1 - Zephyr_2_mf1;
Zephyr_2_T1 = 1.4 * (Zephyr_2_mi1 + Zephyr_2_mi2 + Zephyr_2_mi3) * grav_sea_level;
Zephyr_2_Isp1 = Zephyr2_params{1,8};
%----Using Trajectory Function with inputvariables above
Zephyr_1_Trajectories = Trajectory_TL_0407('Zephyr',[Zephyr_1_Cd1; Zephyr_1_Radius1; Zephyr_1_mi1; Zephyr_1_mf1; Zephyr_1_T1 ; Zephyr_1_Isp1] , [Zephyr_1_Cd2; Zephyr_1_Radius2; Zephyr_1_mi2; Zephyr_1_mf2; Zephyr_1_T2; Zephyr_1_Isp2] , [0; 0; 0; 0; 0; 0], 1, 28.50);
Zephyr_2_Trajectories = Trajectory_TL_0407('Zephyr',[Zephyr_2_Cd1; Zephyr_2_Radius1; Zephyr_2_mi1; Zephyr_2_mf1; Zephyr_2_T1 ; Zephyr_2_Isp1] , [Zephyr_2_Cd2; Zephyr_2_Radius2; Zephyr_2_mi2; Zephyr_2_mf2; Zephyr_2_T2; Zephyr_2_Isp2] , [Zephyr_2_Cd3; Zephyr_2_Radius3; Zephyr_2_mi3; Zephyr_2_mf3; Zephyr_2_T3; Zephyr_2_Isp3] , 2, 34.60);
[Minimum_Values_1 , Index_1] = min(Zephyr_1_Trajectories, [], 1);
[Minimum_Values_2 , Index_2] = min(Zephyr_2_Trajectories, [], 1);
Optimal_Zephyr_1_Result = Zephyr_1_Trajectories(Index_1(10),:);
Optimal_Zephyr_2_Result = Zephyr_2_Trajectories(Index_2(10),:);
%-------------------------------------------------------------------------
%This code runs the Trajectory function once for when the optimal
%trajectory is found and graphs it
Optimal_Zephyr_1_Trajectory = Optimal_Trajectory('Zephyr',[Zephyr_1_Cd1; Zephyr_1_Radius1; Zephyr_1_mi1; Zephyr_1_mf1; Zephyr_1_T1 ; Zephyr_1_Isp1] , [Zephyr_1_Cd2; Zephyr_1_Radius2; Zephyr_1_mi2; Zephyr_1_mf2; Zephyr_1_T2; Zephyr_1_Isp2] , [0; 0; 0; 0; 0; 0], 1, 28.50,Optimal_Zephyr_1_Result);
Optimal_Zephyr_2_Trajectory = Optimal_Trajectory('Zephyr',[Zephyr_2_Cd1; Zephyr_2_Radius1; Zephyr_2_mi1; Zephyr_2_mf1; Zephyr_2_T1 ; Zephyr_2_Isp1] , [Zephyr_2_Cd2; Zephyr_2_Radius2; Zephyr_2_mi2; Zephyr_2_mf2; Zephyr_2_T2; Zephyr_2_Isp2] , [Zephyr_2_Cd3; Zephyr_2_Radius3; Zephyr_2_mi3; Zephyr_2_mf3; Zephyr_2_T3; Zephyr_2_Isp3] , 2, 34.60,Optimal_Zephyr_2_Result);
Optimal_Zephyr_1_Trajectory = Optimal_Zephyr_1_Trajectory(1:Minimum_Values_1(8), :);
Optimal_Zephyr_2_Trajectory = Optimal_Zephyr_2_Trajectory(1:Minimum_Values_2(8), :);
scales1 = Optimal_Zephyr_1_Result(1:7);
scales2 = Optimal_Zephyr_2_Result(1:7);
plotTrajZ('Zephyr',[Zephyr_1_Cd1; Zephyr_1_Radius1; Zephyr_1_mi1; Zephyr_1_mf1; Zephyr_1_T1 ; Zephyr_1_Isp1] , [Zephyr_1_Cd2; Zephyr_1_Radius2; Zephyr_1_mi2; Zephyr_1_mf2; Zephyr_1_T2; Zephyr_1_Isp2] , [0; 0; 0; 0; 0; 0], 1, scales1);
plotTrajZ('Zephyr',[Zephyr_2_Cd1; Zephyr_2_Radius1; Zephyr_2_mi1; Zephyr_2_mf1; Zephyr_2_T1 ; Zephyr_2_Isp1] , [Zephyr_2_Cd2; Zephyr_2_Radius2; Zephyr_2_mi2; Zephyr_2_mf2; Zephyr_2_T2; Zephyr_2_Isp2] , [Zephyr_2_Cd3; Zephyr_2_Radius3; Zephyr_2_mi3; Zephyr_2_mf3; Zephyr_2_T3; Zephyr_2_Isp3] , 2, scales2);
save('Trajectory_TL_0407_Workspace.mat');
%-----------Imbeded Function ---------------------------------------------
function Optimal_Result = Optimal_Trajectory(name, step1, step2, step3, mission, launch_latitude, Optimals)
%% Earth and Launch Site Inputs
%% Earth and Launch Site Inputs
mu = 3.986e14; %m^3/s^2 Gravitational Parameter of Earth
g0 = 9.80665; %m/s^2 Local Gravity at S.L.
R_earth = 6378000; %m Radius of Earth
h0 = 7640; %m Scale Height (Wikipedia: https://en.wikipedia.org/wiki/Scale_height)
L0 = launch_latitude; %deg, N Latitude of Launch Site
rho0 = 1.225; %kg/m^3 Air Density at S.L.
v_ls = 465.1*cos(deg2rad(L0)); %m/s Speed of Launch Site
if mission == 1
final_alt = 500000; %m Final Orbit
rf = final_alt+R_earth; %m Distance btwn Final Orbit Alt and Center of Earth
inc = 60; %deg Inclination
elseif mission == 2
final_alt = 550000; %m Final Orbit
rf = final_alt+R_earth; %m Distance btwn Final Orbit Alt and Center of Earth
inc = 95; %deg Inclination
else
warning('This mission does not exist in this simulation!')
end
%% Initial Conditions
t0 = 0; %s %Initial Time
dt = 1; %s %Change in Time
gamma0 = pi/2; %rad %Initial Flight Path Angle
gamma_dot0 = 0; %rad/s %Initial Change in Flight Path Angle
drag0 = 0; %N %Initial Drag
x0 = 0; %m %Initial Downrange Distance
xdot0 = 0; %m/s %Initial Downrange Speed
h00 = 0; %m %Initial Vertical Distance (Launch Site Altitude)
hdot0 = 0; %m/s %Initial Vertical Speed
v0 = 0; %m/s %Initial Speed
q0 = 0; %Pa %Initial Dynamic Pressure
%Label_Results = ['Stage 1 Throttle', 'Stage 2 Throttle' , 'Stage 3 Throttle', 'Pitch-Kick' , 'Check', 'Delta-V to Circularize'];
%Results = (Label_Results);
Results = [];
Record = [];
%% Simulation
scale_factor_1 = Optimals(1);
scale_factor_2 = Optimals(2);
scale_factor_3 = Optimals(3);
pitch_kick = Optimals(4);
mleft_1 = Optimals(5);
mleft_2 = Optimals(6);
mleft_3 = Optimals(7);
%% Array Initialization
t = zeros(600, 1);
t(1) = t0;
gamma_dot = zeros(600, 1);
gamma_dot(1) = gamma_dot0;
gamma = zeros(600, 1);
gamma(1) = gamma0;
xdot = zeros(600, 1);
xdot(1) = xdot0;
x = zeros(600, 1);
x(1) = x0;
hdot = zeros(600, 1);
hdot(1) = hdot0;
h = zeros(600, 1);
h(1) = h00;
rho = zeros(600, 1);
rho(1) = rho0;
drag = zeros(600, 1);
drag(1) = drag0;
g = zeros(600, 1);
g(1) = g0;
v = zeros(600, 1);
v(1) = v0;
q = zeros(600, 1);
q(1) = q0;
%% LV Configuration Data
stage1_Cd = step1(1); %Stage 1 Coefficient of Drag
stage1_Radius = step1(2); %m %Stage 1 Radius
stage1_mi = step1(3)+step2(3)+step3(3); %kg %Stage 1 Initial Mass (Stage l Struct, Stage 1 Fuel, Stage 2 Struct, Stage 2 Fuel)
stage1_S = pi*stage1_Radius^2; %m^2 %Stage 1 Cross-Sectional Area
stage1_Thrust = scale_factor_1*step1(5); %N %Stage 1 Thrust
stage1_Isp = step1(6); %s %Stage 1 Isp
stage1_mdot = stage1_Thrust/(stage1_Isp*g0); %kg/s %Stage 1 Mass Flow
stage1_mf = step1(4)+step2(3)+step3(3); %kg %Stage 1 Final Mass (Stage 1 Struct, Stage 2 Fuel, Stage 2 Struct)
stage2_Cd = step2(1); %Stage 2 Coefficient of Drag
stage2_Radius = step2(2); %m %Stage 2 Radius
stage2_mi = step2(3)+step3(3); %kg %Stage 2 Initial Mass (Stage l Struct, Stage 1 Fuel, Stage 2 Struct, Stage 2 Fuel)
stage2_mf = step2(4)+step3(3); %kg %Stage 2 Final Mass (Stage 1 Struct, Stage 2 Fuel, Stage 2 Struct)
stage2_S = pi*stage2_Radius^2; %m^2 %Stage 2 Cross-Sectional Area
stage2_Thrust = scale_factor_2*step2(5); %N %Stage 2 Thrust
stage2_Isp = step2(6); %s %Stage 2 Isp
stage2_mdot = stage2_Thrust/(stage2_Isp*g0); %kg/s %Stage 2 Mass Flow
stage3_Cd = step3(1); %Stage 3 Coefficient of Drag
stage3_Radius = step3(2); %m %Stage 3 Radius
stage3_mi = step3(3); %kg %Stage 3 Initial Mass (Stage l Struct, Stage 1 Fuel, Stage 2 Struct, Stage 2 Fuel)
stage3_mf = step3(4); %kg %Stage 3 Final Mass (Stage 1 Struct, Stage 2 Fuel, Stage 2 Struct)
stage3_S = pi*stage3_Radius^2; %m^2 %Stage 3 Cross-Sectional Area
stage3_Thrust = scale_factor_3*step3(5); %N %Stage 3 Thrust
stage3_Isp = step3(6); %s %Stage 3 Isp
if stage3_mi ~= 0
stage3_mdot = stage3_Thrust/(stage3_Isp*g0); %kg/s %Stage 3 Mass Flow
else
stage3_mdot = 0; %kg/s %Stage 3 Mass Flow
end
a0 = (stage1_Thrust/stage1_mi) - g0*sin(gamma0); %m/s^2 %Initial Acceleration
a = zeros(600, 1);
a(1) = a0;
m = zeros(600, 1);
m(1) = stage1_mi;
count = 1; %Begin Count Here
while gamma(count) > deg2rad(1)
t(count+1) = t(count)+1;
if m(count) > stage1_mf+(mleft_1*(stage1_mi-stage1_mf))%The Order of calculating these terms are important
v(count+1) = v(count) + a(count)*dt;
m(count+1) = m(count) - stage1_mdot*dt;
h(count+1) = h(count) + hdot(count)*dt;
x(count+1) = x(count) + xdot(count)*dt;
if h(count) >= 400 && h(count) <= 600
gamma(count+1) = -pitch_kick + gamma(count) + gamma_dot(count)*dt;
else
gamma(count+1) = gamma(count) + gamma_dot(count)*dt;
end
rho(count+1) = rho0*exp(-h(count+1)/h0);
drag(count+1) = .5*stage1_Cd*stage1_S*rho(count+1)*v(count+1)^2;
g(count+1) = g0/((1+(h(count+1)/R_earth))^2);
gamma_dot(count+1) = -((g(count+1)/v(count+1)) - (v(count+1)/(R_earth+h(count+1))))*cos(gamma(count+1));
a(count+1) = stage1_Thrust/m(count+1) - drag(count+1)/m(count+1) - (g(count+1)*sin(gamma(count+1)));
xdot(count+1) = v(count+1)*cos(gamma(count+1))*R_earth/(R_earth+h(count+1));
hdot(count+1) = v(count+1)*sin(gamma(count+1));
q(count+1) = drag(count+1)/(stage1_Cd*stage1_S);
elseif m(count) > stage2_mi && m(count) <= stage1_mf+(mleft_1*(stage1_mi-stage1_mf))
ttemp = t(count);
for j = 1:5
v(count+1) = v(count) + a(count)*dt;
m(count+1) = stage2_mi;
h(count+1) = h(count) + hdot(count)*dt;
x(count+1) = x(count) + xdot(count)*dt;
gamma(count+1) = gamma(count) + gamma_dot(count)*dt;
rho(count+1) = rho0*exp(-h(count+1)/h0);
drag(count+1) = .5*stage2_Cd*stage2_S*rho(count+1)*v(count+1)^2;
g(count+1) = g0/((1+(h(count+1)/R_earth))^2);
gamma_dot(count+1) = -((g(count+1)/v(count+1))-(v(count+1)/(R_earth+h(count+1))))*cos(gamma(count+1));
a(count+1) = (-drag(count+1))/m(count+1) - (g(count+1)*sin(gamma(count+1)));
xdot(count+1) = v(count+1)*cos(gamma(count+1))*R_earth/(R_earth+h(count+1));
hdot(count+1) = v(count+1)*sin(gamma(count+1));
q(count+1) = drag(count)/(stage2_Cd*stage2_S);
t(count+1) = ttemp + j;
end
elseif m(count) > stage2_mf+(mleft_2*(stage2_mi-stage2_mf))
v(count+1) = v(count) + a(count)*dt;
m(count+1) = m(count) - stage2_mdot*dt;
h(count+1) = h(count) + hdot(count)*dt;
x(count+1) = x(count) + xdot(count)*dt;
gamma(count+1) = gamma(count) + gamma_dot(count)*dt;
rho(count+1) = rho0*exp(-h(count+1)/h0);
drag(count+1) = .5*stage2_Cd*stage2_S*rho(count+1)*v(count+1)^2;
g(count+1) = g0/((1+(h(count+1)/R_earth))^2);
gamma_dot(count+1) = -((g(count+1)/v(count+1))-(v(count+1)/(R_earth+h(count+1))))*cos(gamma(count+1));
a(count+1) = (stage2_Thrust-drag(count+1))/m(count+1) - (g(count+1)*sin(gamma(count+1)));
xdot(count+1) = v(count+1)*cos(gamma(count+1))*R_earth/(R_earth+h(count+1));
hdot(count+1) = v(count+1)*sin(gamma(count+1));
q(count+1) = drag(count+1)/(stage2_Cd*stage2_S);
elseif m(count) > stage3_mi && m(count) <= stage2_mf && stage3_mi ~= 0
ttemp = t(count);
for j = 1:5
v(count+1) = v(count) + a(count)*dt;
m(count+1) = stage3_mi;
h(count+1) = h(count) + hdot(count)*dt;
x(count+1) = x(count) + xdot(count)*dt;
gamma(count+1) = gamma(count) + gamma_dot(count)*dt;
rho(count+1) = rho0*exp(-h(count+1)/h0);
drag(count+1) = .5*stage3_Cd*stage3_S*rho(count+1)*v(count+1)^2;
g(count+1) = g0/((1+(h(count+1)/R_earth))^2);
gamma_dot(count+1) = -((g(count+1)/v(count+1))-(v(count+1)/(R_earth+h(count+1))))*cos(gamma(count+1));
a(count+1) = (-drag(count+1))/m(count+1) - (g(count+1)*sin(gamma(count+1)));
xdot(count+1) = v(count+1)*cos(gamma(count+1))*R_earth/(R_earth+h(count+1));
hdot(count+1) = v(count+1)*sin(gamma(count+1));
q(count+1) = drag(count+1)/(stage3_Cd*stage3_S);
t(count+1) = ttemp + j;
end
elseif m(count) > stage3_mf+(mleft_3*(stage3_mi-stage3_mf)) && stage3_mi ~= 0
v(count+1) = v(count) + a(count)*dt;
m(count+1) = m(count) - stage3_mdot*dt;
h(count+1) = h(count) + hdot(count)*dt;
x(count+1) = x(count) + xdot(count)*dt;
gamma(count+1) = gamma(count) + gamma_dot(count)*dt;
rho(count+1) = rho0*exp(-h(count+1)/h0);
drag(count+1) = .5*stage3_Cd*stage3_S*rho(count+1)*v(count+1)^2;
g(count+1) = g0/((1+(h(count+1)/R_earth))^2);
gamma_dot(count+1) = -((g(count+1)/v(count+1))-(v(count+1)/(R_earth+h(count+1))))*cos(gamma(count+1));
a(count+1) = (stage3_Thrust-drag(count+1))/m(count+1) - (g(count+1)*sin(gamma(count+1)));
xdot(count+1) = v(count+1)*cos(gamma(count+1))*R_earth/(R_earth+h(count+1));
hdot(count+1) = v(count+1)*sin(gamma(count+1));
q(count+1) = drag(count)/(stage3_Cd*stage3_S);
else
v(count+1) = v(count) + a(count)*dt;
m(count+1) = m(count);
h(count+1) = h(count) + hdot(count)*dt;
x(count+1) = x(count) + xdot(count)*dt;
gamma(count+1) = gamma(count) + gamma_dot(count)*dt;
rho(count+1) = rho0*exp(-h(count+1)/h0);
drag(count+1) = 0;
g(count+1) = g0/((1+(h(count+1)/R_earth))^2);
gamma_dot(count+1) = -((g(count+1)/v(count+1))-(v(count+1)/(R_earth+h(count+1))))*cos(gamma(count+1));
a(count+1) = -(g(count+1)*sin(gamma(count+1)));
xdot(count+1) = v(count+1)*cos(gamma(count+1))*R_earth/(R_earth+h(count+1));
hdot(count+1) = v(count+1)*sin(gamma(count+1));
q(count+1) = 0;
end
if count >= 3000
break
else
count = count + 1;
end
end
%% Hohmann Transfer
r_periapsis = h(count) + R_earth;
r_apoapsis = rf;
eccentricity_transfer = (r_apoapsis - r_periapsis)/(r_apoapsis + r_periapsis);
if inc < 90
delta_v_circularization = sqrt(mu/r_periapsis) - (v(count) + v_ls);
else
delta_v_circularization = sqrt(mu/r_periapsis) - (v(count) - v_ls);
end
delta_v_1 = sqrt(mu/r_periapsis) * (sqrt((2*r_apoapsis)/(r_periapsis+r_apoapsis))-1);
delta_v_2 = sqrt(mu/r_apoapsis) * (1-sqrt((2*r_periapsis)/(r_periapsis+r_apoapsis)));
delta_v_total = delta_v_1 + delta_v_2 + delta_v_circularization;
Optimal_Result = [t,v,m,h,x,gamma,a,q];
end