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DecMPC_Analytical.m
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DecMPC_Analytical.m
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%% Decentralized MPC Algorithm for Spacecraft Rendezvous
% =========================================================================
% AA277 | Luke Neise, Samuel Low, Michael Ying, Tamas Kis
clc; clear all; close all;
dt = 60; % Dynamics time step
a = 6725000; % Semi-major axis [m]
N = 5; % MPC prediction horizon size
u_lb = -3.0; % ΔV lower bound [m/s]
u_ub = 3.0; % ΔV upper bound [m/s]
Q = 0.1*eye(6); % State error cost matrix
R = 0.001*eye(3); % Control effort cost matrix
P = N^2*eye(6); % Terminal state error cost matrix
% Because the semi-major axis is a much larger numerical quantity than all
% the other angular variables, the optimization weighting should scale it
% down by the inverse arc length constant for relative weighting.
Q(1,1) = 0.2*N / (0.0175 * a);
P(1,1) = 0.2*N / (0.0175 * a);
%% Exterior Online Simulation
% =========================================================================
% Set up our spacecraft and reference
% Elements: [ a, ex, ey, inc, argp, nu ]
xk1 = [a+25000, 0.002, 0.002, deg2rad(45.05), deg2rad(45.05), 1*pi/180]';
xk2 = [a-25000, -0.002, -0.002, deg2rad(45.00), deg2rad(45.00),-1*pi/180]';
xkR = [a, 0, 0, deg2rad(45.025),deg2rad(45.025), 0]';
% Setup some arrays for plotting the final results.
xF1a = [];
xF2a = [];
xFRa = [];
uF1a = [];
uF2a = [];
dv1a = [0];
dv2a = [0];
% Initialize some simulation time
duration = 86400;
timeaxis = linspace( 0, duration, (duration/dt)+1 );
% Initialize arrays for run time
t_runtime = zeros( 1, length(timeaxis)-1 );
t_runtimesum = zeros( 1, length(timeaxis)-1 );
% Main for loop.
for k = 0 : 1 : round(duration/dt)
% Run the main DMPC program
tic;
[xF1, xF2, xFR, uF1, uF2] = run_DecMPC( xk1, xk2, xkR, dt, ...
N, Q, R, P, ...
u_lb, u_ub );
% Update the states in external simulation
xF1a(:,k+1) = xF1(:,1); % Take only the first element.
xF2a(:,k+1) = xF2(:,1); % Take only the first element.
xFRa(:,k+1) = xFR(:,1); % Take only the first element.
uF1a(:,k+1) = uF1(:,1); % Take only the first element.
uF2a(:,k+1) = uF2(:,1); % Take only the first element.
xk1 = nonlinear_dynamics( xF1(:,1), uF1(:,1), dt );
xk2 = nonlinear_dynamics( xF2(:,1), uF2(:,1), dt );
xkR = nonlinear_dynamics( xFR(:,1), zeros(3,1), dt );
% Update the Delta V values for SC1 and SC2
dv1 = sum(abs(uF1(:,1)));
dv1a(end+1) = dv1a(end) + dv1;
dv2 = sum(abs(uF2(:,1)));
dv2a(end+1) = dv2a(end) + dv2;
% Update the run-time
t_runtime(k+1) = toc;
if k+1 > 1
t_runtimesum(k+1) = t_runtimesum(k) + t_runtime(k+1);
else
t_runtimesum(k+1) = t_runtime(k+1);
end
end
% Loop through all the states again now that we have the final orbit and
% perform the conversion to RTN states
RTN_pos1 = [];
RTN_pos2 = [];
RTN_vel1 = [];
RTN_vel2 = [];
for k = 1 : 1 : round(duration/dt)
oe1 = xF1a(:,k);
oe2 = xF2a(:,k);
oeR = xFRa(:,k);
% RTN1 = elements_to_RTN( oeR, oe1 );
% RTN2 = elements_to_RTN( oeR, oe2 );
RTN1 = elements_to_RTN( oe1, oe2 );
RTN2 = RTN1;
RTN_pos1(:,k) = RTN1(1:3)';
RTN_pos2(:,k) = RTN2(1:3)';
RTN_vel1(:,k) = RTN1(4:6)';
RTN_vel2(:,k) = RTN2(4:6)';
end
% converts time to hours
timeaxis = timeaxis/3600;
% converts semi-major axes to kilometers
xF1a(1,:) = xF1a(1,:)/1000;
xF2a(1,:) = xF2a(1,:)/1000;
% converts angles to degrees
xF1a(4,:) = xF1a(4,:)*(180/pi);
xF2a(4,:) = xF2a(4,:)*(180/pi);
xF1a(5,:) = xF1a(5,:)*(180/pi);
xF2a(5,:) = xF2a(5,:)*(180/pi);
xF1a(6,:) = xF1a(6,:)*(180/pi);
xF2a(6,:) = xF2a(6,:)*(180/pi);
% converts RTN positions to kilometers
RTN_pos1 = RTN_pos1/1000;
%% Saving data
% time [h]
t = timeaxis;
% semi-major axes [km]
a1 = xF1a(1,:);
a2 = xF2a(1,:);
% x-component of eccentricity vector [-]
ex1 = xF1a(2,:);
ex2 = xF2a(2,:);
% y-component of eccentricity vector [-]
ey1 = xF1a(3,:);
ey2 = xF2a(3,:);
% inclination [deg]
i1 = xF1a(4,:);
i2 = xF2a(4,:);
% RAAN [deg]
Om1 = xF1a(5,:);
Om2 = xF2a(5,:);
% argument of latitude [deg]
u1 = xF1a(6,:);
u2 = xF2a(6,:);
% control effort [m/s]
ctrl1 = uF1a;
ctrl2 = uF2a;
% cumulative Delta-V [m/s]
dV1 = dv1a;
dV2 = dv2a;
% RTN positions [km]
RTN_2wrt1 = RTN_pos1;
% trims all arrays so they are same length (not sure why they are
% different?) and packages them into a structure
DecMPC_analytical.t = t(1:1440);
DecMPC_analytical.a1 = a1(1:1440);
DecMPC_analytical.a2 = a2(1:1440);
DecMPC_analytical.ex1 = ex1(1:1440);
DecMPC_analytical.ex2 = ex2(1:1440);
DecMPC_analytical.ey1 = ey1(1:1440);
DecMPC_analytical.ey2 = ey2(1:1440);
DecMPC_analytical.i1 = i1(1:1440);
DecMPC_analytical.i2 = i2(1:1440);
DecMPC_analytical.Om1 = Om1(1:1440);
DecMPC_analytical.Om2 = Om2(1:1440);
DecMPC_analytical.u1 = u1(1:1440);
DecMPC_analytical.u2 = u2(1:1440);
DecMPC_analytical.ctrl1 = ctrl1(:,1:1440);
DecMPC_analytical.ctrl2 = ctrl2(:,1:1440);
DecMPC_analytical.dV1 = dV1(1:1440);
DecMPC_analytical.dV2 = dV2(1:1440);
DecMPC_analytical.RTN_2wrt1 = RTN_2wrt1(:,1:1440);
% saves simulation data
save('data/DecMPC_analytical.mat','DecMPC_analytical');
%% Primary Decentralized MPC Optimization Process (Analytical).
% =========================================================================
function [xf1, xf2, xfR, uf1, uf2] = run_DecMPC( xk1, xk2, xkR, dt, ...
N, Q, R, P, u_lb, u_ub)
% Input: xk1, xk2, are 6xN matrices detailing the states of SC1/2
% Initialize MPC global and local variables.
u10 = zeros(3,N); % Initialize RTN control ΔV for SC1 [m/s]
u20 = zeros(3,N); % Initialize RTN control ΔV for SC2 [m/s]
uf1 = u10;
uf2 = u20;
xf1 = xk1(:,1);
xf2 = xk2(:,1);
xfR = xkR(:,1);
mu = 3.986004415e14;
Rinv = inv(R);
% Re-initialize (re-update) the initial state to length 1.
xf1 = xf1(:,1); % Take just the first vector.
xf2 = xf2(:,1); % Take just the first vector.
xfR = xfR(:,1); % Take just the first vector.
% Populate the states of both spacecraft first.
for i = 1:N
xf1(:,i+1) = nonlinear_dynamics( xf1(:,i), uf1(:,i), dt );
xf2(:,i+1) = nonlinear_dynamics( xf2(:,i), uf2(:,i), dt );
xfR(:,i+1) = nonlinear_dynamics( xfR(:,i), zeros(3,1), dt );
end
% (Re)-populate states based on control from previous iteration.
for i = 1:N
xf1(:,i+1) = nonlinear_dynamics( xf1(:,i), uf1(:,i), dt );
xf2(:,i+1) = nonlinear_dynamics( xf2(:,i), uf2(:,i), dt );
% Now compute the B matrix in the state transition for SC1.
sma1 = xf1(1,i);
inc1 = xf1(4,i);
lat1 = xf1(6,i);
n1 = sqrt(mu/(sma1)^3);
B1 = [ 0 n1 0 ;
sin(lat1)/(sma1*n1) 2*cos(lat1)/(sma1*n1) 0 ;
-cos(lat1)/(sma1*n1) 2*sin(lat1)/(sma1*n1) 0 ;
0 0 cos(lat1)/(sma1*n1) ;
0 0 sin(lat1)/(sma1*n1*sin(inc1)) ;
-2/(sma1*n1) 0 -cot(inc1)*sin(lat1)/(sma1*n1) ];
% Now compute the B matrix in the state transition for SC2.
sma2 = xf2(1,i);
inc2 = xf2(4,i);
lat2 = xf2(6,i);
n2 = sqrt(mu/(sma2)^3);
B2 = [ 0 n2 0 ;
sin(lat2)/(sma2*n2) 2*cos(lat2)/(sma2*n2) 0 ;
-cos(lat2)/(sma2*n2) 2*sin(lat2)/(sma2*n2) 0 ;
0 0 cos(lat2)/(sma2*n2) ;
0 0 sin(lat2)/(sma2*n2*sin(inc2)) ;
-2/(sma2*n2) 0 -cot(inc2)*sin(lat2)/(sma2*n2) ];
% Compute the horizon-stacked transitions for SC1 and SC2.
A1 = eye(6);
A2 = eye(6);
A1(6,1) = (-3*n1*dt/(2*sma1));
A2(6,1) = (-3*n2*dt/(2*sma2));
A1stack = A1;
A2stack = A2;
for m = 1:(N-i)
A1stack = A1stack * A1;
A2stack = A2stack * A2;
end
% Compute the optimal control vector for SC1
x1_0_term = -1 * B1' * Q * ( xf1(:,i+1) - xfR(:,i+1) );
x1_F_term = -1 * B1' * A1stack * P * (xf1(:,end)-xfR(:,end));
uf1(:,i) = x1_0_term + x1_F_term;
uf1(:,i) = Rinv * uf1(:,i);
% Compute the optimal control vector for SC2
x2_0_term = -1 * B2' * Q * ( xf2(:,i+1) - xfR(:,i+1) );
x2_F_term = -1 * B2' * A2stack * P * (xf2(:,end)-xfR(:,end));
uf2(:,i) = x2_0_term + x2_F_term;
uf2(:,i) = Rinv * uf2(:,i);
% Take the saturation if the control input exceeds.
uf1 = max( uf1, u_lb );
uf1 = min( uf1, u_ub );
uf2 = max( uf2, u_lb );
uf2 = min( uf2, u_ub );
% Recompute next state
xf1(:,i+1) = nonlinear_dynamics( xf1(:,i), uf1(:,i), dt );
xf2(:,i+1) = nonlinear_dynamics( xf2(:,i), uf2(:,i), dt );
end
end