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quadrotor_dynamics.m
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quadrotor_dynamics.m
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function [sys,x0,str,ts] = quadrotor_dynamics(t,x,u,flag, quad, x0, groundflag)
% Flyer2dynamics lovingly coded by Paul Pounds, first coded 12/4/04
% A simulation of idealised X-4 Flyer II flight dynamics.
% version 2.0 2005 modified to be compatible with latest version of Matlab
% version 3.0 2006 fixed rotation matrix problem
% version 4.0 4/2/10, fixed rotor flapping rotation matrix bug, mirroring
% version 5.0 8/8/11, simplified and restructured
% version 6.0 25/10/13, fixed rotation matrix/inverse wronskian definitions, flapping cross-product bug
warning off MATLAB:divideByZero
global groundFlag;
% New in version 2:
% - Generalised rotor thrust model
% - Rotor flapping model
% - Frame aerodynamic drag model
% - Frame aerodynamic surfaces model
% - Internal motor model
% - Much coolage
% Version 1.3
% - Rigid body dynamic model
% - Rotor gyroscopic model
% - External motor model
%ARGUMENTS
% u Reference inputs 1x4
% tele Enable telemetry (1 or 0) 1x1
% crash Enable crash detection (1 or 0) 1x1
% init Initial conditions 1x12
%INPUTS
% u = [N S E W]
% NSEW motor commands 1x4
%CONTINUOUS STATES
% z Position 3x1 (x,y,z)
% v Velocity 3x1 (xd,yd,zd)
% n Attitude 3x1 (Y,P,R)
% o Angular velocity 3x1 (wx,wy,wz)
% w Rotor angular velocity 4x1
%
% Notes: z-axis downward so altitude is -z(3)
%CONTINUOUS STATE MATRIX MAPPING
% x = [z1 z2 z3 n1 n2 n3 z1 z2 z3 o1 o2 o3 w1 w2 w3 w4]
%INITIAL CONDITIONS
n0 = [0 0 0]; % n0 Ang. position initial conditions 1x3
v0 = [0 0 0]; % v0 Velocity Initial conditions 1x3
o0 = [0 0 0]; % o0 Ang. velocity initial conditions 1x3
init = [x0 n0 v0 o0]; % x0 is the passed initial position 1x3
groundFlag = groundflag;
%CONTINUOUS STATE EQUATIONS
% z` = v
% v` = g*e3 - (1/m)*T*R*e3
% I*o` = -o X I*o + G + torq
% R = f(n)
% n` = inv(W)*o
% Dispatch the flag.
%
switch flag
case 0
[sys,x0,str,ts]=mdlInitializeSizes(init, quad); % Initialization
case 1
sys = mdlDerivatives(t,x,u, quad); % Calculate derivatives
case 3
sys = mdlOutputs(t,x, quad); % Calculate outputs
case { 2, 4, 9 } % Unused flags
sys = [];
otherwise
error(['Unhandled flag = ',num2str(flag)]); % Error handling
end
end % End of flyer2dynamics
%==============================================================
% mdlInitializeSizes
% Return the sizes, initial conditions, and sample times for the
% S-function.
%==============================================================
%
function [sys,x0,str,ts] = mdlInitializeSizes(init, quad)
%
% Call simsizes for a sizes structure, fill it in and convert it
% to a sizes array.
%
sizes = simsizes;
sizes.NumContStates = 12;
sizes.NumDiscStates = 0;
sizes.NumOutputs = 12;
sizes.NumInputs = 4;
sizes.DirFeedthrough = 0;
sizes.NumSampleTimes = 1;
sys = simsizes(sizes);
%
% Initialize the initial conditions.
x0 = init;
%
% str is an empty matrix.
str = [];
%
% Generic timesample
ts = [0 0];
if quad.verbose
disp(sprintf('t\t\tz1\t\tz2\t\tz3\t\tn1\t\tn2\t\tn3\t\tv1\t\tv2\t\tv3\t\to1\t\to2\t\to3\t\tw1\t\tw2\t\tw3\t\tw4\t\tu1\t\tu2\t\tu3\t\tu4'))
end
end % End of mdlInitializeSizes.
%==============================================================
% mdlDerivatives
% Calculate the state derivatives for the next timestep
%==============================================================
%
function sys = mdlDerivatives(t,x,u, quad)
global a1s b1s groundFlag
%CONSTANTS
%Cardinal Direction Indicies
N = 1; % N 'North' 1x1
E = 2; % S 'South' 1x1
S = 3; % E 'East' 1x1
W = 4; % W 'West' 1x1
D(:,1) = [quad.d;0;quad.h]; % Di Rotor hub displacements 1x3
D(:,2) = [0;quad.d;quad.h];
D(:,3) = [-quad.d;0;quad.h];
D(:,4) = [0;-quad.d;quad.h];
%Body-fixed frame references
e1 = [1;0;0]; % ei Body fixed frame references 3x1
e2 = [0;1;0];
e3 = [0;0;1];
%EXTRACT ROTOR SPEEDS FROM U
w = u(1:4);
%EXTRACT STATES FROM X
z = x(1:3); % position in {W}
n = x(4:6); % RPY angles {W}
v = x(7:9); % velocity in {W}
o = x(10:12); % angular velocity in {W}
%PREPROCESS ROTATION AND WRONSKIAN MATRICIES
phi = n(1); % yaw
the = n(2); % pitch
psi = n(3); % roll
% rotz(phi)*roty(the)*rotx(psi)
R = [cos(the)*cos(phi) sin(psi)*sin(the)*cos(phi)-cos(psi)*sin(phi) cos(psi)*sin(the)*cos(phi)+sin(psi)*sin(phi); %BBF > Inertial rotation matrix
cos(the)*sin(phi) sin(psi)*sin(the)*sin(phi)+cos(psi)*cos(phi) cos(psi)*sin(the)*sin(phi)-sin(psi)*cos(phi);
-sin(the) sin(psi)*cos(the) cos(psi)*cos(the)];
%Manual Construction
% Q3 = [cos(phi) -sin(phi) 0;sin(phi) cos(phi) 0;0 0 1]; % RZ %Rotation mappings
% Q2 = [cos(the) 0 sin(the);0 1 0;-sin(the) 0 cos(the)]; % RY
% Q1 = [1 0 0;0 cos(psi) -sin(psi);0 sin(psi) cos(psi)]; % RX
% R = Q3*Q2*Q1 %Rotation matrix
%
% RZ * RY * RX
iW = [0 sin(psi) cos(psi); %inverted Wronskian
0 cos(psi)*cos(the) -sin(psi)*cos(the);
cos(the) sin(psi)*sin(the) cos(psi)*sin(the)] / cos(the);
if any(w == 0)
% might need to fix this, preculudes aerobatics :(
% mu becomes NaN due to 0/0
error('quadrotor_dynamics: not defined for zero rotor speed');
end
%ROTOR MODEL
for i=[N E S W] %for each rotor
%Relative motion
Vr = cross(o,D(:,i)) + v;
mu = sqrt(sum(Vr(1:2).^2)) / (abs(w(i))*quad.r); %Magnitude of mu, planar components
lc = Vr(3) / (abs(w(i))*quad.r); %Non-dimensionalised normal inflow
li = mu; %Non-dimensionalised induced velocity approximation
alphas = atan2(lc,mu);
j = atan2(Vr(2),Vr(1)); %Sideslip azimuth relative to e1 (zero over nose)
J = [cos(j) -sin(j);
sin(j) cos(j)]; %BBF > mu sideslip rotation matrix
%Flapping
beta = [((8/3*quad.theta0 + 2*quad.theta1)*mu - 2*(lc)*mu)/(1-mu^2/2); %Longitudinal flapping
0;];%sign(w) * (4/3)*((Ct/sigma)*(2*mu*gamma/3/a)/(1+3*e/2/r) + li)/(1+mu^2/2)]; %Lattitudinal flapping (note sign)
beta = J'*beta; %Rotate the beta flapping angles to longitudinal and lateral coordinates.
a1s(i) = beta(1) - 16/quad.gamma/abs(w(i)) * o(2);
b1s(i) = beta(2) - 16/quad.gamma/abs(w(i)) * o(1);
%Forces and torques
T(:,i) = quad.Ct*quad.rho*quad.A*quad.r^2*w(i)^2 * [-cos(b1s(i))*sin(a1s(i)); sin(b1s(i));-cos(a1s(i))*cos(b1s(i))]; %Rotor thrust, linearised angle approximations
Q(:,i) = -quad.Cq*quad.rho*quad.A*quad.r^3*w(i)*abs(w(i)) * e3; %Rotor drag torque - note that this preserves w(i) direction sign
tau(:,i) = cross(T(:,i),D(:,i)); %Torque due to rotor thrust
end
%RIGID BODY DYNAMIC MODEL
dz = v;
dn = iW*o;
dv = quad.g*e3 + R*(1/quad.M)*sum(T,2);
% vehicle can't fall below ground
if groundFlag && (z(3) > 0)
z(3) = 0;
dz(3) = 0;
end
do = inv(quad.J)*(cross(-o,quad.J*o) + sum(tau,2) + sum(Q,2)); %row sum of torques
sys = [dz;dn;dv;do]; %This is the state derivative vector
end % End of mdlDerivatives.
%==============================================================
% mdlOutputs
% Calculate the output vector for this timestep
%==============================================================
%
function sys = mdlOutputs(t,x, quad)
%TELEMETRY
if quad.verbose
disp(sprintf('%0.3f\t',t,x))
end
% compute output vector as a function of state vector
% z Position 3x1 (x,y,z)
% v Velocity 3x1 (xd,yd,zd)
% n Attitude 3x1 (Y,P,R)
% o Angular velocity 3x1 (Yd,Pd,Rd)
n = x(4:6); % RPY angles
phi = n(1); % yaw
the = n(2); % pitch
psi = n(3); % roll
% rotz(phi)*roty(the)*rotx(psi)
R = [cos(the)*cos(phi) sin(psi)*sin(the)*cos(phi)-cos(psi)*sin(phi) cos(psi)*sin(the)*cos(phi)+sin(psi)*sin(phi); %BBF > Inertial rotation matrix
cos(the)*sin(phi) sin(psi)*sin(the)*sin(phi)+cos(psi)*cos(phi) cos(psi)*sin(the)*sin(phi)-sin(psi)*cos(phi);
-sin(the) sin(psi)*cos(the) cos(psi)*cos(the)];
iW = [0 sin(psi) cos(psi); %inverted Wronskian
0 cos(psi)*cos(the) -sin(psi)*cos(the);
cos(the) sin(psi)*sin(the) cos(psi)*sin(the)] / cos(the);
% return velocity in the body frame
sys = [ x(1:6);
inv(R)*x(7:9); % translational velocity mapped to body frame
iW*x(10:12)]; % RPY rates mapped to body frame
%sys = [x(1:6); iW*x(7:9); iW*x(10:12)];
%sys = x;
end
% End of mdlOutputs.