Home » Posts tagged 'Aerospace Engineering'
Tag Archives: Aerospace Engineering
AGI STK 10 MATLAB INTERFACE: Satellite Ground Track
close all; clear all; clc
AGI STK, is as a software package from Analytical Graphics, Inc.(AGI) that allows to perform complex analyses of ground, sea, air, and space missions. More information can be found in AGI website. https://www.agi.com To create this code we used educational code samples from % https://www.agi.com/resources/ % Establish the connection AGI STK 10 try % Grab an existing instance of STK 10 uiapp = actxGetRunningServer('STK10.application'); catch % STK is not running, launch new instance % Launch a new instance of STK10 and grab it uiapp = actxserver('STK10.application'); end %get the root from the personality %it has two... get the second, its the newer STK Object Model Interface as %documented in the STK Help root = uiapp.Personality2; % set visible to true (show STK GUI) uiapp.visible = 1; % From the STK Object Root you can command every aspect of the STK GUI % close current scenario or open new one try root.CloseScenario(); root.NewScenario('SatelliteGroundTrack'); catch root.NewScenario('SatelliteGroundTrack'); end % Set units to utcg before setting scenario time period and animation period root.UnitPreferences.Item('DateFormat').SetCurrentUnit('UTCG'); % %set units to epoch seconds because this works the easiest in matlab % root.UnitPreferences.Item('DateFormat').SetCurrentUnit('EPSEC'); % Set scenario time period and animation period root.CurrentScenario.SetTimePeriod('25 May 2013 12:00:00.000', '26 May 2013 12:00:00.000'); root.CurrentScenario.Epoch = '25 May 2013 12:00:00.000'; % Create satellite satObj = root.CurrentScenario.Children.New('eSatellite', 'SmallSats1'); % Propagate satellite satObj.Propagator.InitialState.Representation.AssignClassical(... 'eCoordinateSystemJ2000', 6750, 0.1, 53.4, 0, 0, 0); % CoordinateSystem, Semimajor Axis, Eccentricity, Inclination, % Arg. of Perigee, RAAN, Mean Anomaly satObj.Propagator.StartTime = '25 May 2013 12:00:00.000'; satObj.Propagator.StopTime = '25 May 2013 15:00:00.000'; satObj.Propagator.Propagate; % Get Latitude, Longitude for the satellite over the course of the mission. LLAState = satObj.DataProviders.Item('LLA State').Group.Item('Fixed'); Elems = {'Time';'Lat';'Lon'}; satStartTime = satObj.Propagator.EphemerisInterval.FindStartTime; satStopTime = satObj.Propagator.EphemerisInterval.FindStopTime; Results = LLAState.ExecElements(satStartTime, satStopTime, 10, Elems); time = cell2mat(Results.DataSets.GetDataSetByName('Time').GetValues); Lat = cell2mat(Results.DataSets.GetDataSetByName('Lat').GetValues); Long = cell2mat(Results.DataSets.GetDataSetByName('Lon').GetValues);
Plot
figure(1); hold on; axis([0 360 -90 90]); load('topo.mat','topo','topomap1'); contour(0:359,-89:90,topo,[0 0],'b') axis equal box on set(gca,'XLim',[-180 180],'YLim',[-90 90], ... 'XTick',[-180 -120 -60 0 60 120 180], ... 'Ytick',[-90 -60 -30 0 30 60 90]); image([-180 180],[-90 90],topo,'CDataMapping', 'scaled'); colormap(topomap1); ylabel('Latitude [deg]'); xlabel('Longitude [deg]'); title('Satellite Ground Track'); scatter(Long,Lat,'.r');
Ground track generated by AGI STK10
Relative Motion of Satellites, Numerical Simulation
This example shows how to use the state vectors of spacecraft A and B to find the position, velocity and acceleration of B relative to A in the LVLH frame attached to A. We numerically solve fundamental equation of relative two-body motion to obtain the trajectory of spacecraft B relative to spacecraft A and the distance between two satellites. For more details about the theory and algorithm look at Chapter 7 of H. D. Curtis, Orbital Mechanics for Engineering Students,Second Edition, Elsevier 2010
clc; clear all; % Initial State vectors of Satellite A and B RA0 = [-266.77, 3865.8, 5426.2]; % [km] VA0 = [-6.4836, -3.6198, 2.4156]; % [km/s] RB0 = [-5890.7, -2979.8, 1792.2]; % [km] VB0 = [0.93583, -5.2403, -5.5009]; % [km/s] mu = 398600; % Earth’s gravitational parameter [km^3/s^2] t = 0; % initial time dt = 15; % Simulation time step [s] dT = 4*24*3600; % Simulation interval [s] % Using fourth-order Runge–Kutta method to solve fundamental equation % of relative two-body motion F_r = @(R) -mu/(norm(R)^3)*R; VA = VA0; RA = RA0; VB = VB0; RB = RB0; ind = 1; while (t <= dT) % Relative position hA = cross(RA, VA); % Angular momentum of A % Unit vectors i, j,k of the co-moving frame i = RA/norm(RA); k = hA/norm(hA); j = cross(k,i); % Transformation matrix Qxx: QXx = [i; j; k]; Om = hA/norm(RA)^2; Om_dt = -2*VA*RA'/norm(RA)^2.*Om; % Accelerations of A and B,inertial frame aA = -mu*RA/norm(RA)^3; aB = -mu*RB/norm(RB)^3; % Relative position,inertial frame Rr = RB - RA; % Relative position,LVLH frame attached to A R_BA(ind,:) = QXx*Rr'; % A Satellite k_1 = dt*F_r(RA); k_2 = dt*F_r(RA+0.5*k_1); k_3 = dt*F_r(RA+0.5*k_2); k_4 = dt*F_r(RA+k_3); VA = VA + (1/6)*(k_1+2*k_2+2*k_3+k_4); RA = RA + VA*dt; % B Satellite k_1 = dt*F_r(RB); k_2 = dt*F_r(RB+0.5*k_1); k_3 = dt*F_r(RB+0.5*k_2); k_4 = dt*F_r(RB+k_3); VB = VB + (1/6)*(k_1+2*k_2+2*k_3+k_4); RB = RB + VB*dt; R_A(ind,:) = RA; R_B(ind,:) = RB; time(ind) = t; t = t+dt; ind = ind+1; end r_BA = (R_BA(:,1).^2+R_BA(:,2).^2+R_BA(:,3).^2).^0.5;
close all; figure(1); hold on; plot3(R_A(:,1),R_A(:,2),R_A(:,3),'r'); plot3(R_B(:,1),R_B(:,2),R_B(:,3),'y'); title('Satellites orbits around earth'); legend('Satellite A','Satellites B'); xlabel('km');ylabel('km'); % Plotting Earth load('topo.mat','topo','topomap1'); colormap(topomap1); % Create the surface. radius_earth=6378; [x,y,z] = sphere(50); x =radius_earth*x; y =radius_earth*y; z =radius_earth*z; props.AmbientStrength = 0.1; props.DiffuseStrength = 1; props.SpecularColorReflectance = .5; props.SpecularExponent = 20; props.SpecularStrength = 1; props.FaceColor= 'texture'; props.EdgeColor = 'none'; props.FaceLighting = 'phong'; props.Cdata = topo; surface(x,y,z,props); hold off; figure(2); plot3(R_BA(:,1),R_BA(:,2),R_BA(:,3),'k'); title('The trajectory of spacecraft B relative to spacecraft A'); xlabel('km');ylabel('km');zlabel('km'); figure(3); plot(time/3600,r_BA); title('Distance between two satellites'); xlabel('hour');ylabel('km') min_r = min(r_BA); max_r = max(r_BA); fprintf('Max distance between two satellites %6.4f km \n',max_r); fprintf('Min distance between two satellites %6.4f km \n',min_r);
Max distance between two satellites 13850.3054 km Min distance between two satellites 262.0271 km
Relative Motion of Satellites
clc; clear all; close all;
% This example shows how to use the state vectors of spacecraft A and B % to find the position, velocity and acceleration of B relative % to A in the LVLH frame attached to A. For more details about the theory and % algorithm look at Chapter 7 of H. D. Curtis, Orbital Mechanics for % Engineering Students,Second Edition, Elsevier 2010
Input
State vectors of Satellite A and B
RA = [-266.77, 3865.8, 5426.2]; % [km] VA = [-6.4836, -3.6198, 2.4156]; % [km/s] RB = [-5890.7, -2979.8, 1792.2]; % [km] VB = [0.93583, -5.2403, -5.5009]; % [km/s] % Earth gravitational parameter mu = 398600; % [km^3/s^2]
Algorithm
hA = cross(RA, VA); % Angular momentum of A % Unit vectors i, j,k of the co-moving frame i = RA/norm(RA); k = hA/norm(hA); j = cross(k,i); % Transformation matrix Qxx: QXx = [i; j; k]; Om = hA/norm(RA)^2; Om_dt = -2*VA*RA'/norm(RA)^2.*Om; % Accelerations of A and B,inertial frame aA = -mu*RA/norm(RA)^3; aB = -mu*RB/norm(RB)^3; % Relative position,inertial frame Rr = RB - RA; % Relative position,LVLH frame attached to A R_BA = QXx*Rr'; % Relative velocity,inertial frame Vr = VB - VA - cross(Om,Rr); % Relative velocity,LVLH frame attached to A V_BA = QXx*Vr'; % Relative acceleration, inertial frame ar = aB - aA - cross(Om_dt,Rr) - cross(Om,cross(Om,Rr))- 2*cross(Om,Vr); % Relative acceleration,LVLH frame attached to A a_BA = QXx*ar'; fprintf('Position of B relative to A in LVLH frame attached to A \n'); fprintf('R_BA = [%4.2f %4.2f %4.2f] km \n\n', R_BA); fprintf('Velocity of B relative to A in LVLH frame attached to A \n'); fprintf('V_BA = [%6.4f %6.4f %6.4f] km/s \n\n', V_BA); fprintf('Acceleration of B relative to A in LVLH frame attached to A \n'); fprintf('a_BA = [%8.8f %8.8f %8.8f] km/s^2 \n', a_BA);
Position of B relative to A in LVLH frame attached to A R_BA = [-6701.22 6828.28 -406.24] km Velocity of B relative to A in LVLH frame attached to A V_BA = [0.3168 0.1120 1.2470] km/s Acceleration of B relative to A in LVLH frame attached to A a_BA = [-0.00022213 -0.00018083 0.00050590] km/s^2
Orbital Inclination Change
Transfer from a LEO 350 km circular orbit with 53.4 deg inclination to a Geostationary Equatorial Orbit(GEO)
clear all; clc; close all;
Input
R_LEO = 6378 + 350; % km R_GEO = 42164; % km mu = 398600; % km^3/s^2 incl = 53.4; % deg
Method 1: Hohman transfer from LEO to GEO and after inclination change
Rp = R_LEO; Ra = R_GEO; e = (Ra - Rp)/(Ra + Rp); % transfer orbit eccentricity a = (Ra + Rp)/2; % transfer orbit semimajor axis V_LEO = (mu/R_LEO)^0.5; Vp = (2*mu/R_LEO - mu/a)^0.5; Va = (2*mu/R_GEO - mu/a)^0.5; V_GEO = (mu/R_GEO)^0.5; dV_LEO = abs(Vp - V_LEO); dV_GEO = abs(V_GEO - Va); dV_Hoff = dV_GEO + dV_LEO; % Inclination change at GEO dV_incl = 2*V_GEO*sind(incl/2); dV_total1 = dV_incl + dV_Hoff; fprintf('Method 1: Hohman transfer from LEO to GEO and after inclination change\n'); fprintf('dV_Hoff = %6.2f [km/s] \n',dV_Hoff); fprintf('dV_incl = %6.2f [km/s] \n',dV_incl); fprintf('dV_total = %6.2f [km/s] \n',dV_total1);
Method 1: Hohman transfer from LEO to GEO and after inclination change dV_Hoff = 3.87 [km/s] dV_incl = 2.76 [km/s] dV_total = 6.64 [km/s]
Method 2: Inclination change in LEO and after Hohman transfer to GEO
dV_incl = 2*V_LEO*sind(incl/2); dV_total2 = dV_incl + dV_Hoff; fprintf('\nMethod 2: Inclination change in LEO and after Hohman transfer to GEO\n'); fprintf('dV_incl = %6.2f [km/s] \n',dV_incl); fprintf('dV_Hoff = %6.2f [km/s] \n',dV_Hoff); fprintf('dV_total = %6.2f [km/s] \n\n',dV_total2); fprintf('Method 2/Method 1 = %6.2f \n',dV_total2/dV_total1);
Method 2: Inclination change in LEO and after Hohman transfer to GEO dV_incl = 6.92 [km/s] dV_Hoff = 3.87 [km/s] dV_total = 10.79 [km/s] Method 2/Method 1 = 1.63
Kepler Equation Solver Without Transcendental Function Evaluations
clc;clear all;
tic; j = 1; for M = 0:0.005:pi i = 1; for e = 0:0.005:1 c3 = 5/2 + 560*e; a = 15/c3*(1-e); b = -M/c3; y = (b^2/4+ a^3/27)^0.5; x = (-b/2 + y)^(1/3) - (b/2 + y)^(1/3); c15 = 3003/14336 + 16384*e; c13 = 3465/13312 - 61440*e; c11 = 945/2816 + 92160*e; c9 = 175/384 -70400*e; c7 = 75/112 + 28800*e; c5 = 9/8-6048*e; x2 = x^2;x3 = x2*x;x4 = x3*x; x5 = x4*x;x6 = x5*x;x7 = x6*x; x8 = x7*x;x9 = x8*x; x10 =x9*x; x11 = x10*x; x12 = x11*x; x13 = x12*x; x14 = x13*x; x15 = x14*x; f = c15*x15 + c13*x13 + c11*x11 + c9*x9 + c7*x7 + c5*x5 + c3*x3 +... 15*(1 - e)*x -M; f1 = 15* c15 *x14 + 13* c13 *x12 + 11* c11 *x10 + 9* c9 *x8 + 7* c7 *x6 + ... 5 *c5 *x4 + 3 *c3 *x2 + 15*(1 - e); f2 = 210* c15 *x13 + 156* c13 *x11 + 110* c11 *x9 + 72* c9 *x7 + ... 42* c7 *x5 + 20* c5 *x3 + 6* c3 *x; f3 = 2730*c15 *x12 + 1716* c13 *x10 + 990* c11 *x8 + 504* c9 *x6 +... 210* c7 *x4 + 60* c5 *x2 + 6* c3; f4 = 32760* c15 *x11 + 17160* c13 *x9 + 7920* c11 *x7 + 3024* c9 *x5 +... 840* c7 *x3 + 120* c5 *x; f5 = 360360* c15 *x10 + 154440* c13 *x8 + 55440* c11 *x6 + 15120* c9 *x4 ... + 2520* c7 *x2 + 120* c5; f6 = 3603600* c15 *x9 + 1235520* c13 *x7 + 332640* c11 *x5 + 60480* c9 *x3... + 5040* c7 *x; f7 = 32432400* c15 *x8 + 8648640* c13 *x6 + 1663200* c11 *x4 + 181440* c9 *x2 ... + 5040* c7; f8 = 25945920* c15 *x7 + 51891840* c13 *x5 + 6652800* c11 *x3 + ... 362880* c9 *x; f9 = 1.8162144e9* c15 *x6 + 259459200* c13 *x4 + 19958400* c11 *x2 + ... 362880* c9; f10 = 1.08972864e10* c15 *x5 + 1.0378368e9*c13 *x3 + 39916800* c11 *x; f11 = 5.4486432e10* c15 *x4 + 3.1135104e9* c13 *x2 + 39916800* c11; f12 = 2.17945728e11* c15 *x3 + 6.2270208e9* c13 *x; f13 = 6.53837184e11* c15 *x2 + 6.2270208e9* c13; f14 = 1.307674368e13* c15 *x; f15 = 1.307674368e13* c15; g1 = 1/2; g2 = 1/6; g3 = 1/24; g4 = 1/120; g5 = 1/720; g6 = 1/5040; g7 = 1/40320; g8 = 1/362880; g9 = 1/3628800; g10 = 1/39916800; g11 = 1/479001600; g12 =1/6.2270208e9 ; g13 = 1/8.71782912e10; g14 = 1/1.307674368e12; u1 = -f/f1; h2 = f1 + g1* u1* f2; u2 = -f/h2; h3 = f1 + g1* u2* f2 + g2* u2^2*f3; u3 = -f/h3; h4 = f1 + g1* u3* f2 + g2* u3^2*f3 + g3* u3^3*f4; u4 = -f/h4; h5 = f1 + g1* u4* f2 + g2* u4^2*f3 + g3* u4^3*f4 + g4* u4^4*f5; u5 = -f/h5; h6 = f1 + g1* u5* f2 + g2* u5^2*f3 + g3* u5^3*f4 + g4* u5^4*f5 +... g5* u5^5*f6; u6 = -f/h6; h7 = f1 + g1* u6* f2 + g2* u6^2*f3 + g3* u6^3*f4 + g4* u6^4*f5 +... g5* u6^5*f6 + g6* u6^6*f7; u7 = -f/h7; h8 = f1 + g1* u7* f2 + g2* u7^2*f3 + g3* u7^3*f4 + g4* u7^4*f5 +... g5* u7^5*f6 + g6* u7^6*f7 + g7* u7^7*f8; u8 = -f/h8; h9 = f1 + g1* u8* f2 + g2* u8^2*f3 + g3* u8^3*f4 + g4* u8^4*f5 +... g5* u8^5*f6 + g6* u8^6*f7+ g7* u8^7*f8 + g8* u8^8*f9; u9 = - f/h9; h10 = f1 + g1* u9* f2 + g2* u9^2*f3 + g3* u9^3*f4 + g4* u9^4*f5 +... g5* u9^5*f6 + g6* u9^6*f7+ g7* u9^7*f8 + g8* u9^8*f9 + ... g9* u9^9*f10; u10 = -f/h10; h11 = f1 + g1* u10* f2 + g2* u10^2* f3 + g3* u10^3* f4 + g4* u10^4*f5... + g5* u10^5* f6 + g6* u10^6*f7+ g7* u10^7* f8 + g8* u10^8* f9... + g9* u10^9* f10 + g10* u10^10* f11; u11 = -f/h11; h12 = f1 + g1* u11* f2 + g2* u11^2* f3 + g3* u11^3* f4 + ... g4* u11^4* f5 + g5* u11^5* f6 + g6* u11^6* f7... + g7* u11^7* f8 + g8* u11^8* f9 + g9* u11^9* f10 + g10* u11^10* f11... + g11* u11^11* f12; u12 = -f/h12; h13 = f1 + g1* u12* f2 + g2* u12^2* f3 + g3* u12^3* f4 +... g4* u12^4* f5 + g5* u12^5* f6 + g6* u12^6* f7... + g7* u12^7* f8 + g8* u12^8* f9 + g9* u12^9* f10 +... g10* u12^10* f11 + g11* u12^11* f12 + g12* u12^12* f13; u13 = -f/h13; h14 = f1 + g1* u13* f2 + g2* u13^2* f3 + g3* u13^3* f4 + ... g4* u13^4* f5 + g5* u13^5* f6 + g6* u13^6* f7... + g7* u13^7* f8 + g8* u13^8* f9 + g9* u13^9* f10 + g10* u13^10*f11... + g11* u13^11* f12+ g12* u13^12* f13 + g13* u13^13* f14; u14 = -f/h14; h15 = f1 + g1* u14* f2 + g2* u14^2* f3 + g3* u14^3* f4 + ... g4* u14^4* f5 + g5* u14^5* f6 + g6* u14^6* f7... + g7* u14^7* f8 + g8* u14^8* f9 + g9* u14^9* f10 +... g10* u14^10* f11 + g11*u14^11* f12+ g12* u14^12* f13 + ... g13* u14^13* f14 + g14* u14^14* f15; u15 = -f/h15; x = x + u15; w = x - 0.01171875*x^17/(1 + e); E = M + e*(-16384*w^15 + 61440*w^13 - 92160*w^11+ 70400*w^9 -... 28800*w^7 + 6048*w^5 - 560*w^3 + 15*w); E_nit(i,j) = E; i = i + 1; end j = j + 1; end t_nit = toc; fprintf('Calculation time = %4.6f [s] \n',t_nit);
Calculation time = 2.591335 [s]
Newton Iterative Solver
tic; j = 1; eps = 1E-15; % Tolerance for M = 0:0.005:pi i = 1; for e = 0:0.005:1 En = M; Ens = En - (En-e*sin(En)- M)/(1 - e*cos(En)); while ( abs(Ens-En) > eps ) En = Ens; Ens = En - (En - e*sin(En) - M)/(1 - e*cos(En)); end; E_it(i,j) = Ens; i = i + 1; end j = j + 1; end t_it = toc; fprintf('Calculation time = %4.6f [s] \n',t_it);
Calculation time = 3.520123 [s]
Error = log10(abs(E_it - E_nit)); e = 0:0.005:1; M = (0:0.005:pi); contourf (M,e,Error); xlabel('M - Mean anomaly [rad]'); ylabel('e - Eccentricity'); title('Error in E calculation on Log scale [rad] '); colorbar; REFERENCES Adonis Pimienta-Pe˜nalver and John L. Crassidis†,ACCURATE KEPLER EQUATION SOLVER WITHOUT TRANSCENDENTAL FUNCTION EVALUATIONS, University at Buffalo, State University of New York, Amherst, NY, 14260-4400
Small Satellites 