Orbital Rendezvous

Shuen
/
Orbital Rendezvous
on 10 Nov 2026
7
92
34
3
8992
game.m
function g = game()

g.init   = @initState;
g.update = @updateState;
g.render = @renderFrame;
end

function S = initState(ax)
%% Scales & mission parameters
S.R0  = 7178;  S.V0 = sqrt(398600/S.R0);     % target orbit radius [km], speed [km/s]
S.Rp  = 6378/S.R0;  S.Ra = (6378+120)/S.R0;  % planet, atmosphere interface 
S.aT  = 0.055;                               % thrust acceleration
S.ts  = 2*pi/8;                              % canonical time per game second (8 s/orbit)
S.DVb = 600/(S.V0*1e3);                      % delta-v budget: 600 m/s
S.gR  = 200/S.R0; S.gV = 50/(S.V0*1e3);      % rendezvous gate: 200 km, 50 m/s
S.h   = 2e-3;                                % RK4 step
S.Tlim = 20*2*pi;                            % time limit: 20 orbits

%% Pixel canvas & palette
W = 200; H = 150; cx = 125; cy = 76;         % canvas size, planet centre [px]
S.cx = cx; S.cy = cy;                        % radial offsets from target orbit drawn x3 (rD)
pal = [1 .97 .90; .20 .20 .28; .55 .55 .60; .25 .50 .95; .82 .89 .97; 1 .90 .40; ...
       1 1 1; .95 .38 .32; .10 .45 .90; .97 .55 .10; .47 .80 .76; .30 .62 .60; ...
       .98 .84 .50; .96 .60 .70; .20 .28 .55; 1 .80 .15; .97 .62 .57];
% 1 paper 2 ink 3 grey 4 panel 5 grid 6 window 7 white 8 chaser 9 prograde 10 retrograde 11 ocean 12 ocean-dk 13 land 14 atmosphere 15 target orbit 16 plume 17 predicted orbit
S.pal = pal;
[xx,yy] = meshgrid(1:W,1:H);  X = xx-cx;  Y = cy-yy;  RR = hypot(X,Y);  TH = atan2(Y,X);
B = ones(H,W);
B((mod(xx,10)==0 & mod(yy,2)==0) | (mod(yy,10)==0 & mod(xx,2)==0)) = 5;  % graph paper
Pr = rD(S.Rp);  pl = RR <= Pr;
B(pl) = 11;
B(pl & sin(3*TH + 2*cos(2*TH)).*cos(RR/4) > .4) = 13;      % continents
B(pl & hypot(X+5,Y-5) > Pr) = 12;                          % night-side shading
S.B = B;

% Space station, small crew capsule, burn marker
S.Sst = ['44.......44'
         '44..222..44'
         '44332623344'
         '44..222..44'
         '44.......44'];
S.Sch = ['..2..'
         '.282.'
         '28682'
         '.222.'
         '.3.3.'];
S.Sx  = ['2.2'; '.2.'; '2.2'];
S.q = linspace(0,2*pi,600);  S.cq = cos(S.q);  S.sq = sin(S.q);

%% Graphics objects
S.im = image(ax,'CData',B);
colormap(ax,pal);
ax.XLim = [0.5 W+0.5];  ax.YLim = [0.5 H+0.5];  ax.YDir = 'reverse';  ax.Color = pal(1,:);
% Title: single label on an ink plate
text(ax,W/2,1.5,'ORBITAL RENDEZVOUS','Color',pal(10,:),'BackgroundColor',pal(5,:), ...
    'EdgeColor',pal(10,:),'LineWidth',1.5,'Margin',4, ...
    'FontName','Courier','FontSize',18,'FontWeight','bold', ...
    'HorizontalAlignment','center','VerticalAlignment','top','Clipping','off','HitTest','off');
% Lines: circles, predicted orbit, burn arrow
ring = @(rr) {cx + rr*S.cq, cy - rr*S.sq};
c = ring(Pr);        plot(ax,c{:},'-', 'Color',pal(2,:), 'LineWidth',1.5);    % planet outline
c = ring(rD(S.Ra));  plot(ax,c{:},'--','Color',pal(3,:), 'LineWidth',1);      % atmosphere interface
c = ring(40);        plot(ax,c{:},':', 'Color',pal(15,:),'LineWidth',1.5);    % target orbit
S.lO = plot(ax,NaN,NaN,'-','Color',pal(17,:),'LineWidth',2);                  % predicted orbit
S.lB = plot(ax,NaN,NaN,'-','LineWidth',2.5);                                  % burn arrow

spr = @(Zs) image(ax,'CData',max(double(Zs)-48,1),'AlphaData',double(Zs~='.'),'Visible','off');
S.iX  = spr(S.Sx);                                                            % last-burn marker
S.lP  = plot(ax,NaN,NaN,'-','Color',pal(16,:),'LineWidth',4);                 % exhaust plume
S.iSt = spr(S.Sst);                                                           % station
S.iCh = spr(S.Sch);                                                           % chaser
S.hH  = text(ax,3,3,'','Color',pal(2,:),'FontName','Courier','FontSize',11, ...
             'FontWeight','bold','VerticalAlignment','top','BackgroundColor',pal(1,:),'EdgeColor',pal(2,:),'Margin',3);
S.hB  = text(ax,100,148,'','FontName','Courier','FontSize',14,'FontWeight','bold', ...
             'HorizontalAlignment','center','VerticalAlignment','bottom');
S.hM  = text(ax,44,92,'','Color',pal(2,:),'FontName','Courier','FontSize',11, ...
             'FontWeight','bold','HorizontalAlignment','center', ...
             'BackgroundColor',pal(1,:),'EdgeColor',pal(2,:),'Margin',5);
S.hAp = text(ax,0,0,'Ap','Color',pal(8,:),'FontName','Courier','FontSize',11, ...
             'FontWeight','bold','HorizontalAlignment','center');
S.hPe = text(ax,0,0,'Pe','Color',pal(8,:),'FontName','Courier','FontSize',11, ...
             'FontWeight','bold','HorizontalAlignment','center');
S.st = 0;  S = newGame(S);
end

function S = updateState(S, dt, pressed)
S.u = 0;
if S.st == 0                                         % title screen
  if pressed, S.st = 1; end
elseif S.st == 1                                     % flight
  if pressed
    if S.on, S.on = 0; S.nx = -S.nx;                 % tap: cut engine, alternate
    else,    S.on = 1; S.d = S.nx; S.bp = S.s(1:2);  % tap: burn starts
    end
  end
  if S.on && S.dvU >= S.DVb, S.on = 0; S.nx = -S.nx; end   
  u = S.d*S.on;  S.u = u;
  S.acc = S.acc + dt*S.ts*(1 - .75*(u~=0));          % slow-mo while burning
  n = floor(S.acc/S.h);  S.acc = S.acc - n*S.h;
  h = S.h;  aT = S.aT;
  for k = 1:n                                        % RK4 two-body + thrust
    s = S.s;
    k1 = fd(s,u,aT); k2 = fd(s+h/2*k1,u,aT); k3 = fd(s+h/2*k2,u,aT); k4 = fd(s+h*k3,u,aT);
    S.s = s + h/6*(k1+2*k2+2*k3+k4);  S.t = S.t + h;  S.dvU = S.dvU + abs(u)*aT*h;
    S = tgt(S);
    if S.dR < S.gR && S.dV < S.gV, S.st = 2; S.res = 'RENDEZVOUS - DOCKING GATE REACHED'; break; end
    if norm(S.s(1:2)) < S.Ra,      S.st = 2; S.res = 'BURNED UP IN THE ATMOSPHERE'; break; end
    if S.t > S.Tlim,               S.st = 2; S.res = 'OUT OF TIME'; break; end
  end
  if S.st == 2
    S.on = 0;  S.u = 0;
  end
elseif pressed                                       
  S = newGame(S);  S.st = 1;
end
end

function renderFrame(~, S)
pal = S.pal;  cx = S.cx;  cy = S.cy;  R0 = S.R0;  V0 = S.V0;
%% Message box
if S.st == 0
  S.hM.String = sprintf(['ORBITAL-RENDEZVOUS\nTAP = START BURN, TAP = CUT\n' ...
    'BURNS ALTERNATE: RETRO, PRO, RETRO ...\nREACH STATION: <200 km, <50 m/s\nTAP TO START']);
elseif S.st == 2
  S.hM.String = sprintf('%s\ndV USED %.0f m/s  TIME %.1f ORBITS\nTAP TO RETRY', ...
                        S.res, S.dvU*V0*1e3, S.t/(2*pi));
end
S.hM.Visible = S.st ~= 1;

%% Chaser orbit elements -> predicted orbit, Ap/Pe, phase gain per rev
r = S.s(1:2);  v = S.s(3:4);  rn = norm(r);  hh = r(1)*v(2) - r(2)*v(1);
ev = (v'*v - 1/rn)*r - (r'*v)*v;  e = norm(ev);  om = atan2(ev(2),ev(1));
sma = 1/(2/rn - v'*v);  pa = sma*(1-e);  aa = sma*(1+e);
if e < .95 && sma > 0
  dd = rD(hh^2./(1 + e*cos(S.q-om)));
  set(S.lO,'XData',cx + dd.*S.cq,'YData',cy - dd.*S.sq,'Visible','on');   % predicted orbit
  S.hPe.Position = [cx+(rD(pa)-6)*cos(om), cy-(rD(pa)-6)*sin(om), 0];
  S.hAp.Position = [cx-(rD(aa)+6)*cos(om), cy+(rD(aa)+6)*sin(om), 0];
  set([S.hPe S.hAp],'Visible',e > 2e-3);
  gain = 360*(1 - sma^1.5);                          % deg gained per rev
else
  set([S.hPe S.hAp S.lO],'Visible','off');
  gain = NaN;
end

%% Sprites: last-burn marker, station, burn arrow, plume, chaser
if ~isnan(S.bp(1)), place(S.iX,px(S.bp,cx,cy)); else, S.iX.Visible = 'off'; end
place(S.iSt,px([cos(S.ph); sin(S.ph)],cx,cy));
pc = px(r,cx,cy);  vh = v/norm(v);
dn = S.on*S.d + ~S.on*S.nx;                          % burn direction shown
cl = 9*(dn > 0) + 10*(dn < 0);
L = [4 13];  ls = {':','-'};                         
set(S.lB,'XData',pc(2) + L*dn*vh(1),'YData',pc(1) - L*dn*vh(2), ...
    'Color',pal(cl,:),'LineStyle',ls{S.on+1});
if S.u ~= 0                                          % exhaust opposite thrust
  L = [2 4+3*rand];
  set(S.lP,'XData',pc(2) - L*S.d*vh(1),'YData',pc(1) + L*S.d*vh(2),'Visible','on');
else
  S.lP.Visible = 'off';
end
place(S.iCh,pc);

%% HUD
if dn > 0, lab = {'PROGRADE >>','SPEED UP: RAISE ORBIT, LOSE PHASE'};
else,      lab = {'RETROGRADE <<','SLOW DOWN: LOWER ORBIT, GAIN PHASE'}; end
if S.on, S.hB.String = {['BURNING ' lab{1} ' - TAP TO CUT'], lab{2}};
else,    S.hB.String = {['TAP TO BURN ' lab{1}], lab{2}}; end
S.hB.Color = pal(9 - (dn < 0),:);                  % hint: red (8) retrograde, blue (9) prograde
phd = mod(rad2deg(S.ph - atan2(r(2),r(1))) + 180, 360) - 180;
S.hH.String = sprintf(['AHEAD %8.1f deg\nGAIN  %8.1f deg/rev\nRANGE %8.0f km\n' ...
  'REL V %8.1f m/s\nPERI  %8.0f km\nDV    %8.0f m/s\nORBIT %8.2f'], phd, gain, ...
  S.dR*R0, S.dV*V0*1e3, pa*R0 - 6378, (S.DVb - S.dvU)*V0*1e3, S.t/(2*pi));
end

function S = newGame(S)
c0 = -pi/2;  S.ph0 = c0 + deg2rad(25 + 30*rand);     
S.s = [cos(c0); sin(c0); -sin(c0); cos(c0)];        
S.t = 0; S.dvU = 0; S.nx = -1; S.d = -1; S.on = 0; S.u = 0; S.acc = 0;
S.bp = [NaN; NaN]; S.res = ''; S.sc = 0;
S = tgt(S);
end

function S = tgt(S)                                  
S.ph = S.ph0 + S.t;  S.dR = norm(S.s(1:2) - [cos(S.ph); sin(S.ph)]);
S.dV = norm(S.s(3:4) - [-sin(S.ph); cos(S.ph)]);
end

function k = fd(s,u,aT)                              
k = [s(3:4); -s(1:2)/norm(s(1:2))^3 + u*aT*s(3:4)/norm(s(3:4))];
end

function d = rD(r)                                   
d = 40*(1 + 3*(r-1));                                
end

function p = px(p,cx,cy)                             
r = norm(p);  d = rD(r);
p = [cy - d*p(2)/r, cx + d*p(1)/r];
end

function place(hI,p)                                 
[za,zb] = size(hI.CData);
set(hI,'XData',p(2) + [-1 1]*(zb-1)/2,'YData',p(1) + [-1 1]*(za-1)/2,'Visible','on');
end
Play Game
Preview

Play Game opens this in MATLAB Drive. Click Run this app there to play.