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adaptive_MPC.m
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%% QUADROTOR BALANCING PENDULUM LQR SIMULATION - ROTATIONAL EQUILIBRIUM
%
% MATLAB simulation of the paper A Flying Inverted Pendulum by Markus Hehn
% and Raffaello D'Andrea
%
% Equilibrium point rotating around a constant radius at constant rot. vel
%% INIT
clc
clear
addpath('functions/');
disp('------------------------------------------------------------------');
disp(' ADAPTIVE MPC ');
disp('');
disp('------------------------------------------------------------------');
%% DEFINE CONSTANTS
g = 9.81; % m/s^2
m = 0.5; % kg
L = 0.565; % meters (Length of pendulum to center of mass)
l = 0.17; % meters (Quadrotor center to rotor center)
I_yy = 3.2e-3; % kg m^2 (Quadrotor inertia around y-axis)
I_xx = I_yy; % kg m^2 (Quadrotor inertia around x-axis)
I_zz = 5.5e-3; % kg m^2 (Quadrotor inertia around z-axis)
simTime = 6; % 2 second simulation
h = 0.1; % sampling time
fprintf('Sampling time: %f \n',h');
N = simTime/h;
% define desired set point sequences in terms of Radius and Omega
R_radius_sequence = 2*ones(1,N+1); % meters (radius of turn)
Omega_sequence = 1*ones(1,N+1) + 0.007*(1:N+1); % rad/s (rotational velocity)
OmegaAngle_prev = 0;
%% INITIALIZE VARIABLE SEQUENCES
states_cart = zeros(3,N+1);
states_pendulum = zeros(2,N+1);
states_pendulum_actual = zeros(2,N+1);
euler_angles = zeros(2,N+1);
yaw_angle = zeros(1,N+1);
beta_angle = zeros(1,N+1);
gamma_angle = zeros(1,N+1);
beta_dot_angle = zeros(1,N+1);
gamma_dot_angle = zeros(1,N+1);
reference_trajectory = zeros(2,N+1);
u_tilde = zeros(1,N+1);
v_tilde = zeros(1,N+1);
w_tilde = zeros(1,N+1);
u_actual = zeros(1,N+1);
p_tilde = zeros(1,N+1);
q_tilde = zeros(1,N+1);
p_actual = zeros(1,N+1);
q_actual = zeros(1,N+1);
mu_tilde = zeros(1,N+1);
nu_tilde = zeros(1,N+1);
mu_actual = zeros(1,N+1);
nu_actual = zeros(1,N+1);
x = zeros(12,N+1);
u = zeros(3, N+1);
y = zeros(12, N);
t = zeros(1,N+1);
Vf = zeros(1,N+1);
P = eye(12);
% define initial conditions
% p1 p2 q1 q2 u1 u2 v1 v2 w1 w2 mu nu
x(:,1) = [0.1 0 0 0 0 0 0 0 0 0 0.10 0.1 ]';
%% ITERATE THROUGH EACH TIME STEP
disp('------------------------------------------------------------------');
disp(' Simulating MPC System');
disp('------------------------------------------------------------------');
disp('');
fprintf('Simulation time: %d seconds \n',simTime);
disp('');
u_limit = 1;
for k = 1:N
t(:,k+1) = k*h;
if ( mod(t(k),1) == 0 )
fprintf('t = %d sec \n', t(k));
end
% Obtain new linearized system
Omega = Omega_sequence(k);
R_radius = R_radius_sequence(k);
% get new linearized system matrices and Equilibrium nominal points
[Ad,Bd,Cd,EP] = linearized_ss(L,g,Omega,R_radius,h);
% Define Q and R and determine optimal LQR gain based on new system matrices
R = 0.1*eye(3);
Q = 1*eye(12);
Q(5,5) = 10; Q(7,7) = 10; Q(9,9) = 10;
% [X,eigvals,K] = dare(Ad,Bd,Q,R);
% Determine control action from LQR
% u(:,k) = -K*x(:,k);
% Define Prediction Matrices
% tuning weights
Q = 1*eye(12);
Q(5,5) = 10; Q(7,7) = 10; Q(9,9) = 10; % state cost
R = 0.1*eye(length(Bd(1,:))); % input cost
S = 10*eye(size(Ad)); % terminal cost
[X,eigvals,K] = dare(Ad,Bd,Q,R);
% prediction horizon
N = 18;
Qbar = kron(Q,eye(N));
Rbar = kron(R,eye(N));
Sbar = S;
LTI.A = Ad;
LTI.B = Bd;
LTI.C = Cd;
dim.N = N;
dim.nx = size(Ad,1);
dim.nu = size(Bd,2);
dim.ny = size(Cd,1);
[P,Z,W] = predmodgen(LTI,dim);
H = (Z'*Qbar*Z + Rbar + 2*W'*Sbar*W);
x0 = x(:,k);
d = (x0'*P'*Qbar*Z + 2*x0'*(Ad^N)'*Sbar*W)';
limits = [5*ones(N,1);
1.5*ones(N,1);
0.5*ones(N,1)];
% MPC control action
cvx_begin quiet
variable u_N(3*N)
minimize ( (1/2)*quad_form(u_N,H) + d'*u_N )
u_N >= -limits;
u_N <= limits;
cvx_end
u(:,k) = u_N(1:3); % MPC control action
% Save variables
u_tilde(k) = x(5,k);
v_tilde(k) = x(7,k);
w_tilde(k) = x(9,k);
u_actual(k) = u_tilde(k) + EP.u_0;
p_tilde(k) = x(1,k);
q_tilde(k) = x(3,k);
p_actual(k) = p_tilde(k) + EP.p_0;
q_actual(k) = q_tilde(k) + EP.q_0;
mu_tilde(k) = x(11,k);
nu_tilde(k) = x(12,k);
mu_actual(k) = mu_tilde(k) + EP.mu_0;
nu_actual(k) = nu_tilde(k) + EP.nu_0;
OmegaAngleT = OmegaAngle_prev + Omega*h;
OmegaAngle_prev = OmegaAngleT;
R_uvw_to_xyz = [cos(OmegaAngleT) -sin(OmegaAngleT) 0;
sin(OmegaAngleT) cos(OmegaAngleT) 0;
0 0 1];
% determine x,y,z states of quadrotor
states_cart(:,k) = R_uvw_to_xyz*[u_actual(k); v_tilde(k); w_tilde(k)];
R_pq_to_rs = [cos(OmegaAngleT) -sin(OmegaAngleT);
sin(OmegaAngleT) cos(OmegaAngleT)];
R_euler = R_pq_to_rs;
% determine
euler_angles(:,k) = R_euler*[nu_actual(k); mu_actual(k)];
yaw_angle(k) = 0;
% determine r,s states of pendulum (relative to quadrotor center)
states_pendulum(:,k) = R_pq_to_rs*[p_actual(k); q_actual(k)];
% determine absolute pendulum position
states_pendulum_actual(:,k) = states_pendulum(:,k) + states_cart(1:2,k);
% calculate control input in terms of omega_x,y,z
mu = mu_actual(k);
nu = nu_actual(k);
% determine gamma, beta angles of quadrotor
gamma = -asin(sin(OmegaAngleT)*sin(mu)*cos(nu) - cos(OmegaAngleT)*sin(nu));
beta = asin((cos(OmegaAngleT)*sin(mu)*cos(nu) + sin(OmegaAngleT)*sin(nu))/(cos(gamma)));
beta_angle(k) = beta;
gamma_angle(k) = gamma;
beta_dot_angle(k) = (R_radius*Omega^3*acos(gamma)*(tan(beta)*tan(gamma)*cos(OmegaAngleT) + acos(beta)*sin(OmegaAngleT)))/sqrt(g^2+(R_radius*Omega^2)^2);
gamma_dot_angle(k) = (R_radius*Omega^3*acos(gamma)*cos(OmegaAngleT))/sqrt(g^2+(R_radius*Omega^2)^2);
% determine reference input based on R_radius and Omega sequences
reference_trajectory(:,k) = [cos(OmegaAngleT); sin(OmegaAngleT)]*R_radius;
% Apply control and update state equations
x(:,k+1) = Ad*x(:,k) + Bd*u(:,k);
y(:,k) = Cd*x(:,k);
% Determine Terminal Constraint
Vf(k) = 0.5*x(:,k)'*X*x(:,k);
end
%% PLOT 3D TRAJECTORY
states_trajectory = [states_cart(1:3,:);
euler_angles(1,:);
euler_angles(2,:);
yaw_angle;
states_pendulum(1:2,:)]';
visualize_quadrotor_trajectory_rotating(states_trajectory, reference_trajectory);
%% PLOT RESULTS
other.mu_actual = mu_actual;
other.beta_dot_angle = beta_dot_angle;
other.gamma_dot_angle = gamma_dot_angle;
other.beta_angle = beta_angle;
other.gamma_angle = gamma_angle;
plot_rotational(t,x,u,other);
% stability plots
% figure(123);
% Vf_dif = Vf(2:end) - Vf(1:end-1);
% stairs(Vf_dif);
% hold on
% stairs(Vf);
% grid();
%% FUNCTIONS
function Rt = R_x(y)
Rt = [1 0 0 ;
0 cos(y) -sin(y);
0 sin(y) cos(y)];
end
function Rt = R_y(y)
Rt = [cos(y) 0 sin(y) ;
0 1 0;
-sin(y) 0 cos(y)];
end
function Rt = R_z(y)
Rt = [cos(y) -sin(y) 0;
sin(y) cos(y) 0;
0 0 1];
end
function [Ad,Bd,Cd,EP] = linearized_ss(L,g,Omega,R_radius,h)
% equilibrium constants
q_0 = 0;
zeta_0 = 0.5; % initial zeta_0 estimate
% zeta_0 = sqrt(L^2-q_0^2-p_0^2); % zeta = relative position of pendulum along z-axis (Eqn 8)
p_0 = -(Omega^2*R_radius)/(Omega^2+g/zeta_0);
zeta_0 = sqrt(L^2-q_0^2-p_0^2); % correct
z_0 = Inf; % NOT USED - constant reference altitude
u_0 = R_radius;
v_0 = 0;
w_0 = 0;
mu_0 = atan(-Omega^2*R_radius/g); % nominal euler angle
nu_0 = 0; % nominal euler angle
a_0 = sqrt(g^2 + (R_radius*Omega^2)^2); % nominal thrust
C1 = zeta_0^2/L^2;
C2 = Omega^2 + g*L^2/(zeta_0^3);
C3 = Omega;
C4 = -(p_0/zeta_0)*a_0*sin(mu_0) - a_0*cos(mu_0);
C5 = (p_0/zeta_0)*cos(mu_0) - sin(mu_0);
C6 = Omega^2 + g/zeta_0;
% p1 p2 q1 q2 u1 u2 v1 v2 w1 w2 mu nu
Ac =[ 0 1 0 0 0 0 0 0 0 0 0 0 ; % p1
C1*C2 0 0 2*C1*C3 0 0 0 0 0 0 C1*C4 0 ; % p2
0 0 0 1 0 0 0 0 0 0 0 0 ; % q1
0 -2*C3 C6 0 0 0 0 0 0 0 0 a_0 ; % q2
0 0 0 0 0 1 0 0 0 0 0 0 ; % u1
0 0 0 0 C3^2 0 0 2*C3 0 0 a_0*cos(mu_0) 0 ; % u2
0 0 0 0 0 0 0 1 0 0 0 0 ; % v1
0 0 0 0 0 -2*C3 C3^2 0 0 0 0 -a_0 ; % v2
0 0 0 0 0 0 0 0 0 1 0 0 ; % w1
0 0 0 0 0 0 0 0 0 0 -a_0*sin(mu_0) 0 ; % w2
0 0 0 0 0 0 0 0 0 0 0 0 ; % mu
0 0 0 0 0 0 0 0 0 0 0 0 ]; % nu
% mu_dot nu_dot a
Bc =[ 0 0 0; % p1
0 0 C1*C5; % p2
0 0 0; % q1
0 0 0; % q2
0 0 0; % u1
0 0 sin(mu_0); % u2
0 0 0; % v1
0 0 0; % v2
0 0 0; % w1
0 0 cos(mu_0); % w2
1 0 0; % mu
0 1 0 ]; % nu
% full state feedback
Cc = eye(12);
% descritize system
sysc = ss(Ac,Bc,Cc,[]);
sysd = c2d(sysc,h);
Ad = sysd.A;
Bd = sysd.B;
Cd = sysd.C;
% assign equilibrium parameters to struct
EP.a_0 = a_0;
EP.q_0 = q_0;
EP.p_0 = p_0;
EP.zeta_0 = zeta_0;
EP.mu_0 = mu_0;
EP.nu_0 = nu_0;
EP.u_0 = u_0;
EP.v_0 = v_0;
EP.w_0 = w_0;
EP.z_0 = z_0;
end
%.