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% Initial Conditions
x0 = [3; % 3 radians
0]; % 0 rad/s
% System Dynamics
A = [0 1;
0.01 0];
B = [0;
1];
C = [1 0];
D = 0;
% Control Law
Q = [1 0; % Penalize angular error
0 1]; % Penalize angular rate
R = 1; % Penalize thruster effort
K = lqr(A,B,Q,R);
% Closed loop system
sys = ss((A - B*K), B, C, D);
% Run response to initial condition
t = 0:0.005:30;
[y,t,x] = initial(sys, x0, t);
%% Everything below this is just for the visualization
% Patch drawings
ufo_data = [
0.54 .12;
0.48 0.24;
0.42 0.31;
0.3 0.4;
0.18 0.45;
0.06 0.48;
-0.06 0.48;
-0.18 0.45;
-0.3 0.4;
-0.42 0.31;
-0.48 0.24;
-0.54 .12;
0.54 .12;
0.54 0.06;
0.6 0.06;
0.78 0.03;
0.91 0.01;
0.91 0.03;
1.01 0.03;
1.01 -0.01;
1.1 -0.03;
1.26 -0.12;
1.36 -0.18;
1.40 -0.21;
1.38 -0.24;
1.26 -0.27;
1.1 -0.3;
1.01 -0.32;
1.01 -0.34;
0.91 -0.34;
0.91 -0.32;
0.72 -0.34;
0.48 -0.35;
0.72 -0.6;
0.62 -0.6;
0.38 -0.35;
0.3 -0.35;
-0.3 -0.35;
-0.38 -0.35;
-0.62 -0.6;
-0.72 -0.6;
-0.48 -0.35;
-0.91 -0.32;
-0.91 -0.34;
-1.01 -0.34;
-1.01 -0.32;
-1.1 -0.3;
-1.26 -0.27;
-1.38 -0.24;
-1.40 -0.21;
-1.36 -0.18;
-1.26 -0.12;
-1.1 -0.03;
-1.01 -0.01;
-1.01 0.03;
-0.91 0.03;
-0.91 0.01;
-0.78 0.03;
-0.6 0.06;
-0.54 0.06;
0.54 0.06;
];
thrust = [
0.05 0;
-.05 0;
-.06 -0.1;
-.05 -0.1;
-.06 -0.2;
0 -0.25;
0.06 -0.2;
0.05 -0.1;
0.06 -0.1;
0.05 0];
% yoffset moves the patch up from 0, 0
yoffset = 1;
%% Create Animation
% Rotate to inital condition
ufo_rot = rot(ufo_data, x0(1));
r_thrust_rot = rot(thrust, x0(1));
l_thrust_rot = rot(thrust, x0(1));
% Set up the figure and the subplots
fig = figure('Position', [100 80 670 800]);
h3 = subplot(3, 1, 3);
h3.OuterPosition = [0 0.01 1 .19];
h3a = area(0, 0);
hold on
h3p = plot(0, 0, 'r');
h3.YGrid = 'on';
h3.XGrid = 'on';
h3.LineWidth = 1;
h3.Color = [0 0 0];
h3p.LineWidth = 4;
axis([0 12 -3 3]);
h3.GridColor = 'w';
h3.GridAlpha = 0.5;
text(6.1, 1.5, 'Acceleration (fuel)', 'Color', [1 0 0], 'FontSize', 24)
h2 = subplot(3, 1, 2);
h2.OuterPosition = [0 .2 1 .2];
h2p = plot(0, 0, 'g');
h2p.LineWidth = 4;
h2.YGrid = 'on';
h2.XGrid = 'on';
h2.LineWidth = 1;
h2.GridColor = 'w';
h2.GridAlpha = 0.5;
axis([0 12 -.5 3.5]);
h2.Color = [0 0 0];
text(6.1, 2.5, 'Angular Error (time)', 'Color', [0 1 0], 'FontSize', 24)
h1 = subplot(3, 1, 1);
h1.OuterPosition = [0 .4 1 0.63];
h1.XTick = [];
h1.YTick = [];
patch([-3 3 3 -3], [-1.7 -1.7 3.3 3.3], 'k');
axis([-3 3 -1.7 3.3]);
line([0 0.28], [1 2.99], 'Color', [0.5 0.3 0.3], 'LineWidth', 1);
line([0 0], [1 -1], 'Color', [0.3 0.5 0.3], 'LineWidth', 1);
H = patch(ufo_rot(:, 1), ufo_rot(:, 2) + yoffset, [0.7 0.7 0.7]);
RBT = patch(r_thrust_rot(:, 1), r_thrust_rot(:, 2), 'r');
LBT = patch(l_thrust_rot(:, 1), l_thrust_rot(:, 2), 'r');
RTT = patch(r_thrust_rot(:, 1), r_thrust_rot(:, 2), 'r');
LTT = patch(l_thrust_rot(:, 1), l_thrust_rot(:, 2), 'r');
RBT.FaceAlpha = 0;
LBT.FaceAlpha = 0;
RTT.FaceAlpha = 0;
LTT.FaceAlpha = 0;
RBT.EdgeColor = 'none';
LBT.EdgeColor = 'none';
RTT.EdgeColor = 'none';
LTT.EdgeColor = 'none';
hold on
% Initial time and fuel values
current_time = 0;
fuel = 0;
% Write time and fuel to the screen
text(-2.5, 3, 'Time: sec', 'Color', [1 1 1], 'FontSize', 24,...
'HorizontalAlignment', 'left');
text(-2.5, 2.7, 'Fuel: units', 'Color', [1 1 1], 'FontSize', 24,...
'HorizontalAlignment', 'left');
TME = text(-1.1, 3, num2str(current_time, '%.1f'), 'Color', [0 1 0], ...
'FontSize', 24, 'HorizontalAlignment', 'right');
FUL = text(-1.1, 2.7, num2str(fuel, '%.0f'), 'Color', [1 0 0], ...
'FontSize', 24, 'HorizontalAlignment', 'right');
accel_true = diff(x(:, 2))/(t(2)-t(1));
pause(2)
% Update Animation
for i = 1:5:length(x)-1
% Simulation ending condition
if and(abs(x(i,1)) < 0.011, abs(x(i,2)) < 0.011)
x(i,1) = 0;
pause(1);
% Turn on tractor beam
G = patch([0.1 0.8 -0.8 -0.1], [-0.35 -2.3 -2.3 -0.35] + yoffset, 'y');
G.FaceAlpha = 0.6;
return
end
% Scale acceleration to make thruster size look better fuel units
% just a few digits (asthetics)
accel_scaled = (x(i+1, 2) - x(i, 2));
fuel = fuel + abs(accel_scaled);
accel_scaled = min(2, max(-2, accel_scaled * 1000));
% Scale thruster based on accleration and direction
if accel_scaled > 0
r_scaled(:, 1) = thrust(:, 1) * accel_scaled * 0.5;
r_scaled(:, 2) = thrust(:, 2) * accel_scaled;
l_scaled(:, 1) = thrust(:, 1);
l_scaled(:, 2) = thrust(:, 2);
elseif accel_scaled < 0
r_scaled(:, 1) = thrust(:, 1);
r_scaled(:, 2) = thrust(:, 2);
l_scaled(:, 1) = thrust(:, 1) * -accel_scaled * 0.5;
l_scaled(:, 2) = thrust(:, 2) * -accel_scaled;
else
r_scaled(:, 1) = thrust(:, 1);
r_scaled(:, 2) = thrust(:, 2);
l_scaled(:, 1) = thrust(:, 1);
l_scaled(:, 2) = thrust(:, 2);
end
% Offset thrusters to place correctly on UFO
rb_thrust = r_scaled + [0.96 -0.34];
lb_thrust = l_scaled + [-0.96 -0.34];
rt_thrust = l_scaled * [1 0; 0 -1] + [0.96, 0.03];
lt_thrust = r_scaled * [1 0; 0 -1] + [-0.96, 0.03];
% Rotate thrusters with UFO
rb_thrust_rot = rot(rb_thrust, x(i,1)) + [0 yoffset];
lb_thrust_rot = rot(lb_thrust, x(i,1)) + [0 yoffset];
rt_thrust_rot = rot(rt_thrust, x(i,1)) + [0 yoffset];
lt_thrust_rot = rot(lt_thrust, x(i,1)) + [0 yoffset];
% Update thruster data
RBT.XData = rb_thrust_rot(:, 1);
RBT.YData = rb_thrust_rot(:, 2);
LBT.XData = lb_thrust_rot(:, 1);
LBT.YData = lb_thrust_rot(:, 2);
RTT.XData = rt_thrust_rot(:, 1);
RTT.YData = rt_thrust_rot(:, 2);
LTT.XData = lt_thrust_rot(:, 1);
LTT.YData = lt_thrust_rot(:, 2);
% Turn thrusters off if acceleration is low
if accel_scaled > 2e-1
RBT.FaceAlpha = 1;
LBT.FaceAlpha = 0;
RTT.FaceAlpha = 0;
LTT.FaceAlpha = 1;
elseif accel_scaled < -2e-1
RBT.FaceAlpha = 0;
LBT.FaceAlpha = 1;
RTT.FaceAlpha = 1;
LTT.FaceAlpha = 0;
else
RBT.FaceAlpha = 0;
LBT.FaceAlpha = 0;
RTT.FaceAlpha = 0;
LTT.FaceAlpha = 0;
end
% Rotate ufo
ufo_rot = rot(ufo_data, x(i,1));
H.XData = ufo_rot(:, 1);
H.YData = ufo_rot(:, 2) + yoffset;
h2p.XData = t(1:i);
h2p.YData = x(1:i, 1);
h3p.XData = t(1:i);
h3p.YData = accel_true(1:i);
h3a.XData = t(1:i);
h3a.YData = accel_true(1:i);
current_time = num2str(t(i), '%.1f');
% Update time and fuel
TME.String = current_time;
FUL.String = num2str(fuel*30, '%.0f');
pause(0.0001);
end
function output = rot(data, angle)
output = size(data);
Q = [cos(angle) -sin(angle); sin(angle) cos(angle)];
for i = 1:length(data)
output(i, :) = Q * data(i, :)';
end
end
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